Problem: A cantilever retaining wall has a total height $H = 5.0 \text{ m}$ (from base of footing to top of stem). The soil unit weight $\gamma_s = 18 \text{ kN/m}^3$ and angle of internal friction $\phi = 30^\circ$. The concrete unit weight $\gamma_c = 24 \text{ kN/m}^3$. Wall dimensions: Base width $B = 3.0 \text{ m}$, Toe length $1.0 \text{ m}$, Heel length $1.6 \text{ m}$, Stem thickness $0.4 \text{ m}$, Base thickness $0.5 \text{ m}$. Calculate the Factor of Safety against Overturning ($FS_{OT}$) and Sliding ($FS_{S}$). Assume coefficient of base friction $\mu = 0.5$.

Problem: A cantilever retaining wall supports level backfill with total height $H = 6.0 \text{ m}$ (base to top of stem). Active earth pressure is represented by an equivalent fluid pressure of $5.5 \text{ kN/m}^3$. Determine the Factor of Safety against Overturning ($FS_{OT}$) if the total resisting moment about the toe is $\Sigma M_R = 750 \text{ kN-m/m}$. Ignore the stabilizing effect of passive pressure $P_p$ on the toe.

Problem: A retaining wall is analyzed for bearing pressure on the soil. The resultant vertical force $R_v = 300 \text{ kN/m}$ acts at an eccentricity $e = 0.6 \text{ m}$ from the centerline of the base. Calculate the maximum and minimum soil bearing pressures ($q_{max}$, $q_{min}$) if the base width is $B = 4.0 \text{ m}$. Will tension develop at the heel?

Problem: Following heavy monsoon rains, a 4-meter-tall cantilever retaining wall enclosing a residential property violently failed by sliding and overturning simultaneously. Investigations showed the wall was structurally sound and designed correctly for the active soil pressure of the gravel backfill. However, the weepholes at the base of the wall were completely clogged with silt and roots. Analyze the mechanism of this failure.

Problem: A cantilever retaining wall has a height of $H = 6.0 \text{ m}$. The backfill soil has a unit weight of $\gamma = 18.5 \text{ kN/m}^3$, an angle of internal friction $\phi = 32^\circ$, and slopes upward at an angle $\beta = 10^\circ$. The wall friction angle is assumed to be $\delta = 20^\circ$, and the back face of the wall is vertical ($\alpha = 90^\circ$). Calculate the active earth pressure coefficient ($K_a$) using Coulomb's theory and the total active thrust ($P_a$).

Problem: For the retaining wall in Example 1, a highway is built on the level backfill, acting as a uniform surcharge $q = 15 \text{ kPa}$. Recalculate the overturning moment ($M_O$). ($H = 5.0 \text{ m}$, $K_a = 0.333$, $P_{a,\text{soil}} = 75 \text{ kN/m}$, $M_{O,\text{soil}} = 125 \text{ kN-m/m}$).

Problem: A retaining wall has a lateral active driving force $P_a = 150 \text{ kN/m}$ and a total vertical resisting force $\Sigma V = 240 \text{ kN/m}$. The base friction coefficient is $\mu = 0.4$. A shear key is added, engaging a passive resistance $P_p = 80 \text{ kN/m}$. Verify the Factor of Safety against sliding.

Problem: The heel slab of a retaining wall is $2.0 \text{ m}$ long and $0.5 \text{ m}$ thick. It supports a $4.0 \text{ m}$ high column of backfill soil ($\gamma = 18 \text{ kN/m}^3$). Ignore the upward soil bearing pressure for a conservative design. Calculate the factored ultimate bending moment ($M_u$) at the face of the stem. Concrete $\gamma_c = 24 \text{ kN/m}^3$.

Problem: A retaining wall design includes wind loads. Calculate the sliding factor of safety if the total driving force $P_{horizontal} = 200 \text{ kN/m}$ and resisting force $F_R + P_p = 320 \text{ kN/m}$. The code allows a reduced FS of 1.5 when wind is included.