Problem $3$: An AISI $1006$ HR steel shaft $($solid$)$ is to be designed for $\backslash (90\backslash \%\backslash )$ reliability. The shaft transmits a torque of $340$ Nm through gears $\backslash ($ C $\backslash )$ and $\backslash ($ D $\backslash )$ as shown below. The radial loads on the shaft due to gears $\backslash ($ C $\backslash )$ and $\backslash ($ D $\backslash )$ are $1\mathrm{.}5$ kN and $2$ kN $,$ respectively and they are in the same plane. The shaft is simply supported by bearings at $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash ).$ The torque transfer between gears and the shaft takes place through rectangular keys. The ratio of the diameters $\backslash (($D $/$ d$)\backslash )$ is $1\mathrm{.}2$ and fillet$-$radius to smaller diameter ratio $\backslash (($r $/$ d$)\backslash )$ of $0\mathrm{.}02$ at the bearing locations. The stress concentration factors at the keyways are $5$ for bending and $3$ for torsion. Determine the minimum allowable shaft diameter for the $500$ mm span between the bearings based on a factor of safety of $2$ for the shaft. Use both the static yielding and modified Goodman criteria. $(16$ pts$)$

$\backslash ($ F $\backslash )$