$($a$)$ A relatively soft soil deposit having $C=25k\frac{N}{m^2}$ and $=20$ is proposed to be improved with the help of $500$ mm diameter stone columns, placed at a center$-$to$-$center distance of $2\mathrm{.}0$ meter up to a depth of $6\mathrm{.}0$ meter. The stone columns are arranged in a square grid pattern. After compaction, the stone columns material is likely to have an angle of internal friction of $40,$ Water Table at the site occurs at a depth of $3\mathrm{.}0$ meter from E$,$G$.$L$..$ The dry soil above the G$.$W$.$T$.$ has a unit weight of $16k\frac{N}{m^3}$ and that of the saturated soil below the water table is $19k\frac{N}{m^3}$ For the soil take $Nc=14\mathrm{.}83,Nq=6\mathrm{.}4$ and $N_{}=0.$ Calculate the safe load capacity of the stone column unit cell.

$(10)$