Problem: A cantilever retaining wall has a retained height $H=4.5\text{ m}$ measured from the top of its $0.5\text{ m}$ footing to the top of backfill, so the overall concrete system is $5.0\text{ m}$ from footing bottom to wall top. The base width is $B=3.0\text{ m}$, consisting of a $1.0\text{ m}$ toe, $0.4\text{ m}$ stem, and $1.6\text{ m}$ heel. Use $\gamma_s=18\text{ kN/m}^3$, $\phi=30^\circ$, $\gamma_c=24\text{ kN/m}^3$, and base friction coefficient $\mu=0.50$.

Use the lesson's simplified Rankine model: level homogeneous cohesionless drained backfill, smooth vertical wall, no surcharge, no water pressure, no seismic action, and no credited passive resistance. Check overturning, sliding, resultant location, and full-contact bearing. Bearing capacity is not given, so do not claim a bearing-capacity pass.

Problem: A $6.0\text{ m}$ retained height is represented directly by a triangular equivalent-fluid lateral-pressure distribution with coefficient $w_{eq}=5.5\text{ kN/m}^3$. The base of this pressure diagram coincides with the overturning reference elevation. If $\sum M_R=750\text{ kN}\cdot\text{m/m}$, calculate the service overturning factor of safety. Ignore passive resistance and other lateral actions.

Problem: A service vertical resultant $R_v=300\text{ kN/m}$ acts with eccentricity $e=0.60\text{ m}$ toward the toe of a $B=4.0\text{ m}$ base. Determine whether the middle-third condition is satisfied and calculate the toe and heel contact pressures if full contact applies.

Problem: A wall was designed for drained backfill, but a drainage failure allows water to rise to $H_w=4.0\text{ m}$ behind the wall. Calculate the additional hydrostatic resultant using $\gamma_w=9.81\text{ kN/m}^3$ and explain why it must be added separately to the effective-stress earth-pressure model.

Problem: A vertical wall retains $H=6.0\text{ m}$ of homogeneous cohesionless drained soil with $\gamma=18.5\text{ kN/m}^3$, $\phi=32^\circ$, backfill slope $\beta=10^\circ$, and wall friction $\delta=20^\circ$. Under the angle convention used in the lesson, calculate the Coulomb active coefficient and total thrust. No surcharge, water pressure, or seismic action is included.

Problem: Add a uniform surcharge $q=15\text{ kPa}$ to the drained level backfill of Example 1. Use $H=4.5\text{ m}$ and $K_a=1/3$. For a conservative teaching check, ignore any beneficial vertical surcharge weight over the heel and add only its destabilizing lateral effect. Recalculate the overturning moment and overturning factor of safety using $\sum M_R=390.96\text{ kN}\cdot\text{m/m}$.

Problem: A wall has horizontal driving force $P_H=150\text{ kN/m}$ and service vertical force $\sum V=240\text{ kN/m}$. The base friction coefficient is $\mu=0.40$. A shear-key model predicts a theoretical passive resistance of $80\text{ kN/m}$, but the selected design basis credits only $40\text{ kN/m}$ after considering mobilization and geotechnical uncertainty. Calculate the sliding factor of safety.

Problem: A $2.0\text{ m}$ heel slab is $0.50\text{ m}$ thick and supports $4.0\text{ m}$ of backfill with $\gamma_s=18\text{ kN/m}^3$. For this isolated teaching calculation, ignore upward bearing pressure and other loads and use the stated factored dead-load multiplier $1.2$ only. Concrete unit weight is $24\text{ kN/m}^3$. Determine the factored cantilever moment at the stem face.

Problem: A wall has been verified under static Rankine active pressure for level drained backfill. The site engineer now requests earthquake stability verification. Is it acceptable to reuse the same static $K_a$ and simply label the resulting force “seismic earth pressure”?

Problem: Revisit Example 1 with a uniform $15\text{ kPa}$ surcharge covering the full heel and extending behind the wall. Include both the downward load over the $1.60\text{ m}$ heel and the lateral $K_aqH$ action. Retain the original level, drained geometry, $\mu=0.50$, and service basis.

Problem: In Example 1 let groundwater rise $4.0\text{ m}$ above footing top. Use $\gamma_{\mathrm{sat}}=18\text{ kN/m}^3$, $\gamma_w=9.81\text{ kN/m}^3$, $K_a=1/3$, and an 18 kN/m³ total soil weight over the heel. Ignore uplift and front water only for this controlled teaching case.

Problem: For any selected scenario, express the three cantilever root actions per metre without assigning strength load factors.