$2\mathrm{.}4$ A simply supported U beam is loaded as shown in Figure Q$\mathrm{.}4($a$).$ Its cross section is shown in Figure Q$\mathrm{.}4($b$).$

Figure Q$\mathrm{.}4($a$)$

Figure Q$\mathrm{.}4($b$)$

$($All dimensions are in mm $)$

$($a$)$ Calculate the distance $(\backslash (\backslash$mathbf$\{$y c$\}\backslash ))$ between the neutral $\backslash (\backslash$mathbf$\{$X$\}-\backslash$mathbf$\{$X$\}\backslash )$ axis and the bottom of the beam for the section as shown in Figure Q$\mathrm{.}4($b$).$

$($b$)$ Determine its moment of inertia $(\backslash (\backslash$mathrm$\{$I$\}\_\{\backslash$mathrm$\{$xx$\}\}\backslash ))$ about the neutral X$-$X axis. $(6$ marks$)$

$($c$)$ If the maximum bending stresses in the beam are $\backslash (355\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$mm$\}^\{2\}\backslash ),$ determine the maximum uniformly distributed load $\backslash ($ w $\backslash )$ that the beam can carry. $(9$ marks$)$

$($d$)$ Determine the maximum shear stress in the beam under the uniformly distributed load $\backslash ($ w $\backslash )$ obtained in part $($c$).$