A couple of weeks ago, a friend of mine from college got married in Rome, providing an excellent excuse to finally go visit a city I’ve wanted to see for years. We spent about a week there, doing some heavy tourism despite the unpleasant heat. It’s a gorgeous city, and we didn’t come close to seeing everything it has to offer, but there was one site we made a point of visiting twice: the Pantheon.
The Pantheon, which technically is the “Basilica di Santa Maria ad Martyres,” having been consecrated as a Christian church in the 600′s, was originally built as a temple to all the Roman gods back in the reign of Augustus, but destroyed in one of ancient Rome’s many devastating fires. It was rebuilt around 120 CE, completed in the reign of Hadrian (who decided to confuse future archaeologists by re-using the original massive portico with its inscription crediting the building to Marcus Agrippa), and has remained remarkably intact for nearly 1900 years.
The signature feature of the Pantheon is its giant dome, 43 meters across and 43 meters high, with an 8-meter open “oculus” in the center. The dome is made of concrete, then a relatively new material perfected by the Romans, and to this day is the largest un-reinforced concrete dome in the world.
The dome is a truly awe-inspiring sight, and like most tourists who visit the place, I immediately wondered “How in the world did they make that?” As I’m a great big geek with access to an academic library, this wondering has gone on a good bit longer than most, and led to reading journal articles about computer simulations of the Pantheon’s dome, and to checking out multiple books on Roman architecture and construction techniques. I’ll probably have more to say about that in another post, but for the moment I want to talk about a little basic physics.
You probably won’t be surprised to learn that there’s a great deal of science involved in the construction of something as colossal as the Pantheon, but in the general reductionist spirit of physics, we can break it down to a really simple balance of two forces: gravity pulling the components of the roof down, and forces from the concrete and bricks of the structure trying to keep it up. The Pantheon succeeds because its designers did a brilliant job of playing these off against each other, but an inherently three-dimensional structure like a dome is really complicated, so in the traditional manner of physics analysis, we’ll simplify it further, because the essential idea is captured by another signature feature of Roman architecture, the arch.
The photo above shows a section of the Aqua Claudia, one of the many aqueducts bringing clean water to Rome from 40-odd miles away. This is slightly older than the Pantheon, begun during the reign of Caligula and finished in the reign of Claudius (though it was repaired several times after that), and also an amazing engineering achievement. The water flowed underground for a large portion of its course, but when the ground drops away closer to the city, the channel was supported on this long chain of arches, providing a more gentle drop in elevation that avoids too-rapid flow or backing up.
The arches supporting the aqueduct give it a kind of elegant look, but they might seem like an unnecessary complication for supporting what is, after all, essentially just a straight pipe. The Romans knew, though, that arches are, in fact, the best structure for this kind of thing, providing a sturdy and stable support for the water channel that’s held up for nigh on 2,000 years. And it’s all thanks to physics.
The best way to explain why an arch is a good structure for supporting an aqueduct is to start by explaining why a flat top would be a bad idea. So, let’s think about bridging some gap with just a straight piece of material, like this:
I’ve drawn this as five blocks right next to each other, but it needn’t be an actual division– you can think of a continuous piece of material, if you prefer, and only imagine breaking it into segments for purposes of physics analysis.
If we zoom in on just one segment, there are three forces that matter: a force of gravity pulling down (green arrow), and two forces from the neighboring segments pulling it up to keep it from falling (reddish arrows). If it’s a set of blocks, like the classic Don’t Break The Ice game, these would be frictional forces (and there would need to be compression forces pushing in on either end); if it’s a solid piece of material, they’re internal forces from the bonds holding whatever it is together. If the straight bar isn’t going to fall, these forces must add up to zero, and if you imagine stacking the two red arrows on top of each other, you can see that they’re the same height as the green one.
