---
title: "Stability of cantilever wall, q = 50 kPa"
subject: "Foundation Engineering"
chapter: 3
key: "semester_5/foundation_engineering:stability-cantilever-wall-q50"
source: https://www.bctengineer.com/q/semester_5/foundation_engineering/stability-cantilever-wall-q50
site: BCT Engineering
---

# Stability of cantilever wall, q = 50 kPa

Foundation Engineering · Chapter 3: Lateral Earth Pressure Theories and Retaining Walls

- Asked in: 2081 Baisakh, 8 marks
- Video: [soil mechanics numerical, stability of slopes, active earth pressure, retaining wall numerical](https://www.youtube.com/watch?v=sCqfk9he-HM) — Civil Engineering Exam
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/foundation_engineering/stability-cantilever-wall-q50
- Whole chapter: https://www.bctengineer.com/md/semester_5/foundation_engineering/chapter-3

## Question

Check the overall stability of the cantilever retaining wall shown in figure below. Take the allowable soil pressure as $600\ \text{kN/m}^2$. [Figure: cantilever wall, surcharge 50 kPa on backfill, stem top width 0.30 m, backfill height 4.00 m, backfill $\gamma_1 = 18\ \text{kN/m}^3$, $\phi = 40^\circ$, $\delta = 25^\circ$; heel 1.90 m, base slab thickness 0.45 m, total base width 2.80 m, embedment (toe side) 1.00 m. Dimensions as read from the scan.]

## Short answer

Overturning safety factor is 3.01, sliding safety factor 1.60. Maximum base pressure 173.3 kN/m² is below the allowable 600 kN/m², and eccentricity 0.29 m is within B/6; the wall is stable.

## Answer

**Assumptions (dimensions read from the scan):** stem 4.00 m high and 0.30 m thick, base slab 0.45 m thick (total height 4.45 m), base width 2.80 m, heel 1.90 m so toe $= 2.80-1.90-0.30 = 0.60$ m; embedment 1.0 m (soil over the toe is 0.55 m high). Soil $\gamma = 18$ kN/m$^3$, $\phi = 40^\circ$, concrete 25 kN/m$^3$, base friction $\delta = 25^\circ$, surcharge 50 kPa. Rankine active pressure on a vertical plane through the heel.

#### 1. Active thrust
$$
K_a = \tan^2(45^\circ-20^\circ) = 0.2174
$$
| Part | Force (kN/m) | Arm above base (m) | Moment |
|---|---|---|---|
| Surcharge $K_a qH$ = 0.2174 x 50 x 4.45 | 48.38 | 2.225 | 107.65 |
| Soil $\tfrac12K_a\gamma H^2$ | 38.75 | 1.483 | 57.48 |
| **Total** | **87.13** | | **165.13** |

#### 2. Vertical loads (about the toe)
| Part | Force (kN/m) | Arm (m) | Moment |
|---|---|---|---|
| Stem 0.3 x 4 x 25 | 30.00 | 0.750 | 22.50 |
| Base slab 2.8 x 0.45 x 25 | 31.50 | 1.400 | 44.10 |
| Soil on heel 1.9 x 4 x 18 | 136.80 | 1.850 | 253.08 |
| Surcharge on heel 50 x 1.9 | 95.00 | 1.850 | 175.75 |
| Soil on toe 0.6 x 0.55 x 18 | 5.94 | 0.300 | 1.78 |
| **Total** | **299.24** | | **497.21** |

#### 3. Checks
- **Overturning:** $FS_{ot} = 497.21/165.13 = 3.01$ (>2, safe).
- **Sliding:** $FS_s = 299.24\tan25^\circ/87.13 = 1.60$ (>1.5, safe even without passive resistance; with the passive force on the 1 m embedment, $P_p = \tfrac12K_p\gamma D^2 = 41.4$ kN/m, it rises to 2.08).
- **Base pressure:** $\bar x = (497.21-165.13)/299.24 = 1.110$ m, $e = 1.4-1.110 = 0.290$ m $< B/6 = 0.467$ m, so the whole base is in compression.
$$
q_{max} = 173.3\ \text{kN/m}^2,\qquad q_{min} = 40.4\ \text{kN/m}^2
$$
$q_{max} < q_a = 600$ kN/m$^2$. Safe.

**Answer: the wall is stable.** $FS_{ot} = 3.01$, $FS_{sliding} = 1.60$, $q_{max} = 173.3$ kN/m$^2$ $<600$ kN/m$^2$, $e < B/6$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Where is Rankine active pressure applied for a cantilever wall?
   A. On the stem back face
   B. On a vertical plane through the heel, using the full height including slab
   C. On the toe
   D. At the top
   Answer: B. The thrust acts on a vertical plane through the heel, height 4.45 m; soil above the heel counts as weight. (Key point: Plane for Rankine thrust)

2. Is the wall safe in sliding, and what increases it?
   A. No, FS = 0.5
   B. Yes, FS = 1.60 without passive; passive on the embedment raises it to 2.08
   C. FS is 3.01
   D. Unsafe, redesign
   Answer: B. 299.24 x tan25/87.13 = 1.60 > 1.5; adding Pp = 41.4 gives 2.08. (Key point: Sliding check)

3. How is bearing checked?
   A. q_max = 173.3 compared with allowable 600 kN/m^2
   B. q_min compared with allowable
   C. FS_ot compared with 600
   D. Not required
   Answer: A. The maximum base pressure must not exceed the allowable soil pressure; 173.3 < 600. (Key point: Bearing check)

4. e = 0.290 m and B/6 = 0.467 m. Conclusion?
   A. Tension at the heel
   B. Whole base in compression
   C. Overturning
   D. Sliding failure
   Answer: B. e < B/6 means no tension under the base. (Key point: Middle-third rule)

## Related in Foundation Engineering

- [Stability of cantilever wall, 6 m, q = 35](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Astability-cantilever-wall-q35) — Foundation Engineering, chapter 3
- [Stability of wall with surcharge 25 kN/m²](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Astability-retaining-wall-q25) — Foundation Engineering, chapter 3
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