Problem $1$:

A soil profile is given in Figure $1.$ A uniformly distributed load of $100$ kPa is applied on the ground surface. The average value of the coefficient of volume compressibility of the clay is $\backslash ($ m$\_\{$v$\}=0\mathrm{.}001\backslash$mathrm$\{$kPa$\}^\{-1\}\backslash ),$ the coefficient of consolidation of the clay is $\backslash ($ c$\_\{$v$\}=4\backslash$mathrm$\{$~m$\}^\{2\}/\backslash )$ year. Assume that the secondary consolidation can be neglected.

$(1)$ Calculate the average degree of consolidation for the layer after $5$ years.

$(2)$ Determine the excess water pressure at the center and at the quarter points $($i$.$e$.\backslash ($ z $/$ H$=\backslash )\backslash (0\mathrm{.}5,0\mathrm{.}25\backslash )$ and $0\mathrm{.}75,$ respectively$)$ after $5$ years.

$(3)$ What is the final consolidation settlement of the clay layer induced by the applied surface load?

$(4)$ How much settlement has occurred after $($a$)5$ years and $($b$)15$ years?

$(5)$ If impervious bedrock were immediately below the clay layer, how long would it take for the clay layer to settle $0\mathrm{.}2$ m $?$ How much settlement would occur after $15$ years?