A solid circular steel shaft ABC shown in Figure Q$2$ is securely bonded by a brass sleeve in segment $AB$ for an effective reinforcement. The shaft is fixed at end $A,$ while torques $T_B=2000Nm$ and $T_C($in $Nm)$ are applied at B and C$,$ respectively in the directions indicated. The shear moduli of elasticity of the two materials are $G_b=40{10}^3\frac{N}{m}{m}^2($for brass$)$ and $G_s=80{10}^3\frac{N}{m}{m}^2($for steel$).$

Calculate the:

a$)$ torque $T_C($in $Nm)$ required for a zero angle of twist at end $C.$

b$)$ torques $T_b$ and $T_s($in $Nm)$ in the brass and steel components respectively of segment $AB.$

c$)$ reactive torque $T_A($in $Nm)$ at end $A.$

d$)$ angle of twist, ${}_B$ at point B$.$

e$)$ maximum shear stresses in the two segments and indicate the material with maximum stress.

f$)$ maximum shear strains in the two segments and indicate the material with maximum strain.