An A$36$ hinged rectangular column is $50mm100mm$ in cross$-$section.

$($a$)$ Determine the mode of failure of the compression member if it is $2$ meters.

$($b$)$ Compute its allowable strength.

$($c$)$ Calculate the mode of failure of $4$ the compression member if it is $0\mathrm{.}5$ meter long.

$($d$)$ Obtain the design strength.

The initial compressive force of a steel column can be determined by using $P_e=\frac{{}^{2}EI}{(kl{)}^2}.$ The x $-$ axis has an

unbraced length of $8$ m which is pinned at the top and fixed at the bottom to prevent sidesway. The $y-$axis has

an unbraced length of $4$ m due to the bracing at the mid$-$height. The properties of the column are the following:

$A=8129m{m}^2,I_x=178\mathrm{.}3({10}^6)m{m}^4,I_y=18\mathrm{.}8({10}^6)m{m}^4,F_y=345$MPa

Proportionality Limit $f_s=320$MPa

$($a$)$ Determine the critical$/$maximum slenderness ratio of the column

$($b$)$ Compute the design strength of the column.

$($c$)$ Calculate the initial compressive load of the column.

$($d$)$ Obtain the minimum length of the column without exceeding the proportionality limit$.$

A W$835$ steel column has an unsupported length of $8$ m $.$ Using A ${?}^{}6$ steel, calculate for the ultimate axial

load capacity, $($a$)$ if both ends are fixed $($b$)$ if both ends are pinned

$W8x35$ Properties, $A=6645\mathrm{.}16m{m}^2,I_x=52\mathrm{.}45({10}^6)m{m}^4,I_y=17\mathrm{.}69({10}^6)m{m}^4$

$r_x=88\mathrm{.}9mm,r_y=51\mathrm{.}56mm$

A W$1279$ of A$573$ Grade $60$ steel is used as a compression member. It is $9$ m long, pinned at the top, fixed

at the bottom, and has an additional support in the weak direction at mid$-$height. Properties of the section are

as follows:

$A=14,500m{m}^2,I_x=258\mathrm{.}6({10}^6)m{m}^4,I_y=84\mathrm{.}3750({10}^6)m{m}^4$

Compute the design axial load capacity of the column.