Question $9$:

$($Value: $24$ marks$)$

An excavation was carried out in a stiff clay. A thin clayey seam was found when excavation reached a depth of $H=5m,$ detailed investigation showed that this thin layer with an inclination angle of $=30$ has a relative low shear strength compared with the stiff clay, as shown in Figure $4$ below. The thin layer is very likely to be the potential failure surface.

$($a$)$ If excavation angle $=85($near vertical$)$ determine the factor of safety against sliding along the thin clayey seam.

$($b$)$ Determine the values of $,$ for which the factor of safety against sliding along the soil$-$rock interface is equal to $1\mathrm{.}3$ and $1\mathrm{.}5,$ respectively.

Note: If the thickness of thin seam is neglected, the unit weight of stiff clay is $19\mathrm{.}00k\frac{N}{m^3},$ the ground water table is deeper than $H.$