As you can imagine, this "highest building" problem has been addressed by the best & brightest for the last 5000+ years. I can't come up with the correct reference right now, but the ideal shape for a tall structure made out of a uniform material is a long oval shape (curved at both the bottom & top) & braced with guy wires. It sits on one end point, while the taut wires keep it upright. It isn't particularly pretty! The "waist" has the largest circumference, which is necessary to counteract Euler buckling. You can build a similar tower without the guy wires, but it will be only 1/2 the height, because it sits on its "waist". If you move away from a uniform material, then an ideal structure is fractal. Most TV towers are simple triangular lattice structures. To make one into a fractal, simply replace each beam with another (smaller) triangular lattice. The reference below shows what the result looks like: http://www.london-institute.org/people/farr/fractals.shtml I don't believe that this reference (or any others that I can find) found the optimum number of fractal levels, nor the optimum scaling of each fractal level. There's also no mathematical proof that the resulting structure is optimal. In particular, it may be that a 2-material solution might be much more efficient, where one of the materials is good at compressive loads and the other material is good at tensile loads. Note that these fractal structures only address Euler buckling; you still have the problem of the basic compressive strength of the material, which can be crushed in a non-Eulerian manner. I was trying to come up with a "dynamic programming" (recursive descent) model to compute such things a few years ago, but I got diverted into another project. I have the ".stl" file for this "Farr" structure which I can email you if you would like. ".stl" files are used for 3D printing. So-called "tensegrity" structures have been investigated in great detail by NASA, because it wants to put large, light structures into space, but send them up in a "compressed" manner, and then pull them taut into their final shape. NASA has also looked at origami-like structures for the same reasons. At 01:19 PM 9/21/2014, Warren D Smith wrote:

Suppose we wanted to create an extremely high tower (think of it as a very long vertical rod) by combining the facts that (a) compressive strength of materials is large (b) "buckling" instabilities can be prevented by active control. Another idea is an extremely large-span arch (if a constant-diameter rod is bent into the shape of a "cycloid" curve, then the stresses in this arch are purely compressive). (Of course, it'd be a bummer if you had a power or software failure... or an airplane hit it...) In fact back when I was an undergrad at MIT, some engineering prof was assuring us gullible students this was a revolutionary idea that would completely change structural engineering, and he was procuring big grants to do it.

These are fine fantasies; now let's think a little about reality. (The results of my analysis will suggest said professor was completely full of shit, albeit apparently a successful con-man.)

For a vertical tower, you could keep the stabilization turned on as you built it. However for an arch, I'm not seeing how to do that. So it seems as though an actively-stabilized arch would be pretty much impossible to build. (You could build with reinforcement which later was removed, but not if we are doing this for an enormous arch beyond the size attainable without active stabilization...)

Percy Bridgeman, in his investigations of high pressures, was able to reach up to 5 GPa pressures using high strength steels and 10 GPa pressures (100,000 Kgforce per square cm) using tungsten-carbide/cobalt (6 wt% Co) composite material. Later, 300 GPa pressures (comparable to those at Earth's core) were achieved using "anvils" made of single crystal diamond and 600 GPa using "2 stage" diamond anvils with the second (very small) stage made of "nanocrystalline poly-crystal diamond." Unfortunately, all of these apparatus eventually fail, one reason being "creep." Even at only 0.5 GPa, today's commercial steel high pressure equipment says you have to replace the "Bridgeman seal" every 2 years.

In view of this, I doubt it is possible to maintain non-hydrostatic compressive stresses of more than about 300 MPa (equals 30000 kgforce/cm^2) in any large structure. That stress level would be sufficient to support a vertical steel rod, of uniform diameter, 38 km high. The shrinkage of steel at this stress level is about 1.5 parts per thousand. The infrastructure (sensors and actuators) required to perform the active control would add additional weight -- so in view of that plus the desire for some safety margin, perhaps the upper limit really is only about 20 km.

