P$4\mathrm{.}9$ Rigid beam ABC is supported as shown in Figure P$4\mathrm{.}9.$ The pin connections at $B,C,$ and $D$ are each double shear connections, and the ultimate shear strength of the pin material is $620$ MPa $.$ Tie rod $(1)$ has a yield strength of $340$ MPa $.$ A uniformly distributed load of $w=15k\frac{N}{m}$ is applied to the beam as shown. A factor of safety of $3\mathrm{.}0$ is required for all components. Assume $a=700mm,b=900mm,c=300$ mm $,$ and $d=650mm.$ Determine

$($a$)$ the minimum diameter required for tie rod $(1).$

$($b$)$ the minimum diameter required for the double shear pins at $B$ and $D.$

$($c$)$ the minimum diameter required for the double shear pinat $C.$