Lect.12&13 Consolidation

8/10/2019 Lect.12&13 Consolidation

1/34

Lecture 12 & 13: Consolidation theory

By

Dr. Amir Khan

Soil Mechanics 1 (ENG2001M)


2/34

Learning outcomes:

by the end of this session and the tutorial session, you should be able to:

1. Determine the coefficient of volume change.

2. Determine the pre-consolidation pressure.

3. Estimate the in situ e-log curve.

4. Understand Terzaghis theory of one-dimensional

consolidation.

5. Determine coefficient of consolidation from oedometer test.

a. Casagrandes logarithm of time method.

b. Taylors root of time method.

Consolidation theory


3/34

Text books:

The following books are recommended.

1. Craig (2004). Soil Mechanics.

2. Barnes (2000). Soil Mechanics, Principles and Practice.3. Smith and Smith (1998). Elements of Soil Mechanics.



4/34


The compressibility of the clay can be represented by one of the

following coefficients according to the clay type:

1. The compressibility index, CcThe compression index (Cc) is the slope of the straight line on the e

log plot, and is dimensionless. For any two points on the linear

portion of the plot

2. The coefficient of volume change, mv


5/34


The coefficient of volume change, mv:

is defined as the volume change per unit volume per

unit change in effective stress:

The dimension (units) ofmv

is an inverse of pressure (m2/kN).

m

dV

V

dv

'

The coefficient of volume compressibility (mv), defined as the volume change per

unit volume per unit increase in effective stress (i.e. ratio of volumetric strain to

applied stress). The units of mvare the inverse of stiffness (m2/MN).


6/34

The volume change can be expressed in terms of void ratio or specimenthickness:


''1

11

''

HH

Hvm

d

H

dH

d

V

dV

vm

''1

1

1

1

ee

evm

If, for an increase in effective stress from 0to 1,the void ratio decreases frome0to e1, then


7/34

For one-dimensional consolidation:

H = mv H

where = the uniform increase in effective stress over the layerthickness.


If varies with depth, the settlement of the layer of thickness H isgiven by:

Note: The value of mv for a particular soil is not constant but it

depends upon the stress range over which it is calculated.

H

v dHmH0


8/34

British Standard, BS 1377 specifies that mv

should be calculated for

a stress increment of 100 kN/m2 in excess of the effective

overburden pressure of the in situ soil at the depth of interest.

However, it may also be calculated, if required, for any other

stress range.



9/34

Determination of preconsolidation pressure:

Preconsolidation pressure: is the maximum effective stress that

has been acted on the clay in the past.

Casagrande (1936) proposed as empirical procedure to obtain the

pre-consolidation pressure for an overconsolidated clay from e -log plot.

Note: Whenever possible the preconsolidation pressure for an

overconsolidated clay should not be exceeded in construction.



10/34


Determination of over-consolidation pressure

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

10 100 1000 10000

log '

e

1

6

2

3

5

4

Casagrandegraphicalconstructionmethodforderiving '

max.

max.


11/34

The steps of the proposed procedure for the determination of the

overconsolidation pressure are as follow:

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

2. Determine the point D of the maximum curvature of the

recompression part AB of the curve.

3. Draw a tangent to the curve at point D.

4. Draw a horizontal line from point D.

5. Bisect the angle between the horizontal line and the tangent linethrough D.

6. The vertical through the point of intersection of the bisector and

CB produced, gives the approximate value of the

preconsolidation pressure.



12/34

In-Situ e - log curveDue to the effects of sampling and preparation, the specimen in an

oedometer test will be slightly disturbed.



13/34

Consolidation theoryThe procedure for obtaining in-situ e - log s curve is as

follows:

1. Draw the e - log curve from oedometer test.

2. Determine the preconsolidation pressure c.

3. Draw a line with in-situ void ratio, eo.

4. Determine point E with coordinates log cand eo.

5. Determine point F as the point at which a horizontal line drawn

from e = 0.42 eointersects with the normal consolidation line.

