---
title: "Terzaghi lateral stability, light well"
subject: "Foundation Engineering"
chapter: 9
key: "semester_5/foundation_engineering:well-lateral-stability-light-derive"
source: https://www.bctengineer.com/q/semester_5/foundation_engineering/well-lateral-stability-light-derive
site: BCT Engineering
---

# Terzaghi lateral stability, light well

Foundation Engineering · Chapter 9: Well Foundations

- Asked in: 2080 Bhadra, 3 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/foundation_engineering/well-lateral-stability-light-derive
- Whole chapter: https://www.bctengineer.com/md/semester_5/foundation_engineering/chapter-9

## Question

Derive an expression for lateral stability analysis of light weight well using Terzaghi's simplified approach.

## Short answer

Terzaghi's method for light wells treats the well as a rigid body rotating at depth z0. Net soil reaction kz (k=γ'(Kp−Ka)/FS) acts on width B. Equilibrium yields cubic 2z0³+3hz0²−(1.5hD²+D³)=0. Solve for z0; allowable resisting horizontal force is H=Bk(z0²−D²/2). The well is stable if applied H is smaller.

## Answer

**Terzaghi's simplified approach** for a **light well** (self-weight small, so the base does not resist horizontal force) treats the well as a rigid body in cohesionless soil, rotating about a point O at a depth $z_0$ below the scour level.

**Assumptions:**
- The soil reaction at depth $z$ is the net passive resistance $\sigma = \dfrac{\gamma'z(K_p - K_a)}{FS} = kz$, with $k = \dfrac{\gamma'(K_p - K_a)}{FS}$, acting on the width $B$ of the well.
- Above O the reaction opposes the horizontal force; below O it acts in the same direction as the force.
- The horizontal force $H$ acts at a height $h = M/H$ above the scour level, and $D$ is the depth of embedment.

```text
   H -->  (height h)
   ----|----- scour level
       |  <--- sigma = kz   (above O)
      -O----  rotation point z0
       |  ---> sigma = kz   (below O)
       |_____ D
```

**Derivation**

Horizontal equilibrium:
$$
H = Bk\left(\frac{z_0^2}{2} - \frac{D^2 - z_0^2}{2}\right) = Bk\left(z_0^2 - \frac{D^2}{2}\right)
$$

Moments about the scour level:
$$
Hh = Bk\left(\frac{D^3 - z_0^3}{3} - \frac{z_0^3}{3}\right) = \frac{Bk}{3}\left(D^3 - 2z_0^3\right)
$$

Eliminating $H$ gives the equation for the depth of the point of rotation:
$$
2z_0^3 + 3h\,z_0^2 - \left(\tfrac{3}{2}hD^2 + D^3\right) = 0
$$

Solve for $z_0$, then the allowable equivalent resisting force is $H = Bk\left(z_0^2 - \tfrac{D^2}{2}\right)$. The well is stable if this value is greater than the applied horizontal force.

## Check yourself

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

1. The net soil reaction at depth z is:
   A. sigma = gamma' z K_a
   B. constant with depth
   C. sigma = gamma' z (K_p - K_a)/FS = kz
   D. sigma = gamma' z/FS
   Answer: C. Net passive resistance increases linearly with depth. (Key point: Soil reaction)

2. The well rotates about a point O at depth z_0, so:
   A. Reaction opposes H everywhere
   B. There is no reaction above O
   C. Reaction opposes H above O and acts with H below O
   D. Reaction is constant
   Answer: C. This gives H = Bk(z_0^2 - D^2/2). (Key point: Rotation point)

3. Eliminating H gives z_0 from:
   A. z_0 = D/2
   B. 2z_0^3 + 3h z_0^2 - (1.5 h D^2 + D^3) = 0
   C. z_0^2 = hD
   D. z_0 = h
   Answer: B. Combining horizontal equilibrium and moments about scour level, with h = M/H. (Key point: Equation for z_0)

## Related in Foundation Engineering

- [Methods of lateral stability analysis of well](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Awell-lateral-stability-methods) — Foundation Engineering, chapter 9, asked 2×
- [Circular well 4 m, light and heavy well](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Acircular-well-4m-light-heavy) — Foundation Engineering, chapter 9
- [Cantilever sheet pile, simplified method](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Acantilever-sheet-pile-backfill-3m) — Foundation Engineering, chapter 5

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