Plane Failure Analysis Statement

A $10-$meter$-$wide road is proposed on a hillside where the slope of the hill is $65$ with an estimated height of $45$ meters. This site is prone to slope instability and has experienced sliding during exceptionally heavy rainfall in the past. The sliding surface is a bedding plane striking parallel to the slope crest and dipping at an angle of $36,$ which daylights at both the hill slope and the upper slope. The upper slope dips$12$ at a gentle angle of $13.$

A reconnaissance survey conducted by a geologist identified the presence of a vertical tension crack approximately $16\mathrm{.}3$ meters away from the crest of the hill slope. Furthermore, the bedding plane intersects the upper slope approximately $22\mathrm{.}5$ meters from the crest. However, the depth of the tension crack could not be estimated and requires mathematical or geometrical calculations for its determination. The density of the rock, cohesion, and friction angle along the bedding plane were measured as $26\mathrm{.}4k\frac{N}{m^3},68$kPa, and $22,$ respectively. The density of water is assumed to be $9\mathrm{.}81k\frac{N}{m^3}.$

$4.$ Effect of Slope Face Angle:

Establish a graphical relationship of the factor of safety with slope face angles ranging from $40$ to $90$ for both fully drained and fully saturated conditions$\mathrm{.}8$