On Ice Walking

Just how safe is it?

Today at work I participated in several online meetings, however while we were waiting for colleagues to tune for one of them we found ourselves doing the stereotypically British thing; we talked about the weather.

To be fair the weather was unusual as much of the country, even in the south, had been covered with a blanket of snow. Indeed it had been reported that a portion of the Thames had frozen, which really was an unusual event. This was the predicate for a discussion about whether it would be safe to walk on said ice [you now know this wasn’t written in the summer].

This intrigued me so I set aside a little time that afternoon to try and work out just how safe it would be. This post is about what I found. Now, I should preface what follows by saying that I have absolutely no practical experience of this subject. Everything that follows is a postulation on my part, based on the engineering principles that I believe to be at work. You therefore shouldn’t base your ice fishing trip or curling match on what I have come up with.  

If you must go walking on a frozen lake, I suggest you ask someone who knows what they are talking about and isn’t making it up as they go. Statistically I imagine that such a person is far more likely to have a Canadian accent than a British one. You have been warned!

Ice floats because it is roughly 10% lighter than water. If we are to stand on a sheet of ice we therefore increase its weight making it more likely the ice will sink rather than float. It therefore struck me that the first part of the puzzle ought to be an assessment of how much ice there needs to be to ensure floatation occurs rather than sinking.

A cube of water with sides 1 m long weighs approximately 1000 kg, which means the same cube made of ice must weigh roughly 900 kg. This would imply that for our cube of ice to remain buoyant in water it must carry no more than 100 kg [1000-900].

Now, the minimum recommended thickness of ice for walking is 4 inches; I know because I googled it. That’s equivalent to 100 mm or 0.1 m. 

This means that a block of ice with a square surface that has sides 1 m long, but a thickness restricted to 0.1 m, can carry 10 kg [0.1 x 100]. It follows that if an average person weight 75 kg, then their weight must be spread over a block of ice with an area of 7.5 meters square [75/10] i.e. a square with sides measuring 2.739 m[1].

Everyone knows that ice is a brittle material that fractures rather than bends so that’s quite a large area for load to spread from our feet without something going wrong. I therefore started to think about how the load gets from our feet to cover such a large area and by what mechanisms it could go wrong.

Perhaps our feet would simply punch through the ice like a stiletto heel on soft ground. Such a mechanism would require the ice to shear on a vertical plane passing through the ice, which extends along a perimeter enclosing our feet. The length of the perimeter and the depth of the ice would therefore define the shear plane.

Punching shear yields a stress in the ice equivalent to 0.001 kg[2] acting on every square millimetre of ice on that plane.

The second potential mechanism could result from a shear plane developing across the full width of our notional 7.5 m square block of ice. This time the shear plane being defined by the width and depth of ice. This mechanism yields a stress of 0.0003.

The final mechanism I considered was bending. To transfer load from the feet to the outer edges of our square block the ice must be capable of bending. This is a bit more tricky to calculate, but it results in a stress equivalent to 0.005 kg acting over every square millimetre of ice.

I didn’t really know if any of these figures were significant or not so I got back on google. It turns out that, according to the people who measure such things, ice has a shear strength of 0.06 and a bending strength of 0.07. 

This means that there is a factor of safety against shear failure of 60 [0.06/0.0001] and a factor of 14 against bending failure [0.07/0.005]. I find this quite reassuring, because I don’t actually believe that ice has the strengths I just quoted.

This is not because the diligence of the researchers is at fault; I’m not saying that their work is wrong. What I am saying is that ice is not a manufactured product like steel. It is not made to possess specified properties.

The strength of ice depends on many things. What is the air temperature, did it rise and then fall again during its formation. Is the water fresh or salty? Is there a current or a flow in the body of water that is busy scouring the underside and weakening its structure. Was there snowfall during its formation. Did someone, or something, step on the ice while it was forming thus inducing cracks in its interior.

These and many other issues that I likely haven’t thought of have the potential to change the strength of a given block of ice; its value is therefore not a fixed thing[3]. If I am to walk on ice I would quite like to know that the existence of one or more adverse factors does not completely undermine the strength of the ice I am going to rely on. A factor of 14 sounds good to me. It’s roughly 10 times what I might use for say concrete. That’s about right because I know with far greater certainty what concrete will do.

Obviously if ice melts we are in trouble, but before that point is reached, I have no idea which combination of ice factors might eventually undermine a safety factor of 14. That’s why you shouldn’t take advice about ice walking from someone that’s making it up as he goes.

That being said, my somewhat crude assessment has yielded an interesting conclusion. I started by trying to work out what amount of ice I would require to mobilise to prevent the average person from sinking. My sums therefore exist on just the right side of not sinking, effectively a factor of safety of 1. This assumption eventually yielded a factor of 14 against the ice breaking i.e. in this scenario I am actually more likely to sink than the ice is to break!

[1] I know it would be more realistic to assume a circular perimeter, but I used a square to keep the sums simple. You’ll get over it.

[2] I know that working out stresses in kg is a bit weird, but unless you have a technical background you won’t know what N/mm2 is, or MPa for my European friends, or psi for my American friends. I didn’t want this post to be an explanation of units.

[3] Ice researchers know the strength of ice doesn’t have a fixed value. They provide ranges of values and couch them in temperature limitations and so forth. I picked from the lower end of the scale.