1
Modules 12,13,14 / Topic 7
CONSOLIDATION – A CAUSE OF SETTLEMENT OF FOUNDATIONS
7.1 What is consolidation?
On a saturated soil, if you apply a load, say through a shallow foundation such as
a footing (Fig 7.1), the soil drains - assuming that it is not a closed system and that
facility for drainage exists – to an extent compatible with the magnitude of the load,
continuing to remain saturated throughout this process. As a result the pore space
gets shrunk, its thickness gets reduced and consequently the footing undergoes
settlement. It is, however, not an instantaneous, but a time-dependent process, which
should be duly taken into account in the (geotechnical) design of the footing. The
process of this continuation of drainage and the consequent settlement is called
consolidation, in the sense the soil consolidates itself under the load by letting the soil
particles to come into a closer packing. In the settled position, if the load is removed
and if there is facility for ingress of water into the soil, the soil will swell to some extent
(This is different from the swelling associated with clays in the presence of water
thanks to its mineralogical composition.) without returning to the original position, i.e.
the position that existed before applying the load. This implies that there is no full
recovery, but only partial recovery of compression (settlement) in soils, on the negative
side of consolidation. In other words, consolidation is a partially reversible
phenomenon.
It was Karl Terzaghi who first gave (1925) a theoretical basis for the process of
consolidation, treating it as a transient physical phenomenon. It is indeed one of his
earliest and most significant contributions to modern theoretical Soil Mechanics.
To throw further light on this phenomenon, let us consider a (rubber) sponge with
a large pore space placed in water. If we take it out and apply a pressure, water
squeezes out and the sponge compresses. If the pressure is now removed, the
compression is recovered, and that too instantaneously. This is because the sponge
is a continuous body and not a particulate system which the soil is, thanks to which
the scope of rearrangement of particles to the original state does not exist, which
makes the compression to recover instantaneously unlike in soil where the
corresponding consolidation is a time-dependent phenomenon. Even in sand,
because of high e and therefore high k, consolidation is instantaneous unlike in fine
silt and clay where the time factor can be reckoned in months, if not years.
7.2 The mechanics of consolidation
In order to understand the mechanics of consolidation, which would eventually lead
to a theory for the same, let us consider a body of saturated soil subjected to a load
(pressure) (Fig 7.2) where facility exists for the soil to drain.
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When a saturated soil mass is subjected to an external load, the load is instantly
shared by the grain skeleton or network and the pore water, as we have already seen
in Topic 2. The stress induced in the grain structure is called the intergranular
pressure, 𝑝̅, and that in the pore water as pore pressure u. If the total load (pressure)
is p, we can state,
p = p̅ + u
(7.1)
as per Terzaghi’s effective stress principle, the seminal nature of which we have noted
already in Topic .
In consolidation theory we speak of load increments rather than loads. It is as if
the soil is already consolidated under an existing load, and our effort is to examine
how the soil is going to undergo further consolidation on account of an increment of
load we are adding to the existing load.
Since water is more incompressible (less compressible) than the soil structure, at
the instant of loading, the load is fully carried or borne by the pore water, subjecting it
to a hydrostatic excess pressure (in the sense of over and above the normal
hydrostatic pressure existing in the pore water). But since the pore water tends to
drain under this hydrostatic excess pressure, having the facility for it, as it drains, the
load is slowly shifted and transferred to the soil structure. Since the total pressure p
must remain the same at all time, this results in a progressive decrease of pore
pressure u and a corresponding increase of the intergranular pressure 𝑝̅ . It follows
from the above that, if Δp is the load increment and we reckon t as the time, we can
state,
at t = 0, Δp = u, 𝑝̅ = 0
at any time t, Δ p = u + 𝑝̅
When the process of transfer is complete corresponding to the end of drainage,
t = (theoretically) ∞, Δ p = 𝑝̅ , u = 0
The above process is accompanied by a progressive change in volume of the soil,
which is equal to the volume of water drained since the soil remains saturated
throughout this process. Our essential concern is this volume change because of the
fact that it leads to settlement of foundations transferring loads to the soil.
Thus, in essence, consolidation is a process of time-dependent drainage brought
about by an external load, accompanied by a change in total volume.
