# 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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## This question has an open bounty worth +50 reputation from Sulthan ending in 5 days.

This question has not received enough attention.

Not enough attention and interesting question.

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
Do I understand correctly that you want to address the issue of the stability of a wall without foundings and with little or no bottom friction (ice for the wall, you didn't mention what the ground was like) ? Then of course you need a pretty thick wall to prevent it from falling down, at least in case it is a straight wall. I believe your problem needs a more detailed description to be addressed in a useful way. –  Joce 2 days ago
You should also remember about quality of ice, because if you freeze just normal water, there would be small bubbles inside bricks, so they would be weaker than pure ice. –  user46147 2 days ago
This is an obvious reference to the Wall in Game of Thrones. I am also interested in the possibility of such a construction (I know it's impossible but I want to know why exactly). –  Sulthan 2 days ago

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

We would have to know more about the ice we want to build the wall with.

For example, for ice in icesheets, you have an ice which effectively reaches a plastic region of the stress strain curve at around $0,5 MPa$. I am not a geologist, but I believe that the glaciers can be only thicker than $\sim 50 m$ thanks to it's specific shape and the fact that the surroundings press the glacier back and don't let it crumble (mostly). This would not be the case of a narrow wall, where already at $50 m$ the ice from a natural ice-sheet would crumble away before reaching plasticity (I have a feeling the plasticity will not be caused purely by microscopical structure breakdown but also mesoscopic ice grains breaking each other and moving around).

This study of the US geological survey assesses pragmatically "the crushing strength" of ice and finds that the best ice you find in nature at an ideal temperature has a lower crushing bound at about $\sim 400 \, pounds/inches^2 = 2,8 MPa$.

As Jaime already stated, the main issue is the very lowest bit of the ice carrying the pressure of the whole column above it. This pressure is $$P=\rho h g \approx 1000 kg \cdot m^{-3} \cdot 200 m \cdot 10 m\cdot s{-2} = 2 MPa \, .$$ So, if you do not take just any ice, but optimalize it's properties, you should be able to build wall of any thickness up to the height of $200m$.

But our intuition is rightly not satisfied with the vision of the mighty wall of ice with the thickness of a paper sheet. Such a wall would obviously tip over with the slightest blow of air, not to mention a giant riding a mammoth from far north Beyond the Wall. (I assume it is no accident the height is same as the one of the Wall from Game of Thrones :)

It is obvious that a thicker wall makes it less prone to being made a hole into, but let us concentrate on the issue of stability. First, the mere fact that a thicker wall is heavier makes it harder to tip over to a critical angle where the wall falls on it's own. Second, the larger base causes that the tipping angle is much higher and the beginning torque to start tipping it of is much larger. If you draw a scheme of the wall with a thickness $d$ and a height $l$, it is easy to show that for the tipping angle $\alpha_t$ we have $$tan(\alpha_t) = \frac{d/2}{h/2} \to \alpha_t = arctan \left( \frac{d}{h} \right) \,.$$ Or, the tipping angle $\alpha_t$ is always growing with the ratio $d/h$. Furthermore, the initial torque needed to even start tipping of the wall is also growing with the thickness of the wall. The initial torque $\tau$ would have to be $$\tau = F_{proj} l \,,$$ where $l = \sqrt{(d/2)^2 + (h/2)^2}$ is the distance to the lower edge of the wall from the center of mass and the $F_{proj} = M g d/l$ is the part og the weight of the wall with mass $M$ perpendicular to length $l$. The torque required to even start to tip the wall over then is $$\tau = Mgd \,.$$ So the torque required to tip the wall is simply proportional both to the mass of the wall and the thickness. Considering the fact that the mass is also a function of $d$, we have an even steeper growth of stability of our wall with it's thickness.

But beware! The edge over which the wall would be tipping reaches virtually infinite pressure in the process of the tipping. This is a consequence of the fact that the very lower part of the edge over which we are tipping the wall is carrying the weight of the whole wall. We must thus build the wall so that the required $\tau$ to even start tipping it is never reached by an intruder. You do not even have to use so much ice - the tipping base is made effectively larger e.g. by slightly curving the wall around.

Even if we ensure a large enough base, we must be sure the intruder actually does not exert enough pressure to start crumbling the lower edges of the wall. For example, a very brutal gust such as the one detected in Australia could exert a dynamical pressure of roughly $0.9 MPa$. This blow would be surely fatal, but even weaker ones could crumble the wall due to inhomogeneities and uneven distribution of pressure.

Overall, the $700 ft \sim 200 m$ tall Wall from Game of Thrones is just slightly unrealistic. It is hardly imaginable that the technology and coordination of that time would be able to create such consistently ideal ice as we were considering up to now. If anything, my best guess would be an estimate of an overall maximal pressure $\sim 2 MPa$ leading to say a $100 m$ tall stable wall. Taking into account that even stone structures in the medieval ages such as cathedrals were never higher than around $160 m$, even the $100 m$ wall from ice would be formidable. Obviously, Physics is loosing breath here, reasoning requires that Magic had to be involved.

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