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 Ice Walking

Just how safe is it?

Today at work I participated in several online meetings, however while we were waiting for colleagues to tune for one of them we found ourselves doing the stereotypically British thing; we talked about the weather.

To be fair the weather was unusual as much of the country, even in the south, had been covered with a blanket of snow. Indeed it had been reported that a portion of the Thames had frozen, which really was an unusual event. This was the predicate for a discussion about whether it would be safe to walk on said ice [you now know this wasn’t written in the summer].

This intrigued me so I set aside a little time that afternoon to try and work out just how safe it would be. This post is about what I found. Now, I should preface what follows by saying that I have absolutely no practical experience of this subject. Everything that follows is a postulation on my part, based on the engineering principles that I believe to be at work. You therefore shouldn’t base your ice fishing trip or curling match on what I have come up with.  

If you must go walking on a frozen lake, I suggest you ask someone who knows what they are talking about and isn’t making it up as they go. Statistically I imagine that such a person is far more likely to have a Canadian accent than a British one. You have been warned!

Ice floats because it is roughly 10% lighter than water. If we are to stand on a sheet of ice we therefore increase its weight making it more likely the ice will sink rather than float. It therefore struck me that the first part of the puzzle ought to be an assessment of how much ice there needs to be to ensure floatation occurs rather than sinking.

A cube of water with sides 1 m long weighs approximately 1000 kg, which means the same cube made of ice must weigh roughly 900 kg. This would imply that for our cube of ice to remain buoyant in water it must carry no more than 100 kg [1000-900].

Now, the minimum recommended thickness of ice for walking is 4 inches; I know because I googled it. That’s equivalent to 100 mm or 0.1 m. 

This means that a block of ice with a square surface that has sides 1 m long, but a thickness restricted to 0.1 m, can carry 10 kg [0.1 x 100]. It follows that if an average person weight 75 kg, then their weight must be spread over a block of ice with an area of 7.5 meters square [75/10] i.e. a square with sides measuring 2.739 m[1].

Everyone knows that ice is a brittle material that fractures rather than bends so that’s quite a large area for load to spread from our feet without something going wrong. I therefore started to think about how the load gets from our feet to cover such a large area and by what mechanisms it could go wrong.

Perhaps our feet would simply punch through the ice like a stiletto heel on soft ground. Such a mechanism would require the ice to shear on a vertical plane passing through the ice, which extends along a perimeter enclosing our feet. The length of the perimeter and the depth of the ice would therefore define the shear plane.

Punching shear yields a stress in the ice equivalent to 0.001 kg[2] acting on every square millimetre of ice on that plane.

The second potential mechanism could result from a shear plane developing across the full width of our notional 7.5 m square block of ice. This time the shear plane being defined by the width and depth of ice. This mechanism yields a stress of 0.0003.

The final mechanism I considered was bending. To transfer load from the feet to the outer edges of our square block the ice must be capable of bending. This is a bit more tricky to calculate, but it results in a stress equivalent to 0.005 kg acting over every square millimetre of ice.

I didn’t really know if any of these figures were significant or not so I got back on google. It turns out that, according to the people who measure such things, ice has a shear strength of 0.06 and a bending strength of 0.07. 

This means that there is a factor of safety against shear failure of 60 [0.06/0.0001] and a factor of 14 against bending failure [0.07/0.005]. I find this quite reassuring, because I don’t actually believe that ice has the strengths I just quoted.

This is not because the diligence of the researchers is at fault; I’m not saying that their work is wrong. What I am saying is that ice is not a manufactured product like steel. It is not made to possess specified properties.

The strength of ice depends on many things. What is the air temperature, did it rise and then fall again during its formation. Is the water fresh or salty? Is there a current or a flow in the body of water that is busy scouring the underside and weakening its structure. Was there snowfall during its formation. Did someone, or something, step on the ice while it was forming thus inducing cracks in its interior.

These and many other issues that I likely haven’t thought of have the potential to change the strength of a given block of ice; its value is therefore not a fixed thing[3]. If I am to walk on ice I would quite like to know that the existence of one or more adverse factors does not completely undermine the strength of the ice I am going to rely on. A factor of 14 sounds good to me. It’s roughly 10 times what I might use for say concrete. That’s about right because I know with far greater certainty what concrete will do.

Obviously if ice melts we are in trouble, but before that point is reached, I have no idea which combination of ice factors might eventually undermine a safety factor of 14. That’s why you shouldn’t take advice about ice walking from someone that’s making it up as he goes.

That being said, my somewhat crude assessment has yielded an interesting conclusion. I started by trying to work out what amount of ice I would require to mobilise to prevent the average person from sinking. My sums therefore exist on just the right side of not sinking, effectively a factor of safety of 1. This assumption eventually yielded a factor of 14 against the ice breaking i.e. in this scenario I am actually more likely to sink than the ice is to break!

[1] I know it would be more realistic to assume a circular perimeter, but I used a square to keep the sums simple. You’ll get over it.

[2] I know that working out stresses in kg is a bit weird, but unless you have a technical background you won’t know what N/mm2 is, or MPa for my European friends, or psi for my American friends. I didn’t want this post to be an explanation of units.

[3] Ice researchers know the strength of ice doesn’t have a fixed value. They provide ranges of values and couch them in temperature limitations and so forth. I picked from the lower end of the scale. 

On Hennebique

Understanding an early patent system

In 1892 Francois Hennebique patented his eponymous ferro-cement system, which is today recognised as one of the earliest forms of reinforced concrete. It was used under licence in many countries, including the UK, where Mouchel was the local partner. The Hennebique system was conceived before the era of codified design, so its worth trying to understand its structural load-paths. 