7.2.1 Spring analogy to explain the mechanics of consolidation
The following simple analogy puts the process of consolidation in a clearer
perspective.
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In the analogy we have a container carrying a spring and water (Fig 7.3). The
spring stands for the soil skeleton and water is the pore water. It is tightly closed by a
lid which has a small opening representing the low value of k of the soil. In Fig 7.3a
which corresponds to the initial state, keeping the lid opening closed we place a weight
W on the lid which corresponds to a load increment of Δp. We now remove the cover
on the lid opening and water starts gushing out as shown in Fig 7.3b which represents
an intermediate stage. Fig 7.3c depicts the end of consolidation when drainage stops
and the load W is fully borne by the spring. At this stage which corresponds to the
end of consolidation, the excess hydrostatic pressure due to W (Fig 7.3a) stands fully
dissipated thanks to drainage for which we have provided facility. This figure also
shows the change in volume (height) as consolidation progresses and ends.
At the stage depicted by Fig 7.3c, if we place another weight W1 on top of W, the
process will repeat itself as in Fig 7.3a, b and c until it reaches the end of consolidation
under the pressure increment due to the additional weight W1.
Before leaving this Section it is useful for us to recall that the three essential
prerequisites for consolidation to take place are, 1) loading, 2) saturated soil, and 3)
drainage facility.
7.3 One-dimensional consolidation
By ‘one-dimensional consolidation’ what we mean is that the loading, drainage and
settlement are all confined to one direction only. Fig.7.4 presents such a situation
which is typically met with in foundation engineering. Here we deal with a layered soil
system in which one layer is clay which is sandwiched between sand layers, both
above and below, into which it can drain. From the position of the water table it is clear
that the layers below are saturated. The clay layer will undergo consolidation due to
the load coming through the footing above by draining into the sand layers at top and
bottom.
7.3.1 Terzaghi’s theory of one-dimensional consolidation
The theory applies to a situation described in Fig 7.4. It enables one to predict total
settlement and also the rate of settlement with time.
Assumptions
The theory is built on the following simplifying assumptions.
1. The soil undergoing consolidation is homogeneous.
2. The degree of saturation is 100 %.
3. The elastic compression of the soil grains and pore water is negligible.
4. The change in volume of the soil is equal to the volume of water drained.
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5. Consolidation is one-dimensional and is confined to the vertical (z) direction.
6. The flow of water from the soil follows Darcy’s law.
7. e and k remain constant throughout consolidation. (In reality, as consolidation
progresses, e decreases and so does k. A first order theory will turn to be too
complicated if such a variation is to be accommodated. This assumption is therefore
made in the interest of simplifying the theory and making it more viable for arriving at
solutions.)
Since we have two drainage faces, referring to Fig 7.5, H is the thickness of the
clay layer per drainage face. (This means that, if there were sand only on one side,
and consequently only one drainage face, H would have been the full thickness.)
Referring to Fig 7.5a, drainage and hence consolidation starts at the interfaces of
clay with the two sand layers and progresses inwards. The excess hydrostatic
pressure u is a function of the depth z, besides being a function of time t. Hence we
can state,
u = f (z, t)
which means that u is a function of the two independent variables z and t.
Fig.7.5b shows the variation of u with z at different time intervals t1, t2 and t3 from
the start of consolidation (t = 0) to the end of it (t = ∞). At t = o, u = Δp over the entire
depth whereas at t = ∞, Δp = 𝑝̅ over the entire depth. The curves corresponding to t1,
t2 and t3 depict the progress of consolidation with time and depth. Taking t2 as a typical
time interval, the hatching to the left indicates to what extent consolidation has
occurred in terms of depth and time, with the unhatched part on the RHS of the curve
indicating the remaining part which is yet to consolidate.
7.3.2 The consolidation equation and its solution
The problem of course needs a differential formulation, leading to a partial
differential equation of the following kind,
𝜕2 𝑢
cv.𝜕𝑧 2 =
𝜕𝑢
𝜕𝑡
, where
(7.2)
cv (mm2/s) called the coefficient of consolidation, is a property of the soil undergoing
consolidation, obtained from a laboratory test to be conducted on a sample.
Eq. (7.2) is a partial differential equation of the second order involving two
independent variables z and t, the solution of which must establish the relationship
between u, z and t. The nature of Eq. (7.2) is typically that of a diffusion problem.
(here u undergoes diffusion (by dissipating itself) with time) which is common to all
physical problems of the diffusion type.) Without going into the mathematics of solving
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the differential equation, we proceed to state the result which has been obtained in
terms of two dimensionless parameters U and T, stated as
U = f(T), in which
(7.3)
U is the degree of consolidation (expressed as %) and defined as,
U=
𝑆𝑡
𝑆
x 100
(7.4)
where St is the amount of settlement at time t, and
S is the total settlement, and
T called the time factor is defined as
𝑐
T = 𝐻𝑣2 𝑡
(7.5)
It may be noted that both U and T are dimensionless which enables a generalised plot
between the two which is given in Fig 7.6. The advantage of such a plot is that the
same plot can be used to show the actual variation between St and t for any specific
problem by merely changing the scales of the x and y axes, i.e. without the need for
replotting (see Kurian, 2005: Sec.2.2.1.1). This is because t = T(
𝐻2
𝑐𝑣
) and St = u(
𝑆
100
) where the quantities within brackets are constants applying to a specific problem,
which multiply T and U giving t and St respectively for a specific problem. This subject
will be re-examined later in Sec 7.4.1.
7.4 The consolidation test
The aim of the consolidation test is to determine the consolidation parameters
(properties) of the soil needed in the consolidation (settlement) analysis. This is
accomplished by conducting a consolidation test in the laboratory on a sample of the
soil extracted from the field. This test involves applying a load on the saturated soil
sample and noting the deformation (compression) undergone by the sample with time
until the deformation has virtually ceased under the load which has been maintained
constant from the beginning to the end of the test.
The test as such sounds very simple in principle, but will prove to be not entirely so
particularly when it comes to processing the results.
Apparatus
The apparatus used for the consolidation test is called Oedometer (originally
devised by Terzaghi) or more simply Consolidometer. Its essential components are:
1) Specimen container, 2) Loading frame, and 3) Arrangement for measuring
deformation (settlement).
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Specimen container
There are two types of containers, (a) the fixed ring container, and (b) the floating
ring container (Fig 7.7).
The container is circular in plan. The specimen is a thin soil cylinder of dia. 60 mm
and thickness 20 mm. In the fixed ring type (Fig 7.7a) the specimen is held in a ring
which is fixed on the base. The porous stones placed at the bottom and top of the
specimen provide the drainage faces. There is a cover plate at the top. The unit is
placed on the loading frame which incorporates a lever (yoke) arrangement for
applying the load at the centre of the cover plate. The dial gauge measuring
deformation (vertical compression) also rests on the cover plate. A reservoir of water
communicates with the bottom porous stone which ensures that the specimen remains
saturated throughout the test.
In the floating ring container (Fig 7.7b) the only difference is that the specimen
container is free and not fixed at the base. As a result, the specimen compresses from
both ends unlike in the fixed ring where the compression is one-sided. Consequently
the friction build-up is higher in the fixed case than the floating case, Fig 7.7, which
makes the latter superior because of the lesser part of the applied load taken up by
friction.
The test
The sample is taken in the ring, the container is assembled and placed in the
loading frame and the load is applied. Simultaneously the stop watch is started and
deformation readings are taken at total elapsed times of typically 0, ¼, 1, 2 ¼, 4, 6 ¼,
8, 12 ¼, 16 min. etc. until the dial gauge movement comes to a close irrespective of
whether it takes hours or days. From the time–deformation data so obtained we can
arrive at the following results.
Reduction of results
The test data enables us to determine the following results,
1) cv – the coefficient of consolidation,
2) r – the primary compression ratio, and
3) ef – void ratio corresponding to the end of the test.
Two methods are available for this purpose, 1) the log fitting method (due to A.
Casagrande) and 2) the square root fitting method (due to D. W. Taylor).
7.4.1 The log fitting method
A semi-log plot is made between dial gauge reading to arithmetic scale on the yaxis and time to log scale on the x-axis (Fig.7.8).