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The critical issue, though, is where these forces act. Gravity pulls on every subpart equally, which has the net effect of acting on the center of mass of the segment. The forces from the neighboring segments, though, are contact forces, and act where the segments come together, parallel to the boundary. The end result is an upward push on both ends and a downward pull in the middle, which will tend to stretch the segment down, and that, in turn, causes the whole thing to stretch a little. What happens then depends on the material you’re using. Something like wood or metal has a decent amount of tensile strength, and can stand up to a bit of stretching, but stones, bricks, and concrete have much less tensile strength, and will tend to fracture. At which point the whole thing comes tumbling down. So building a flat-topped aqueduct out of stone is a Bad Idea.
An arch, on the other hand, lets you redirect forces in a way that avoids this tendency to fracture. If we re-imagine our span as a five-brick arch instead, it looks like this:
In this case, each of the blocks (which again, could be discrete bits or imaginary subdivisions of a continuous material) is a trapezoid shape. If we focus in on the top block, the keystone, we see how this changes the dynamics. We still have a green arrow for gravity pulling straight down, and we still have red arrows parallel to the boundary, but the change in shape means we also get a contact force perpendicular to the boundary on each side, pushing up-and-left or up-and-right. For discrete blocks, this is a Newton’s Third Law force– as gravity pulls the keystone down, it presses into the blocks to either side, exerting a force on them, and they press back with an equal and opposite force.
All four of these forces now need to add up to zero to keep the arch stable, and you can see that it works out by stringing the arrows for the contact forces together and finding that they end up matching the force of gravity (in the lower right of the figure). And while we still have forces acting in different places, because they’re directed inward, they don’t stretch the block, but compress it instead. This is great, because stone, bricks, and concrete have tremendous strength in compression, so the block is much more likely to maintain its integrity. (The parallel-to-the-boundary friction-type force arrows are much smaller here, and if you design your arch properly, you can basically make them as small as you like, making the force essentially all compression.)
There is one complication here, that we can see by choosing to look at one of the blocks out near the end (the technical name for these is “voussoirs,” because everything sounds cooler in French):
For this block, I left out the friction-type parallel forces entirely, to simplify things a little. Here, we still have gravity pulling down, but now the two contact forces have different magnitudes. On the upper side, there’s a contact force from the second block up, which is due to both its weight and the push it feels from the keystone. On the bottom side, the force is much larger, because it needs to match the vertical component of both gravity and the push from the rest of the arch.
If we do the stringing-together-arrows thing, though, we can see that these don’t quite match perfectly– there’s a leftover outward force, represented by the red arrow. This needs to be countered by something else– generally great big wall of some sort– and for this reason you’ll sometime hear people talking about how arches will direct vertical loads into horizontal forces.
The key thing here, though, is that all these forces are compression forces. If you’re making an aqueduct, you can just chain arches together ad infinitum, and the outward push of one will provide the inward push needed to hold the next one up, and vice versa. If you’re making a building, you just anchor your arches in a really massive wall, and the Romans didn’t mess around with this sort of thing– the Pantheon walls are about six meters thick. Either way, the arch gives you a great tool for spanning a big space without actually needing to fill the whole thing with bricks.
(In fact, if you look closely at the exterior shot of the Pantheon, you can see that the top level contains “blind arches,” which are very strongly built to hold the weight of the dome, and then bricked up inside. The Romans loved their arches, even when they wanted a solid wall.)
A three-dimensional dome like that on the Pantheon is a little more complicated than a two-dimensional arch, but the basic idea is the same: the weight of the top bit is supported by the push from lower parts of the dome, in a way that minimizes the tension forces concrete can’t really handle, in favor of compression forces that concrete is ideal for. With a dome, you also need to worry about stretching around the circumference as the outward push tries to make the dome stretch, and the builders of the Pantheon played some clever tricks to counteract that, changing to a lighter concrete mix near the top of the dome, and adding massive reinforcing rings toward the outside.
The central ideas that make the whole awe-inspiring structure possible are just basic introductory Newtonian physics, though. Of course, this raises an interesting question, namely how did the Romans know to do all this, given that the central physics principles weren’t clearly articulated for another millennium and a half? That gets further outside my core expertise, thus the checking out of books from the Union College library, so I’ll save that discussion for a future post…