A shaft is to be designed in two sections, Section I and Section II$,$ with diameters D$1,$ and D$2,$ respectively, machined from AISI $4640,$ OQT $1000.$ The shaft is subjected to a load of $3000$ lb$.$ applied at a point between the two sections, such that the length of Section I is a $=10$ in and the length of section Il is b $=20$ in$.$ The total length of the shaft is $30$ in$.$

The shaft supports the load Q at the junction of the two sections. The ends A and B are fixed, preventing any twisting or rotation, but they are designed to support the balancing reactions due to Q without inducing additional moments. The design is for steady loading conditions with a factor of safety N$=3\mathrm{.}0,$ based on the octahedral shear stress theory. Design the shaft system

Please solve the value of D$1$ and D$2,$ and explain the formulas used, and site the book of reference $.$