A $60m$ long and $20m$ deep plane truss has a pin support at A and a roller support at $G.$

a$.$ Calculate the vertical reaction at $A$ as a function of the simultaneously applied loads $F_B,F_C,$ and $F_D.$

b$.$ Calculate the internal axial force in member C$-$D as a function of the simultaneously applied loads $F_B,F_C,$ and $F_D.$

Consider a moving load $($e$.$g$.,$ truck$)$ with mass $10,000kg$ driving across this bridge from A$-$B$-$C$-$D$.$ When its position is $x=0$ at node $A,$ all its weight is applied at node A$.$ When its position is $x=60$ at node $D,$ all its weight is applied vertically downward at node $D.$ The table below shows the equivalent mass applied $($in thousands of $kg)$ at $B,C,$ and $D$ as the truck drives across the bridge from A$-$B$-$C$-$D$.$ You can linearly interpolate between these points.

$\backslash$table$[[x,0,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60],[F_B,0,2,4,6,8,10,8,6,4,2,0,0,0,0,0,0],[F_C,0,0,0,0,0,0,2,4,6,8,10,8,6,4,2,0],[F_D,0,0,0,0,0,0,0,0,0,0,0,2,4,6,8,10]]$

Plug your equations from $($a$)$ and $($b$)$ into Matlab, R$,$ or Excel to answer the following questions:

c$.$ What is the maximum $($i$.$e$.,$ uplift$)$ vertical reaction at A$?$ Where is the truck when this occurs?

d$.$ What is the minimum $($i$.$e$.,$ downward$)$ vertical reaction at A$?$ Where is the truck when this occurs?

e$.$ What is the maximum internal force $($i$.$e$.,$ tensile$)$ in member C$-$D$?$ Where is the truck when this occurs?

f$.$ What is the minimum internal force $($i$.$e$.,$ compression or smallest magnitude of tensile force$)$ in member C$-$D$?$ Where is the truck when this occurs?

Hint: Use your equations to fill out additional rows in the table above for the reaction at A and the internal force in member $C-D.$