A beam, of variable cross$-$section, is fixed at its

ends and loaded with a torque at its midspan

$($point B$).$ From A to B$,$ the beam is of S$1$

geometry, and from B to C$,$ S$2$ geometry. Given

torque $T=2\mathrm{.}0kNm,$ length $L=3\mathrm{.}0m,$ and shear

modulus $G=75,000$MPa, determine:

a$)$ the reactions at A and $C$; and

b$)$ the total angle of twist at $B.$