Problem $2.$ A thin$-$walled beam has the cross$-$section as shown in the figure. The thickness of each flange varies linearly from $\backslash ($ t$\_\{1\}\backslash )$ at the tip to $\backslash ($ t$\_\{2\}\backslash )$ at the junction with the web. The web itself has a constant thickness $\backslash ($ t$\_\{3\}\backslash ).$

$1.$ Find the shear flow distribution.

$2.$ Calculate the distance $\backslash (\backslash$xi$\_\{$S$\}\backslash )$ from the web to the shear center $\backslash ($ S $\backslash ).$

Note: The thicknesses are to be taken as very small in comparison with the other cross$-$sectional dimensions in calculating the section properties.