$-$ For the T$-$beam shown:

a$.$ Locate the position of the centroid and calculate the second moment of area of the section about the centroidal axis $\backslash (\backslash$mathrm$\{$x$\}-\backslash$mathrm$\{$x$\}\backslash ).$

b$.$ Determine the maximum uniformly distributed load $\backslash (\backslash$omega $\backslash )$ that the beam can carry if the compressive and tensile bending stresses are limited to $160\backslash (\backslash$mathrm$\{$N$\}/\backslash$mathrm$\{$mm$\}^\{2\}\backslash )$ and $\backslash (180\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$mm$\}^\{2\}\backslash )$ respectively.

c$.$ Draw the bending stress distribution for the section at the centre of the span when subjected to this maximum uniformly distributed load.