Figure $1$ shows a closed tank which is completely filled with water and connected to a Mercury manometer which can be used to determine the water pressure inside the tank. The step shaped gate ABC $(3$ m wide into the page$)$ is hinged at A and supported by a column CE$.$ If the column were to fail, the gate would rotate about the hinge at A and allow the water to spill. The column is made from timber $($E $=14,000$ MPa, y $=100$ MPa$)$ with a rectangular cross section with dimensions b $($mm$)$ in the x$-$direction and d $($mm$)$ in the y$-$direction, such that d $=4$b as shown in Figure $2.$ The column is pinned at C and E for buckling about the x$-$axis and fixed at C and E for buckling about the y$-$axis. In addition to this, the column is laterally restrained at D in the x$-$direction $($affects buckling about the y$-$axis only$).$ a$)$ Determine the forces acting on the surface AB and BC due to the pressure generated by the water and mercury. By considering moment equilibrium at hinge A$,$ determine the force inthe column CE to keep the gate closed.b$)$ By considering squashing, buckling about x and y axes, find the minimum cross section dimensions b and d required to prevent the column from failing and releasing the water. $($Hint: Express Ixx, Iyy and A in terms of b$.)$ Which mode of failure governs the axial compressive capacity of the column CE$?($Neglect the self$-$weight of the gate.$)$

Figure $1$