--- title: "Active thrust, 7.5 m wall, 3 m sand" subject: "Foundation Engineering" chapter: 3 key: "semester_5/foundation_engineering:two-layer-wall-7-5m-3m-sand" source: https://www.bctengineer.com/q/semester_5/foundation_engineering/two-layer-wall-7-5m-3m-sand site: BCT Engineering --- # Active thrust, 7.5 m wall, 3 m sand Foundation Engineering · Chapter 3: Lateral Earth Pressure Theories and Retaining Walls - Asked in: 2075 Chaitra, 8 marks - Video: [Numerical on Active Earth Pressure in Retaining Wall using Rankine's Theory](https://www.youtube.com/watch?v=AoGO89lIbVk) — Curious Civil Engineer - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/foundation_engineering/two-layer-wall-7-5m-3m-sand - Whole chapter: https://www.bctengineer.com/md/semester_5/foundation_engineering/chapter-3 ## Question A retaining wall of 7.5 m high has two layers of backfill. The soil supported consists of 3 m sand ($\gamma = 18\ \text{kN/m}^3$, $\phi = 35^\circ$) overlying saturated clayey soil ($\gamma = 19.5\ \text{kN/m}^3$, $\phi = 35^\circ$, $c = 16\ \text{kN/m}^2$). The ground water table is at the upper surface of the clay. Make a sketch of the distribution of the active pressure on the wall. Calculate the total earth thrust per meter of the wall and its point of application. Assume that the backfill is horizontal at the surface. ## Short answer Rankine active pressures: sand (Ka=0.271) and saturated clay (Ka=0.271, c=16 kN/m²) with water pressure; tension crack in clay ignored. Total thrust is 139.5 kN/m, acting horizontally at 2.10 m above the base. ## Answer **Assumptions:** smooth vertical wall, horizontal backfill, Rankine theory; sand above the water table is taken at the given unit weight; the clay is saturated, so the effective stress uses $\gamma' = \gamma_{sat} - \gamma_w = 19.50 - 9.81 = 9.69\ \text{kN/m}^3$. **Coefficients** $$ \begin{aligned} K_{a1} &= \tan^2(45^\circ-35^\circ/2) = 0.2710 \\ K_{a2} &= \tan^2(45^\circ-35^\circ/2) = 0.2710,\quad 2c\sqrt{K_{a2}} = 2\times16\times0.5206 = 16.66\ \text{kN/m}^2 \end{aligned} $$ **Pressures (kN/m$^2$)** - Sand, top: $0$. - Sand, bottom ($z=3$ m): $p = K_{a1}\gamma H_1 = 0.2710\times18\times3 = 14.63$. - Clay, top: $\sigma_v' = 18\times3 = 54.00$, so $p = 0.2710\times54.00 - 16.66 = -2.02$. - Clay, bottom ($z=7.5$ m): $\sigma_v' = 54.00 + 9.69\times4.5 = 97.60$, so $p = 0.2710\times97.60 - 16.66 = 9.79$. - Water pressure at the base: $u = 9.81\times4.5 = 44.15$ (zero at the water table). The effective pressure at the top of the clay is **negative** (-2.02), so a tension crack is assumed; the negative part is ignored. The pressure becomes zero at $z_0 = 4.5\times2.02/(9.79+2.02) = 0.77$ m below the water table (interface) and the clay diagram is a triangle from there. ```text 0 --| |\ sand | \ | \ WT -|---14.63 | eff -2.02 clay | \ | \ eff 9.79 base|-----> + water 44.15 ``` **Forces per metre run and lever arms about the base** | Part | Force (kN/m) | Arm above base (m) | Moment (kN m/m) | |---|---|---|---| | Sand triangle | 21.95 | 5.500 | 120.73 | | Clay, triangle (below crack) | 18.26 | 1.243 | 22.69 | | Water pressure | 99.33 | 1.500 | 148.99 | | **Total** | **139.53** | | **292.41** | $$ \bar y = \frac{\sum M}{\sum F} = \frac{292.41}{139.53} = 2.10\ \text{m above the base} $$ **Answer: total thrust $P = 139.5$ kN/m (water included), acting horizontally at 2.10 m above the base of the wall.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Pressure at the top of the clay is -2.02 kN/m2. What does this indicate? A. Water suction B. Tension, so a tension crack is assumed and the negative part ignored C. Passive pressure D. An error to be rounded to zero everywhere Answer: B. Soil cannot carry tension; the negative part is ignored. (Key point: Negative pressure) 2. The effective clay pressure becomes zero at about what depth below the interface? A. 0.77 m B. 4.5 m C. 2 m D. 0 Answer: A. z0 = 4.5 x 2.02/(9.79+2.02) = 0.77 m. (Key point: Zero pressure depth) 3. Water pressure thrust over 4.5 m is: A. 99.33 kN/m at 1.5 m above the base B. 44.15 kN/m C. 198.7 kN/m D. 29.43 kN/m Answer: A. 0.5 x 44.15 x 4.5 = 99.33, acting at 4.5/3 = 1.5 m. (Key point: Water thrust) 4. The total thrust and its position are: A. 139.5 kN/m at 2.10 m above base B. 102 kN/m at 1.86 m C. 133.6 kN/m at 2.39 m D. 139.5 kN/m at 3.75 m Answer: A. Sum F = 139.53, sum M = 292.41, ybar = 2.10 m. (Key point: Result) ## Close variants set in other exams - [Active thrust, 7.5 m wall, 4.5 m sand](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Atwo-layer-wall-7-5m-4-5m-sand) — Foundation Engineering, chapter 3, asked 4× - [Active thrust, 7.5 m wall, 5 m sand](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Atwo-layer-wall-7-5m-5m-sand) — Foundation Engineering, chapter 3, asked 4× - [Active thrust, 6 m wall, sand over clay](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Atwo-layer-wall-6m-sand-clay) — Foundation Engineering, chapter 3, asked 4× ## Related in Foundation Engineering - [Active thrust with surcharge and water table](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Awall-6m-surcharge-water-table) — Foundation Engineering, chapter 3 - [Active pressure, sloping backfill and water table](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Awall-sheet-pipe-sloping-backfill) — Foundation Engineering, chapter 3 - [Active pressure, 8 m wall, table of layers](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Awall-8m-two-layer-table) — Foundation Engineering, chapter 3 - [Stability of gravity wall, two layers](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Astability-gravity-wall-two-layer) — Foundation Engineering, chapter 3 - [Passive thrust on retaining wall back](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Awall-passive-thrust-two-layer) — Foundation Engineering, chapter 3 ## Related in other subjects - [Clay over sand, effective stress at depths](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aclay-sand-profile-effective-stress-depths) — Soil Mechanics, chapter 6