$5.$ The section of an I$-$section vertical steel strut $(\backslash (\backslash$mathrm$\{$E$\}=200\backslash$mathrm$\{$GN$\}/\backslash$mathrm$\{$m$\}^\{2\},\backslash$sigma$\_\{\backslash$mathrm$\{$Y$\}\}=300\backslash$mathrm$\{$MN$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash ))$ of web and flange thickness $10$ mm is twice as deep $(\backslash (=2$ B $\backslash ))$ as it is wide $($B$).$ The strut is of length $6$ m and is subject to a compressive load of $90$ kN applied through the centroid of the section. If the beam is built in at the bottom and free at the top, determine the required width $($B$)$ and depth $(2$B$)$ of the strut to avoid failure by buckling. Determine the compressive stress in the beam at the Euler load, expressed as a percentage of the yield stress.

Cr $1$

Repeat the calculation for the case where the beam is pinned at the top and still built$-$in at the bottom. Calculate the slenderness ratio of the beam. Determine down to what slenderness ratio buckling remains the limiting failure mode for this steel. Hence what is the shortest length of strut $($of this particular section$)$ to which Euler theory applies?