6. The line EF represents the in-situ virgin consolidation line.

7. The in-situ recompression line can be approximated by a straight

line GH parallel to the mean slope of the recompression curve.


14/34

Consolidation theoryDegree of consolidation

It represents the progress of the consolidation process for an element

of soil at a depth z under a particular stress increment and is defined as:

1eoe

eoezU

where;

eo= initial void ratio before the start of consolidation.e1= void ratio at the end of consolidation

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

Two types of clays: normally consolidation consolidation coefficient (Cc) and

over consolidated coefficient of volume change (mv))

Final settlement could be after 10 or 20 years.


15/34


Assuming a linear e

curve over the stress range in question, it can beexpressed as:

Or noting that 1 = o + ui = + u and 1 = - u duringconsolidation:

o

ozU

''

''

1

ii

iz

u

u

u

uuU

1

Pore water pressure .wZ Sand

Clay increase pore water

pressure due to low permeability

after time it will decrease and

reach the initial line.

uuo

Area under curve

Total excess pressurei

u

u


16/34

Terzaghis theory of one-dimensional consolidation:In 1925, Terzaghi presented this theory for the first time and most

practical work on the prediction of settlement rates is now based on

the differential equation that he evolved.



17/34

Terzaghis theory of one-dimensional consolidation:

The main assumptions made in the theory are:

1. The soil is homogenous and saturated.

2. The solid particles and water are incompressible.

3. Strains are small.

4. The coefficient of permeability and the coefficient of volume

compressibility remain constant throughout the process.

5. Darcys law is valid at all hydraulic gradient.

6. Water flow and compression are one-dimensional.

7. There is a unique relationship, independent of time, between void

ratio and effective stress.



18/34

The theory relates the following three quantities:

1. The excess pore water pressure, u.

2. The depth, z, below the top of the clay layer.

3. The time, t, from the instantaneous application of a total

stress increment.


mu

t

k u

zv

w

2

2

u

tc

u

zv

2

2


19/34


20/34

Consolidation theorySolution of the consolidation equation:

In the governing equation, z is measured from the top of the

clay.

Complete drainage is assumed at both upper and lower surfaces.

The thickness of the layer is taken as 2d.

The initial pore water pressure

ui= at t = 0

where;ui= initial excess pore water pressure, uniform over the whole depth.

M = 0.5 (2m+1) where m is a positive integer varying from 0 to

Tv= Cvt / d

2

is a dimensionless number called the time factor.d = drainage path length.


21/34

The boundary and initial conditions can be expressed mathematically as:

1. when z = 0, u = 0

2. when z = 2d, u = 0

3. when t = 0, u = ui

An analytical solution for the consolidation equation, which satisfies the

boundary and initial conditions, can be obtained as:


vTMm

m

i ed

Mz

M

uu

2

0

sin2


22/34

where;

ui

= initial excess pore water pressure, uniform over the whole depth.

M = 0.5 (2m+1) where m is a positive integer varying from 0 to

Tv= Cvt / d2is a dimensionless number called the time factor.

d = drainage path length.

The progress of consolidation can be shown by plotting a series of

curves of u against z for different times, t. Such curves are called

Isochrones.

The form of ischrones depends upon the initial distribution of excess

pore water pressure and the drainage conditions at the boundary of

the clay layer.



23/34


A layer for which both upper and lower boundaries are free-draining is described as anopen layer. For an open layer:

A layer for which only one boundary is free-draining is described as

ahalf-closed layer. For a half-closed layer, d = the thickness of the

layer.

dthickness of the layer

2

Pore water pressure .wZ Sand

Clay thickness of layer

t = 2d

Isochrones drainage of water

uo


24/34


Examples of isochrones

a. Constant initial distribution of ui

and for open and closed layer

a. Triangular initial distribution of

uiand for open layer

b. Triangular initial distribution of

uiand for closed layer

drainage from both sides


25/34


Relationship between average degree of consolidation andtime factor

Tv = Cv t / d2 is a dimensionless number called the time

factor. d = drainage path length, the coefficient of

consolidation (Cv)

After drainage (U) = 0.25, the amount of settlement will be25% w.r.t total settlement.


26/34


In practice, the average degree of consolidation, U, over the depth of

the layer as a whole is of interest and the consolidation settlement

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

For settlement at time t

Ht= U x H

H (value of settlement)

Over consolidated '. .v vH m H


27/34

Determination of Coefficient of Consolidation, Cv, from

oedometer test

There are two commonly used methods for determination of Cv from

laboratory one-dimensional consolidation test.

Both methods are graphical and based on fitting the experimental

settlement-time results to the theoretical U Tvcurve.

In the theoretical relationship U is equivalent to the settlement and Tv

is equivalent to time.



28/34


a. Casagrandes Logarithm of Time MethodFrom the laboratory oedometer test results, deformation of the

sample against the logarithm of time is plotted. Then,

Three zones:

Initial compression Primary consolidation

Secondary compression


29/34


1. Extend the straight-line portions of primary and secondary

consolidations to intersect at C representing 100 % consolidation.

2. Select two times t1and t2at the starting part of the curve so that t2

= 4 t1and measure the vertical distance between them.

3. Set off an equal distance above the first point A to fix the point as

corresponding to U = 0.

4. The point corresponding to U = 50 % can be located in the midway

between asand a100points.

5. For 50 % average degree of consolidation, Tv= 0.196

The value of d is taken as half the average thickness of the sample for

the particular pressure increment.

50

2196.0

t

dCv


30/34


b. Taylors Root of Time Method

From the laboratory oedometer test results, deformation of the sample

against the square root of time is plotted.

F G

1. The point D corresponding to U = 0 can

be obtained by producing back the

linear part of the curve to the ordinate

at zero time.

2. The time corresponding to 90%

consolidation is found by determining

the point E at which FE = 1.15 FG.

3. For u = 90 %, Tv= 0.848.


31/34

The following compression readings were obtained in an oedometer test on a

specimen of saturated clay (Gs = 2.73):

Consolidation theoryWorked example 3

Pressure (kN/m2) Dial gauge after 24 hours (mm)0 5.000

54 4.747

107 4.493

214 4.108

429 3.449

858 2.608

1716 1.676

3432 0.737

The initial thickness of the sample was 19.0 mm and at the end of the test, the

water content was 19.8 %. Plot the e-log curve and determine thepreconsolidation pressure. Determine the value of mv for the stress increments

100-200 kN/m2

and 1000-1500 kN/m2

. What is the value of Cc for the latterincrement.


32/34


985.0445.054.0

445.0565.6263.419

19

54.01

263.4

1

263.4737.0000.5

54.073.2198.0

1

1

o

final

sc

e

eee

ee

eee

H

e

H

e

H

Gwe


33/34

P (kN/m2) Reading (mm) H (mm) e e1

0 5.000 0 0 0.985

54 4.747 5.0-4.747= 0.253 0.0264 0.9586

107 4.493 0.507 0.0530 0.932

214 4.108 0.892 0.0932 0.892

429 3.449 1.551 0.1620 0.823

858 2.608 2.392 0.2499 0.73511716 1.676 3.324 0.3473 0.6377

3432 0.737 4.263 0.4454 0.5396

eee o 1

He

eH

He

10447.0

)1(

1

HHHo



34/34


Worked example 3

Relationship between void ratio and log v

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

10 100 1000 10000

log '

e

1

6

2

3

5

4

Preconsolidation pressure = 290 kN/m2

Date post:	02-Jun-2018
Category:	Documents
Upload:	lkhdasou
View:	217 times
Download:	0 times

Lect.12&13 Consolidation

Documents