QUESTION: Hi! I was reading about the sizes and size limits (about the radius or diameter) for O'Neill Cylinders and I came across to 3.2km radius for a steel cylinder and in one site I read that the maximum safe diameter for one made of titanium is 14km and one site refered the 14km as the radius and both sites said that the 23km radius/diameter is the largest possible without safety factors. And then I read Gerald O'Neill's calculations that said that maximum diameter for a cylinder made of titanium would be 12 miles which translates to just around 19 and half kilometer diameter. So which of these assertions is correct, the 14km diameter, the 14km radius or the 19,5 km diameter?
ANSWER: Ummmm...you neglected a few things!
You said 23km radius/diameter...which is is, radius or diameter?
What thickness is the material? What is the atmospheric pressure inside? What is the mass/unit surface area of human civilization inside that these cylinders are supposed to support?
All of these factors are very important. Also the type of steel and grade of titanium. Also the rotation rate, are they supposed to provide 1 standard Earth gravity? People can live with lower gravity and lower air pressure just fine, or we wouldn't have Denver (the air pressure more than the gravity). Are you supposed to make assumptions about these parameters? If you just want the last sentence answered as a question, then you need to answer these questions (thickness, gravitation from spin, pressure, surface density) and choose just one material.
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QUESTION: OK Let's make thickness 5 meters, gravity 1g, pressure 90% of what we have on Earth at sea level and the civilization inside would be 50 million population and they would live in multi story buildings maximum height 300m. I don't know anything about the mass/unit factor you mentioned so if these all figures are needed could you think of some mass/unit of surface area?
There are still so many factors, like what you'd use for the window sections to let sunlight in. Also you never picked a material at all, so we'll go with the ridiculous choice of titanium because it's lighter and stronger. You propose to support 300 meters of buildings on only 5m of titanium at 1g. That alone seems unlikely, even with lightweight materials like carbon fiber you're talking about buildings in the 15x atmosphere range for the pressure that their weight would exert on the surface. Even with only 0.9 atm and the weight of titanium itself (no people) you get 2 atmospheres from the weight of the spinning titanium and the atmospheric pressure. Using Barlow's formula for bursting pipes as an approximation and assuming total rupute (ultimate strength of titanium), I get 1.3 km diameter for the rupture diameter of this structure. This ignores the window sections, which presumably would need to have enough tensile strength to keep the titanium segments from flying apart. Making the titanium thicker (see the pressure section, above) won't help much, as you'll be adding weight to the walls. Also, there's only so far you can lower the buildings the people live in and so much mass you can remove before the people can't sustain themselves. And you need about 8 million metric tons of titanium per kilometer of tube, so I'm not sure where you get the tube material in the first place...or how you get it to space.
In short, leave these to science fiction. The engineering challenges of our biggest projects in history are staggering...but not staggering on this scale.