$($a$)$ i$)$ To find the total and effective vertical stresses with depth, we need to consider

capillary water rising to $1$ meter above the water table in the partially$-$saturated sand

layer. This means we have to account for the weight of the soil and the weight of the

water.

ii$)$ After calculating the stresses, we need to sketch them on a diagram showing stress

versus depth. This includes the total stress, effective stress, and pore water pressure.

$($b$)$ i$)$ At a depth of $12$ meters, we need to determine the principal stresses. For this, we

use the lateral stress ratio $(K$ which is the ratio of horizontal stress to vertical stress.

Given $K=0\mathrm{.}6,$ we can calculate the principal stresses.

ii$)$ We also need to find the orientation of the planes where the shear stress is

maximum. This involves determining the angles of these planes relative to the

horizontal plane and sketching them.

iii$)$ At the same depth $(12$ meters$),$ we calculate the maximum shear stress and the

shear$-$to$-$normal stress ratio.

iv$)$ Finally, we plot Mohr's circle to represent the stress state at this depth. This includes

indicating the points representing the principal stresses and the maximum shear stress.