Question $1$

A developer wants to build a gravity retaining wall as part of new housing development. The geometry, and soil, wall and interface properties are given in Figure Q$1.$

$($a$).$ By considering the limiting equilibrium of a series of potential active sliding wedges extending from the bottom corner of the wall, determine the minimum horizontal force that the wall needs to provide to maintain the stability of the soil. You may solve the problem either graphically by means of force vector diagrams, or algebraically by resolving forces. Appy a factor of $1\mathrm{.}25$ to tan ${}^{\text{'}}($but do not apply any other design measures such as overdigs or surcharges$).$

$[25$ marks$]$

$($b$).$ Draw a free body diagram of the retaining wall, labelling on to it the forces acting on the structure and their lines of action. Use this to check the stability of the wall against sliding and topping around its front corner $($the point A in Figure Q$1).$ As before, you should apply a factor of safety of $1\mathrm{.}25$ to $tan\text{'}.$ State clearly any assumptions that you need to make.

$[18$ marks$]$

$($c$).$ Convert the forces acting on the base of the wall into vertical, horizontal and moment loads, $V,H$ and $M,$ all acting through the centre point of the base of the structure. Determine the force $V_{\text{m}\text{a}\text{x}},$ and then use the combination of forces and moments and $($:$V_{\text{m}\text{a}\text{x}}$