On Pentre Ifan

Some observations about Dolmen

This evening I watched a documentary presented by Alice Roberts. I don’t think I have watched a programme presented by Dr Roberts that hasn’t been interesting, and this one was no different, except that the most interesting thing wasn’t strictly the topic. The documentary was about the bluestones of stonehenge, but it strayed ever so slightly to Pentre Ifan in Wales were I was introduced to Dolmens. These are Neolithic structures, which predate Stonehenge. There are lots of them around the world and some are thought to be around 7,000 years old. The Dolmen at Pentre Ifan is a spectacular example, whose massive capstone appears to float above the tips of three standing stones. I expect that’s why I noticed it and why Alice Roberts chose that example.

 

Archaeologists believe that Dolmens were ancestral tombs, because scattered human remains are often found between the standing stones. Some also believe that they were originally buried beneath a mound of earth or smaller stones.

Interesting as these conjectures are I can’t get past the graceful form of the structure. I suspect that archaeologists, for the most part, take for granted that the monument stands, while going to great lengths to try and understand how it got there and how it was built. I think I understand why this is, but there is a certain irony that for such an old object so much more effort is expended on the temporal than the permanent [1].

To me the fact that Pentre Ifan is standing at all is more interesting and is the subject of this post. Maybe on another occasion I’ll fall in to line with everyone else and have a go at speculating how such structures are built, but not today.

I should of course pause to note that I have never seen a Dolmen in person, nor do I really know anything about them, other than tonight’s brief introduction. I am keen to remedy that and will aim to visit some examples when the covid-19 lock-down has ended, however that isn’t going to stop me from donning my engineering hat and doing some speculation of my own. I shall be doing so based on a few images I have pulled off the internet; what could possibly go wrong?

I am going to start with the assumption that the stones are igneous rocks, because they have that appearance and because there are outcrops in the the relevant part of Wales. This ought to make the rock relatively strong; though like any rock it will be brittle and weak in tension.

There is evidence of horizontal fractures in the capping stone and equivalent vertical features in the standing stones. Structurally this is not a terribly efficient arrangement. A stronger arrangement would be to align planes of weakness in the standing stones horizontally so that they are squashed together. For the capping stone it would have been better to align them vertically so as to avoid separation due to shear flow generated by bending forces.

I suspect that this was not done, because creating the great slabs of stone required cleaving them from the parent rock by exploiting the noted weaknesses. Without them stone-age workmen would have had difficulty creating such slabs with primitive tools.

The next thing I notice is that the capping stone appears to be fatter at one end than the other and that the soffit appears to have been cleaved as it progresses towards the thin end. I suspect that it was not originally so.

The fat end is supported by two standing stones, while the thin end is carried by one. In the short axis the fat end of the capping stone can bridge laterally between two close supports. It can possibly do this in direct shear and without inducing bending.

Conversely, in the longitudinal direction the capping stone must span almost 5 meters between the single and double support. This almost certainly results in it experiencing bending. As has been seen in prior posts bending causes the top surface of a beam to experience compression and the soffit to experience tension. 

In order for this to happen a beam will also experience a laminar shear flow in the horizontal direction. This can be understood by imagining a beam divided into horizontal slices. For tension to be experienced on the soffit while compression is experienced on top it is necessary for the imaginary slices to slip past each other.

The consequence of these actions appears to be evident in the structure. Since stone does not deal well with tension it is inevitable that small cracks must have developed in the soffit. There would also have been lateral movement along the rock’s natural horizontal weaknesses. It is conceivable that together these effects led to the soffit spalling, however it is more likely that they were abetted by other effects too.

 

Rainwater will have soaked through the top surface of the slab and migrated under gravity to the soffit. While moisture would evaporate quickly from the top the soffit would remain in the shade helping to keep the stone damp and wet. Persistent dampness will have weakened the rock structure and freeze thaw action would have exploited the many small cracks and natural weaknesses. Eventually the fractured rock would spall until it arrived at a horizontal plane of weakness whereupon the process would start again.

Perhaps another aggravating factor would be expansion and contraction due to the cycle of heating and cooling. Since only the top surface is exposed to the sun there is likely to be a thermal gradient in the capping stone as it warms. During the day the top of the stone would expand relative to the soffit and thereby start to close some of the soffit cracks. Conversely, during the night it would start to contract and thereby re-open the soffit cracks. Thus, by repetition the soffit would slowly be fatigued and further cracks induced.

When considering thermal effects it is worth noting that having only three small points of contact between the capping and standing stones is probably beneficial, because the soffit is free to articulate. More severe cracking would be much more likely if the top surface were free to expand and contract while the soffit was held in place by more restrictive contact. 

