Figure $1$ shows a cast iron roller assembly driven by a gear. The roller extends over $80\%$ of length a and is centered in that dimension. The roller occupies $95\%$ of the exposed shaft length between the bearing faces. The shaft is steel with $S_y=427$MPa and $S_{ut}=745$MPa. For the data shown in Table$1,$ find the safety factor against fatigue failure for a shaft diameter of $40$ mm at point C $($middle of the roller$).$

Assumptions:

The finish is machined, reliability is $90\%,$ and the shaft is at room temperature.

One half of the torque $(T)$ is taken out at the right$-$hand key and the other half at the lefthand key $($it means that at the middle of the roller only one half of the torque is present$).$

Note that the given magnitudes of the radial force shown acting on gear in this problem is not necessarily consistent with a load associated with the given torque for any real gear of reasonable pressure angle.

Table $1$

$\backslash$table$[[l(cm),a(cm),b(cm),P$ or $p(N)or(\frac{N}{c}m),T_{\text{m}\text{i}\text{n}}(N.m),T_{\text{m}\text{a}\text{x}}(N.m)$