A solid round shaft is to be made from the AISI $4340$ steel of Fig. $8\mathrm{.}32.$ It must resist a bending moment of

$,$ with a safety factor of two against yielding. Also, a half$-$circular surface crack of depth

may be present, and a safety factor of three against brittle fracture is needed. Some combinations of yield strength and fracture toughness from Fig.

are given in Table P$8\mathrm{.}42.$

$($a$)$ For material heat treated to a yield strength of

$,$ what shaft diameter is required to resist yielding if the possible crack is ignored? Also, what shaft diameter is required to resist fracture due to the

crack, where the possibility of yielding is ignored?

$($b$)$ Which value from $($a$)$ should be chosen to avoid failure by either cause?

$($c$)$ Repeat $($a$)$ for the additional combinations of yield strength and fracture toughness given. What combination of yield strength and shaft diameter gives the most efficient design, such that the diameter, and thus the weight, is minimized?

$($d$)$ Repeat the previous analysis for

and for

$.$ Is the choice of a yield strength sensitive to the crack size that might be present