---
title: "Derive 1D primary consolidation settlement"
subject: "Soil Mechanics"
chapter: 9
key: "semester_4/soil_mechanics:settlement-equation-1d-primary"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/settlement-equation-1d-primary
site: BCT Engineering
---

# Derive 1D primary consolidation settlement

Soil Mechanics · Chapter 9: Compressibility of Soil

- Asked in: 2076 Baisakh, 3 marks
- Video: [Consolidation Settlement Equations](https://www.youtube.com/watch?v=JdtTIaVcgsc) — Thomas Weaver
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/settlement-equation-1d-primary
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-9

## Question

Derive the general equation for the calculation of settlement from one-dimensional primary consolidation.

## Short answer

For a clay layer of thickness H and initial void ratio e0, primary consolidation settlement is Sc = Δe/(1+e0) H. Equivalently Sc = mv Δσ' H. For normally consolidated clay, Sc = Cc H/(1+e0) log10((σ0'+Δσ')/σ0').

## Answer

Consider a clay layer of thickness $H$ and initial void ratio $e_0$ under a stress increase $\Delta\sigma'$, so that the void ratio decreases by $\Delta e$ (one-dimensional compression, no lateral strain).

```text
  before: Vs + Vv0 = 1 + e0     after: Vs + Vv = 1 + e0 - Δe
  |  H   |                      | H - ΔH |
```

For a unit volume of solids, the initial total volume is $1 + e_0$ and the reduction in volume is $\Delta e$. Since the area is constant, the vertical strain equals the volumetric strain:

$$
\frac{\Delta H}{H} = \frac{\Delta V}{V} = \frac{\Delta e}{1+e_0}
$$

$$
\boxed{S_c = \Delta H = \frac{\Delta e}{1+e_0}\,H}
$$

In terms of the compressibility:

- With $a_v = \dfrac{\Delta e}{\Delta\sigma'}$: $\;S_c = \dfrac{a_v}{1+e_0}\Delta\sigma'\,H = m_v\,\Delta\sigma'\,H$
- With the compression index $C_c$ (NC clay): $\Delta e = C_c\log_{10}\dfrac{\sigma'_0 + \Delta\sigma'}{\sigma'_0}$, so

$$
S_c = \frac{C_c\,H}{1+e_0}\log_{10}\frac{\sigma'_0 + \Delta\sigma'}{\sigma'_0}
$$

where $\sigma'_0$ is the effective overburden pressure at the middle of the layer.

## Check yourself

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

1. In 1D compression the strain is
   A. dH/H = de/(1+e0)
   B. dH/H = de/e0
   C. dH/H = e0/(1+de)
   D. dH = de x e0
   Answer: A. For a unit volume of solids, the initial volume is 1+e0 and reduction is de, so strain = de/(1+e0). (Key point: Strain relation)

2. The general settlement equation is
   A. Sc = (de/e0) H
   B. Sc = (de/(1+e0)) H
   C. Sc = (1+e0) H/de
   D. Sc = de H
   Answer: B. Sc = dH = (de/(1+e0)) H. (Key point: Settlement formula)

3. For NC clay, the settlement in terms of Cc is
   A. Cc H /(1+e0) log10((s0 + ds)/s0)
   B. Cc H e0 log10(s0/ds)
   C. Cc H ln(ds)
   D. Cc (1+e0) H
   Answer: A. Since de = Cc log10((s0+ds)/s0), Sc = Cc H/(1+e0) log10((s0+ds)/s0). (Key point: Using Cc)

4. Where is sigma'0 evaluated in the layer?
   A. At the top
   B. At the bottom
   C. At the middle of the layer
   D. At the water table
   Answer: C. sigma'0 is the effective overburden at the middle of the layer. (Key point: Overburden location)

## Related in Soil Mechanics

- [Derive Terzaghi 1D consolidation theory](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aterzaghi-1d-consolidation-derivation) — Soil Mechanics, chapter 9, asked 2×
- [Primary settlement, sand over clay, 110 kPa](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aprimary-settlement-sand-clay-110kpa) — Soil Mechanics, chapter 9
- [Consolidation test, 90% time, k, Cc](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aconsolidation-test-50-90-percent-time) — Soil Mechanics, chapter 9
- [Settlement with preconsolidation 125 kPa](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aprimary-consolidation-settlement-preconsolidated) — Soil Mechanics, chapter 9
- [Final settlement and settlement at 0.5 year](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Afinal-settlement-clay-time-half-year) — Soil Mechanics, chapter 9
- [Methods to accelerate consolidation](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aaccelerating-consolidation-methods) — Soil Mechanics, chapter 9, asked 5×

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