Theory and related problems on soil consolidation.

1. .
Lecture Note on Consolidation
Prepared by-
Md. Hasan Imam.
Lecturer, Department of Civil Engineering.
UITS.

2. 1
Consolidation is a process by which soils decrease in volume. According to Karl von
Terzaghi "consolidation is any process which involves a decrease in water content of
saturated soil without replacement of water by air." The process opposite to consolidation
is called swelling, which involves an increase in the water content due to an increase in
the volume of the voids.
Consolidation may be due to one or more of the following factors:
1. External static loads from structures.
2. Self-weight of the soil such as recently placed fills.
3. Lowering of the ground water table.
4. Desiccation.
Normally Consolidated and Overconsolidated Clays:
A clay is said to be normally consolidated if the present effective overburden pressure p0
is the maximum pressure to which the layer has ever been subjected at any time in its
history, whereas a clay layer is said to be overconsolidated if the layer was subjected at
one time in its history to a greater effective overburden pressure, pc, than the present
pressure, p0. The ratio pc/ p0 is called the overconsolidation ratio (OCR).
Overconsolidation of a clay stratum may have been caused due to some of the following
factors:
1. Weight of an overburden of soil which has eroded
2. Weight of a continental ice sheet that melted
3. Desiccation of layers close to the surface.
Experience indicates that the natural moisture content, wn, is commonly close to the
liquid limit, WL, for normally consolidated clay soil whereas for the overconsolidated
clay, wn is close to plastic limit wp .

3. 2
Coefficient of compressibility:
We can plot the field e-p curves from the laboratory test data. The weight of a structure or
of a fill increases the pressure on the clay stratum from the overburden pressure p00 to the
value p0) + Δp shown in Figure. The corresponding void ratio decreases from e0 to e.
Hence, for the range in pressure from p0 to (p0) + Δp ) , we may write
Figure: void ratio – pressure diagram.
Coefficient of volume compressibility:
Compressibility is the aptitude of the soil to be deformed. Coefficient of volume
compressibility represents the compression of the clay per unit of original thickness due
to a unit increase of the pressure.
Terzaghi’s 1D Consolidation:
Assumptions:
 The soil medium is completely saturated
 The soil medium is isotropic and homogeneous
 Darcy’s law is valid for flow of water
 Flow is one dimensional in the vertical direction
 The coefficient of permeability is constant

4. 3
 The coefficient of volume compressibility is constant
 The increase in stress on the compressible soil deposit is constant (∆σ=constant)
 Soil particles and water are incompressible
One dimensional theory is based on the following hypothesis
1. The change in volume of soil is equal to volume of pore water expelled.
2. The volume of pore water expelled is equal to change in volume of voids.
3. Since compression is in one direction the change in volume is equal to change in
height
The increase in vertical stress at any depth is equal to the decrease in excess pore water
pressure at the depth
This is Terzaghi’s one dimensional consolidation equation
Cv = Co- efficient of consolidation
Co- efficient of consolidation:
Where k = co efficient of permeability.
.e0 = initial void ratio
av = co- efficient of compressibility
Solution of consolidation equation:
A solution of the Terzaghi’s one dimensional consolidation equation for the mentioned
boundary conditions using Fourier series is given by

5. 4
Construction of e- log σ’ curve:
Note that at the end of the test for each incremental loading the stress on the specimen is
the effective stress, σ ‘. Once the specific gravity of the soil solids, the initial specimen
dimensions, and the specimen deformation at the end of each load has been determined,
the corresponding void ratio can be calculated. A typical void ratio vs. effective pressure
relationship plotted on semilogarithmic graph paper is shown in figure below.
Determination of Pre-consolidation Pressure:
The preconsolidation pressure from an e versus log σ ‘ plot is generally determined by a
graphical procedure suggested by Casagrande (1936), as shown in Figure 6.14b. The
steps are as follows:
1. Visually determine the point P (on the upper curved portion of the e versus log σ ‘plot)
that has the maximum curvature.
2. Draw a horizontal line PQ.
3. Draw a tangent PR at P.
4. Draw the line PS bisecting the angle QPR.
5. Produce the straight-line portion of the e versus log σ ‘plot backward to intersect PS at
T
6. The effective pressure corresponding to point T is the preconsolidation pressure σc. In
the field, the overconsolidation ratio (OCR) can be defined as

6. 5
Compression index Cc:
The slope of the e vs. log σ ‘ plot for normally consolidated soil is referred to as the
compression index Cc.
Skempton's Formula:
Skempton (1944) established a relationship between Cc, and liquid limits for remolded
clays as
Cc= 0.007 (LL - 10)
where, liquid limit LL is in percent.
Terzaghi and Peck Formula:
Based on the work of Skempton and others, Terzaghi and Peck (1948) modified the
skempton’s formula applicable to normally consolidated clays of low to moderate
sensitivity as , Cc = 0.009 (LL -10)
Calculation of one-dimensional consolidation settlement:
The basic principle of one-dimensional consolidation settlement calculation is
demonstrated in Figure shown below. If a clay layer of total thickness Ht is subjected to
an increase of average effective overburden pressure from σ0‘ to σ1 ‘ , it will undergo a
consolidation settlement of ΔHt. Hence the strain can be given by

7. 6
Figure: calculation of Δe from e-log σ’ curve for NC clay (a) and OC clay (b) & (c).

8. 7
Determination of Coefficient of consolidation:
For a given load increment, the coefficient of consolidation Cv can be determined from
the laboratory observations of time versus dial reading. There are several procedures
presently available to estimate the coefficient of consolidation, some of which are
described below.
Logarithm - o f - time method:
The logarithm-of-time method was originally proposed by Casagrande and Fadum (1940)
and can be explained by referring to Figure shown below
1. Plot the dial readings for specimen deformation for a given load increment against time
on semilog graph paper as shown in Figure shown below.
2. Plot two points, P and Q, on the upper portion of the consolidation curve, which
correspond to time t1 and t2, respectively. Note that t2 = 4t1.
3. The difference of dial readings between P and Q is equal to x. Locate point R, which is
at a distance x above point P.
4. Draw the horizontal line RS. The dial reading corresponding to this line is d 0, which
corresponds to 0% consolidation.
5. 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.
6. 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.

9. 8
-----------------------------------------------------0 ----------------------------------------------------
√ t90
√ t90
√t90
√t90

10. 9
Problems related to consolidation:
Problem#01:
Problem#02:

11. 10
Problem# 03:
Problem #04:

12. 11
References:
1. Soil Mechanics, T. W. Lambe and R. V. Whitman, Wiley Book Company, 1969.
2. An Introduction to Geotechnical Engineering, Robert D. Holtz and William D. Kovacs,
Prentice - Hall Book Company,1984.
3. Geotechnical Engineering - Soil Mechanics, John N. Cernica, Wiley Book Company,
1995.
4. Engineering Properties of Soils and Their Measurement, Joseph E. Bowles, 4th Ed.,
McGraw - Hill Book Company, 1992.