3 ms·

>are you sure that thermal expansion imparts enough force against 175 miles of steel in order to push it all the way to the station? Nope. That's why you may n

>are you sure that thermal expansion imparts enough force against 175 miles of steel in order to push it all the way to the station? Nope. That's why you may need to push it! :) Assume the rollers have the Crr of a passenger train wheel (0.0020). If each section weighs 193 tonnes (23 mm thick by 2.23m diameter by 100' by 8 g/cm³), then each roller will provide ~400 N of friction. Over 175 miles (worst case) that works out to 3.56 meganewtons of resistance to motion. For a thin-walled cylinder, buckling stress is modeled using the large deflection approximation[1] as: σ_x = 0.238 Et/r Where σ_x is the bucling pressure, E is the material's Young's modulus, t is the wall thickness, and r is the radius. Plugging in 190 GPa, 20 mm, and 1.125 m yields… 58 meganewtons, or about 16 times as strong as the predicted frictional force I guess they may not need active control after all! But what about curves? Won't it result in a lot of sideways force? Needs more calculations… [1] The weaker of the two. http://www.dtic.mil/dtic/tr/fulltext/u2/a801283.pdf http://www.dtic.mil/dtic/tr/fulltext/u2/a801283.pdf

sillysaurus3 12y ago
Sweet calculations! Thanks! Check out my above comment again, since I was apparently writing out an "EDIT 2" while you were doing these awesome calculations. Basically, what do you think about the fact that it's not a long, straight tube? The rollers would need to dynamically bend the steel tube in order to carry the expansion all the way to the station, wouldn't they? Is that possible? Even if there aren't any rollers, the tube still needs to re-bend itself somehow. But how? That would mean steel curves near the endpoints would move a whopping 150 meters in total. So it seems like part of the tube manufactured as a curve in summer would need to become a straight length of pipe by winter.
- schiffern 12y ago
  Heh, I love that we thought of two different but related problems simultaneously! First yours, since it's much cooler. The minimum curve radius is 3.67 km, or a length difference of 18 mm between the left and right side over the length of one pylon. Assuming that all sections start as straight tubes, how much sideways force would that apply on the pylon? For that we need to calculate the "flexural stiffness"[1], which for a hollow cylinder is given as: EI = E pi/4 (D⁴ - d⁴) Where E is Young's modulus, and D and d are the outer and inner diameter. Therefore the bending moment[2] M is: M = EIκ = E pi/4 (D⁴ - d⁴)κ Where κ is the curvature, equal to 1/r. Finally, muddling through the equations on the bending moment article (and treating the tube as a 200' beam with a load in the center) yields: f = M / (100')²/(200') [not sure about this one] A whopping 300 tonnes•f of lateral force on the pylons! Probably doable, but it means if your tubes are constructed straight your pylons are going to be buttressed at a ~60° angle. Also, note that this only applies to the slip distance (~150m, or three pylons), since the tube sections well within the curve will never move enough to be in a non-curved section of track, so they can be built curved from the beginning. Also you can vary the curvature smoothly[3], which can substantially reduce the forces involved. For example, if you eased into the curve over 30 pylons and it only moved by 3, the sideways force would be reduced by 10x (since the curvature is always "about right"). Further calculations here are left as an exercise to the interested reader. :D As for my own objection… Even on the sharpest turns successive pylons will only vary from straight line by 0.15°, so the lateral forces caused by expansion and contraction shouldn't exceed 9 tonnes force. Piddly by comparison. [1] http://www.explorelifeonearth.org/cursos/CB-18%20simple%20structs%3B%20EI,%20Euler.pdf http://www.explorelifeonearth.org/cursos/CB-18%20simple%20st... [2] https://en.wikipedia.org/wiki/Bending_moment#Example https://en.wikipedia.org/wiki/Bending_moment#Example [3] i.e. the curve doesn't start suddenly, but gradually; this is how actual tracks/roads are laid out. https://en.wikipedia.org/wiki/Euler_spiral https://en.wikipedia.org/wiki/Euler_spiral TL;DR as long as you ease into the curves (which is good for comfort anyway), it looks totally doable even without motorized rollers.
  - sillysaurus3 12y ago
    Do you have an email address we could continue this at? I'm going to need a little while to get back into a happy mindset after a rather unfavorable exchange upthread. I'm really looking forward to picking your brain though. Thanks for chatting with me.
    - schiffern 12y ago
      Sent. Thanks for the convo!

sillysaurus3 12y ago

Sweet calculations! Thanks! Check out my above comment again, since I was apparently writing out an "EDIT 2" while you were doing these awesome calculations. Basically, what do you think about the fact that it's not a long, straight tube? The rollers would need to dynamically bend the steel tube in order to carry the expansion all the way to the station, wouldn't they? Is that possible? Even if there aren't any rollers, the tube still needs to re-bend itself somehow. But how? That would mean steel curves near the endpoints would move a whopping 150 meters in total. So it seems like part of the tube manufactured as a curve in summer would need to become a straight length of pipe by winter.

schiffern 12y ago
Heh, I love that we thought of two different but related problems simultaneously! First yours, since it's much cooler. The minimum curve radius is 3.67 km, or a length difference of 18 mm between the left and right side over the length of one pylon. Assuming that all sections start as straight tubes, how much sideways force would that apply on the pylon? For that we need to calculate the "flexural stiffness"[1], which for a hollow cylinder is given as: EI = E pi/4 (D⁴ - d⁴) Where E is Young's modulus, and D and d are the outer and inner diameter. Therefore the bending moment[2] M is: M = EIκ = E pi/4 (D⁴ - d⁴)κ Where κ is the curvature, equal to 1/r. Finally, muddling through the equations on the bending moment article (and treating the tube as a 200' beam with a load in the center) yields: f = M / (100')²/(200') [not sure about this one] A whopping 300 tonnes•f of lateral force on the pylons! Probably doable, but it means if your tubes are constructed straight your pylons are going to be buttressed at a ~60° angle. Also, note that this only applies to the slip distance (~150m, or three pylons), since the tube sections well within the curve will never move enough to be in a non-curved section of track, so they can be built curved from the beginning. Also you can vary the curvature smoothly[3], which can substantially reduce the forces involved. For example, if you eased into the curve over 30 pylons and it only moved by 3, the sideways force would be reduced by 10x (since the curvature is always "about right"). Further calculations here are left as an exercise to the interested reader. :D As for my own objection… Even on the sharpest turns successive pylons will only vary from straight line by 0.15°, so the lateral forces caused by expansion and contraction shouldn't exceed 9 tonnes force. Piddly by comparison. [1] http://www.explorelifeonearth.org/cursos/CB-18%20simple%20structs%3B%20EI,%20Euler.pdf http://www.explorelifeonearth.org/cursos/CB-18%20simple%20st... [2] https://en.wikipedia.org/wiki/Bending_moment#Example https://en.wikipedia.org/wiki/Bending_moment#Example [3] i.e. the curve doesn't start suddenly, but gradually; this is how actual tracks/roads are laid out. https://en.wikipedia.org/wiki/Euler_spiral https://en.wikipedia.org/wiki/Euler_spiral TL;DR as long as you ease into the curves (which is good for comfort anyway), it looks totally doable even without motorized rollers.
- sillysaurus3 12y ago
  Do you have an email address we could continue this at? I'm going to need a little while to get back into a happy mindset after a rather unfavorable exchange upthread. I'm really looking forward to picking your brain though. Thanks for chatting with me.
  - schiffern 12y ago
    Sent. Thanks for the convo!