I tried to find in the internet some scientific explanation and calculation and looks like it is difficult. I found some calculation for house from standard bricks and it gives $170~\mathrm{m}$ for Ultimate tensile strength of brick $3~\mathrm{MPa}$. Formula is

$$ h = \dfrac{\sigma}{\rho g} $$

where $\sigma$ is Ultimate tensile strength.

this formula is explained here

As I see here there is kind of steel with Ultimate tensile strength $2600~\mathrm{MPa}$ and according to this formula it can be 32 km!

As I understand this formula for square or cylindrical shape, like the same width everywhere. But what if we make it in the shape of Eiffel Tower or kind of hyperbolic and etc? Or shape is does not matter and maximum height will be the same?

Or maybe if we build it from sticks with the same shape as diamond crystal structure we can build it up to 100 km?

Is there any well know way (formula) to calculate maximum height of the tower of the complicated shapes?

UPDATE: As I was told, I should use Compressive strength instead of Ultimate tensile strength. It looks reasonable. In this case calculation will be the same, only for steel I found value not 2600 MPa, but 300 MPa, but I can take another material from here with the similar value 2600. and if I take diamond with Compressive Strength 17000 MPa it will give 480 km.

UPDATE2/ANSWER: Looks like I found answer by myself with help of all your valuable comments. If I use assumptions like Total gravitational force to the basement less or equals breaking force ($\sigma$S) where $\sigma$ - compressive strength and S - area in square meters, I get this formula for cylinder

$$ h \leq \dfrac{\sigma}{\rho g} $$

some numbers for cylinder:

Steel (300 MPa): 3.75 km

Granite (300 MPa, but less density than steel): 11.5 km

Diamond (17000 MPa): 480 km

ABS Plastic (65 MPa): 6.5 km

Strongest concrete (80 MPa): 3.2 km

Carbon epoxy (up to 1500 MPa): 100 km

but for real building we have to divide it to 2 or 3 to have some "factor of safety". In this case only diamond and carbon epoxy can be used.

For cone

$$ h \leq \dfrac{1}{3} \dfrac{\sigma}{\rho g} $$

numbers will be 3 time more than for cylinder.

For other shape this condition should be met

$$ \sigma \geq \dfrac{\rho g V}{S} $$

I tried to calculate Exponential cone like this Tower with 100 km height from carbon epoxy and factor of safety 3

which is Solid of revolution of this function

$$ f(x) = r e^{-α x} $$

where r is basement radius and α is kind of cone steepness. Volume can be calculated by formula from wikipedia article Solid of revolution Looks like for this exponential cone it is possible to build tower of any high from any material, but for materials with low compressive strength, if we take basement r=1 km, desired height 100 km for example, last 70% of the tower it will be very thick (like $10^{-5}$ meters). Of course this kind of needle is not possible to build and it does not make any sense to build. if we accept that final radius at maximum height 100 km equals 0.5 meter, the basement radius for different materials will be like this.

r without "factor of safety"

Steel (300 MPa): 400 km

Granite (300 MPa, but less density): 42 m

Diamond (17000 MPa): 0.56 m with "factor of safety" 5

ABS Plastic (65 MPa): 1.5 km

Strongest concrete (80 MPa): 5000 km

Carbon epoxy (up to 1500 MPa): 0.51 m

if we think about "factor of safety" equals 3, as I understand it is standard for this kind of things, we get this numbers

Steel (300 MPa): $10^{14}$ km

Granite (300 MPa, but less density): 230 km

Diamond (17000 MPa): 0.56 m with "factor of safety" 5

ABS Plastic (65 MPa): 5500 000 km

Strongest concrete (80 MPa): $10^{17}$ km

Carbon epoxy (up to 1500 MPa): 2 m

In this case we can really build a space elevator with high 100 km from 3 materials Granite, Diamond, Carbon epoxy. Even yearly production of Carbon epoxy will be enough to build it :)

This is exact picture for tower with 100 km height from carbon epoxy and factor of safety 3 (all axes in meters)

enter image description here

Of course I do not consider wind, and all other things. With precise engineering calculation might be it will not be possible.

