Problem $2$:

A deposit of clay $)=(18k\frac{N}{m^3}$ is $10$ m thick, and apparently drained on both the top and the bottom. The coefficient of consolidation for the clay is $1\mathrm{.}7{10}^{-4}c\frac{m^2}{s}$ from the laboratory test. The test also indicated $C_c=0\mathrm{.}3,C_r=0\mathrm{.}08,e_0=0\mathrm{.}72$ and ${}_p^{\text{'}}=130$kPa. The groundwater level is assumed at ground surface $($i$.$e$.,0$ m $).$ Divide the clay layer into five layers with equal thickness $($i$.$e$.,2$ m per layer$)$ to calculate settlement. Assume the stress increase due to applied load are $100$kPa,$93$kPa,$83$kPa,$75$kPa and $68$ kPa at depth $1m,3m,5m,7m,$ and $9$ m $,$ respectively.

$($a$)$ What is the ultimate consolidation settlement under the applied load?

$($b$)$ How much settlement would you expect to occur in $5$ years?

$($c$)$ Assuming there is a thin sand seam at a depth of $8$ m below the ground surface, what would be the ultimate consolidation settlement and how much settlement would you expect to occur in $5$ years under this condition?