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The plot starts as a curve (convex upwards), continues and changes curvature
towards the bottom where it joins a near straight part. It resembles Fig.7.6 as it only
involves a change of scale of the axes (see Sec.7.6.1). Three zones are identified on
the curve:
1) Initial compression – the compression undergone before the start of the process of
actual consolidation – can be mainly due to the removal of the entrapped air,
2) Primary compression or consolidation – due to the drainage of pore water, and
3) Secondary compression – due to plastic flow or gradual structural adjustment of
particles under the imposed load.
In order to mark these zones the following procedure is followed. Mark the distance
on the y-axis between 0.1 and 0.4 on the x-axis. Set this distance above the curve at
t = 0.1 min. This point is taken as d0 which marks the beginning of primary
compression, i.e. consolidation. d100 which marks the end of consolidation is obtained
by noting the point of intersection of the two straight lines. df is the final reading noted
in the test. d100 to df is the secondary compression. di is actually the initial reading
obtained in the test corresponding to 0 time. Since in the log scale 0 occurs at - ∞, it
cannot be marked on the time axis (see Kurian, 2005: App.E.7). di to d0 is therefore
the initial compression. Half way between d0 and d100, d50 and the corresponding t50
are marked.
Results
Coefficient of consolidation
𝑐𝑣
T=
𝐻2
𝑡, where H is half the sample thickness.
Therfore
cv =
𝑇
𝑡
𝐻 2 , to be calculated corresponding to 50% primary compression
𝑇
= 𝑡50 x 𝐻 2
=
50
0.197
𝑡50
. 𝐻2
𝐻 may be tasken as the average test thickness =
(7.6)
𝐻𝑖 + 𝐻𝑓
2
Primary compression ratio
r=
Void ratio
𝑑0 − 𝑑100
𝑑𝑖 − 𝑑𝑓
(7.7)
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𝑉𝑓
ef =
𝑉𝑠
=
𝐴(2𝐻𝑓 −2𝐻𝑠 )
𝐴.2𝐻𝑠
To determine 2Hs
Ws = Vs.𝛾𝑠 where Ws is the oven dry weight of the specimen obtained at
the end of test,
= A.2Hs.G𝛾𝑤
from which
2Hs =
𝑊𝑠
(7.8)
𝐴𝐺𝛾𝑤
7.4.2 The square root fitting method
Here, as in the log fitting method, dial gauge readings are plotted on the y-axis in
the arithmetic scale, but unlike in the former, square root of time is plotted on the xaxis in arithmetic scale (this is the ‘square scale’ – see Kurian, 2005: App.E.11). The
plot is shown in Fig 7.9. This method is based on the finding that the variation of
compression is nearly linear with the square root of time.
Against time t0 we have di on the y-axis. The straight (linear) part of the plot is
extended upwards to give d0. The straight portion is extended downwards and at 1.15
times its x-intercept an inclined line is drawn joining d0. Its point of intersection with
the curved part of the plot gives d90 and the corresponding t90. The initial, primary and
secondary compressions are marked in the figure.
Results
1)
𝑇
cv = 𝑡90 . 𝐻 2
90
=
2)
r =
3)
ef =
0.848
𝑡90
where H =
𝐻𝑖 + 𝐻𝑓
2
x 𝐻2
(7.9)
10
(𝑑0 − 𝑑90 )
9
(7.10)
𝑑𝑖 − 𝑑𝑓
2𝐻𝑓 − 2𝐻𝑠
2𝐻𝑠
𝑊
where 2Hs = 𝐴.𝐺.𝛾𝑠
𝑤
(7.11)
Scaling of time between laboratory and field
It is of interest to note that the expression for cv enables one to directly estimate
the time taken for a specific degree of consolidation to occur in the field from the
corresponding laboratory result. Since cv is the same in the laboratory and the field
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(being a soil property), for the same load (pressure), as also T for a given U, it is seen
that the time taken in the field is obtained by multiplying the time taken in the laboratory
by the square of the ratio of the thickness of the field soil and the thickness of the
laboratory sample.
By way of example, if the field thickness is 1.4 m against a laboratory thickness of 20
mm, and if the time taken in the laboratory for, say a degree of consolidation of 90%,
is 3 hours, the same will take (
1400 2
)
20
times the laboratory time, which works out as
1.68 years. If the field soil has only one drainage face, the corresponding factor is
(
1400 2
) , resulting in a
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field time of 2.24 years. One can also extrapolate the time needed
for one value of U in the field from the laboratory result (time) for another value of U,
using the respective values of T.
