A massive soil wedge on a natural hillslope is resting on a distinct planar sliding surface (e.g., a heavily weathered bedrock interface). The weight of the soil wedge and any structures above it exerts a driving shear stress ($\tau_d$) of $25 \text{ kPa}$ down the slope. Geotechnical testing of the sliding surface indicates it has a cohesive strength ($c$) of $10 \text{ kPa}$ and a normal stress ($\sigma_n$) of $40 \text{ kPa}$. The angle of internal friction ($\phi$) is $20^{\circ}$. Calculate the Factor of Safety ($FS$) against sliding.

Re-evaluate the slope from Problem 1 after a heavy, prolonged rainstorm. The water table has risen significantly, saturating the soil above the sliding surface. The physical weight of the soil wedge increases slightly, raising the driving shear stress ($\tau_d$) to $28 \text{ kPa}$. More importantly, the groundwater introduces a positive pore water pressure ($u$) of $15 \text{ kPa}$ on the sliding plane. The normal stress ($\sigma_n$) from the soil weight remains $40 \text{ kPa}$, and the soil properties ($c = 10 \text{ kPa}$, $\phi = 20^{\circ}$) are unchanged. Calculate the new Factor of Safety ($FS$).

A steep rock cut for a highway exposes a massive, unstable wedge of limestone bounded by intersecting joint planes. The wedge weighs $5,000 \text{ kN}$. A kinematic analysis determines that the wedge will slide out of the face along a critical plane dipping at $45^{\circ}$. The joint surface has zero cohesion ($c = 0$) and a friction angle ($\phi$) of $30^{\circ}$.

Without support, the driving force down the plane ($W \cdot \sin(45^{\circ})$) is $3,535 \text{ kN}$, and the resisting frictional force ($W \cdot \cos(45^{\circ}) \cdot \tan(30^{\circ})$) is $2,041 \text{ kN}$, resulting in a disastrous $FS$ of $0.58$.

Engineers propose installing deeply anchored, highly tensioned rock bolts horizontally ($0^{\circ}$ dip) into the rock face to stabilize the wedge. The bolts will apply a total horizontal restraining force ($T$) of $3,000 \text{ kN}$ pushing the wedge firmly back into the slope. Calculate the new Factor of Safety.

A major four-lane highway is constructed through a deep, near-vertical road cut in heavily jointed granite situated in a high-altitude alpine environment experiencing frequent freeze-thaw cycles during spring and autumn.

A multi-span concrete highway bridge is constructed across a wide, fast-flowing river with a bed composed of loose sand, gravel, and cobbles. The bridge is supported by large, cylindrical concrete piers founded deeply in the riverbed sediment.