Remind me never to cross abridge you designed.

If enough people tell you that buckling is a vital issue, will you beleive it or just put your fingers in your eards and yell "nye nyer, nye nyer nyer, I can't hear you!"

The piano wire example is a profound example of buckling.

How do you push a piece of string?

If your nominally perfect 150 mile tower is only 48" thick, then it will easily be bent out true by tiny forces. You mention differential heating - the sunlit side will expand and the cold side contract. Congrat's you've built the world's tallest thermocouple...

The coefficient of thermal expansion of steel is 1.3 * 10^-5/K
The sunlit side of your tower is going to be many tens of degrees warmer than the dark side - even given Steel's good thermal conductivity. Let's imagine a 50 K difference:

50*1.3* 10^-5 *250 km = 162.5m

So the sunlit side of your tower will stretch to be 162.5 m (533 ft) longer than the dark side.

With two sides of the tower only 48" apart, your tower will develop a curve.

How curved will it be? Well, you'll like this...

For two surfaces only 48" (1.22 m) apart, the maximum difference in circumference that can accumulate is 2 * pi * dR for 360 degrees of curvature. (there's a famous mathematical question based on this about adding 3 m to a steel belt round the earth and finding that although we have added what seems an infinitesimal amout, the belt still stands proud by 1m all round the earth).

In your case, you can accomodate 7.66 m of extra length by throwing your tower into a complete circle. For 162.5 m, you need over 21 complete turns of a spiral!

(clearly, this is nonsense, since then different parts of the tower will thern become shaded and lit and the shape will change. Basically, your tower will writhe and wriggle like a worm, under the influence of sunlight alone.

Once curved, it will buckle and collapse.

As I mentioned to Raul (I think it was he), the ellipticity of the Moon's orbit gives a misfit between its average rotation rate and its rate of sweeping out its elliptical orbit. We call this "libration".

You can see that a "fixed" point on the earth facing side of the moon actually rolls by around 5 degrees from the actual earth-facing vector at any moment, subjecting you to off-line tidal forces which will also tend to bend the tower. Even the solar wind will represent a significant lateral force over the length of the tower.

(anything about 10 times longer than its width begins to bend "easily" and by about 100:1 it gets very floppy. Your tower is 200,000:1, worse than a sheet of tissue paper)

So, you will need a baseplate to take the thrust, AND a web of lateral guys (in tension) or an Eiffel-tower arrangement of stays (in compression) to maintain verticality.

The Eiffel tower was built with cast iron, which is good in compression but poor in tension because it is a coarsely crystalline form of steel.

Many materials fail in tension earlier than they fail in compression because they contain defects. For cement, it's the grains of sand and cement, and the gaps between them. Materials scientists can prove this:-

If you take a thin bar of cement and stretch it, it will snap. If you cut a notch in the side, it snaps earlier, depending on how big the notch is. Big nothes thin and weaken the material and it breaks easier. BUT if the notch is tiny (~ 1mm) it has no effect. This tells us that the material ALREADY contains defects in the size range of 1 mm and adding more does nothing. For cast iron, those critical defects are ~ 0.1mm, but still there.

With modern quality steels, tensile strength is good and a tensile guy is lighter than a compressive stay, since you don't have to make it thick enough and heavy enough to avoid buckling itself. In practice, a spider's web of guys would stabilise each other and collectively stabilise the central spine.

Also, if you're going to use this tower as a practical elevator, you need an unencumbered path to run the cars up and down. It's easier to put these inside a hollow tube, than outside a thin spine and have to rig the guys on "bridges" over the tracks.

In any case, weight for weight a hollow tube is always stronger than a solid rod - in compression, since it has more resistance to buckling.

And, all engineers design safety factors to allow for likely abnormal events, like a sudden stop or crash of a fully-loaded but defective elevator car. SInce these neccesarily travel off-centre, this would impose a turning tirque on your tower at the same instant as potentially damaging local supports.

You might even want to be 9-11 proof! Read what happened to the space elevators in Kim Stanley Robinson's "Mars" trilogy!