Load increments
Our discussion of the consolidation test was so far confined to a single load stage
or rather, load increment. Very often it will be necessary for us to conduct the test for
several load increments so that we can determine cv, r and ef for different load
increments, by either of the fitting methods, and plot each result against the load
(pressure). Among them the variation of e with p is of special importance. As we
expect, e decreases with p. An important finding has been that if we plot e vs. log p
(or p to log scale) a major part of the plot is found to be linear (Fig 7.10). The slope of
the linear part is of special interest to us as we need it for the determination of
settlement of structures on soils undergoing consolidation. This slope is called
compression index (Cc). We observe that, since both e and log p are dimensionless,
Cc is a dimensionless quantity.
If we want to know the slope of the e-p curve (which is not linear) at any point (slope
of the tangent to the point) from Cc which is the slope of the e-log p curve, we can get
it as follows.
log10 p = loge p x log10 e
𝑑
𝑑𝑝
= 0.435 loge p
1
𝑙𝑜𝑔10 𝑝 = 0.435. 𝑝
𝑑 𝑙𝑜𝑔10 𝑝 =
Cc =
0.435
𝑝
. 𝑑𝑝
−𝑑𝑒
𝑑 𝑙𝑜𝑔10 𝑝
increases)
-de = Cc .d log10 p
= 𝐶𝑐 .
0.435
𝑝
. 𝑑𝑝
(negative sign because e decreases as log p
10
Therefore
−
𝑑𝑒
𝑑𝑝
=
0.435
𝑝
. 𝐶𝑐
(7.12)
which is the slope of the e-p curve at any given value of p. It is called the coefficient
of compressibility (av). As noted, it is specific to a given value of p. It has the dimension
L2/F.
7.5 Normally loaded and preloaded soils
In terms of the consolidation history of the soil, we distinguish between normally
loaded (or normally consolidated) and preloaded (or preconsolidated) soils.
A clay is said to be normally consolidated if the present pressure to which it is
subjected is greater than all the past pressures. On the other hand, a clay is said to
be preconsolidated if it has been subjected to a load in the past higher than the present
load. The maximum pressure to which it has been subjected is called the
‘preconsolidation pressure’.
The e-log p curve obtained from the consolidation test enables us to identify
whether a soil is normally consolidated or preconsolidated. In the e-log p curve a
preloaded clay will have a flatter curved part before it merges into the straight inclined
part (Fig 7.11). The curved initial part relates to the preconsolidated state and the
straight line part that follows pertains to the normally consolidated state, the transition
point indicating the maximum preconsolidation pressure.
A. Casagrande has suggested a simple graphical procedure for identifying the
maximum preconsolidation pressure from the e-log p curve. According to this
empirical approach, the point of maximum curvature (which corresponds to the
minimum radius of curvature) is identified visually (point a). The tangent to the curve
is drawn at this point as also a horizontal line. The angle between these lines is
bisected. The straight part of the curve is extended upwards and its point of
intersection with the bisector is noted (point b). Avertical is drawn through b which
gives the maximum preconsolidation pressure on the x-axis.
7.5.1 Causes of preconsolidation
Some of the causes of preconsolidation of a soil are the following:
1) removal of a pre-existing overburden by erosion,
2) melting of glacial ice, the long presence of which at an earlier epoch,
overconsolidated the soil below,
3) flood plains subjected to periodic deposits subsequently undergoing evaporation
and desiccation,
4) long term lowering of ground water levels by pumping or construction activities
could preload clays after restoration of the ground water levels.
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7.6 Computation of settlement by consolidation
Fig 7.12 shows a normally consolidated saturated clay of thickness H (full)
undergoing one-dimensional consolidation. Consider an element of soil at mid-depth
of this clay layer. Let the original effective intergranular pressure at this point due to
the overburden be 𝑝̅ and let the increment of pressure at this point be Δp, which causes
consolidation. Let the original void ratio of the layer be e0.
