d $=50$mm$,$ L$1=\mathrm{.}8$m$,$ L$2=\mathrm{.}6$m$,$ L$3=\mathrm{.}6$m$,$ P$1=2$kN$,$ P$2=2\mathrm{.}5$kN$,$ P$3=\mathrm{.}5$kN$,$ P$4=3$kN$,$ T$1=5$kN$*$m$,$ T$2=2\mathrm{.}5$kN$*$m$,$ X $=20\%$

A section of a crankshaft, made from hot$-$rolled steel, is subjected to a combination of loads as illustrated below. The loads are applied along the shaft's centroidal axes, and the dimensions are measured from one centroidal axis to another. The circular shaft has a diameter of $$d$$mm$.$

Given that the strain at failure is $0\mathrm{.}5$ and Qyt$=$Qyc$=500$MPA, determine whether point F can withstand these loads.

a$)$If the shaft's volume decreases by X$\%$ while maintaining the same material, length, and outer diameter, calculate the inner diameter of the resulting hollow cross$-$section.

b$)$For the new hollow cross$-$section, determine whether point F can withstand the applied loads.

c$)$Evaluate how the following parameters change$($reduce or increase in terms of percentage$)$afterthe volume reduction:$1-$Axial stiffness $($k$=$AE$/$L$)2-$Flexural stiffness $($k$=$IE$/$L$3)3-$Torsional stiffness $($K$=$GJ$/$L$)4-$Manufacturing cost$5-$Carbon emissions from manufacturing