Contextual Analysis

It was very windy on Thursday and I wanted pizza. There’s a good shop two blocks away, and if you get two slices they use a small box rather than a bag. As I was walking back to the office, it quickly became clear that I had to use both hands to carry the box, even though the box, with the pizza, weighed less than a pound.

The wind gusts were causing far more loading on the box than its weight. A code-level gust might actually cause a pressure on the box of 40 pounds as well as suction on the windward side, but we were not close to that hurricane-force wind. For the sake of argument, maybe the wind pressure was 5 or so pounds with equivalent suction. (The pressure goes by the square of the wind speed, so if the wind speed was half of the code hurricane then the pressure would be one-quarter as much.) Wind can only exert pressure perpendicular to a solid surface, and because I’m not a complete barbarian I was trying to carry the box on the level, so the worst case scenario was a gust downwards, where I’d have the combined wind pressure, back-side suction, and weight totaling at around 11 pounds. That’s still not very much.

If I carried the box using both hands, the cardboard formed a funny-looking beam spanning between my hands. That beam was more or less the condition known as uniformly loaded and simply-supported. Each hand would carry half the load vertically (again, assuming the worst case condition was the wind pushing down) and no other force. If I carried the box with one hand (as I did in the elevator photo above) then it was a cantilever beam, and the hand holding it had to carry the full weight (still not so much) but also provide moment restraint to keep the box level. My hand’s contact consisted primarily of my thumb pressing down on top and the other four fingers pressing up on the bottom. Eleven pounds times half the box width, or about six and a half inches gives me a moment of 71.5 pound-inches; resolving that into a couple at my fingers means dividing by the box depth or about 2 inches, giving a force to be resisted (in one direction at my thumb, in the opposite direction at the other fingers) of about 36 pounds. But that force is parallel to the top and bottom surfaces of the box, so it’s resisted by the friction between my fingers and the cardboard. My usual reference books don’t give friction coefficients for skin on cardboard, but if we want to assume there’s a pretty good amount of friction, with a μ value of 0.67, then the 36 pound force would require me to clench my hand so that my thumb was pressing down with a force of 54 pounds and my fingers pressing up with the same force. According to NASA, the average grip strength for men is about 130 pounds and maxes out when the object gripped is about 2-1/2 inches wide, or roughly the size of my box. So 54 pounds of grip is certainly possible, but not a minor effort.

But, of course, the box isn’t very strong against crushing loads. If I tried to grip it with anything approaching 54 pounds of grip, the top and bottom deformed inwards. This helped me get the required moment capacity by providing some mechanical interlock between my hand and the box in addition to the friction. It also weakens the box “beam” that I was relying on.

In other words, my perception was correct that it was a lot easier to carry the box with both hands, even though the total weight and wind pressure of 11 pounds is easily within my one-hand capacity. Of course, the analysis doesn’t include aerodynamic flutter, so it’s not complete.

I may have over-thought this issue.

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