for the circular motion one.....

read HCV chapter " Forces" so that u r not shocked when i say that tension is an electromagnetic force produced due to forces between electrons and protons of bodies involed.

If this is true, then the part considered in the problem has electrons, protons and all other stuff in all direction (both sides of the point C in the ring) which exert forces on point C as and when the point C tries to bulge out due to centrifugal force produced while rotation.

 

and what is the problem.........
in pulley sums also u consider tension in both direction along the string, dont you.......???