I had recently made a clamp-type shaft coupler and was impressed by the amount of clamping force it could exert. This started me wondering how to actually calculate the total inwards radial force exerted by a clamp like this, or any similar clamping device like a hose clamp, barrel lid clamp etc.
After a bit of trial and error, I came up with two methods. The simple method uses the fact that the radius of the band decreases by a factor of 2π less than the circumference, so the total inward force will be a factor of 2π greater than the tension in the band. The rigorous method imagines the band wrapped around an n-sided polygon and works out the total force exerted on all the corners of the polygon. As the number of sides tends to ∞, the polygon approximates a circle, giving the same result as the simple method.
The latter method was interesting, since I ended up with an expression which tended to ∞*0 (the product of infinity and zero) as n tended to ∞! Fortunately, L'Hôpital's Rule can be used to solve this problem.
I've put all the calculations into a LaTeX PDF, available here, since LaTeX is so good at representing equations.