A uniform shaft with fixed ends at A and $D$ is subjected to external torques of magnitude $T_0$ and $2T_0,$

as shown below. The diameter of the shaft is $d,$ and its shear modulus is $G.$

$($a$)$ Determine the internal torques $T_1,T_2$ and $T_3$ in each segment respectively. $(10$ points$)$

$($b$)$ Determine the maximum shear stress in each segment. $(10$ points$)$

$($c$)$ Determine the angle of rotation of the shaft at joint $B,{}_B.(10$ points$)$