A long natural slope is inclined at $\beta = 20^\circ$. The soil is a silty sand with $c' = 0$, $\phi' = 32^\circ$, and $\gamma_{sat} = 19.5 \text{ kN/m}^3$. Determine the factor of safety against sliding if steady groundwater seepage occurs parallel to the slope and emerges at the ground surface.

A long natural slope consists of dry sand with a unit weight ($\gamma$) of $17 \text{ kN/m}^3$ and an effective angle of internal friction ($\phi'$) of $32^\circ$. The sand has no cohesion ($c' = 0$). Determine the maximum angle ($\beta$) at which the slope will remain stable with a Factor of Safety of 1.5.

A finite slope is to be excavated in a saturated clay. The clay has an undrained shear strength ($s_u = c$) of $45 \text{ kPa}$, an angle of internal friction $\phi = 0^\circ$, and a unit weight $\gamma = 19 \text{ kN/m}^3$. The slope angle is $\beta = 45^\circ$. Using Taylor's stability chart, for $\phi=0$ and $\beta=45^\circ$, assuming a deep hard stratum (depth factor $D$ is large), the stability number $N_s$ is approximately $0.18$. Determine the maximum critical height ($H_c$) of the slope, and the actual allowable height ($H$) for a Factor of Safety of 1.5.

A vertical cut is made in a stiff clay deposit. The clay has a unit weight of $20 \text{ kN/m}^3$, an undrained cohesion ($c$) of $30 \text{ kPa}$, and $\phi = 0$. Determine the theoretical maximum depth of a tension crack ($z_c$).

A new housing development required the grading of a steep natural hill composed primarily of overconsolidated, stiff clay. The engineers cut a 12-meter high slope at an angle of 40 degrees.

An existing highway cut through a weathered rock and soil matrix began showing signs of superficial sliding and tension cracks near the crest.