The deck is supported by the two columns with square cross$-$section of width $45$ mm $.$ Column $\backslash ($ A B $\backslash )$ is pinned at $\backslash ($ A $\backslash )$ and fixed at $\backslash ($ B $\backslash ),$ whereas $\backslash ($ C D $\backslash )$ is pinned at $\backslash ($ C $\backslash )$ and $\backslash ($ D $\backslash ).$ Both columns are made from Douglas Fir wood with Young's modulus $\backslash ($ E$=13\mathrm{.}1\backslash$mathrm$\{$GPa$\}\backslash ).$ Note that walls are preventing the deck to move sideways.

$($a$)$ If the center of gravity $\backslash ($ G $\backslash )$ of the load is $\backslash ($ d$=1\mathrm{.}9\backslash$mathrm$\{$~m$\}\backslash )$ away from point $\backslash ($ A $\backslash ),$ determine the maximum weight $\backslash ($ W$\_\{\backslash$text $\{$cr $\}\}\backslash )$ of the load that can be supported by deck before one of the columns buckles. Which of the columns would buckle at that critical weight?

$($b$)$ If the distance $\backslash ($ d $\backslash )$ can be varied between $0$ and $5$ m $,$ what is the maximum possible value of $\backslash ($ W$\_\{\backslash$text $\{$cr $\}\}\backslash )?$ What is the corresponding distance $\backslash ($ d $\backslash )?$