A structural column with a C$-$shaped cross$-$section is used as a support element in a pedestrian bridge. The column is made of steel and has the following dimensions: flange width $=150mm,$ web height $=300mm,$ flange thickness $=10mm,$ and web thickness $=10mm.$ The column is $4$ meters in length and is fixed at both ends. The modulus of elasticity $(E)$ of the steel is taken as $210$GPa.

Given the dimensions of the C$-$section, calculate the moment of inertia about the axis perpendicular to the web $($considering the web as the vertical part of the section$).$

Then, calculate the critical load $($Pcr$)$ for the column.

Assume the column is perfectly straight before loading and that both ends are fixed. $[5$ Marks$]$