A solid circular shaft, made of a material with a $225$MPa yield stress, lies in the $x-z$

plane, fixed at one end and loaded with force $P30$ to the $y-$axis at the free end.

Determine the maximum value of $P$ if the shaft is designed according to the following

theories of failure;

a$)$ Maximum Principal Stress;

b$)$ Maximum Principal Strain;

c$)$ Maximum Shear Strain Energy.

Assume E $=207$ GPa; Poisson's Ratio $=0\mathrm{.}3$

$(4\mathrm{.}98$kN$,4\mathrm{.}88kN,4\mathrm{.}80kN$