--- title: "Stability of gravity wall, two layers" subject: "Foundation Engineering" chapter: 3 key: "semester_5/foundation_engineering:stability-gravity-wall-two-layer" source: https://www.bctengineer.com/q/semester_5/foundation_engineering/stability-gravity-wall-two-layer site: BCT Engineering --- # Stability of gravity wall, two layers Foundation Engineering · Chapter 3: Lateral Earth Pressure Theories and Retaining Walls - Asked in: 2080 Baisakh, 12 marks - Video: [Geotechnical-Factor of Safety Against Sliding on Retaining Wall](https://www.youtube.com/watch?v=F2golrHWDO4) — Civil Engineering Academy - Page: https://www.bctengineer.com/q/semester_5/foundation_engineering/stability-gravity-wall-two-layer - Whole chapter: https://www.bctengineer.com/md/semester_5/foundation_engineering/chapter-3 ## Question A concrete gravity retaining wall of height 6 m retains two layers of horizontal backfill soil. The upper layer is 3 m deep cohesionless soil of unit weight $20\ \text{kN/m}^3$ and angle of internal friction $32^\circ$ and the bottom layer is c-$\phi$ soil of unit weight $22\ \text{kN/m}^3$, cohesion $30\ \text{kN/m}^2$ and angle of internal friction $27^\circ$. There is water table at the depth of 2 m from the ground surface. The back of the retaining wall is smooth and vertical with the top width 1.2 m and bottom width 5 m. Check the stability of the retaining wall. Take unit weight of concrete as $25\ \text{kN/m}^3$. ## Short answer FS against overturning = 4.87, sliding = 1.79; eccentricity 0.53 m (< B/6), maximum base pressure ≈ 88 kN/m²; bearing capacity is adequate; the wall is stable. ## Answer **Assumptions:** smooth vertical back; the sloping face is on the front (toe) side, so the wall is a rectangle $1.2\times6$ m plus a triangle $3.8\times6$ m; no embedment or passive resistance; water in the backfill acts at the wall back and uplift acts under the base. $\gamma_w = 9.81$ kN/m$^3$, and the unit weights given are taken as saturated below the water table. #### 1. Active pressure (Rankine) $$ K_{a1} = \tan^2(45^\circ-16^\circ) = 0.3073,\quad K_{a2} = \tan^2(45^\circ-13.5^\circ) = 0.3755,\quad 2c\sqrt{K_{a2}} = 36.77 $$ - Upper layer, 0-2 m (dry/moist): $p$ at 2 m $= 0.3073\times20\times2 = 12.29$ kN/m$^2$. - Upper layer, 2-3 m (submerged, $\gamma'=10.19$): $\sigma_v'=40+10.19 = 50.19$, $p = 15.42$ kN/m$^2$. - Lower layer (c-$\phi$): top $p = 0.3755\times50.19-36.77 = -17.92$; base $\sigma_v' = 86.76$, $p = -4.19$. Both are **negative**, so the c-$\phi$ soil gives no effective active pressure (tension zone neglected). - Water pressure: $0$ at 2 m depth, $9.81\times4 = 39.24$ kN/m$^2$ at the base. | Part | Force (kN/m) | Arm above base (m) | Moment | |---|---|---|---| | Triangle 0-2 m | 12.29 | 4.667 | 57.35 | | Rectangle 2-3 m | 12.29 | 3.500 | 43.02 | | Triangle 2-3 m | 1.57 | 3.333 | 5.22 | | Water 0.5 x 39.24 x 4 | 78.48 | 1.333 | 104.64 | | **Total $\sum H$** | **104.63** | | **210.23** | #### 2. Wall weight and uplift | Part | Force (kN/m) | Arm from toe (m) | |---|---|---| | Rectangle 1.2 x 6 x 25 | 180.0 | 4.400 | | Triangle 0.5 x 3.8 x 6 x 25 | 285.0 | 2.533 | | Uplift (taken uniform, $39.24\times5$) | -196.2 | 2.500 | | **$\sum V$ (net)** | **268.8** | | $M_r = 180\times4.4+285\times2.533 - 196.2\times2.5 = 1023.5$ kN m/m. #### 3. Checks - **Overturning:** $FS_{ot} = 1023.5/210.2 = 4.87$ (>2). Safe. (Without uplift it would be 7.20.) - **Sliding:** base friction $\delta = \tfrac23\times27^\circ = 18^\circ$, adhesion $c_a = \tfrac23\times30 = 20$ kN/m$^2$: $$ FS_s = \frac{\sum V\tan18^\circ + c_aB}{\sum H} = \frac{268.8\times0.3249 + 20\times5}{104.6} = 1.79\ (>1.5) $$ Safe. - **Eccentricity and base pressure:** $\bar x = (1023.5-210.2)/268.8 = 3.026$ m, $e = |2.5-3.026| = 0.526$ m $< B/6 = 0.833$ m, so no tension. The resultant lies toward the heel: $$ q_{max} = 87.7,\ q_{min} = 19.9\ \text{kN/m}^2\ (\text{effective, after uplift}) $$ - **Bearing capacity** (Vesic factors for $\phi=27^\circ$: $N_c=23.94$, $N_\gamma=14.47$; $B' = 5-2e = 3.95$ m, surface footing, $\gamma'=12.19$): $$ q_u = cN_c + \tfrac12\gamma' B' N_\gamma = 30\times23.94 + 0.5\times12.19\times3.95\times14.47 = 1067\ \text{kN/m}^2 $$ $FS_b = 1067/q_{max}$ is well above 3. Safe. **Answer:** $FS_{ot} = 4.87$, $FS_{sliding} = 1.79$, $e = 0.53$ m $< B/6$, $q_{max} \approx 88$ kN/m$^2$ with a large bearing margin. The wall is **stable** (assuming the stated adhesion and uplift conditions). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why does the lower c-phi layer add no effective active soil thrust? A. Its pressure Ka*sigma - 2c sqrt(Ka) is negative at top and base (tension) B. Its Ka is zero C. It is dry D. Water cancels it Answer: A. Both ends are negative (-17.92 and -4.19), so no active pressure is used. (Key point: c-phi layer pressure) 2. What effect does uplift under the base have on stability checks? A. It increases the net vertical force B. It reduces net vertical load and resisting moment, lowering FS C. It only affects the horizontal force D. None Answer: B. Uplift 196.2 kN/m reduces sum V to 268.8 and FS_ot from 7.20 to 4.87. (Key point: Water and uplift) 3. How is base sliding resistance computed here? A. Sum V tan(phi) only B. Sum V tan(delta) + c_a B, with delta = 2/3 phi and c_a = 2/3 c C. Ph x B D. c B only Answer: B. FS = (268.8 x 0.3249 + 20 x 5)/104.6 = 1.79. (Key point: Sliding resistance) 4. What does e = 0.526 m < B/6 = 0.833 m tell us? A. Resultant is in the middle third, so no tension at the base B. Wall will overturn C. Bearing has failed D. Sliding is safe Answer: A. Resultant within the middle third means whole base is compressed (q_max 87.7, q_min 19.9). 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