$($s$)$ A counc pith metric grade $9\mathrm{.}8$ bolt with a hexagonal nut assembly is used to keep two mastined parts of a machine together as shown in the figure below with a maximum exlemal had Not holt and portion of member. $\backslash ($ P$=20\backslash$mathrm$\{$kN$\}\backslash ).$ The major diameter of the bolt is $14$ mm $.$ Both the tolt and bottom member are made of steel, while the top plate is made of wrought alumunum alloy $($A$/2024-$O$).$ Assume the bolts are reusable. Use $\backslash ($ E$\_\{$S t$\}=200\backslash$mathrm$\{$GPa$\}\backslash )$ and $\backslash ($ E$\_\{$A $1\}=62\backslash$mathrm$\{$GPa$\}\backslash )$

a$)$ Determine the stiffnesses of the bolt and members and fraction of load acting on the bolt $\backslash ($ C $\backslash )$

b$)$ All the static and fatigue factors of safety using Gerber theory?