It would therefore seem that there is a good explanation as to why the capping stone is fatter at one end than the other, and all else being equal, by what mechanism it will eventually fail.

The standing stones appear to carry the capping stone effortlessly. The fact that they do so with such small points of contact would suggest the compressive strength of the stone must be relatively high. That said, it is assumed that the contact surfaces could not have been prepared to a modern standard and therefore the distribution of load will not be entirely even. This will have allowed load concentrations to be formed which may over time pry and fracture the rock locally.

This is important because compression in the standing stones will cause lateral bursting forces to develop due to passion’s ratio [this is another concept we have seen before]. Within the body of the stone compressive stress is generally low and therefore the bursting forces have little effect, however where load is concentrated stresses are higher. If such stresses coincide with a vertical plane of weakness it could encourage the bursting forces to form a split in the rock. As with the capping stones this could be exacerbated, either by moisture penetration, or by bending induced by one side of the rock being warmed faster than the other. There does seem to be some evidence of such processes at work on the surface of the standing stones.

Another facet of the pictured Dolmen is how it maintains lateral stability. It is self evident that there is no rotational resistance at the junctions between the capping stone and its supports, therefore we must conclude that the standing stones must cantilever from ground level. Since the capping stone bears heavily upon them there will be sufficient friction generated to share lateral load between the standing stones according to their stiffness.

Lateral loads would come from the wind and to a lesser extent thermal effects. There also appears to be a slight incline to the capping stone, which would imply there is a resultant lateral load, due to the stone’s self-weight, to be resisted.

Something else that is structurally relevant, though I am unsure whether it was intended, is the orientation of the standing stones. The two stones at one end are orientated perpendicular to the single stone at the other. Strength being proportional to the cube of depth this arrangement presents the full depth of at least one stone in each orthogonal direction, thus maximising cantilever action in both.

It is also interesting that the end supported by two stones is aligned with the noted incline to the caping stone, thus maximising resistance to the permanent lateral load. The possibility that this was intended is intriguing.

A different explanation would be that since one end of the structure has two supports, and the other just one, there could have been differential settlement. Assuming this were the case the narrow supports would again have been beneficial, because they would have allowed the capping stone to rotate and find a new point of equilibrium. Alternatively, more substantial supports would have likely led to fracture. I have no idea what the bearing strata beneath the standing stones is like, but in the absence of further evidence the mechanism for differential settlement seems plausible.

Of course it is also possible that the fractured soffit could have contributed to creating the observed incline too.

While the depth of embedment of the standing stones is not clear from viewing the surface it seems reasonable to assume that it must be substantial to ensure there is sufficient passive resistance to prevent overturning or sliding of the stones. In a uniform soil it would also be reasonable to assume that bearing pressure would increase with depth.

On paper it is perhaps possible that the stones could be mounted near the ground surface with stability being maintained by their shear size and mass. In the real world this does not seem plausible as the surface of the ground is prone to become waterlogged, there is also the possibility of frost action. Either of these effects could be sufficient to topple the stones.

Two further practical matters exist. Firstly, a shallow footing would be vulnerable to digging near the base when bones were to be buried. 

Secondly, and perhaps more importantly, the processes of standing large stones on end without a crane, or other modern equipment, would seem to make it necessary to tip them into a hole. If said hole were then packed tight with backfill it would lock the stones in place allowing them to behave as cantilevers.

Here I run the risk of getting into the question of how the stones were erected and I said I wasn’t going to do that. I best stop here.

[1] I base this thought on working with a few archaeologists and the number of documentaries there are about erecting Stonehenge, rather than any proper search of the archaeological literature.

On Pendentives

A tale of two domes

Perhaps two of the most important domes from antiquity are those belonging to the Pantheon in Rome and the Hagia Sofia of Istanbul. Although the latter is to be found in modern day Turkey, it is of course from the era of Byzantine rule in Constantinople and is therefore of Roman origin.

From their outward appearance the two domes would appear to be similar, indeed how different can the construction of a dome be? Further inquiry would reveal that the materials used to construct the two domes are different. The Pantheon is made of unreinforced concrete, while the Hagia Sofia’s dome is of masonry. Indeed the Pantheon remains the world’s largest unreinforced concrete dome to this day.

This difference is interesting, but in my view is not fundamental to the way in which these two revolutionary structures work. I am perfectly aware of the Pantheon’s oculus, its coffered soffit, the variation in concrete mix over its height, all of which were designed to save weight. These are important details, which are both interesting and worthy of study, maybe I will write about them in some future point, however they are not directly relevant to the subject of this post.