An element of soil of unit plan area within the consolidating layer is shown in Fig
7.12b. If we take the volume of solids Vs as 1 (unity), the void space can be shown as
e0, since Vv = Vs .e0 = e0 Let the void space undergo a decrease of Δe due to
consolidation. Hence the strain undergone by the element =
Δ𝑒
1+ 𝑒0
. This gives the
settlement as
Δ𝑒
S = H 1+ 𝑒
(7.13)
0
Eq. (7.13) is general and can be used to calculate settlement wherever the original
void ratio and the change in void ratio are known.
Reverting to consolidation and referring to Fig 7.12c,
𝑒 −𝑒
1
𝑐𝑐 = 𝑙𝑜𝑔 0 − 𝑙𝑜𝑔
𝑝1
̅
𝑝
Δ𝑒
𝑝
log( ̅1 )
=
𝑝
Since p1 = p0 + Δp, we can state
𝑝̅ + Δ𝑝
Δe = cc log(
𝑝̅
)
Hence from Eq. (7.13), it follows that,
𝐶
S = 1+𝑐𝑒 . H.log10 (1 +
0
Δ𝑝
𝑝̅
)
(7.14)
This is the well known equation for the settlement by consolidation of a normally loaded
soil.
The time it takes for this settlement to occur is our next concern. In the mean time
it may be noted that settlement would be the same even if there is only one drainage
face; only, the time for S to take place will be much higher than with two drainage
faces.
The presence of e0 in the denominator of Eq (7.14) may look anomalous because
of which it appears the higher the e0, the lesser the settlement, contrary to what we
would normally expect! A closer examination would, however, reveal that this is not
true. If we look at the parent equation (Eq.7.13) and substitute Δe with (e0 – e),
𝑒 −𝑒
0
S = H ( 1+
)
𝑒
0
(7.15)
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This equation gives settlement values when the same soil at different values of e0 are
all consolidated to the same final e value. That is to say, e is constant and e0 is a
variable. Using common symbols, Eq. (7.15) can be put down as,
𝑥−𝑐
y = k ( 𝑎 + 𝑏𝑥 )
(7.16)
Kurian (2005: Sec. 2.2.1.2) has shown that y (S) increases with x (e0) as per this
equation (Fig.7.13), removing any doubt that may arise in the mind of the student in
this regard.
7.6.1 Variation of S with Δp
Referring to Eq. (7.14), in a given problem, Cc, e0 and H and 𝑝̅ are constants; as
such, what causes settlement S is Δp. Hence it shall be of interest for us to examine
the variation of S with Δp. Fig 7.14 shows that S-Δp variation is of the same form as
x-log x.
On the x-axis, starting with (1 +
Δ𝑝
𝑝̅
Δ𝑝
) we obtain ( 𝑝̅ ) by subtraction of 1, and Δ𝑝 by
multiplying the latter with the constant 𝑝̅ . On the y-axis, S is obtained by multiplying
log(1 +
Δ𝑝
) by a constant which is (
𝑝̅
𝐶𝑐
1+ 𝑒0
).H. (One may note that subtraction or
multiplication does not alter the linearity of the scales.)
The Δp-S curve is extended in the figure on the LHS for negative values of Δp.
If Eq. (7.14) is plotted for negative values of Δp, we get ‘swelling’ which is the
opposite of compression which we get for positive values of Δp. The same is, however,
shown dotted since Eq.(7.14) is not valid for negative values of Δp, the reason being
that consolidation is not a reversible phenomenon. Even if it may be partially
reversible, it does not follow Eq. (7.14).
7.6.2 Combined influence of p and 𝛥p
Consider the situation depicted by Fig 7.15 in which two identical layers of clay
(same Cc, e0 and H) but located at different depths. As depth increases, while Δp
decreases very fast with depth, 𝑝̅ increases linearly with depth. The combined result
Δ𝑝
is a still faster decrease of ( 𝑝̅ ) and its consequent influence on settlement.
7.6.3 Compression index from liquid limit
In the case of normally loaded sedimented clays of low sensitivity, A. W. Skempton
has attempted to relate Cc with the liquid limit wL by a simple equation which has been
accepted as quite reliable for settlement calculations. The relationship is,
Cc = 0.009 (wL – 10)
(7.17)
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where wL is expressed as a percentage.
