---
title: "Max height of unsupported clay excavation"
subject: "Foundation Engineering"
chapter: 3
key: "semester_5/foundation_engineering:max-unsupported-height-clay"
source: https://www.bctengineer.com/q/semester_5/foundation_engineering/max-unsupported-height-clay
site: BCT Engineering
---

# Max height of unsupported clay excavation

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

- Asked in: 2081 Bhadra, 4 marks
- Video: [Maximum depth of unsupported excavation in Soil](https://www.youtube.com/watch?v=GSvU11gpuWE) — Civil Engineering - NPTEL to COMPETITIVE EXAM
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/foundation_engineering/max-unsupported-height-clay
- Whole chapter: https://www.bctengineer.com/md/semester_5/foundation_engineering/chapter-3

## Question

Derive a relation for the maximum height of unsupported excavation in clayey soil.

## Short answer

For a vertical cut in clay (φ=0) at active state, total thrust Pa = ½γHc² − 2cHc. Setting Pa=0 yields maximum unsupported height Hc = 4c/γ (or 4cu/γ) before cracking. With tension cracks and safety factor, practical height is smaller.

## Answer

For a vertical cut in clay with $\phi = 0$, cohesion $c$ and unit weight $\gamma$, Rankine's active pressure at depth $z$ is
$$
p_a = \gamma z - 2c
$$
since $K_a = 1$. The pressure is zero at $z_0 = 2c/\gamma$ and tensile above it.

For an unsupported (vertical) cut, the soil is stable up to the height at which the **total active thrust over the full height is zero**; i.e. the compressive force below balances the tension above, so no support is needed:
$$
\begin{aligned} P_a &= \int_0^{H_c}(\gamma z - 2c)\,dz = \tfrac12\gamma H_c^2 - 2cH_c = 0 \\ H_c &= \frac{4c}{\gamma} \end{aligned}
$$

**Result:** $H_c = 4c/\gamma$ (for $c = c_u$ in undrained condition, $H_c = 4c_u/\gamma$). Using a factor of safety, the safe height is $H = H_c/F$.

Note: because soil cannot carry tension, tension cracks of depth $2c/\gamma$ form; considering them, the practical safe height is smaller (the theoretical value with the cracks is $2.67c/\gamma$ by Terzaghi's analysis), so a factor of safety of 2 or more is used.

## Check yourself

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

1. For phi = 0 clay, active pressure at depth z is:
   A. gamma z + 2c
   B. gamma z - 2c
   C. gamma z / 2c
   D. 2c - gamma z^2
   Answer: B. Ka = 1, so pa = gamma z - 2c. (Key point: Active pressure in clay)

2. The critical height Hc is found by setting:
   A. Pa over full height = 0
   B. pa at the top = 0
   C. Pa = gamma H
   D. z0 = H
   Answer: A. Total thrust over the full height is zero: 1/2 gamma Hc^2 - 2c Hc = 0. (Key point: Condition for Hc)

3. The resulting maximum height is:
   A. 2c/gamma
   B. 4c/gamma
   C. c/gamma
   D. 8c/gamma
   Answer: B. Hc = 4c/gamma. (Key point: Result)

4. Considering tension cracks, the practical safe height is:
   A. Larger
   B. Smaller (Terzaghi: about 2.67c/gamma), so FS of 2 or more is used
   C. Infinite
   D. Unchanged
   Answer: B. Cracks reduce the stable height. (Key point: Effect of cracks)

## Related in Foundation Engineering

- [Skempton's findings on clay bearing capacity](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Askempton-net-safe-bc) — Foundation Engineering, chapter 6
- [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×
- [Circular footing on stiff clay, water table](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Acircular-footing-stiff-clay) — Foundation Engineering, chapter 6
- [Cantilever sheet pile, 8 m excavation](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Acantilever-sheet-pile-8m-excavation) — Foundation Engineering, chapter 5
- [Cantilever sheet pile, 5 m excavation](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Acantilever-sheet-pile-5m-sand) — Foundation Engineering, chapter 5
- [Active pressure on wall, cohesive soil](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Acohesive-wall-pressure-diagram) — Foundation Engineering, chapter 3

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- [Irrigation interval for clay loam soil](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Ainterval-clay-loam-fc25) — Irrigation and Drainage Engineering, chapter 2
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