# Engineering The Tallest Snowman

It’s that most magical time of the year (again), and what better way to celebrate, than a bit of seasonal engineering? Over the last few years I’ve already discussed the miraculous building loading that is Santa Claus and provided you all with an arguably dangerous Christmas Tree modeller; so for 2014 I’ve decided to tackle that fundamental engineering question:

How tall can you build a snowman?

## Material Specifications

All this adds up to a material where the question isn’t “will it ever fail?” but “how long do you want it to stand up before it fails; in seconds, preferably…”

You might be surprised to learn that there’s a lot of research undertaken into the field of snow construction by various governmental institutions. Alas, this article isn’t going to become a Snowden (pun entirely intended) exposé into a dearth of secret projects to weaponise giant snow people à la Gundam Wing; the mundane truth is: If you want to understand an avalanche, you have to understand snow.

As a material, however, it appears to exist primarily to laugh in the face of structural models. I have now read more than I’ll ever need to know about the engineering properties of snow, and, quite frankly, it makes geotechnical parameters look like works of axiomatic precision. The problem stems from the fact that snow is just loosely packed (starting at around 10-30% the density of water) layers of tiny crystalline ice particles.

This means that it is a highly compressible material (in fact, you can simply crush it until the density approaches 80-90% of water, when it officially becomes ice), that is highly sensitive to temperature (it can melt/disaggregate and re-freeze/bond into new structures through sublimation and sintering) and behaves like everything from a linear-elastic solid to a visco-plastic liquid… To be honest, after losing a few hours of my life to reading about the wonder of snow mechanics; I was yearning for a simple material- like cake!

The thing is, unlike the civil engineering standards of concrete and steel, all these troubles add up to a material where the question isn’t just “will it ever fail?” but “how long do you want it to stand up before it fails; in seconds, preferably…”. If there are any experts in snow engineering who would like to chip in, at this point, that would be perfect- but until then; here are the mechanical properties I’ve deciphered for densely packed snow at sub-zero temperatures that will remain “true” for a given value of time:

• Compressible strength: 3 MPa
• Shear strength: 1 MPa
• Tensile strength: 1 MPa
• Poisson’s ratio: 0.3
• Low creep strength*: 70 kPa
• Density: 7 kN/m3

*The low creep strength is the limiting stress at which the strain rate remains constant; the snow would not start life in a state of failure. It is, however, so low that I doubt we’ll achieve much with it.

## The World’s Largest Snowman

The world’s tallest snowman was actually a woman, 32.21m tall, and built by the residents of Bethel, Maine, USA. Right of the bat, however, I’m going to disqualify this entry; somewhat ironically, on the grounds of Architecture.

You see, Bethel (which I assumed they called her) the giant snow woman already encompasses the most efficient engineering form. As I discussed when attempting a similar feat with cake- whenever you’re going for “the world’s tallest”, a pyramid is pretty much the best way to build high for those to whom the words ‘structural efficiency‘ are just a set of random sounds. And although a snow man is hardly anatomically correct, a cone just doesn’t cut it in my book.

We can’t keep building pyramids…

Just for completeness, however: Confined snow under compression tends towards ice. Noting previous research into this topic, the compressive strength of ice is around 8.8MPa. As a rough guide, the loading distribution of a pyramid allows you to build three times higher than the equivalent tower. We can also take an estimate that the snow changes linearly from light and fluffy, to solid ice; essentially making the average density 60% that of water:

$h_{max} = 3 \times 8.8 \text{MPa} / ({60 \% \times 10 \text{kN/m}^3})=4.4\text{km}$

As that’s a sizeable fraction of the diameter of the earth, and around about the thickness of the Antarctic ice sheets (so logically, we should be able to go even higher!); I can’t shake the feeling that maybe Maine should have put a bit more effort in…

## Failure Mechanisms

A proper snowman is two balls, one a top the other. The bottom ball is required, by Euclid (et al), to be 1.618 times the size of the top; or else our mega-structure will end up offending sensitive eyes. To ensure a good foundation, we’ll allow a zone the same diameter as the head to form a cylindrical pad for the base of the snowman. A similar proportion of bearing area will be provided between the head and the body:

Being Brunel doesn’t make snowmen, but if it did…

At a glance, this gives us two inter-related failure mechanisms (which highlight why the conical-woman is cheating): Either the sides of the body fall off in shear, or it cracks apart as the head pushes through the body and causes it to expand (a rather fatal example for a use of Poisson’s ratio!).

## How Big Can You Build A Snowman?

So I’ve waffled on enough; time to take a guess. I mean, er, report my findings after a detailed period of analysis*…

*Ed: It was at this point in the article that I realised I’d picked a stupidly difficult piece of geometry to calculate for! For the sake of simplicity our structure is two spheres in volume; the bearing areas are perfect circular planes and are connected to form an internal core. We’ll negate accessories loading.

Sample shows tensile failure; try again…

The first failure mechanism is caused by the weight of the head cleaving the body in two. As the body ‘squishes’ (technical term) it begins to bulge, and creates tensile forces along the centre of the body ball; exceeding the 70kPa limiting creep stress at 27.5m.

However we’re being a little unfair here, the compressive force decreases the strain rate as our snow packs up to much stronger ice. I’m therefore moved to say that, as long as you think of our snowman as a transient structure, we can use the full elastic tensile limit of 1MPa. From some very rough calculations (lit. unfounded gut-feeling) I’d say it’d last a good few hours in this condition; especially with a bit of maintenance.

Test displays shear failure- try again…

Then comes shear. It’s the sections hanging off the supported bit of our snowman that are causing the shear; so we’ll say that’s everything not within the blunt cone formed between the base and the neck. This means that the weight of the head just transfers in bearing down to the base, but ~60% of the body sphere volume is supported in shear; hanging off the middle. Unexpectedly, we can reach about 130m until the shear passes the limiting creep stress.

No snow men were hurt- except this one…

As the capacity in shear is based on cohesion, it is very sensitive to creep strain- but If you don’t mind the snowman existing for only a couple of seconds, we can push the shear utilisation up to the elastic limit too. So finally, we come to the final failure- compression. Remember I talked about the wonder of snow becoming ice and getting incredibly strong last time? Well, that’s only helpful if you don’t care about the shape (i.e. it becoming a pyramid). So we’re limited to our 3MPa before the snow starts to rupture and crush (n.b. because of the poisons ratio of 0.3, the tensile limit still isn’t exceeded)- which happens at around 320m.

So there you have it, if you want a snowman that’ll last you through the night, stop at 27.5m, if you just want to prove a point- feel free to wind it all the way up to 320m; just don’t come complaining to me when it collapses before you get a photo…

Just for my sadist enjoyment, I’ve decided to attach the spreadsheet containing my workings this year.

I shall look forward to all the comments pulling apart my methodology, choice of values and formatting choices.

Merry Christmas, one and all.

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