If a ring is spinning with a relativistic tangential velocity, assuming it can withstand the centripetal force and remains rigid, will length contraction cause a force to squeeze the ring? Or does the whole thing get smaller without physically applying a force?
Provided the force exists, I imagine the force would be in the opposite direction as the centripetal force and at some speed, they would cancel out.
A similar question crossed my mind a few years ago, and I finally concluded that the ring perforce shatters. This is indeed a modified version of Ehrenfest paradox which can question the formal resolution to this anomaly for a rotating disc:
Any rigid object made from real materials that is rotating with a transverse velocity close to the speed of sound in the material must exceed the point of rupture due to centrifugal force, because centrifugal pressure can not exceed the shear modulus of material.
Because, contrary to a Born-rigid disc, we can always rotate a ring so that the centrifugal acceleration/force acting on the ring approaches zero, while its tangential speed approaches $c$ if, and only if, the ring's radius is large, while its angular velocity is small enough so that $v=r\omega\approx c$ and $a=r\omega^2\approx 0$. (For instance, assume $r=10^{20}\space m$ and $\omega \approx 3×10^{-12}\space$ $rad/s$.)
It is worth, too, reviewing the Wiki Talk Page of the mentioned Wiki article.
