QUESTION $4$

$[20]$

$4\mathrm{.}1$ Two C$180$x$70$ channel sections are arranged back to back with their webs held rigidly in contact with their webs vertical. A $40614039$ parallel flange I section is then rigidly attached $($symmetrically$)$ to the upper flanges of the channel sections, such that the flange of the l$-$beam is fixed to the flanges of the channels as shown in the figure betow.

Working from $O-O,$ determine the:

$4\mathrm{.}1\mathrm{.}1$ Position of the centroid of the cross section

$4\mathrm{.}1\mathrm{.}2$ Second moment of area about the $x-x$ axis and the $y-$y axis through the centroid

$(8)$

$4\mathrm{.}2$ A short hollow cylindrical column with $200$ mm external and $120$ mm internal diameters carries a vertical eccentric load of $400$ kN $.$ Find the maximum eccentricity $($h$)$ of the load if the allowable stresses are $60$ MPa $($compressive$)$ and $25$ MPa $($tension$).$

$(10)8$