---
title: "Active pressure on wall, cohesive soil"
subject: "Foundation Engineering"
chapter: 3
key: "semester_5/foundation_engineering:cohesive-wall-pressure-diagram"
source: https://www.bctengineer.com/q/semester_5/foundation_engineering/cohesive-wall-pressure-diagram
site: BCT Engineering
---

# Active pressure on wall, cohesive soil

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

- Asked in: 2081 Baisakh, 6 marks
- Video: [Rankine Active and Passive Pressures on Retaining Wall in Civil Engineering | Cohesive Soil Mass](https://www.youtube.com/watch?v=mcdowuAFAuo) — Soil Mechanics and Engineering Geology
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/foundation_engineering/cohesive-wall-pressure-diagram
- Whole chapter: https://www.bctengineer.com/md/semester_5/foundation_engineering/chapter-3

## Question

Draw an earth pressure diagram for a purely cohesive soil ($\gamma_t$, $\phi = 0$, $c$) supported by a rigid retaining wall of height $H$ at active condition. Find the magnitude and line of action of this active earth pressure force per unit length of the wall from the base of the retaining wall.

## Short answer

Active pressure p=γz−2c yields tension -2c at top, zero at z0=2c/γ, then a triangle of height H−z0 with base γH−2c. Neglecting tension, Pa = ½(γH−2c)(H−z0) acting at a distance (H−z0)/3 above the base.

## Answer

For a purely cohesive soil, $\phi=0$, $K_a = 1$, so Rankine's active pressure at depth $z$ is
$$
p_a = \gamma_t z - 2c
$$

**Pressure diagram**
- At the top ($z=0$): $p_a = -2c$ (tension).
- Zero at depth $z_0 = 2c/\gamma_t$.
- At the base ($z=H$): $p_a = \gamma_t H - 2c$.

```text
 z=0  |<-2c            (tension, ignored)
      |
 z0   |----+ 0         z0 = 2c/gt
      |     \
      |      \
 H    |-------+ gt*H - 2c
```

Soil cannot carry tension, so a tension crack forms to depth $z_0$ and the tensile part is ignored. The pressure then acts only on a triangle of height $(H - z_0)$ with base value $\gamma_t H - 2c = \gamma_t(H - z_0)$.

**Magnitude of active force**
$$
\begin{aligned} P_a &= \tfrac12(\gamma_t H - 2c)(H - z_0) = \tfrac12\gamma_t (H - z_0)^2 \\ &= \tfrac12\gamma_t H^2 - 2cH + \frac{2c^2}{\gamma_t} \end{aligned}
$$

**Line of action** is at the centroid of the triangle, one-third of its height above the base:
$$
\bar{y} = \frac{H - z_0}{3} = \frac{H - 2c/\gamma_t}{3} \ \text{(from the base)}
$$

If the tension zone were not neglected (no crack), $P_a = \tfrac12\gamma_t H^2 - 2cH$, but this under-estimates the thrust on the wall, so the crack is allowed for in design.

## Check yourself

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

1. The pressure is zero at depth:
   A. 2c/gamma
   B. c/gamma
   C. gamma H
   D. H/3
   Answer: A. z0 = 2c/gamma_t. (Key point: Crack depth)

2. With the crack allowed for, Pa =
   A. 1/2 gamma H^2 - 2cH
   B. 1/2 gamma (H - z0)^2
   C. gamma H - 2c
   D. 1/2 gamma H^2
   Answer: B. Triangle of height H - z0 and base gamma(H - z0) gives 1/2 gamma (H - z0)^2. (Key point: Active force)

3. Line of action above the base is:
   A. H/3
   B. (H - z0)/3
   C. H/2
   D. z0/3
   Answer: B. Centroid of the triangle is one third of its height (H - z0) above the base. (Key point: Line of action)

4. Ignoring the crack (taking tension) results in:
   A. Over-estimated thrust
   B. Under-estimated thrust, so cracks are allowed for
   C. Correct thrust
   D. Zero thrust
   Answer: B. Without the crack Pa = 1/2 gamma H^2 - 2cH, which under-estimates thrust. (Key point: Ignoring cracks)

## Related in Foundation Engineering

- [Wall movement versus earth pressure plot](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Awall-movement-pressure-plot) — 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
- [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, 3 m sand](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Atwo-layer-wall-7-5m-3m-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 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

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