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 Ductility

The benefit of elastoplastic materials

One of the most useful structural properties a material can have is ductility. In order to understand why this is so it is first necessary to explain some related concepts. We will start with stress and strain.

Stress is a measure of load intensity; more precisely it is the ratio of force divided by area. This concept can be visualised by considering how it is that someone can lie down on a bed of nails without being harmed, yet if the same person were to tread on a single nail it would pierce their foot. The reason a bed of nails causes no harm is because the person’s weight is spread over the cumulative area of many nails whereas a single nail concentrates a person’s weight over a very small area i.e. the bed of nails applies low stress, while the single nail applies high stress.

When a structural member is exposed to stress it will change in length. If the stress is tensile it will become longer and if it is compressive the reverse is true. If the material is uniform then the elongation or shortening is spread evenly along the length of the member. For example, a member that is half as long will change in length by half as much. The ratio of elongation or shortening per unit length is defined as strain.

Stress and strain are of course related; higher stress will cause higher strain. A useful way of expressing this relationship is to plot a stress-strain graph. The convention is to measure stress on the vertical axis and strain on the horizontal axis.

If a material is ductile, like steel, the relation of stress to strain will be directly proportional resulting in a straight line on our graph [1]. The inclination of the straight line is know as the elastic modulus, for reasons we will see shortly.

There will, however, come a point where strain will increase more quickly than stress and the graph is no longer a straight line. The proportional limit of the material has thus been breached. Soon after this point the graph will become horizontal, which means that strain will increase without a corresponding increase in stress. This threshold is known as the yield stress.

Beyond yield the internal structure of the material will begin to change at an atomic level. This process is known as strain hardening and it results in additional strength, and therefore higher stress, in return for further strains.

Eventually the stress-strain curve will again flatten leading to increased strains, but this time with decreasing stress. The point where this occurs is know as the ultimate stress.

Increased strain accompanied by decreasing stress is a rather curious effect. Why would increasing strain result from a reduced stress? There is of course a rational answer to this question. 

When any material is stretched there is a corresponding narrowing to facilitate the stretch. Generally this is too small to notice, however beyond the point at which the stress-strain curve flattens the effect becomes more pronounced and ‘necking’ occurs.

If we were to calculate true stress, based on a necked [reduced] cross-section, rather than continuing with a nominal stress, based on the original cross-section, then the stress-strain plot would in fact continue to grow. After a period of necking fracture will eventually occur.

Returning to the beginning of our stress-strain plot we can modify our approach. This time instead of continually increasing the stress applied to our test member we can load it to a known stress and then unload it again. We can then increase the stress to a slightly higher value and then unload again. This sequence of loading and unloading may be repeated.

When we do this we find that as long as we remain below the proportional limit the unloaded member will fully recover its original shape and will follow the straight line portion of the stress-strain graph. Beyond the proportional limit a full recovery will not be made; there will be some residual strain left behind.

When the member fully recovers its shape it is said to be elastic. The point at which it loses its elasticity is known as the elastic limit. Beyond the elastic limit materials are considered to be plastic. For many materials, like steel. The proportional limit, yield stress and elastic limit are relatively closer together and are in practise treated as if they were the same. For some materials, such as rubber, the elastic limit lies well beyond the proportional limit.

When large permanent strains start to occur within the plastic zone the term plastic flow is adopted. 

There are several reasons why ductility, incorporating both elastic and plastic behaviour, is important. We shall discuss one of them below and save another for the next post.

Supposing we wished to design a series of floor beams for a new building. The owners would not be terribly happy if they were to permanently deform and sag while the building was in use. For this reason we would want them to remain within the elastic range. We would therefore perform our design based on limiting stresses in the beams to the yield stress of the material.

This would satisfy the requirements for every day use of the building, however supposing an unforeseen or extreme event occurred, which caused the floor to become overloaded. In such circumstances you would not wish the floor beams to fail suddenly and without warning, for this would inevitably lead to a loss of life. 

Self-evidently it would be preferable for there to be a visible warning that something was wrong so that the occupants could vacate the floor safely and remedial action could be taken. This opportunity is provided by plastic behaviour in the floor beams. Although permanent deformation will occur within the plastic range the floor will at least remain stable and safe.

