Problem $2$: A $5m$ thick brown silty sand fill layer with a total unit weight $=$

$19\mathrm{.}62k\frac{N}{m^3}($use the same unit weight above and below water table for this

soil$)$ was placed over a $15m$ thick layer of compressible normally consolidated

gray silty clay. Underlying the clay is a brown sandy gravel. Before placing

the fill, the water table was $3m$ above the clay surface $($after placing the $5m$

fill, the water table is at $2m$ below the top of the fill$).$ Assume the settlement

of the fill and sandy gravel is small compared to the settlement of the silty clay

layer. The normally consolidated silty clay has an initial void ratio $=1\mathrm{.}1,C_c=$

$0\mathrm{.}36,C_{}=0\mathrm{.}06,C_v=0\mathrm{.}86\frac{m^2}{y}r,$ and saturated unit weight $=14\mathrm{.}91k\frac{N}{m^3}.$

Calculate:

The consolidation settlement of the silty clay layer due to the new fill

assuming the fill is applied simultaneously at $t=0$

Compute the time rate of settlement $($present results in graphical format of

settlement vs$.$ time$)$ assuming the fill is applied simultaneously at $t=0$

Plot the effective stress with depth of the silty clay layer at average degree

of consolidation of $50\%.$ For this, select $10$ points across the depth of the

silty clay layer $($at depths of $5m,6\mathrm{.}5,8,..$etc$)$

If the fill was applied linearly over $6$ months, estimate the settlement at $3$

months and after $2$ years.

For the case in No$.4,$ compute the time rate of settlement $($present results

in graphical format of settlement vs$.$ time$)$ and compare with results in No$.$

Plot both of the same graph.