  • $\begingroup$ If you make an exponential shape, you can build higher (by a logarithmic multiplier), but ultimately the 1/r potential of gravity will kick in and then you will be making planets and stars. Those can be much, much larger... but they won't rely on the tensile strength any longer, but ultimately be stabilized by radiation, at least for a while. The better concepts for really tall structures on Earth are probably the launch towers/loops: en.wikipedia.org/wiki/… $\endgroup$ – CuriousOne Aug 8 '16 at 13:14
  • $\begingroup$ Thanks, but what means this 1/r rule? Can you give some examples? Also if we are talking about tower with 100-300 km high, will we reach this 1/r thing or it is much higher? $\endgroup$ – Zlelik Aug 8 '16 at 13:17
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    $\begingroup$ This is the exact question that people working on space elevators are trying to answer. But with unlimited resources, you could make a pyramid as high was you want. Problem is, at a certain point you stop calling it a building and start calling it a mountain $\endgroup$ – Jim Aug 8 '16 at 13:26
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    $\begingroup$ Obligatory XKCD $\endgroup$ – RedGrittyBrick Aug 8 '16 at 13:49
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    $\begingroup$ The ultimate tensile strength is irrelevant, because the material at the base of the building is in compression, not tension. The compressive strength of most materials is bigger than the ultimate tensile strength. As an extreme example, the tensile strength of unreinforced concrete is zero for practical purposes, but concrete is a very good construction material when loaded so that it is compressed $\endgroup$ – alephzero Aug 8 '16 at 13:59

If you make a pile of bricks, you need to talk about strength in compression, not tension. Bricks and stone are much stronger in compression than tension.

This had a strong influence on architecture. The Parthenon is a lot of closely spaced columns because you can't lay a stone over a long gap. The bottom is under tension and fails.

When the arch was discovered, it became possible to open up the interior of buildings. An arch is under compression over its entire length. Medieval cathedrals took advantage of this.

Modern materials make taller buildings possible. Steel is strong in tension. More designs become possible. Wind loads and earthquakes matter. They try to push a building over. The tallest buildings are stretching the limits of materials.

Mountains are limited by the strength of the crust of the Earth. A mountain that is too tall will slowly be shortened as the crust sags.

The crust is moving, so it isn't as simple as that. Mt. Everest is still rising as the Indian subcontinent rams into Asia.

The tallest mountain is the island of Hawaii. It is still growing because it is an active volcano.

I don't know that either of these push the limits of crust. Mountains may not get big enough to do that because nothing causes them to grow without limit, or because they erode about as fast as they grow.

But there are areas in Scandinavia that used to be under glaciers that are rising.

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    $\begingroup$ Well, many good words, but I need numbers. You tells "Modern materials make taller buildings possible." What does it mean taller by your opinion? and I would like to consider only ideal case for now, no wind, no earthquakes and etc. Just gravity, material properties and specific tower shape. I am basically looking for formula or tool to calculate max high in my ideal assumptions. $\endgroup$ – Zlelik Aug 8 '16 at 14:11
  • $\begingroup$ Sorry. Words are all I have. Each building is designed and built separately. It takes a lot of thought and work for an architect. It is not just one formula. If you want to know more, maybe there is an architecture site? $\endgroup$ – mmesser314 Aug 8 '16 at 14:34
  • $\begingroup$ You did not get my point. I do not ask about exact architectural project. I am looking for some theoretical physics things like formula I mentioned in the question. But this formula works only for cylinder or cube. I am interesting in something similar but for complicated shape like cone, exponential cone or something built from sticks. My formula is explained here talkingphysics.wordpress.com/2011/09/08/… $\endgroup$ – Zlelik Aug 8 '16 at 15:06
  • $\begingroup$ Looks like this Article will fully answer my question :) "The Tsiolkovski Tower Re-Examined" adsabs.harvard.edu/abs/1999JBIS...52..175L $\endgroup$ – Zlelik Aug 8 '16 at 15:23
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    $\begingroup$ Funny you just found it. $\endgroup$ – Bob Bee Aug 9 '16 at 4:03

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