Have you ever thought about how a dome stays rooted in place? The foundation, of course, but there’s more to it than you might think.
When constructing buildings out of brick and metal, the purpose of the foundation is to resist the downward force of all those heavy materials as well as the lateral forces of the wind. A dome is a little different though.
Domes weigh very little relative to a building—this means the foundation needs to deal with different stressors. In the case of domes, the foundation also needs to account for something completely counter-intuitive: upwards force. Wait, so do we mean domes are trying to rise up off the ground and into the sky?
Well it would actually never happen, but yes, that’s what domes need to be engineered against. Think about it. Domes, especially in the winter, are usually heated—meaning the air inside of a dome is warmer than the air outside. Another system not dissimilar to this exists in the form of a hot air balloon. The only difference is that we want a hot air balloon to float away—not so much with a dome.
But just how much force are we talking here? It’s actually not too tough to figure out with a little bit of science and math.
First, we need to calculate just how much air is inside of a dome. For argument’s sake, let’s consider the Toronto FC’s training dome in Toronto, Ontario.
This dome is a training space for Toronto’s MLS soccer team to train in during the winter months. The last thing they want is to have dozens of highly paid athletes floating off into the sunset.
Let’s assume that it’s a cold day at -15?C outside, but the inside of the dome is a comfortable 20°C (that’s about 5°F and 68°F). The TFC Academy dome’s dimensions are 364’ long by 242’ wide and 75’ tall. Converting to metric (which makes the math a bit easier) we have a dome that’s 111m X 74m X 23m.
A dome is, more or less the shape of half of an ellipsoid. OK, not quite, but for our purposes (and in the interest of not making things super complicated) it’s an assumption that will get us pretty close. All we need to do is take an ellipsoid, slice it down the middle like a sub sandwich, and we (almost) have a dome.
The volume of an ellipsoid is V = (4πabc)/3, where a, b, and c are the distance outwards from the centre of the ellipsoid. For our dome, these would be length divided in half (55.5m), width divided in half (37m) and the height (23m). These values give us an internal volume for the ellipsoid of 198,000 m3 (if you want to check for yourself, Google-ing “ellipsoid volume” results in a pretty useful calculator). Since a dome is only the top of the sub sandwich, the TFC dome has a volume of (roughly) 99,000 m3.
The next equation is one you might remember from high school chemistry: the ideal gas law, PV=nRT. Without getting into all the math here (you can do it yourself if you’d like), we can assume the density of the air inside the dome to be 1.2051 km/m3 and outside 1.3685 km/m3 (assuming the same atmospheric pressure inside and outside of the dome.)
Finally, we can calculate the net buoyant force of the dome in these conditions. By figuring out the difference in density (1.3685 km/m3-1.2051 km/m3=0.1634 km/m3) and multiplying that by the volume (99,000 m3), we end up with 16,176.6 kg of upwards force! Of course, the weight of the dome and anything attached to it will counteract this upwards force, but it is still a considerable amount of force pushing the dome upwards.
This is why it’s important for us to over-engineer the foundation to account for all this upwards force. We also help counteract this force with cables to provide even more strength as needed for domes in areas with higher winds and even lower temperatures.
The last thing a team needs is for their air dome to float away!