To do this we must think of Hennebique’s creation, not as a beam, but as a truss made of composite materials. This may seem like an odd thing to do, but it is necessary to explain how stresses are distributed throughout the section. This approach also, as we shall see, highlights several weaknesses in Hennebique beams.

To make sense of this analogy we need to remember that concrete is strong in compression, but weak in tension. Conversely, wrought iron is equally strong in both. It follows that the key to visualising the load-path is see tension where there is iron and compression where it is absent. 

That said, before we look at the Hennebique system itself it is useful to remind ourselves of the alternatives that were available at the time.

The picture below shows a brick jack arch floor, which was conceived as a fire proof system, although given the exposure of the iron flange on the soffit it is more correctly described as non-combustible. The load paths for a jack arch floor are straightforward. The brick arch spans laterally and is supported on iron beams spanning into the page. There are tie rods evident so that arch spreading is contained and only vertical load is transferred by the brickwork.

 An improvement on this design was to replace the brickwork by fully encasing the iron beams with concrete. In principle this would certainly improve the fire resistance of the floor and make it more durable. At least it would have if the concrete didn’t include breeze, which sometimes contained unburnt coke, and sulphates, which could form a mild acid in the presence of water.

This form of construction is known as filler joist construction. It was very common, particularly in the early twentieth century, and many examples remain today. The load path is essentially the same as for the jack arch, except the arch form must be imagined within the body of the concrete, because it would have been much simpler to create a flat soffit than a curve.

The filler joist floor therefore remains a rudimentary structure; there is no composite behaviour between the iron and concrete.

This relationship was changed when it was realised that an inverted T-shape would be more efficient because the concrete fill could resist the compressive stress to which the top iron flange had been subject. The bottom flange was still required to resist tension and the web transferred load between the two by resisting shear.

 

This was an important breakthrough, because the floor was now a composite system, which shared load between concrete and iron. At that time wrought iron was exceedingly expensive and therefore this was much more than an analytical curiosity.

Hennebique’s genius was to make two further steps; or perhaps two and half. Firstly, he replaced the bottom flange of the beam with round iron bars. These were easier to make than a T-beam and had a greater surface area than a single flange with which to bond with the concrete.

 

Secondly, he did away with the iron web, which transferred load between the top and bottom of the beam. To understand the way in which he did this it is helpful to look at other systems that were common at the time.

The next image shows a trussed girders taken from a carpentry manual written in the 1860’s. I have previously written a longer post about this topic, however the key issue in this instance is the way in which timber at the top of the section is used in compression and iron rods are used in tension. Short compression stools are used to transfer load between the two.

  

As can be seen in the following image Hennebique uses exactly the same load path for his concrete system. It is reasonably straightforward to see the tension elements, highlighted in red, however the compression parts, highlighted in blue, must be imagined within the body of the concrete, much as the compression arch is imagined within the filler joist system. I do not mean imagined in the sense that the load path does not really exist. Imagined only in the sense that not all of the concrete in the beam is contributing to the load path.

When Hennebique tested his system he found that though it was successful it did not work as well as it ought; diagonal cracks formed with increasing frequency towards the end of the beam. Such cracks are related to the interaction of shear and bending forces. From Hennebique’s writing I am not entirely sure that he fully understood this mechanism, though nevertheless he found an effective solution. That may be because he found it difficult to describe or I have found it difficult to follow his writing.

 

I think Hennibque believed that there were longitudinal shear forces parallel to the length of the beam, which were causing the failure he had observed in testing. He thought that by intercepting these longitudinal stresses with vertical stirrups of bent iron he could improve the strength of his beam. While such stresses do exist there are also vertical shear stresses, which means that Hennebique’s thinking, if this was what he thought, is incomplete.

Nevertheless, Hennebique’s solution did work and cracking was avoided, though perhaps not wholly for the reasons he thought. We can understand why by referring to the next image, which shows a modern understanding of shear transfer; again red represents tension and blue compression.

We can see in this example that the load path is a fully formed truss with the stirrups and iron bars resisting tension and the top chord and diagonals resisting compression. This modern understanding highlights one reason, beyond a lack of clarity in Hennebique’s writing, that I think Hennebique did not wholly understand the load path.

The case we have thus far examined has a single span with tension at the bottom and compression at the top, however if we were to add additional spans the relationship we have established reverses at the support, with tension at the top and compression at the bottom. In such circumstances we encounter a problem with Hennebique’s stirrups. At mid span they are hooked around the tension bars, but at the support they are open at the top and can be pulled clear. Thus, for multi-span beams the Hennebique system is less efficient than single spans. Had he fully grasped the load-path I am sure Hennebique would have corrected this. Perhaps, as shown above, his tests were all conducted on single spans.

Another shortfall of the Hennebique system is with the tension bars themselves. It will not have escaped the notice of readers with a keen eye that Hennebique’s tension bars have hooks at the end. These were, I am sure, intended to improve the transfer of load between the iron bars and the concrete. At face value this was a sensible measure, because at failure the bars were simply pulled through the concrete. This meant a premature bond failure before the iron had reached yield. This happened because, unlike modern bars, which are deformed to improve the bond, Hennebique’s bars had a smooth surface.

While seemingly a good idea the noted hooks did not really work, though we can forgive Hennebique for this, because the reason why is really rather complicated. In simple terms the bond on the bars must fail before the hooks can be mobilised. Why this happens is perhaps a subject for a different post.

Notwithstanding these shortfalls, which are made with the benefit of hindsight, Hennebique’s system was ultimately very successful and marks him out as a significant figure in the pantheon of structural engineering.

While it is true that Hennebique was not the only person to develop a patent system for reinforced concrete; his was perhaps the most successful. This was probably due to the licensing system he operated marking him out as a great businessman as well as a great engineer.