Problem $1(25$ pts$)$

A low$-$carbon cylindrical steel shaft is designed with a diameter of $30$ mm $.$ It is subjected to an axial load $P,$ a bending moment $M,$ as well as a torque $T,$ as shown in Figure $1.$ Yielding initiates when the following is applied: $P=60kN,M=300$ N$.$m and $T=500$ N$.$m$.$

Determine the axial load, the maximum bending stress as well as the maximum torsional stress in the shaft.

Given the values found in $(1),$ use a Mohr's circle to calculate the principal stresses.

Determine the yield stress ${}_Y$ assuming the maximum shear stress criterion applies.

Figure $1$