Elephants in Stilletos

I was intending to post this yesterday but my brief intoduction to Inktober ended up taking a lot longer than planned, so I postponed this one…

Yesterday’s prompt word for Inktober was “Pressure”. As usual, I thought about several possible avenues for interpreting this prompt, one of which was the scientific definition of pressure as force over area. This reminded me of a fun fact I learned while I was at school, namely that a woman in stiletto heels will do more damage to a wooden floor than an elephant, because although she weighs a lot less than the elephant (assuming average-sized women and elephants), all her weight is concentrated over a very small area compared to the elephant’s feet.

While thinking about this yesterday, it occurred to me that this was all based on the assumption (to be fair, probably a fairly safe one) that the elephant isn’t also wearing stiletto heels. This set me off on a rare foray into cartoon-style illustration:

Probably not one of my best sketches ever, but it was quite fun to make. I also took a bit of time to do my own calculations to verify the assertion that a woman in stilettos exerts more force than a barefoot elephant.

To do this, I looked up a few figures and estimated a few others.

According to an article appearing in the Independent in 2017, the average weight for a woman in the UK is 11 stone. That’s slightly higher than I expected, but I decided to go with that figure. Converting to metric units, that’s close enough to 70kg, so we can use that for our woman’s weight.

Except that it’s actually her mass, since weight is a force (gravity) acting on a massive object (i.e. an object that has mass, not necessarily a particularly large one) and is dependent on the strength of the gravitational field it’s in. We actually need the weight for our calculation (as pressure is force/area), but that’s easy enough to calculate from the mass. Newton’s 2nd law says that force is mass times acceleration (F=ma if you like equations, as I do) and in this case the acceleration is that due to gravity. That varies from place to place around the world but it’s roughly 9.81 metres per second squared. For my rough calculations, I decided that a nice round figure of 10m/s2 would do fine. So our average woman weighs about 700 Newtons.

I didn’t make a note of where I found the figures for an elephant but apparently a female African Bush Elephant weighs on average around 3 tons. I’m not sure if that’s supposed to be long tons or short tons, but either way I decided that just calling it 3 metric tonnes (3000kg) would be close enough. Again, that’s actually the elephant’s mass (everyday language tends to be shockingly imprecise when it comes to such things), and her weight would be 30,000N using the same figure of 10m/s2 for the acceleration due to gravity. Incidentally, I decided that since our woman is (by definition) female, I’d go with a female elephant too (they tend to be a bit smaller than the males) and since I tend to think of African savannas before African forests or any part of India when thinking of elephants, I opted for an African Bush Elephant (a species that’s generally somewhat bigger than the the other two varieties).

That bit was relatively easy. Working out the areas was slightly more problematic, especially for the woman in stilettos. You will probably be relieved to hear that I don’t have any stilettos in my own shoe collection, and I was too lazy to go out and find a woman with high heels so I could measure the surface area of her heels and toes together or figure out how much of her weight would be concentrated on each part of her foot. For the initial calculation, at least, I wanted to work on the assumption that both the woman and the elephant would be standing with their weight evenly distributed across all their legs (that sounds a bit weird for the woman – obviously “all” is just “both” in her case!). A bit of online research revealed that stiletto heels usually have a diameter of no more than one centimetre, but I couldn’t find anything out about the area of the front part of the foot that would be in contact with the ground and presumably bear its share of the weight. I settled for a rough estimate of about 1cm2 for the surface area of each heel and 50cm2 for the surface area of the toe/ball of each foot. For convenience I tweaked the latter down to 49cm2, giving a total surface area of 100cm2 for both feet (heels and toes combined).

The elephant’s foot size was actually a bit easier to determine. Apparently a typical African Elephant has feet between 40 and 50cm in diameter. I decided to give the woman a bit of a helping hand by assuming our elephant had relatively small feet (hence providing less area to spread the weight) and therefore a 40cm diameter, or 20cm radius which, if we assume that the feet are circular, gives a surface area of about 1250cm2 per foot or 5000cm2 for all four feet.

To ensure our final units are correctly expressed as Pascals, or Newtons per square metre, it’s handy at this stage to convert those areas into square metres rather than square centimetres. The woman, standing with both feet firmly on the floor is putting all her 700N of weight through 0.01m2 of the floor, while the elephant’s 30,000N is being spread across 0.5m2 with the net result that the woman is exerting 70,000N/m2 or 70kPa of pressure on the floor, while the elephant is exerting only 60,000N/m2 or 60kPa. So our average woman is indeed liable to do a bit more damage to our delicate wooden floor than our average elephant, though the figures are actually quite close.

The difference gets more pronounced if they both put all their weight on a smaller area. I’m not sure how practical it would be to rest all your weight on one heel while wearing stilettos (mind you, I’m not convinced it’s very practical to wear stilettos in the first place) but suppose she’s able to do so, our woman is now channeling 700N through an area of just 1cm2 or 0.0001m2 which makes for 7MPa of pressure (that’s 7 Megapascals, 7 million Pascals or 7×106Pa if you’re not afraid of scientific notation – it’s definitely much more convenient than long trails of zeroes at either end of your numbers). Assuming that it’s enough of a challenge for our elephant to stand on just one foot, without going up on her toes or heels, she would be putting 30kN through 0.125m2, which amounts to 2.4×105Pa, which is 240kPa or 0.24MPa – significantly less than the woman on one heel.

Since my cartoon was based on the idea that an elephant wearing stilettos would do more damage to the floor than a woman in stilettos, I couldn’t leave this set of calculations without considering the pressure exerted by our elephant if she were to don a set of stiletto heels. Presumably these would have to be custom made and I’ve no idea how big they would be, nor whether she’d wear them on all four feet or just two, so let’s assume that the heels themselves culminate in points the same size as the woman’s ones, i.e. 1cm2 each and the elephant has somehow managed to contrive to stand with all her 30kN of weight bearing down on just one of these heels. That would make for a pressure of 3×108Pa, or 300MPa. As we would expect, our elephant in stilettos would do considerably more damage to any floor than our woman. It’s probably just as well that elephants are not, as far as I’m aware, in the habit of wearing stiletto heels.

I should probably add that it’s been a good few years since I last did this sort of calculation, so I hope I haven’t made any major mistakes with my units or figures, or any assumptions that are too crazy (apart from the basic premise itself, perhaps). Still, I’m fairly confident, at least that the claim made by my cartoon is fundamentally correct:

An average-sized woman in stiletto heels exerts more pressure on the floor than an average-sized elephant…

… unless, of course, the elephant is also wearing stilettos!

(Magnus Forrester-Barker, 2021-10-09)