The precision shaft with $50$ mm diameter made with Q&T steel at $315\backslash$deg C having S $1030$ MPa, S$-820$ MPa is subjected to alternating $($reversed$)$ bending loading, and commercially ground surfaces.

$($a$)$ Estimate the S$-$N curve for the bar and plot it using $0,9$ for fatigue strength fraction $($f$).$

$($b$)$ Draw modified Goodman diagram using the estimated endurance limit$.$