Future construction - a theoretical estimate?

[technobot]

Suppose we had the technology to produce artificial diamond beams and carbon nanotube rope cheaply enough to use in large quantities. If we use the diamond for compressive load, and the CNT ropes for tensile load, how tall a skyscraper could we build using these materials, in theory? Is there any stronger material or combination of materials known today?

 

[russ_watters]

I haven't actually done the calculations, but since people seem to think that we could build a space elevator with carbon nanotubes, the answer is that there is no theoretical limit to the height of a building until you get up to 30,000 miles or so, whereupon the rotation of the Earth starts to pull the building out of the ground.

 

[technobot]

Yes, I've thought of the space elevator as well. But I would like to focus more on the skyscraper problem (sorry if it wasn't highlighted enough). I would think that the problems for a skyscraper would be different, since it is not suspended and hence doesn't gain as much from centrifugal forces, at least until it reaches some minimum height..

I am not a civil engineer, but the two major problems that I'm aware of are the building supporting it's own weight, and it resisting lateral wind forces (and earthquakes). A better shape and some internal damping apparatus may help with the lateral forces, but ultimately the materials should place some limits of their own...

 

[TVP45]

Generally, the first limiting factors you would have to look at is buckling, wind loading, and soil/bedrock loading. In the case of a skyscraper, there would be some building code issues as well.

 

[technobot]

Hmm... forgot about bedrock.. I suppose with a deep enough foundation that can be resolved. Of course, one would also need the technology to build that deep. and I guess you'd have to count the foundation depth as part of the total structure height as far as structural integrity is concerned. So the higher one wants to build, the deeper the foundations, and the deeper the foundations, the less height above ground...

I guess the most basic height limit estimate can be obtained from calculating how much of its own weight can diamond carry.. Then one would have to factor in other influences. I'll try to make that calculation later. In the mean time - what else would I need to calculate?

 

[Danger]

My belief (or maybe just hope) is that things will turn 180.
Building underground is the proper way to go.

 

[technobot]

Since ancient times our species has always looked toward the sky. For that reason I find it hard to believe we will cease our attempts to build ever higher. However, we may do both. In any case, this is a philosophical discussion which is not the concern of this thread. If you wish to pursue it, you are welcome to open a separate thread wherever it is appropriate, and post a link here.

Now, for the promised calculations. But first, a few properties of diamond that will be needed:

density [http://en.wikipedia.org/wiki/Diamond" ]:
d = 3.5 gr/cc = 3500 kg/m^3
compressive strength [http://link.aip.org/link/?JAPIAU/50/2763/1" ]:
Sc = 4.1 Mbars = 4.1e11 Pa
young modulus [http://en.wikipedia.org/wiki/Young%27s_modulus" ]:
E = ~1000 GPa = ~1.0e12 Pa

Treating the skyscraper simplistically as a single solid diamond column, we get (Pf = load to failure):

Pf*A = m*g = d*V*g = d*h*A*g
h = Pf/(d*g) = Sc/(d*g)
h = 4.1e11 Pa / (3500 kg/m^3 * 9.8 m/s^2) = 11,953,352 m = ~12000 km.

Indeed seems almost unlimited... Of course, in reality, the structural skeleton would have to carry more than it's own weight, so if we assume that the skeleton is say 5% of the total weight, this reduces the maximum limit to about 600 km. Still a lot, given that 100 km is considered outer space.

Now on to buckling. From [http://en.wikipedia.org/wiki/Buckling" ]:

h = { 2.5 * E * r^2 / (d*g) }^1/3

where r is the radius of a solid round column of the given material. Substituting for diamond, we get:

r = 50 m ==> h = ~5.7 km
r = 100 m ==> h = ~9.0 km
r = 500 m ==> h = ~26.3 km
r = 1000 m ==> h = ~41.8 km
r = 5000 m ==> h = ~122.1 km

Clearly, this is a much more limiting factor. However, as much of the building is air, the density may actually be lower, which would increase this limit. Using a conical shape rather than a cylindrical one would raise the limit further, and probably would also be more stable.

I've also found that there may be more suitable materials, since diamond is rather fragile. These include multi-crystal diamond (as opposed to single-crystal), http://en.wikipedia.org/wiki/Aggregated_diamond_nanorods" . They seem to have similar strength and hardness, but not as much technical data on them.

Now, how do I estimate the wind load limits? Oh, and I haven't factored in compression of the building under it's own weight - that would reduce the height somewhat...