44
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Module 3:
Lecture - 6 on Compressibility
and Consolidation
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Contents
 Stresses in soil from surface loads;
 Terzaghi’s 1-D consolidation theory;
 Application in different boundary conditions;
 Ramp loading;
 Determination of Coefficient of consolidation;
 Normally and Over-consolidated soils;
 Compression curves; Secondary consolidation;
 Radial consolidation;
 Settlement of compressible soil layers and
 Methods for accelerating consolidation settlements.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Preconsolidation pressure
It is the previous maximum effective stress to which the
soil has been subjected in the past.
Normally consolidated: A soil is called NC if the
present effective overburden pressure is the maximum
to which the soil has ever been subjected, i.e. σ′present
≥ σ′past maximum
Over consolidated: A soil is called OC if the present
effective overburden pressure is less than the
maximum to which the soil was ever been subjected
in the past , i.e. σ′present < σ′past maximum
 In the natural condition in the field, a soil may be
either normally consolidated or overconsolidated.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
A soil in the field may become overconsolidated
through several mechanisms:
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Preconsolidation pressure
 Whenever possible the preconsolidation pressure for an
overconsolidated clay should not be exceeded in construction.
 Compression will not usually be great if the effective vertical
stress remains below σc′ only if σc′ is exceeded compression
will be large.
 In the field, the overconsolidation ratio (OCR) can be
defined as:
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Preconsolidation pressure
OCR = 1  Normally Consolidated Soil
OCR > 1  Over Consolidated Soil
OCR < 1  Under Consolidated Soil
For e.g., Recently deposited soils either
geologically or by man
(The soil has not yet come to
equilibrium under the weight of the
overburden load; PWP would be excess
of hydrostatic)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Example problem
The laboratory consolidation data for an undisturbed clay
sample are as
follows:
e1 = 1.1, σ1′ = 95 kPa
e2 = 0.9, σ2′ = 475 kPa
a) Calculate the coefficient of volume compressibility, mv
b) What will be the void ratio for a pressure of 600 kPa ?
(Note: σc′ < 95 kPa)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Example problem
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Determination of Coefficient of Consolidation
 The coefficient of Consolidation Cv is the only term in
the consolidation equation that takes into account soil
properties which govern the rate of consolidation.
The coefficient of consolidation (Cv) signifies the
rate at which a saturated clay undergoes 1dimensional consolidation when subjected to an
increase in pressure.
A knowledge of Cv is essential for predicting the
rate of primary consolidation settlement.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Determination of Coefficient of Consolidation
Several methods are available for obtaining Cv
These methods compare characteristic features of
the theoretical time factor, T, and the degree of
consolidation, U, relationship with time-compression
data obtained in the laboratory.
The square root of time-fitting method (root t method)
proposed by Taylor (1948) and the logarithm of timefitting method (log t method), also called Casagrande's
method (Casagrande and Fadum 1940), are the most
widely used methods in practice and are considered as
standard methods.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Determination of Coefficient of Consolidation
The root t method yields Cv values larger than those
obtained from Casagrande's log t method
a) Plot the dial readings for specimen deformation for a
given load increment against time on semi-log graph paper.
b) Plot two points, A and B, on the upper portion of the
consolidation curve, which correspond to time t1 and t2,
respectively. Note that t2 = 4t1
c) The difference of dial readings between A and B is equal to x.
Locate point R, which is at a distance x above point A.
d) Draw the horizontal line RS. The dial reading corresponding to
this line is d0, which corresponds to 0% consolidation.
e) Project the straight-line portions of the primary consolidation
and the secondary consolidation to intersect at T . The dial reading
corresponding to T is d100, i.e., 100% primary consolidation.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Determination of Coefficient of Consolidation
f) Determine the point V on the consolidation curve that
corresponds to a dial reading of (d0 +d100)/2 = d50 . The
time corresponding to point V is t50, i.e., time for 50%
consolidation.
g) Determine Cv from the equation T = Cvt/H2. The value of T for
Uav = 50% is 0.197
H = Hdr/2
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Casagrande’s Log time method
d0
R
S
d50
d100
V
For d50, t50 =10.2 min.
