Let's break down the problem step by step: ### Given Data - **Wall height (h):** 18 ft (above base), 2 ft (below base) - **Base width (b):** 13 ft - **Top width:** 1 ft - **Toe:** 1.5 ft - **Heel:** 1.5 ft - **Thickness of base:** 1 ft - **Soil Properties (behind wall):** - φ = 34°, γ = 120 pcf (above base) - **Soil Properties (below base):** - φ = 32°, γ = 110 pcf We will **ignore wall self-weight** unless otherwise asked (since material is not specified). Typically, you'd add the weight of the wall for a more accurate calculation. --- ## 1. **Calculate Lateral Earth Pressure (Active)** We use **Rankine's Active Earth Pressure**: \[ K_a = \tan^2 \left(45^\circ - \frac{\phi}{2}\right) \] For φ = 34°: \[ K_a = \tan^2(45^\circ - 17^\circ) = \tan^2(28^\circ) \approx (0.5317)^2 \approx 0.282 \] \[ P_a = \frac{1}{2} K_a \gamma h^2 \] \[ P_a = \frac{1}{2} \cdot 0.282 \cdot 120 \cdot (18)^2 \] \[ = 0.141 \cdot 120 \cdot 324 \] \[ = 0.141 \cdot 38,880 \] \[ = 5,482 \text{ lb/ft (per ft length of wall)} \] **Point of action**: \( h/3 \) from base (i.e., 6 ft above base). --- ## 2. **Weight of Soil Above Heel (Resisting Weight)** Dimension: 1.5' (heel) × 18' (height) × 1 ft (length) \[ W_{heel} = 1.5 \times 18 \times 120 = 3,240 \text{ lb/ft} \] **Distance from toe:** \( 1.5/2 = 0.75 \) ft from back face + 1 ft (thickness at top) = 1.75 ft from back face; so from toe: \( 1.5 + 1.5 + 1 = 4 \) ft (total base width left to right). ## 3. **Weight of Soil Above Toe** Dimension: 1' (toe) × 2' (height) × 1' (length) \[ W_{toe} = 1 \times 2 \times 110 = 220 \text{ lb/ft} \] --- ## 4. **Weight of Soil Above Base (Middle Section)** Dimension: 10' (middle) × 1' (base thickness) × 110 = 1,100 lb/ft --- ## 5. **Weight of Wall (if concrete, γ ≈ 150 pcf)** Assume wall thickness is average (1.5 ft heel + 1 ft stem + 1.5 ft toe) = 4 ft × height 18 ft × γ (if needed). --- ## 6. **Moments Calculation (Overturning & Resisting)** - **Overturning Moment, \(M_{OT}\):** \( P_a \times \frac{h}{3} \) - **Resisting Moment, \(M_{R}\):** \( \sum W_i \times x_i \) --- ## 7. **Factor of Safety Calculations** ### **A. Overturning** \[ FS_{OT} = \frac{M_{R}}{M_{OT}} \] ### **B. Sliding** \[ FS_{sliding} = \frac{\text{Resisting force}}{\text{Sliding force}} \] Resisting force = \( (\sum W) \tan \phi \) ### **C. Bearing Capacity** Check eccentricity \( e \), ensure resultant falls within middle third. Bearing pressure: \[ q_{max} = \frac{W}{B} \left(1 + \frac{6e}{B}\right) \] \[ q_{min} = \frac{W}{B} \left(1 - \frac{6e}{B}\right) \] --- ## 8. **Plug in Numbers** ### **A. Overturning** - \( P_a = 5,482 \text{ lb/ft} \) - Overturning moment arm = 6 ft - \( M_{OT} = 5,482 \times 6 = 32,892 \text{ lb-ft} \) - Resisting moment (sum moments of all weights about toe). --- ### **Weight Calculations** - Heel soil: \( W_{heel} = 3,240 \) lb/ft, arm = 1.5 ft (from toe) - Toe soil: \( W_{toe} = 220 \) lb/ft, arm = 12.5 ft (from toe) - Base soil: \( W_{base} = 1,100 \) lb/ft, arm = 7.5 ft (from toe) \[ M_{R} = (3,240 \times 1.5) + (220 \times 12.5) + (1,100 \times 7.5) = 4,860 + 2,750 + 8,250 = 15,860 \text{ lb-ft} \] --- ### **B. Sliding** Total weight resisting = \( 3,240 + 220 + 1,100 = 4,560 \) lb/ft Resisting force = \( 4,560 \times \tan 32^\circ \approx 4,560 \times 0.624 = 2,847 \) lb/ft Sliding force = \( P_a = 5,482 \) lb/ft \[ FS_{sliding} = \frac{2,847}{5,482} \approx 0.52 \] --- ### **C. Bearing Capacity** Total weight = \( 4,560 \) lb/ft Eccentricity, \( e = \frac{M_R - M_{OT}}{W} = \frac{15,860 - 32,892}{4,560} = -3.74 \) ft (negative means resultant is towards the heel). Check \( e < B/6 = 13/6 = 2.17 \) ft. If not, there's a risk of tension at the toe. --- ## **Summary Table** | Failure Mode | Factor of Safety | Criteria | |------------------|------------------|----------------| | Overturning | \( = \frac{M_R}{M_{OT}} = \frac{15,860}{32,892} = 0.48 \) | > 2.0 | | Sliding | \( = 0.52 \) | > 1.5 | | Bearing Capacity | (Check \( e \)) | \( e < B/6 \) | --- ## **Conclusion** - **FS against overturning and sliding are both < 1**, which is **unsafe**. The wall needs to be redesigned (increase base width, add key, increase wall weight, etc.). - **FS against bearing capacity** is also questionable due to excessive eccentricity. **If you want to include the wall weight, or need a more detailed or different calculation, please specify the wall material or any additional data!** --- **Let me know if you need calculations including the weight of the concrete wall or any other specifics!**