The first thing to note is that classical domes, like later gothic structures, are what I would describe as gravitational or compressive equilibrium structures. That is to say that their structural adequacy is dependent on their shape and not on materials science. This is possible because actual stresses are compressive and sufficiently low that, providing equilibrium is maintained, material strength is unimportant. This makes sense because materials science, at least in the modern sense of stresses and strains, did not exist when they were built. 

Those familiar with the structures in question will no doubt be aware of known cracking in both domes, which might be taken to suggest that there is in fact some material science going on, however as we shall see this is not the case.

To understand the primary difference between the Pantheon and Hagia Sofia, perhaps it is first necessary to explain how a generic dome works. In section domes behave in a similar manner to arches, because their curved profile exerts both vertical and lateral thrust at the seating[1]. Domes are of course unlike arches in the sense that they are 3D structures. This means that the aforementioned vertical thrusts are expressed as compressive meridional stresses extending from the crown of the dome to it’s base. The lateral thrusts push outwards in all directions generating a circumferential or hoop stress that cause domes to spread. It is the way in which the meridional and circumferential stresses are resisted that makes the difference.

Like barrel vaulted structures from the classical period of history the Pantheon is supported on heavy walls that follow the profile of the roof structure, in this case a cylinder, in order to buttress the roof against spreading. Some descriptions I have read speculate a stepped thickening observed at the dome’s base is designed to provide a circumferential tie. Maybe their authors have done more research than me and have data to support this view, however I am disinclined to adopt it based solely on my own intuition that the tensile capacity of concrete, albeit Roman concrete, is too low. Also, if tension were present it would imply materials science is at work to provide the required equilibrium, which is philosophically less satisfying.

It occurs to me that a more elegant solution, which maintains the idea of gravitational equilibrium, would be one where the purpose of the steps was to increase weight at the head of the supporting wall in order to push the dome’s thrust line back into the supporting walls. In essence it would behave, at least in my estimation, like the pinnacle atop a flying buttress.

The dome of the Hagia Sofia is different. It does not find support from heavy buttress walls. Rather it straddles the four corners of a vast open space into which light and air may flood. In character it is a medieval structure whose load paths concentrate the dome’s weight into a carefully defined masonry skeleton. The invention that makes this possible are the inverted triangular masonry panels, known as pendentives, that are located over the four supporting piers. As they spread outwards from their apex a series of four arches are formed on the dome’s perimeter. Together these elements funnel load into the supporting piers where the in plane arch thrusts are buttressed.

It is evident however that there remains unbalanced thrusts perpendicular to the apex of each pendentive arch. Equilibrium is restored by hemispherical domes that lean in the opposite direction to the dome’s thrust in one direction and buttresses in the other.

And so it is that the dome at Hagia Sofia represents a transition between heavy buttress walls and gothic cathedrals of the later medieval period, whose structures had a more clearly defined order of primary and secondary elements and a more sophisticated understanding of load paths.

Now, to the aforementioned cracks in both domes. Many seasoned observers hold the view that these are the result of past seismic events and differential settlements. Indeed the original dome at Hagia Sofia is known to have collapsed during an earthquake leading to the present cupola being constructed with a higher profile in order to reduce the magnitude of lateral thrusts. 

Nevertheless it would appear that in spite of movement to both structures gravitational equilibrium has been restored. They remain stable, or at the very least, are moving very slowly.

[1] For further information I have written several prior posts relating to the behaviour arch structures.

On San Petronio

Gravitational equilibrium & the square cube law

‘Gulliver’s Travels’ is a classic of English Literature written by Jonathan Swift in 1726. It is intended to be a satire of human nature and ‘travellers’ tales’. In the book its protagonist visits the fictional countries of Lilliput and Brobdingnag. The citizens of the former are 12 times smaller than Gulliver and in the latter they are 12 times bigger.

Of course every reader of the book knows that Lilliput and Bribdingnag are fictional, but perhaps fewer might realise that they are necessarily fiction in any possible world that follows the same physical laws as our own. The reason for this is described by the square cube law, which was first attributed to Galileo.

According to this law the size of things cannot be indefinitely scaled up, because the physical properties of objects change as they increase in size. Consider a cube with sides one unit long. If we were to double the length of each side then the surface area of each face will increase from 1 to 4 units. The volume of the cube, and by extension the quantity of stuff from which it is made, increases from 1 to 8 units i.e. doubling the linear dimensions causes the surface area to be squared and the volume of stuff to be cubed.

