Figure $3$ shows a $70$ mm diameter solid shaft made out of steel whose modulus of rigidity is

$85,000\frac{N}{m}{m}^2.$ The shaft is completely fixed at one end and subjected to the distributed and

concentrated torsional loadings shown.

Essentially the shaft is subjected to a distributed torque of $1200N\frac{m}{m}$ for the first $1050mm$ of

its length, to a concentrated torque of $600Nm$ at a distance of $1050$ mm from the fixed end and

to another concentrated torque of $350Nm$ at the free end. The torque directions are as shown

in Figure $3.$

$($a$)$ Sketch the torque diagram for the shaft giving the torque values at the salient points.

$(12$ marks$)$

$($b$)$ Calculate the maximum value for torsional shear stress and state where it occurs.

$($c$)$ Calculate the angle of twist at the free end.

$(12$ marks$)$

Figure $3$