Consolidation theory

Outlines

7.1 Introduction Craig Page # 227

7.2 The oedometer test Craig Page # 227

7.3 Consolidation settlement: one-dimensional method Craig Page # 235

7.6 Degree of consolidation Craig Page # 244

7.7 Terzaghi’s theory of one-dimensional consolidation Craig Page # 245

7.8 Determination of coefficient of consolidation Craig Page # 252

Introduction

Consolidation is the gradual reduction in volume of a fully saturated soil of low permeability due to drainage of some of the pore water.

The process of swelling on other hand is the reverse of consolidation, is the gradual increase in volume of a soil under negative excess pore water pressure.

In the field, when the stress on a saturated clay layer is increased-for example, by the construction of a foundation-the pore water pressure in the clay will increase.

Because the hydraulic conductivity of clays is very small,

some time will be required for the excess pore water pressure to dissipate

and the increase in stress to be transferred to the soil skeleton,

if is a surcharge at the ground surface over a very large area, the increase in total stress at any depth of the clay layer will be equal to

Introduction

However, at time t = 0 (i.e., immediately after the stress is applied), the excess pore water pressure at any depth u will equal , Or (u = )

Hence, the increase in effective stress at time t = 0 will be

= - u

Theoretically, at time t=, when all the excess pore water pressure in the clay layer has dissipated as a result of drainage into the sand layers, u = 0 (at time t=)

Then the increase in effective stress in the clay layer is

= - u = - 0=

This gradual increase in the effective stress in the clay layer will cause

settlement over a period of time and is referred to as consolidation.

The oedometer test

The detail of the test will be in the laboratory,

However the following can be measured during the test

Voids ratio variation with applied stresses

For each load increment

Water content measured at end of test = w1

Void ratio at end of test = e1 = w1Gs (assuming Sr = 100%)

Thickness of specimen at start of test = H0

Change in thickness during test =H

Void ratio at start of test = e0 = e1+e

In the same way e can be calculated up to the end of any increment

period

Dry weight measured at end of test = Ms (i.e. mass of solids)

Thickness at end of any increment period = H1, Area of specimen = A

Equivalent thickness of solids = Hs = Ms/AGsw

Void ratio,

Compressibility characteristics

Typical plots of void ratio (e) after consolidation against

effective stress () for a saturated clay are shown in Figure

The coefficient of volume compressibility (mv),

Mv: The volume change per unit volume per unit increase in

effective stress. The units of mv are the inverse of pressure

(m2/MN).

The volume change may be expressed in terms of either

void ratio or specimen thickness. If, for an increase in

effective stress from 0 to 1, the void ratio decreases from

e0 to e1, then

Note: The value of mv for a particular soil is not constant but depends on the

stress range over which it is calculated.

Preconsolidation pressure

the maximum effective vertical stress that has acted on the clay in the

past, referred to as the preconsolidation pressure (c).

Casagrande construction for estimating the preconsolidation pressure

consists of the following steps:

1. Produce back the straight-line part (BC) of the curve.

2. Determine the point (D) of maximum curvature on

the recompression part (AB) of the curve.

3. Draw the tangent to the curve at D and bisect the

angle between the tangent and the horizontal through

D.

4. The vertical through the point of intersection of the

bisector and CB produced

5. gives the approximate value of the preconsolidation

pressure.

Compression and recompression indices

The compression index, Cc

is the slope of the straight-line portion

(the latter part) of the loading curve, or

The swelling(=recompression index),

Cs=Cr is the slope of the unloading

(reloading) portion of the e-Iog curve.

In most cases, the value of the Cs=Cr is 1/4 to 1/5 of the Cc.

In-situ e–log curve

Consolidation Settlement : One-dimensional Method

sc = consolidation settlement.

Cc compression index

eo void ratio

H=Hc Height of clay layer

’o initial overburden stress

f’=o’+ finial effective stress

Primary Consolidation Settlement

Normally consolidation Soil

)'

'log(H

e1

CrS

o

f

oc

)'

'log(H

e1

CcS

o

f

oc

Over consolidated Soil

I. ’o<’f<’p

II. ’o<’p<’f

)'

'log(H

e1

Cc)

'

'log(H

e1

CrS

p

f

oo

p

oc

1 Cr

1

Cc

’p

Log ’

e

Degree Of Consolidation, Uz

For an element of soil at a particular depth z in a clay layer the progress of the consolidation process under a particular total stress increment can be expressed in terms of void ratio as follows:

Or in term of effective stress as

Note

Then the Uz can be give as

where

Uz = degree of consolidation, at a particular instant of time, at depth z (0Uz 1),

and e0 =void ratio before the start of consolidation,

e1 = void ratio at the end of consolidation and

e= void ratio, at the time in question, during consolidation.

Terzaghi’s Theory of One-dimensional Consolidation

The assumptions made in the theory are:

1. The soil is homogeneous.

2. The soil is fully saturated.

3. The solid particles and water are incompressible.

4. Compression and flow are one-dimensional (vertical).

5. Strains are small.

6. Darcy’s law is valid at all hydraulic gradients.

7. The coefficient of permeability and the coefficient of volume compressibility remain constant throughout the process.

8. There is a unique relationship, independent of time, between void ratio and effective stress.

Terzaghi’s Theory of One-dimensional Consolidation

The theory relates the following three quantities.

1. The excess pore water pressure (ue).

2. The depth (z) below the top of the clay layer.

3. The time (t) from the instantaneous application of a

total stress increment.

The differential equation of consolidation,

cv being defined as the coefficient of consolidation, suitable unit

being m2/year.

Note: k and mv are assumed as constants, cv is constant during

consolidation.

Solution of the consolidation equation

At zero time, therefore, the increment will be carried entirely by

the pore water, i.e. the initial value of excess pore water pressure

(ui) is equal to and the initial condition is

The upper and lower boundaries of the clay layer are assumed to

be free-draining,

The solution is

Where T=Ti=TV= dimensionless time factor, and m counter

Solution Take the form of Isochrones

Average degree of consolidation (U=Uave=Ui)

In practical problems it is the average degree of consolidation (U) over the

depth of the layer as a whole that is of interest, the consolidation settlement at

time t being given by the product of U and the final settlement.

The solution of the above equations can be given the following empirical

relations.

Determination Of Coefficient Of Consolidation

The value of cv for a particular pressure increment in the oedometer test

can be determined by comparing the characteristics of the experimental

and theoretical consolidation curves, the procedure being referred to as

curve-fitting.

Casagrande Method (d=Have)

The root time method (due to Taylor)

Examples