In other words the rate at which the volume and weight of an object increase with size is greater than the rate at which its linear dimensions and surface area increase.

Conversely, the strength of things, normally expressed as a limiting stress, is independent of size [1]. It follows that as things become larger, while retaining their original proportions, they will eventually reach a point where they can no longer support their own weight. This places an upper limit on how big things, including people, can get.

This is a vexing problem for modern day architectural students, who are surprised to learn that the model they have spent hours building does not prove that the structural gymnastics their design requires are viable in the real world.

There is of course an exception to the square cube law, though not a true exception. The square cube law does not cease to work, rather it is not discernible within the normal range of scaling for certain objects.

An example of such an object from the natural world would be a mountain. An equivalent structure form the man-made world would be a pyramid. Both structures come from a class of things that share three key ingredients. Firstly, they are both made of stone, which is a natural material with a very high compressive strength and low tensile strength. Secondly, they are both compression structures that assiduously avoid tension. Thirdly, they are stocky and solid structures. Not solid in the sense that they are strong, though that is indeed true, but solid in the sense that they are not hollow. The implication of being stocky and solid is that they are not prone to buckle, as a slender or thin walled structure is. A second implication of stockiness is a large cross-section, which implies low stress.

These then are the ingredients for subverting Galileo’s square cube law and within the class of things which have these ingredients there is a group of structures that does so with a style and panache that is difficult to surpass. They laugh in the face of the square cube law.

I am of course talking about gothic structures; those great stone cathedrals of the late medieval period with their quadripartite vaults and flying buttresses. I am often asked, mainly by my dad, how it was that medieval masons created such structures in the absence of modern structural theory. The stock answer is that they used rules of thumb, but for me there has to be more to it than that.

How on earth did they manage to design structures, using rules of thumb, that modern analysis shows us to have near perfect proportions. Rules of thumb are intended to apply generally, but must necessarily be derived from the particular. The further a design departs from the particular the less useful a rule of thumb is. 

Gothic structures do have similarities, but there is also great variety in their design, which ought to make rules of thumb less helpful.

Perhaps the answer to this dilemma is to be found in a process of trial and error. While there was undoubtedly a role for trial and error I am not convinced that it played a central role, at least not in the commonly understood sense. I hold this view for several reasons:

Firstly, most cathedrals took hundreds of years to build and though master masons may have worked on several, they would not, in their own lifespan, have time to learn all the necessary forms by trial and error. This brings us back to rules of thumb. Secondly, while there is evidence of design evolution over time there is relatively little evidence of major failures, which is odd given the innovative, and often spectacular, designs adopted. Furthermore, when known failures occurred they were often, though not always, associated with abnormal events like earthquakes or phenomena external to the structure like differential settlement.

We therefore have much evidence of trying daring new things, but scant recorded evidence of failure [2]. This is not entirely a surprise, because cathedrals are expensive and their proprietors were not stupid. Master masons would not have been in charge for long if their structures kept collapsing.

To find a satisfactory explanation I think we need to return to the square cube law. If a structure could be designed to subvert the square cube law then a successful pattern would be successful at any scale. It follows that if you could demonstrate a particular form of structure would stand using a small model made of wood then it would also stand if it was scaled up to full size. 

I do not know whether masons understood that they were subverting the square cube law. I suspect that they didn’t, but I do think that they understood perfectly well the load-paths and principles that were necessary to keep a stone structure in equilibrium. I think they fully understood that tension was the enemy and that gravity must be harnessed to maintain compression in all parts of the structure. 

They were specifically designing structures to achieve gravitational equilibrium and they were doing it by experimentation with scale models. They then used rules of thumb, derived from these models, to scale up their findings to full size structures.

 

Happily for the masons this methodology was just perfect for avoiding the square cube law, whether they knew it or not, and in this way spectacular and innovative designs could be realised without needing to know anything about materials science, stresses or strains.

While this theory involves a heavy dose of speculation it is not without evidential support. For example, we know that Antonio Vicenzo the designer of San Petronio church in Bologna commissioned a model of brick and plaster at circa one eighth scale. It was around 19m long and 6m high. I expect that it was used to convey the design to its proprietors, but I don’t think it is too great a leap to posit that it might also have played a role in the church’s structural design.

[1] I accept that all bets are off at the atomic scale, but we don’t normally consider atomic forces when designing building structures, bridges and the like.

[2] I know that absence of evidence is not evidence of absence, nevertheless evidence is sometimes notable by its absence, particularly when there might be a reasonable expectation to find some.

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 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.