In contrast, the highest tower yet built is 830 meters in Dubai. This "Burj Khalifa" is a non-guyed skyscraper containing residences and hotels. Its cross-section is a 3-pointed star at base about 170 meters wide. (Some other recent tall buildings have parts with separate towers linked by girder/walkways and with wind-turbines attached to those girders which can provide perhaps up to 20% of building's electrical needs.) It is mainly made of steel-reinforced concrete and appears to be a technical success but economic failure. Stresses caused by the blowing of the wind cause the top to sway about 2 meters horizontally, and there is a swinging "tuned mass damper" inside the tower near the top which is something of a semi-active control to prevent resonant swaying. As far as I understand this device is not controlled by any intelligent mechanism, but it is an intentionally designed dynamic non-rigid mechanism which I suppose could be considered as an "analog computer." Also the exterior of the building is intentionally shaped in funny ways that computer simulations of wind flow think make it unlikely to get into a destructive resonance with large wind vortices. Previously the highest were radio masts made of metal periodic "lattices" (triangles) with guy wires and dampers. Specifically the previous record holder was the "Warsaw radio mast": 646 meters high, had equilateral triangle base 5 meters on each side, and weighed about 420 metric tons, of which the guy wires were about 80 tons. It was completed in the early 1970s but collapsed in 1991. The immediate cause of the collapse was a maintenance error, but it suffered from metal fatigue caused by wind stresses, and from difficulty trying to keep it painted. This tower had 3 vertical steel tubes linked by latticework. Each of the 3 main tubes was 245mm in diameter with wall thickness varying from 8 to 34 mm. Thus the cross-sectional area of the tower near the base was 3 * 245 * pi * 34 mm^2 = 78.5 cm^2 for a compressive stress of 420000 kgforce / 78.5 cm^2 = 5350 kgforce/cm^2 = 53 MPa.

So... first of all, passive prevention of buckling, via guy wires, only cost about 20% of the Warsaw tower weight. If you also count all the latticework, then maybe we are up to 50%. If active control cost more weight than that, what would be the point? Well, I suppose there are circumstances where it is infeasible to guy a structure... and the mass of the guy wires presumably grows quadratically or faster with tower height while the mass of an active control system might be hoped to grow only linearly...

In a speed-V wind, assuming drag coefficient of 1, the total torque near bottom on a height-H vertical tube of radius R, is about H^2*R*V^2 kg*sec^2/m^5. Note the H^2, quadratic, growth term, indicating that linear-growth hope is eventually dead meat.

If V=200 km/hr = 55.556 meter/sec and H=650 meter and R=0.245 meter, we find torque = 320 million newton*meters. If this is 10^9 in all (3 tubes) then the force on one tube that needs to be applied to counter this torque is about 2*10^8 newtons which is 2*10^7 kgforce which is 20000 metric tons of force. This far exceeds the weight of the tower supported by that tube at only 420/3=140 metric tons. So actually, maximum wind-caused bending stresses far exceed the gravitational stresses, hence the guy wires are very important.

Active control could be done by measuring torque (bending stress) on the rod, and applying compensating torques. In this way, the bending stresses everywhere could permanently be kept below some constant. Glued-on silicon and metal-foil strain gauges could be used for this, but you need temperature compensation and compensation for stresses caused by the solidification of the glue. The correct tower shape is not necessarily exactly vertical if, e.g. one side gets hotter. For example if one of the three Warsaw tubes got 5C hotter than the other two tubes, it would expand 65 parts per million which would be 4.2 cm of extra tower height. This would bend the tower with a radius of curvature of about 77 km, causing the top of the tower to move 2.7 meters horizontally away from vertical. I believe that the highest winds also were expected to cause about that same amount of bend. Thus the correct shape is not "vertical"; it is "the one which keeps all bending stresses bounded."

With the 3-tube Warsaw-type design, torques could be applied by pulling cables between the 3 tubes. If there were no wind then I would say this is feasible. However, counteracting the torques from wind seems to be to require far too large torques to be feasibly applied in this way. In short removing the guy wires from the Warsaw mast and using active control instead, would be asinine.

SUMMARY: I'm not seeing advantages for active stabilization of structures the size of the largest yet built -- up to 800 meters -- and in fact the whole idea seems to be ridiculous in view of wind loads.

But if we were building on the moon (vacuum, no wind) then it might be the best way to build huge towers.