# Compressive force on a cylinder wrapped with a string

Stevenpd
Hello everyone,

I'm working with a design team to take quick and accurate measurements of the middle upper arm circumference in third world countries to assess malnutrition in children. One of the issues in taking measurements is people pulling variably on the tape measure around the arm. My team and I were trying to figure out if we made a device that applied the same amount of tension to a band around the arm, would it cause a different compressive force for different sized arms (out intuition is yes because the force would be spread out over a larger area). The real question is if the difference in compressive force is significant in affecting measurements.

So we tried to calculate the compressive force of a band around a cylinder with one end of the band fixed and the other end pulling with a tangential force to the cylinder and weren't able to find any equations defining this. The closest we got was circumferential hoop stress.

Anyone have any clue how we might calculate this?

Mentor
With a perfect cylinder of radius r and internal tension F in the ideal band, the compression force everywhere (as force per circumference length) is ##\frac{F}{2r}##.
The reduced circumference in real arms due to this force... well, no idea how to estimate this. I think you can get some formula based on real arms. But you have to calibrate your scale "measured arm circumference" -> "nutrition state" anyway, so this should not be a problem.

Studiot
In the days when surveyors used steel tapes or bands for accurate measurement they had available constant tension devices for holding the tape. These measurement were normally linear measurements.

One point about this is the tape material was steel. This could be tensioned elastically so it recovered its original untensioned length upon release.

Cheaper more robust tapes of fabric or plastic were also used for rougher work. If regularly tensioned in this way they would gradually stretch (creep) thus loose their calibration.

Gold Member
A thought probably of little value... I'd have thought that the pressure applied by the tape would cause the flesh to be 'grooved in' under the tape and to bulge out either side of the tape. A much broader band (I'm thinking of, say, 75 mm) rather than a tape of width 15 mm (say), might lessen this effect.

Stevenpd
Really appreciate the help everyone. One of my professors was nice enough to help us work it out and then afterwards he found it already exists online: http://en.wikipedia.org/wiki/Capstan_equation

Thanks again for all the help!

Gold Member
mfb: I find F/r rather than 2F/r for what it's worth – which may not be much, because I'm prone to slips!

Mentor
mfb: I find F/r rather than 2F/r for what it's worth – which may not be much, because I'm prone to slips!

I find F/r also. Of course, these results apply if there is no friction force under the tape, and the tension is constant around the circumference. If you want to take into account the deformation of the arm resulting from the loading applied by the tape, this is possible by doing a finite element structural analysis on the flesh. Of course, the calculation would have to take into account the geometry of each individual's arm. Also, you can minimize the frictional force by lubricating the tape.

Mentor
Oh, you are right, F/r. And indeed, with an ideal band, no friction and so on.