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Exercise $4$: Failure criteria in Analysis Design: Fatigue

A steel shaft of diameter d is subjected to the following loading:

Axial load that varies from a minimum $F_{\text{m}\text{i}\text{n}}$ to a maximum $F_{\text{m}\text{a}\text{x}}$

Bending Moment which fluctuates between $M_{\text{m}\text{i}\text{n}}$ and $M_{\text{m}\text{a}\text{x}}$

$1/$ Define the relation between d $F_{\text{m}\text{i}\text{n}}{F}_{\text{m}\text{a}\text{x}}{M}_{\text{m}\text{i}\text{n}}{M}_{\text{m}\text{a}\text{x}}$ and also ${}_u($Tensile Strength$),{}_a($Endurance Limit$),$ S$.$F$.($security Factor$)$ in order to prevent failure by fatigue

$2/$ simplify the equation defined in $1/$ considering completely reversed bending moment and completely reversed axial force

$3/$ Simplify the equation defined in $1/$ considering constant axial force F and completely reversed bending moment $M.$