7.6.4 Rate of settlement (settlement with time)
In the majority of design problems we are only concerned with the total settlement
(S) irrespective of the time it will take for the same to develop. It is, however, ideal if
we can plot time (t) vs. settlement (St) – both to arithmetic scales – in a specific
problem. Fig 7.16 is such an example where St (positive) is plotted downwards as per
normal practice in geotechnical engineering. Since, theoretically, full settlement takes
infinite time, S will form an asymptote to the t-St curve as seen in Fig 7.16.
From Fig 7.6 which plots T(log scale) vs. U both of which are dimensionless
quantities, we can obtain t and St for a specific problem as (see Eqs. 4 and 5),
t=
𝑇𝐻 2
𝑐𝑣
𝑈
and St = 100 . 𝑠
(7.18)
(7.19)
It is important to note that in Eq.(7.18) H is the thickness of the layer per drainage
face, whereas in Eq.(7.14) which we use for calculating S, H is the total thickness
irrespective of the number of drainage faces.
7.7 Settlement correction for construction loading
The theory of consolidation assumes that the load is placed instantaneously and it
remains constant with time. When it comes to construction, however, say for a
building, the load increases from zero to the full value over the period of time it takes
for construction to be completed. Fig.7.17a assumes a linear increase for construction
loading which will remain constant from the end of construction. This assumption is
notwithstanding the fact that there is an excavation phase (for foundations) prior to
superstructure construction during which the loading is negative.
Terzaghi and Gilboy have advanced an empirical graphical method for the local
correction of the original time-settlement curve drawn on the assumption of full loading
at 0 time, over the period of the triangular construction loading.
Fig.7.17b starts with the original t-S curve assuming full loading at time 0. Let t1 be
the construction period. At t1 a vertical is dropped intersecting the curve at A1. Let ot
be any time interval between o and ot1. Mark t/2 and drop a vertical at t/2 intersecting
the original curve at C. Draw a horizontal line at C which intersects the line t 1A1 at D.
Join OD. OD intersects the vertical drawn at t at E. tE is the corrected settlement at t
which is therefore a point on the corrected curve. This procedure can be repeated for
more points on ot1 until the corrected curve OEB is obtained under ot1.
It is identified that as regards point t1, the above procedure results in t1B as the
corrected settlement at t1. The rest of the corrected curve is completed for point t1 and
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beyond, by a marginal adjustment, by offsetting AB = A 1B1 = A2B2, etc. until the
corrected curve merges with the original curve.
The assumption behind the above method is,
corrected settlement at t (tE) = full settlement at t/2 for a load at t which is
(ot/ot1) x full load at t1
(Note from Fig 7.17a that
tf
t1 g
=
Ot
Ot1
)
t
Accordingly, 2 c = t1 D
tE
t1
=
D
t
Ot
Ot1
Therefore, tE = 2 c x
Ot
Ot1
7.8 Two and three dimensional consolidation
All what we have examined in detail so far applied to the simple case of onedimensional consolidation. We shall therefore close this Topic by taking a cursory
look at the subject of two and three dimensional consolidation.
Two dimensional consolidation
2-dimensional consolidation is confined to the x-z plane, as in the case of the cross
section of an earthen dam (Fig 7.18).
In this case, u = f (x, z, t)
The corresponding partial differential equation is,
𝜕2 𝑢
cv ( 𝜕𝑥 2 +
𝜕2 𝑢
𝜕𝑧 2
)=
𝜕𝑢
(7.20)
𝜕𝑡
Three dimensional consolidation
All physical field problems where consolidation takes place in the 3-dimensional xy-z space constitute examples of 3-dimensional consolidation.
In this case, u = f (x, y, z, t)
The corresponding partial differential equation is,
𝜕2 𝑢
cv (𝜕𝑥 2 +
𝜕2 𝑢
𝜕𝑦 2
+
𝜕2 𝑢
𝜕𝑢
𝜕𝑧 2
𝜕𝑡
)=
(7.21)
Two and three dimensional consolidation problems being far more complex for
generating closed-form theoretical solutions, today’s geotechnical fraternity would
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increasingly resort to computer assisted numerical methods which can even
accommodate inhomogeneity and anistropy of the medium undergoing consolidation.