Question $4.$ BOLT DESIGN $-$ FATIGUE LOADING

A class $8\mathrm{.}8,M273$ ISO bolt has rolled threads and is used to clamp two rigid members together so that $k_c=6k_b.$ There is an external force fluctuating rapidly between $0$ and $100$ kN $.$ The construction and precision of manufacture are such that there is a possibility of slight bolt bending. Bolt material exhibits and idealized elastic, perfectly$-$plastic stress$-$strain curve.

Answer the following two questions for two different values of initial tension: $(1)$ a regular value of $75$ kN $,$ and $(2)$ the limiting value of ${}_{\text{y}\text{i}\text{e}\text{l}\text{d}}{A}_t$ with ${}_{\text{y}\text{i}\text{e}\text{l}\text{d}}=660$MPa.

a$)$ Make a force$-$versus$-$time plot showing the fluctuating external load, bolt force and clamping force for each of the two values of initial tension. Begin the plot with initial tightening, show three load cycles and conclude with load off.

b$)$ Analyze bolt failure. Consider both separation of parts and eventual fatigue failure for each of the two values of initial tension. Provide safety factors when applicable.