$3.$ A pin$-$jointed truss structure is shown in Figure Q$3$a$.$ All joints can be considered fully pinned for buckling in any direction.

I. Show that the compressive force in members AB and AG is $\backslash (3\backslash$boldsymbol$\{$P$\}\backslash )$ and $\backslash (\backslash$sqrt$\{2\}\backslash$boldsymbol$\{$P$\}\backslash ),$ respectively. $($Hint: Consider the method of section analysis.$)$

$[4/20$ marks$]$

II$.$ Determine the maximum $\backslash (\backslash$boldsymbol$\{$P$\}\backslash )$ load that can be applied at E if a section, as shown in Figure Q$3$b$,$ is used for compressive members. Ignore the effect of joint failure of the truss and tensile failure of members.

$[10/20$ marks$]$

III. Consider only the member AB $.$ The channel section is to be replaced with a circular hollow section of an external diameter of $50$ mm and an internal diameter of $\backslash (\backslash$boldsymbol$\{$d$\}\_\{\backslash$boldsymbol$\{$i$\}\}\backslash ).$ Determine $\backslash (\backslash$boldsymbol$\{$d$\}\_\{\backslash$boldsymbol$\{$i$\}\}\backslash )$ so that a load of $\backslash (20\backslash$mathrm$\{$kN$\}(=\backslash$boldsymbol$\{$P$\})\backslash )$ can be safely applied at E$.$ Material properties and end conditions remain the same.

$[6/20$ marks$]$

A $-$ viuss seculial alea

$\backslash ($ r $\backslash )-$ radius of gyration

$\backslash (\backslash$boldsymbol$\{$A$\}=2590\backslash$mathrm$\{$~mm$\}^\{2\}\backslash )$