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 Lunes & Cracks

Conserving St Peter’s Basilica

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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