Question $5(20$ marks$)$

A steel column that is fixed at the base and free at the top is constructed of a wide$-$

flange section $($Figure Q$5).$ The column height is L$=2$m$.$ The yield stress is $\backslash$sigma $\_($y$)=250$

MPa and the modulus of elasticity is E$=200$GPa. For the given cross section, A$=3,472$

mm$^(2),$I$\_($y$)=13\mathrm{.}72\backslash$times $10^(6)$mm$^(4),$ and I$\_($z$)=4\mathrm{.}51\backslash$times $10^(6)$mm$^(4).$

$($a$)$ Due to the design need, the compressive force P acting at the top of the column

has an eccentricity e$=50$mm$.$ Determine what is the maximum allowable load

P$\_($allow$?)?$

$(14$ marks$)$

$($b$)$ If the compressive force P is applied without any eccentricity, determine the

maximum allowable concentric load. Comment on the effect of eccentricity

based on the comparison of two load capacities.

$(6$ marks$)$

Hint: The secant formula for a column subjected to an eccentric compressive load is

$\backslash$sigma $\_($max$)=($P$)/($A$)[1+($ec$)/($r$^(2))$sec$(($L$\_($e$))/(2$r$)\backslash$sqrt$(($P$)/($EA$)))]$

Figure Q$5$