Problem $2(40$ Points$)-$

The figure below shows a countershaft $(20$ cm diameter$)$ with $2$ V $-$belt pulleys with all elements separated equidistant at a spacing of $100-0\mathrm{.}5Zcm.$ A motor powers pulley diameter$)$ through a belt, with the tension at the tighter side being $3+(\frac{Z}{20})kN$ while the loose side is $2$ kN $.$ The power is transmitted through the shaft and delivered to the belt on pulley diameter$).$ For pulley B$,$ assume that the belt tension on the loose side at B is $30$ percent of the tension on the tight side. Assume the shaft runs at a constant speed and the bearings act as simple supports.

Z is a variable based on the last two digits of your student number. For instance, if your student number is $45436764,Z$ will be $64,$ so the downward force at pulley $C$ will be $3+(\frac{64}{20})=6\mathrm{.}2kN,$ and the length of a beam section will be $68$ cm $.$

a$)$ Determine the unknown tensions in the belt on pulley $B.$

b$)$ Find the reaction forces at the bearings.

c$)$ Draw shear force and bending moment diagrams on the $x-y$ plane.

d$)$ Where is the maximum bending moment located? Determine the bending stress and the torsional shear stress at the location of maximum bending.

e$)$ At the point of maximum bending moment, determine the principal stresses and the maximum shear stress.