$4.\backslash ([25\backslash \%]\backslash )$ A site is to be developed for construction of a one$-$story steel structure. The site is underlain by $6$ m of soft, saturated clay $($Ground Water Table is at ground surface$),$ which in turn is underlain by impervious rock. A $1\mathrm{.}5$ m of freely draining fill is to be placed on the site to raise the ground surface. The structure is relatively flexible and can withstand several inches of differential settlement. So$,$ it is decided early in the design phase that the maximum allowable total settlement is $5$ cm $.$ The fill has a total unit weight of $\backslash (19\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$ The soft clay is normally consolidated with the following properties $\backslash (\backslash$left$(\backslash$gamma$\_\{\backslash$mathrm$\{$t$\}\}=16\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}^\{3\},\backslash$mathrm$\{$~S$\}=100\backslash \%\backslash$right$.\backslash ),\backslash (\backslash$mathrm$\{$Gs$\}=\backslash )\backslash (2\mathrm{.}7,\backslash$mathrm$\{$Cc$\}=0\mathrm{.}78,\backslash$mathrm$\{$C$\}\_\{\backslash$mathrm$\{$R$\}\}=0\mathrm{.}09\backslash )).$ Assume that the water table stays at the interface between the clay and fill during consolidation and assume that the weight of the building is negligible compared to the weight of the fill.

$($a$)$ Compute the total consolidation settlement of the soft clay caused by placement of the $1\mathrm{.}5$ m of fill. For simplicity, use only one clay layer in your settlement calculation.

$($b$)$ Calculate the time required until only $5$ cm of consolidation settlement remains. $($This would be the time required after fill placement before construction of the building could begin$).$ The coefficient of consolidation for the clay $(\backslash (\backslash$mathrm$\{$c$\}\_\{\backslash$mathrm$\{$v$\}\}\backslash ))$ is $0\mathrm{.}03\backslash (\backslash$mathrm$\{$m$\}^\{2\}/\backslash )$ day. Assume that the clay layer is free draining only at its top.