T
Cv
= 0.197(2.06/2)2/(10.2 x 60) cm2/min
t50
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Taylor’s root time method
a) Plot the dial reading and the corresponding squareroot-of-time √t
b) Draw the tangent DQ to the early portion of the plot.
c) Draw a line DR such that OR = 1.15OQ.
d) The abscissa of the point E (i.e., the intersection of DR
and the consolidation curve) will give √t90 (i.e., the
square root of time for 90% consolidation).
e) The value of T for Uav = 90% is 0.848. So,
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Taylor’s root time
method
The
theoretical
curve U vs √Tv is a
straight line upto 60%
consolidation
and
the abscissa of curve
at 90% consolidation
is 1.15 times the
abscissa
of
an
extension
of
the
straight line.
The upper part of U
vs√Tv is almost a
straight line, whose
slope is 2/√π
AC = √Tv =√0.848 for
U = 90%
AB = 0.9 * 2/√π
Hence AC /AB = 1.15
Q R
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Determination of Cv
Log time method makes use of early (primary
consolidation) and later time responses (secondary
compression) and in comparison the root time method
only utilizes early time response, which is expected to be
a straight line.
Root time method should give good results except
when non-linearity's arising from secondary compression
cause substantial deviations from the expected straight
line. Most pronounced for clayey soils with organic
content (i.e. Marine clay).
Hence Log time method most suited for clays and root
time method for silts.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Early stage log t method
The early stage log t
method (Robinson and
Allam, 1996), an extension
of the logarithm-of-time
method, is based on
specimen
deformation
against log-of-time plot.
a) Follow the logarithm-oftime
method
to
determine d0.
b) Draw a horizontal line DE
through d0. Then draw a
tangent through the point of
inflection F.
d0 D
G
E
F
(After Robinson and Alam, 1996)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Early stage log t method
c) The tangent intersects line DE at point G. Determine
the time t corresponding to G, which is the time at Uav
= 22.14 %.
For Uav = 22.14% Tv = 0.0385
In most cases, for a given soil and pressure range, the
magnitude of Cv determined using the logarithm-of-time
method provides lowest value. The highest value is
obtained from the early stage log t method.
oThis is because the early stage log t method uses the earlier part of
the consolidation curve, whereas the logarithm-of-time method
uses the lower portion of the consolidation curve.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Early stage log t method
When the lower portion of the consolidation curve is
taken into account, the effect of secondary
consolidation plays a role in the magnitude of Cv.
Several investigators have also reported that the Cv
value obtained from the field is substantially higher
than that obtained from laboratory tests conducted
using conventional testing methods (i.e., logarithm-oftime and square-root-of-time methods).
Hence, the early stage log t method may provide a
more realistic value of field.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Example problem (After Craig, 2004)
An 8m depth of sand overlies a 6m layer of clay, below
which is an impermeable stratum; the water table is 2m
below the surface of the sand. Over a period of 1 year, a
3m depth of fill (unit weight 20 kN/m3) is to be dumped
on the surface over an extensive area. The saturated unit
weight of the sand is 19 kN/m3 and that of the clay is 20
kN/m3; above the water table the unit weight of the sand
is 17 kN/m3.
For the clay, the relationship between void ratio and
effective stress (units kN/m2) can be represented by the
equation
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
and the coefficient of consolidation is 1.26m2/year.
(a)Calculate the final settlement of the area due to
consolidation of the clay and the settlement after a
period of 3 years from the start of dumping.
(b) If a very thin layer of sand, freely draining, existed
1.5m above the bottom of the clay layer, what would
be the values of the final and 3-year settlements?
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution
Since the fill covers a wide area, the problem can be
considered to be one-dimensional. The consolidation
settlement will be calculated in terms of Cc,
considering the clay layer as a whole, and therefore
the initial and final values of effective vertical stress at
the centre of the clay layer are required.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution
In the calculation of the degree of consolidation 3
years after the start of dumping, the corrected value
of time to allow for the 1-year dumping period is:
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution
b) The final settlement will still be 182mm (ignoring the
thickness of the drainage layer): only the rate of
settlement will be affected. From the point of view of
drainage there is now an open layer of thickness 4.5m
(d = 2.25 m) above a half-closed layer of thickness
1.5m (d = 1.5 m)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay