Module 10 · Lesson 10.2

How much: the e–log σ′ idealisation

Compression and recompression indices, preconsolidation stress, and the primary settlement equation.

Why this matters

To predict how much a clay will settle we need its stress–compression behaviour. Plotted as void ratio against the logarithm of effective stress, that behaviour straightens into lines whose slopes — $C_{c}$ and $C_{r}$ — are all we need. The soil's stress history, marked by the preconsolidation stress, decides which slope applies.

Load a clay in the laboratory (the oedometer test, Module 12) and plot void ratio $e$ against $lo g_{10} σ^{'}$ . Two features appear:

A steep, straight virgin compression line — the path a clay follows when it is loaded beyond any stress it has felt before. Its slope is the compression index $C_{c}$ .
A much flatter recompression/swelling line — the path followed on unloading and reloading within the past maximum stress. Its slope is the recompression index $C_{r}$ (with $C_{r} ≪ C_{c}$ , typically $C_{r} \approx 0.1$ – $0.2 C_{c}$ ).

The stress at the corner between them — the largest effective stress the soil has ever carried — is the preconsolidation stress $σ_{c}^{'}$ .

Compression index

C_{c} = - \frac{Δ e}{Δ ( lo g _{10} σ ^{'} )}

What it calculates: the slope of the virgin compression line on the e–log σ′ plot

C_c: compression index (dimensionless)
$Δ e$: change in void ratio over one part of the line
$σ^{'}$: vertical effective stress (kPa)

In plain terms: A larger Cc means a more compressible clay. Soft, high-plasticity clays have the largest Cc and settle the most.

A clay is normally consolidated (NC) if its present effective stress equals the largest it has ever carried ( $σ_{0}^{'} = σ_{c}^{'}$ , so the overconsolidation ratio $OCR = σ_{c}^{'} / σ_{0}^{'} = 1$ ). It is overconsolidated (OC) if it has been more heavily loaded in the past — by ground since eroded, ice long melted, or a water table that has since risen — so $σ_{c}^{'} > σ_{0}^{'}$ and $OCR > 1$ .

This matters enormously. An OC clay loaded within its past maximum follows the flat recompression line and settles little; push it past $σ_{c}^{'}$ and it drops onto the steep virgin line and settles far more. Predicting settlement therefore begins with finding $σ_{c}^{'}$ .

From first principles

Primary consolidation settlement from the compression index

We want to show: Turn the slope of the e–log σ′ line into the settlement of a clay layer of thickness H.

The layer shortens because its voids shrink. The e–log σ′ line tells us how much the void ratio drops for a given stress increase; converting that void-ratio drop into a thickness change, over the layer, gives the settlement.

Try it

Consolidation settlement and time

How much a clay layer settles, and how that settlement unfolds. Stress history and drainage both matter.

Layer thickness H4.0 m

Compression index C_c0.40

In-situ stress σ'₀80 kPa

Preconsolidation σ'c120 kPa

Load increment Δσ'100 kPa

Coefficient c_v2.0 m²/yr

Drainage

Time t3.0 yr

Fixed for clarity

Initial void ratio e₀ = 0.9; recompression index Cr = 0.15·Cc.
OCR = σ'c/σ'₀; loading past σ'c drops the clay onto the steep virgin line.
Drainage path H_dr = H/2 (double) or H (single); time scales with H_dr².

OCR = σ'c/σ'₀: 1.50
Final primary settlement s∞: 171 mm
At t — degree U: 98 %
At t — settlement: 167 mm
Time to 50% t₅₀: 0.4 yr
Time to 90% t₉₀: 1.7 yr

The load pushes past σ'c: a small recompression part plus a large virgin part — most of the settlement is virgin compression.

Worked example

Settlement of an overconsolidated clay that crosses σ′c

Given

Clay layer 4 m thick, e₀ = 0.9
Cc = 0.40, Cr = 0.06
In-situ vertical effective stress σ′0 = 80 kPa
Preconsolidation stress σ′c = 120 kPa (OCR = 1.5)
A wide fill raises the stress by Δσ′ = 100 kPa, so σ′f = 180 kPa

Find

The primary consolidation settlement.

Assumptions

One-dimensional compression under the wide fill; drained final state so Δσ′ = Δσ.

Practice

A normally consolidated clay layer is 3 m thick with e₀ = 0.8 and Cc = 0.3. A wide load raises the vertical effective stress from 100 kPa to 200 kPa. What is the primary consolidation settlement, in mm?

Summary

On an e–log σ′ plot, clay follows a steep virgin line (slope Cc) and a flat recompression line (slope Cr).
The preconsolidation stress σ′c is the largest effective stress the soil has ever carried; OCR = σ′c/σ′0.
Primary settlement s = Cc·H/(1+e₀)·log₁₀(σ′f/σ′0), using Cr below σ′c and splitting at σ′c if the load crosses it.
An overconsolidated clay settles little until the load pushes it past σ′c.

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint