A retaining wall is to be constructed at a level ground site at a depth of $1\mathrm{.}9m$ below the existing ground level. The subsurface soil conditions are shown in Table $1.$ The groundwater table is located at the ground surface. The SPT N$-$values shown in the table are corrected $N$ values. It is proposed to treat the ground up to $7m$ depth by deep soil mixing technique using $600mm$ diameter lime$-$cement columns. Assume an end bearing mobilization factor of $0\mathrm{.}5.$

$($a$)$ Determine the ultimate load carrying capacity of the individual DM column $($lime$-$cement column$),$ if the load on the DM column due to the retaining wall and the soil is estimated as $90kN,$ estimate the factor of safety against bearing failure.

$($b$)$ If the DM columns are arranged in a square pattern at a spacing of $1m$ center to center, what is the safe bearing capacity of the DM column treated ground? Assume a bearing capacity mobilization factor of $0\mathrm{.}5$ and the factor of safety obtained in $($a$).$

Table $1$: Subsoil conditions for the construction of boundary wall

$\backslash$table$[[($m$)\_($ Depth $),$SPT N value,Soil type,$\backslash$table$[[$Fines content$],[(\%)]],\backslash$table$[[$Bulk unit weight$],[$q$,($kN$/$m$^(3))]]],[0\mathrm{.}3,1,$Clayey silt,$96,18],[1\mathrm{.}3,1,$Clayey sand,$46],[2\mathrm{.}3,2],[3,2],[4,0,42],[6,3],[7,3,$Silty sand,$37],[8,10],[9\mathrm{.}5,8],[10,8,34,19]]$