A round bar $($shown in Figure Q$1)$ with a diameter, $D=40mm$ has a groove with radius, $r=4mm$ machined into it to a diameter, $d=31mm.$ The solid bar, which is made of steel with an ultimate strength $S_{ut}=1400$MPa, is subjected to a maximum and minimum torsional load of $1000N.m$ and $300$ N$.$m$,$ respectively.

Note: Use the related tables, formulas and concepts to answer the following questions:

$(1\mathrm{.}1)$ Formulate to find fatigue limit $($Endurance limit$)$ with the least possible reliability goal.

$(1\mathrm{.}2)$ Apply the DE$-$Modified Goodman fatigue failure theory to estimate factor of safety.

$(1\mathrm{.}3)$ Confirm whether this theory guards against yielding and if it does not, which theory would you recommend instead.