A straight prismatic circular shaft $AD$ has diameter $d$ and length $L.$ Its left end is fixed while its

right end is twisted by ${}_D.$ A uniformly distributed torque, $t$ per unit length, is applied on

segment $AB,$ and a concentrated torque, $T_c,$ on cross$-$section $C.$ The shaft undergoes linear

elastic deformation with a homogeneous shear modulus, $G.$ The lengths of segments $AB$ and $BC$

are $L_{AB}$ and $L_{BC},$ respectively. Please i$)$ calculate the maximum shear stress at an arbitrary cross

section $x$ as a piecewise function of $x,$ ii$)$ calculate the angle of twist at an arbitrary cross section

$x$ as a piecewise function of $x.$