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Problem $1$

$(15$ marks$)$

The figure shows a rotating shaft machined from a steel bar having the following minimum properties:

Ultimate tensile strength $=690$MPa; Yield strength $=580$MPa.

The shaft has the following properties at the critical point $($A$)$:

Reliability requirement $=99\%,$

$,$ Fatigue stress concentration factor in bending $=2\mathrm{.}0$

Fatigue stress concentration factor in axial loading $=1\mathrm{.}6$

Fatigue stress concentration factor in torsion $=1\mathrm{.}5$

$($a$)$ If the shaft is subjected to a steady torsional stress of $40$ MPa $,$ an alternating bending stress of $60$ MPa, and a steady tensile stress of $25$ MPa at the critical point, determine the factor of safety guarding against fatigue using the modified Goodman criterion, expected life in number of cycles and the factor of safety guarding against first$-$cycle yielding.

$($b$)$ If the same shaft is subjected to an alternating bending stress of $150$ MPa only, estimate the factor of safety guarding against fatigue using the Goodman criterion and expected life in number of cycles.