$8\mathrm{.}28$ Solid circular shafts made of $18-$Ni maraging steel $($vacuum melted$)$ are subjected in

service to bending, with a moment $M=15\mathrm{.}0kN*m.$ Half$-$circular surface cracks, as in

Fig. $8\mathrm{.}17($d$),$ may exist in the part. From nondestructive inspection, it is expected that no

cracks larger than $a=5\mathrm{.}0mm$ are present.

$($a$)$ What shaft diameter is required to resist yielding with a safety factor of $2\mathrm{.}0$ if no

crack is present?

$($b$)$ For an inspection$-$size crack, what shaft diameter is required to resist brittle fracture

with a safety factor of $3\mathrm{.}5?$

$($c$)$ What shaft diameter should actually be used?