Relationship between belt tension and pressure it exerts

[Oseania]

TL;DR Summary

If I wrap a belt around my waist and pull, what is the relationship between the pulling force and exerted pressure to my waist?

I have following setup where a non-elastic belt is wrapped around an elastic object, like leather belt around waist. When I pull the belt with force F, it tightens around the elastic object and the belt exerts force to the elastic object. I would like to know what is the relationship between the pulling force F and the exerted pressure (or force) to the elastic object.

 

[ergospherical]

Take a small element of arc which subtends an angle of $2\theta$. If the force per unit length is denoted $p$, and the radius $r$, then the net force on the element in the radial direction is\begin{align*}
F = 2\theta r p - 2T\sin{\theta} \sim 2\theta r p - 2T \theta
\end{align*}For equilibrium, $T = rp$.