# How wide does a wall of ice need to be to stay in place?

Let us say that we have unlimited manpower to construct a huge wall of water ice e.g. 200 m tall (700 feet). -and that the wall is placed in a climate, where the temperature never (for your purpose) goes above freezing point. The wall is to have fairly steep sides, so let us assume a perfect rectangular cross section. A very narrow wall would obviously shatter at the bottom. A very wide wall would approach a natural ice cliff side and thereby be possible.

• How wide does the wall need to be?
• What properties of ice is relevant to calculate that?
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It was a fine question, but the extra part was a bit much. Hans-Peter, perhaps it would be better if you post your extra questions separately. I've edited them out for you as a convenience. (You may revert the edit if you like, of course, but I think it did improve the question.) –  David Z Nov 12 '12 at 19:21
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## 1 Answer

Any cross section of your wall is supporting the weight of all the wall above it. In a first approximation, every cross section will be in a state of pure axial compression. The most heavily solicited cross section will be the one at the very bottom, which will be supporting a compressive pressure of $\rho h g$, where $\rho$ is the density of the ice, $h$ the height of the wall, and $g$ the acceleration of gravity. You could compare that value to the compressive strength of ice, and use that to determine whether your wall will crumble or not.

Your biggest problem would be to figure out what parameters to use, which apparently are very dependent on how the ice has been formed, what temperature it is in... Just taking a look at the phase diagram of ice, makes it clear that its going to have a complex behavior. It could be turn out that the compression force causes a phase transition of the ice at the bottom of the wall, so that you will then have to consider the top and the bottom sections separately. I have found a couple of very old references, here and here, which I have only skimmed diagonally. But that it is possible to write close to a 100 pages on the mechanical properties of ice kind of tells the story...

As an aside note, if you are willing to sacrifice perfectly vertical walls, having a wall with width growing as $A e^{by}$, where $y$ is vertical distance from the top of the wall, will have every cross section of it standing the exact same compressive pressure. This means that you are using all of our ice to its maximum load bearing capacity, plus avoiding different pressures creating sections with different phases.

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Don't you think that before it would crumble from vertical pressure, the wall would fall over to the side? –  user4696 Nov 13 '12 at 9:57
ρhg ~ 1000kg/m^3 * 200m * 10m/s^2 = 2000000kg/m/s^2 = 2MPa. So I think that it is safe to assume, that all of the ice will be in phase I. -but what is the minimum width of a 200 m wall? –  Hans-Peter E. Kristiansen Nov 13 '12 at 22:34
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