It is fairly obvious how this principle will work if the floor beams are made of steel, but perhaps less so for concrete. After all concrete is not a ductile material. It is weak in tension and fails explosively in compression.

The first problem is solved by casting steel reinforcement into the concrete in zones where the concrete is in tension. The reinforcing bars, rather than the concrete, resist the applied tensile stresses.

The addition of steel reinforcing bars also provides the means to deal with concrete’s lack of ductility. The reinforcement is deliberately designed to have a smaller capacity in tension than the concrete has in compression. This way as the reinforced concrete bends the reinforcing bars in the tension zones will reach their elastic limit before the concrete reaches its brittle limit in the compression zones. This principle is known as under-reinforcement and it is key to all reinforced concrete design. 

Thus reinforced concrete becomes a ductile material.

[1] In this example we are of course assuming the application of tensile stress

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. 

On Compressive Membranes

System behaviour in fire

Something that isn’t often appreciated is that when a building is designed to achieve a 90 minute fire rating it doesn’t mean the building is designed to remain standing for 90 minutes. This might seem odd, however if we were to pose the question what size of fire is to be resisted for 90 minutes, it is immediately obvious that there is complexity involved. 

In reality a 90 minute fire rating means that the fire resisting components in the building have been tested in a furnace against a standardised fire lasting for 90 minutes. This allows the relative behaviour of different materials to be tested, however it tells us absolutely nothing about how long a real building will survive subject to a real fire. 

This is in part because fire tests treat standardised components individually, however real components are not a standard size and they act as part of a system not as individual elements. If we are being really picky we might also argue that real fires are different to standardised furnace tests.

It follows that if the structural behaviour of a system subjected to fire can be understood then this can be harnesses en lieu of the rather crude prescriptive approach, which is normally applied.

The floors of many modern buildings are constructed by casting a thin slab of concrete on a corrugated metal deck. The concrete is reinforced using a light steel mesh and the metal deck spans between down-stand steel beams. Often metal studs are welded to the top of the steel beams and are embedded in the concrete. This is known as composite construction, because the steel and concrete act together.

To protect the steel from fire the prescriptive approach is to coat it with a fire resting coating, normally intumescent paint. Intumescent paint swells when it gets hot forming an insulating layer, which prevents the steel from over-heating. Without this insulation layer steel looses significant strength and stiffness at approximately 500 degrees.

If, however, the structural system is taken into account many of the beams may not require intumescent paint. This is beneficial because intumescent paint is expensive. 

In the example shown there are primary beams joining the columns together to form a series of identical bays with two secondary beams in each bay. If we were to suppose that the secondaries are unprotected then we can begin to think about the load-path in a hypothetical fire.

As the fire becomes increasingly hot the secondaries will also become hot and will start to loose strength and stiffness. Eventually they will have little residual capacity. When this happens the floor will begin to sag and instead of supporting the concrete floor the beams will hang from it due to the embedded studs. This process is likely to be accelerated by thermal expansion which causes the beams to buckle as they push against their supports.

Conversely the primary beams, located on the column lines, are protected and will remain unaffected by the ensuing fire. They continue to form a rigid frame around each structural bay. As the floor sags it begins to tug on the primary frame simultaneously pulling each side of the bay towards the middle. Much of this work is being done by the light reinforcing mesh embedded in the floor.

This effect causes a compressive ring to be set up in the concrete at the perimeter of each bay. This ring starts to resist the floor’s tugging and allows a point of equilibrium to be reached where the weight of the hanging floor is balanced by the compressive force in the concrete ring. Although the floor has displaced significantly it has not collapsed and has therefore maintained its integrity. The fact that it has displaced significantly is not materially important, as the sole aim is survival. After a major fire a building would not expect to survive completely unaffected.

This load-path means that some of the primary beams must carry additional load, which was previously supported by the unprotected secondaries. This is acceptable, because in the fire case it is permissible for the additional load to be absorbed by the their factor of safety.

It is also worth noting that the required load assumed for most buildings is in fact much greater than the load the floors will ever see. This means the actual factor of safety is normally higher than is assumed in the cold design.

This form of system behaviour is known as a compressive membrane and I have used it successfully to assess the fire resistance of buildings on several occasions. It is a more rational approach to fire safety than the rather arbitrary prescriptive approach, which has been used historically.