The results obtained from an oedometer test on a sample of clay are given in Table $3$ below:

Table $3$

$\backslash$table$[[\backslash$sigma $\_($v$)($kPa$),160,200,320,640,480,320],[$Void ratio e$,1\mathrm{.}138,1\mathrm{.}089,0\mathrm{.}983,0\mathrm{.}830,0\mathrm{.}842,0\mathrm{.}857]]$

a$)$ Plot a graph of the void ratio against the logarithm of the vertical effective stress. Indicate the one$-$dimensional normal compression line on the graph.

b$)$ Calculate the pre$-$consolidation pressure and determine the slopes of the one$-$dimensional normal compression and unloading$/$reloading lines.

c$)$ A $15$ m thick layer of the clay from parts a$)$ and b$)$ lies below a layer of sand of $15$ m depth from the ground surface. The water table is at the ground surface. The saturated unit weight of both soil layers is $20$ kN$/$m$3.$ A new office building will be constructed at this site. The building will have a raft foundation which will increase the vertical effective stress in the clay layer by $45$ kPa throughout its depth. A soil sample with an initial height of $20$mm was taken from the mid$-$depth of the clay layer and was tested in the oedometer apparatus; the results obtained are given in Table $3.$ Assuming the unit weight of water is equal to $10$kN$/$m$3,$ determine the final settlement due to the consolidation of the clay.