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Assuming that there are no constraints on cost, land to develop, or supply of materials, is there a theoretical limit on how tall a structure can be built? What are the main physical constraints that limit the tallness of structures?

I emphasize structure because i don't want to limit the discussion to skyscrapers or other structures that would also have biological (human) occupancy constraints.

Update:

Based on the comments, I want to further impose the constraint that the structure must support its own weight. I realize that this question is a bit theoretical in scope and that real engineers always have to work with cost/material/land constraints in practice. I'm just wondering, if we eliminated those particular constraints, is the tallness of a self-supporting structure infinitely unbounded, theoretically?

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    $\begingroup$ According to Randall (yes we're on a first-name basis), we probably can't go higher than 2 or 3 kilometers. That specific point isn't well-sourced for me to flesh it out right now though. $\endgroup$ Mar 5, 2015 at 18:40
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    $\begingroup$ So you are asking, "How tall can a solid lump of material be built?" Or are you asking, "How tall can something be while still touching the ground?" and what do you mean by "built?" does it have to go from the ground up? This all maters because the answer is either going to be "space elevator" or "X feet of titanium". The question is too broad. $\endgroup$
    – hazzey
    Mar 5, 2015 at 19:01
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    $\begingroup$ What's the point in asking engineers if you're going to throw most of the real, practical constraints on the problem out the window? $\endgroup$
    – Air
    Mar 5, 2015 at 19:02
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    $\begingroup$ Comments aren't a good medium for this conversation but I'd be happy to continue this in chat tomorrow. $\endgroup$
    – Air
    Mar 6, 2015 at 3:16
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    $\begingroup$ Perhaps if you asked about what factors could prevent a structure from reaching a given height the question could be less broad. $\endgroup$
    – HDE 226868
    Mar 6, 2015 at 3:34

2 Answers 2

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Pedantic Answer First

You specifically eliminated most constraints that would make this anything other than an theoretical exercise. You did specify that it had to be "built" though. By building something, you have to physically move the top piece into place. This means that the entire structure will have been slightly taller than the final condition, i.e. the last "block" would have been "placed". This also implies that "something" was used to place the last "block".

Other Things We Are Throwing Out the Window

  • Gravity decreases as the distance away from the Earth's center increases.
  • We don't need to worry about pesky things like wind, so lateral force is out.
  • The foundation for this structure is adequate. This means that the Earth is made out of something stronger than our structure.
  • We aren't allowed to do anything that lightens our building. No thrusters, balloons, etc.
  • Buckling is not a problem. This structure can be any shape that we want.

To the Question at Hand

This question from physics.SE gets right to the issue. We need the material that has the best strength to density ratio. We need a strong strong material that is light.

From that question, diamond is our material. Also from the ceramics chart here.

A Diamond Tower Then

We now just need to see how tall a tower of diamond can be before the bottom layer crushes.

Properties:

$\gamma = 3.515 g/cm^3\\ \text{Strength} = 225 GPa$

If Wolfram is doing my math correctly for me, the answer is:

$$6,530 Km$$

Or about the radius of the Earth.

But Wait! There is More!

So far this has been about a solid tower that is the same shape the whole way through. The limiting factor was the stress on the bottom layer. What if we make the bottom wider than the top? Why not a pyramid of diamond? We can make this taller!

So a Pyramid of Diamond

A pyramid gets us taller. What limits the size of the base? Nothing, because we have no limits. So as the base gets larger, it eventually gets so large that it wraps around and touches itself from the other side of the Earth (sorry everyone.)

But we don't stop there. Spheres are self supporting, so let's turn this structure into a sphere that covers all of the Earth. We now can go even higher!

What have we created?

A Star Made of Diamond

Astronomers found a star that is made entirely of diamond. But why stop there? The only thing that limits a star's size is the mass that turns it into a Black Hole. We can argue about stopping here or whether larger black holes still count, but you see where we are going...

The Largest Structure

We have now seen that the largest/tallest structure is star made of diamond. (With a squishy Earth center.)

What Went Wrong?

So how did we get from "Tallest Structure?" to "Star Made of Diamond"? We didn't have any limits on what was possible. When you throw the constraints out the window, anything is possible.

We are no longer in the realm of Engineering. We are now in Theoretical Physics.

Engineering is using science to work within constraints.

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I think a proper answer must explore what the basic physics behind really tall buildings are. The most important one is that if you look at the construction from the top down, dividing it in thin horizontal layers like stories, each layer must be able to support everything that is above it. That's probably a fairly undisputed and basic constraint — but it leads us straight to fundamental limitations, including the reason for the fact that we cannot simply build a tower to space.

The load a given solid cross-section of a given material can carry depends solely on its surface area. The reason is simple: Material strength is simply the pressure it can sustain before it starts to give. Pressure is force per surface area, so that the overall force, or load, a solid layer can sustain or carry grows linearly with the surface area.

Unfortunately, we have just described exponential growth: Layer n must have the strength to sustain the sum of the weight of all the layers above it. That weight, if all layers are of the same height, is simply the sum of all their surface areas A1 + A2 + ... + An-1. If each building block can just carry a little more than one block, it is a classic binary tree. After a top layer of A1 = 1, and maybe A2 = 1 as well:

  1. A3 = A1 + A2 = 2
  2. A4 = A1 + A2 + A3 = 4
  3. A5 = A1 + A2 + A3 + A3 = 8, etc.

The surface area of each new layer which is needed to increase the tower height by one layer grows exponentially.

What we see is bascically a "standing rocket equation". We can improve our materials — if each block can carry the weight of 1000 blocks, the factor is 1.001 instead of 2; we would have a doubling every 700 layers or so. If we use cubic meter blocks and want to reach 100km or 100,000m, we'd have 100,000/700 = 142 doublings, a number with 43 decimal places. That's more rock than the volume of the Earth.

If we think that we want to limit the base of a tower to 1 square kilometer, we can have about 13,800 stories of 1 meter (still with our assumption that one block can carry 1000 others, which is pretty optimistic). That order of magnitude aligns with the highest mountains, even though they have a much wider base (probably because they do not consist of steel reinforced concrete). If we allow 10 square kilometers instead, we only get in 3 or 4 more doublings, or less than 2100 extra meters. Such are the limitations of exponential growth.

This is the most basic of limitations for tower height. At some point you'll have to deal with your foundations and the building ground which cannot carry the load of the building (in effect, you can think of your "exponential pyramid" extending into the ground which is fine if the ground is steel reinforced concrete ;-) ). A building as heavy as a mountain would also induce long-term geological effects, for example slowly sink into the crust like mountains or large glaciers do. But these are secondary effects which are consequences of the exponential mass increase with height, which is the actual showstopper.

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