Problem $5.(15$ points$)$

A thin$-$walled circular section beam has a diameter of $200mm$ and is $2m$ long; it is firmly restrained against rotation at each end. A torque of $15kNm$ is applied to the beam at its mid$-$span point. If the maximum shear stress in the beam is limited to $200\frac{N}{m}{m}^2$ and the maximum angle of twist to $2,$ calculate the minimum thickness of the beam walls. Take $G=25000\frac{N}{m}{m}^2.$