Problem $\backslash$#$3$: $(35$ points$)$ A column of length $\backslash ($ L$=3\backslash$mathrm$\{$~m$\}\backslash )$ is free at its upper end, fixed at its base, and subjected to a compressive load $\backslash ($ P$=450\backslash$mathrm$\{$kN$\}\backslash )$ that is offset from the centroid of the cross section by an eccentricity $\backslash ($ e$=10\backslash$mathrm$\{$~mm$\}\backslash )$ in the $\backslash ($ y $\backslash )-$direction. The column has a Young's modulus $\backslash ($ E$=\backslash )220$ GPa and a yield stress $\backslash (\backslash$sigma$\_\{$Y$\}=230\backslash$mathrm$\{$MPa$\}\backslash ).$ Both the upper portion of the column and its I$-$shaped cross section are shown in Figure $3.$

The I$-$shaped cross section has two flanges $($parallel to the z$-$axis$)$ that can be approximated as rectangles of length $100$ mm $($in the $\backslash ($ z $\backslash )-$direction$)$ and thickness $10$ mm $($in the $\backslash ($ y $\backslash )-$direction$),$ and it has one web $($parallel to the $\backslash ($ y $\backslash )-$axis$)$ connecting the two flanges that can be approximated as a rectangle of length $180$ mm $($in the $\backslash ($ y $\backslash )-$direction$)$ and thickness $10$ mm $($in the $\backslash ($ z $\backslash )-$direction$).$ The total length of the cross section in the $\backslash ($ y $\backslash )-$direction is therefore $200$ mm $.$

The $\backslash ($ x $\backslash )-$axis measures distance along the length of the column. In this problem, consider buckling in the $\backslash ($ x$-$y $\backslash )$ plane, only.

a$)$ Draw a clear picture of the cross$-$section with all the dimensions clearly labeled. Calculate the moments of inertia $\backslash ($ I$\_\{$y y$\}\backslash )$ and $\backslash ($ I$\_\{$z z$\}\backslash )$ of the cross section about its centroidal axes. If you use composite areas, your picture should make the sub$-$areas clear.

b$)$ Calculate the load $\backslash ($ P$\_\{$c r i t$\}\backslash )$ that will cause buckling in the $\backslash ($ x$-$y $\backslash )$ plane.

c$)$ For the values $\backslash ($ P$=450\backslash$mathrm$\{$kN$\}\backslash )$ and $\backslash ($ e$=10\backslash$mathrm$\{$~mm$\}\backslash )$ prescribed, find the magnitude $\backslash (\backslash$sigma$\_\{\backslash$max $\}\backslash )$ of the maximum compressive stress in the column, and describe clearly where it occurs. Will the column yield due to this value of $\backslash ($ P $\backslash )?$

d$)$ For the values $\backslash ($ P$=450\backslash$mathrm$\{$kN$\}\backslash )$ and $\backslash ($ e$=10\backslash$mathrm$\{$~mm$\}\backslash )$ prescribed, find the magnitude $\backslash ($ v$\_\{\backslash$max $\}\backslash )$ of the maximum deflection in the $\backslash ($ y $\backslash )-$direction, and describe clearly where it occurs. Figure $3$