$(20$ Marks$)$

A solid $40$ mm diameter steel cylinder with a tensile yield strength $S_{yt}=120$MPa is loaded at its free end with a moment $M_x=450N-m$ and an axial centric load $P_x,$ as shown in the figure.

a$)$ Draw element $H$ indicating the stresses acting on it along with the proper axes orientation.

b$)$ Assuming the material is homogeneous and isotropic use Mohr's circle to determine the maximum value of $P_x$ which can be applied to the cylinder so that failure will not occur if;

a$)$ shear stress $($using Tresca$)$ controls $P_{x.}$

b$)$ yield stress controls $P_x$

NOTE: Marks will only be awarded for a solution using Mohr's circle. Calculations must be based upon the geometry of your circle. No marks will be awarded if you use the stress transformation equations.