Musing the other day with friends about the gigantic structures Albert Speer had envisioned for Third Reich Berlin, I began to think about the structural issues of a giant dome. I suppose the limit to a dome supported by steel trusses is roughly the same as any lengthy truss. But what about a stone dome? Would it be possible to fit stone pieces such that gravity locks the pieces into place to support each other like an arch does? Or does the fitting of one piece to the next require close tolerances that grow impossible as the span grows larger?
A stone arch works because it manages to keep the stones in compression, and there are no tensile loads. This requires balancing forces to push inward from the outside - which is the reason for such constructs as flying buttresses, or requirements for lots of wall each side of a simple arch. A steel truss in contrast is usually an element that is on both compression and tension, and resists bending and buckling because it resists both. A steel arch could in principle be designed in much the same manner as a stone one - and it could end up with only compressive loads. But you still need to constrain the arch, so you would need tensile bands around the arches. Which steel is good for. The
Your ultimate limit is the usual question. Given the strength of a material and its density, there is a maximal sized structure you can make. For stone you are limited by its compressive strength, and your ability to get enough material in the lower levels to cope with the forces and constrain the structure. As we know, there is a maximum size you can make a mountain for this reason. Eventually, no mater what, the mass of the structure exceeds the strength of the stuff at the bottom and it can’t get any higher. For a dome (or arch - a dome is just a 3D arch) the internal compressive loads will be high due to lateral loads as well. But given arbitrary construction technology, you could probably build a dry fitted stone dome of geological proportions. It would more resemble a squat mountain with a dome carved inside the middle.
For very large domes, you would probably not use a solid construction, but a skeleton carrying a light weight skin according to the constructions of Buckminster Fuller
An issue with any large structure is that common materials are fairly homogenous on scales of inches. But not on a scale of miles. With the result that a dome of stone blocks a mile across will be far weaker than an equivalent dome magically carved out from a single piece of homogenous stone the same size.
As well, the larger the structure the more opportunity there is for local dynamic loads to aggregate across a large area into a load concentration at some essentially random spot. So the bigger the structure the more overbuilt it needs to be. Which eventually turns into a prohibitive example of the law of diminishing returns.
This is overcome in common construction by using built-up wooden beams. An LVL beam (laminated veneer lumber) is far stronger then a single piece cut beam, to the point that much lower grades of lumber can be used in the LVL since defects cancel each other out.
I don’t know about beams measured in miles.
You are right about stone, and it doesn’t have to be all that big. I have seen stone steps that fail when being installed due to a hidden weak vein.
Dennis
Whenever I imagine a domed city. I have large buildings properly spaced to support the dome. They also serve as places to install ventilation control. Maintenance facilities. They collect and direct water that falls on the dome to artificial lakes and reservoirs. Lately I envision the buildings also having electrical installations to collect power from solar panels on the dome.
Never just a giant self supporting dome above all the city. Rings of buildings in descending height. Useful in many ways.
Here is a link to the Wikipedia entry on the Pantheon:
Short version: The Pantheon is the largest unsupported (Roman) concrete dome in the world, however it’s not quite what you asked.
hewre is a link to an article about the dome of Hagia Sophia:
Short version: The dome is built by brick and mortar, it is 31, 24 mt (102 ft 6 in) diameters and is 55, 6 mt (182 ft 5 in) high. (copied from the website)
Again, not dressed stone against dressed stone as I understand you to be asking.
So my problem is that I’m imagining stone as having virtually limitless strength in compression. But, in fact, dressed stone pieces around the base of an enormous dome—even one 50 meters across—would be subject to such enormous pressures that a piece would delaminate or fail in some other way.
Do I have that right?
Here is a link to the Wikipedia entry on the Pantheon:
Short version: The Pantheon is the largest unsupported (Roman) concrete dome in the world, however it’s not quite what you asked.
hewre is a link to an article about the dome of Hagia Sophia:
Short version: The dome is built by brick and mortar, it is 31, 24 mt (102 ft 6 in) diameters and is 55, 6 mt (182 ft 5 in) high. (copied from the website)
Again, not dressed stone against dressed stone as I understand you to be asking.
Yes. Not just stone, but bricks and other masonry. If you ever get the chance to look inside those tall smoke chimneys at a factory, the walls are very thick at the bottom just to support the weight from above. 3 to 4 feet thick at times.
They were built in stages, no more than about 6 feet at a time and then several days delay. Any faster and the mortar will not be cured enough to support the next layers.
Skyscrapers as we know them could not be built until iron structures became practical.
Dennis