The solid shaft $($ri $=0)$ and collar shown in the figure are to be press fit together. The shaft is manufactured with a diameter d$=50\mathrm{.}040\mathrm{.}009$ mm and the collar with an internal diameter D$=50\mathrm{.}000\mathrm{.}009$ mm$.$ The remaining dimensions are ro$=32\mathrm{.}5$ mm and R$=25$ mm$.$

a$)$ Assuming normally distributed, natural tolerances on d and D$,$ determine the mean and standard deviation of the radial interference $,$ which is defined stochastically as follows $(5$ points$)$

b$)$ It has been determined that the press fit assembly is only acceptable if the radial interference lies in the range mm$.$ Given your solution to part a$),$ what percentage of assemblies will be acceptable?

c$)$ Suppose that the elastic moduli of both the shaft and the collar are normally distributed such that E $=$ N$(207,5)$ GPa. Given your results of part a$),$ find the mean and standard deviation of the maximum shear stress in the collar, which is given stochastically as max $=$ E $2$R $.$ In so doing, treat the nominal radius R$=25$ mm as deterministic.

d$)$ Suppose the shear yield strength is normally distributed such that Ssy $=$ N$(120,6\mathrm{.}5)$ MPa. Given your results of part c$),$ determine the reliability against yielding.