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Module 5
CONSOLIDATION (Lectures 27 to 34)
Topics
1.1 FUNDAMENTS OF CONSOLIDATION1.1.1 General Concepts of Onedimensional Consolidation
1.1.2 Theory of OneDimensional Consolidation
1.1.3 Relations of and for Other Forms of Initial Excess Pore WaterPressure Distribution
1.1.4 Numerical Solution for OneDimensional Consolidation
Consolidation in a layered soil
1.1.5 Degree of Consolidation under TimeDependent Loading
1.1.6 Standard OneDimensional Consolidation Test and Interpretation
1.1.7 Preconsolidation pressure.
Compression index
Effect of sample disturbance on the e vs. log cirve1.1.8 Calculation of onedimensional consolidation settlement
1.1.9 Calculation of coefficient of consolidation from laboratory test results
Logarithmoftime method
Squarerootoftime method
Sus maximum slope method
Sivaram and Swamees computational method
1.1.10Secondary Consolidation
1.1.11Constant RateofStrain consolidation Tests
Coefficient of consolidation
Interpretation of experimental results
1.1.12ConstantGradient Consolidating Test
Interpretation of experimental results
1.1.13OneDimensional Consolidation with Visoelastic Models
1.2 CONSOLIDATON BY SAND DRAINS

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1.2.1 Sand Drains
1.2.2 FreeStrain Consolidation with no Smear
1.2.3 EqualStrain Consolidation with no Smear
1.2.4 Effect of Smear Zone on Radial Consolidation
1.2.5 Calculation of the Degree of Consolidation with Vertical and Radial
Drainage
1.2.6 Numerical Solution for Radial Drainage
PROBLEMS

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Module 5
Lecture 27
Consolidation1
Topics
1.1 FUNDAMENTS OF CONSOLIDATION1.1.1 General Concepts of Onedimensional Consolidation
1.1.2 Theory of OneDimensional Consolidation
According to Terzaghi (1943), a decrease of water content of a saturated soil without replacement of the
water by air is called a process of consolidation. When saturated clayey soilswhich have a low coefficient
of permeabilityare subjected to a compressive stress due to a foundation loading, the ore water pressure willimmediately increase; however, due to the low permeability of the soil, there will be a time lag between the
application of load and the extrusion of the pore water and, thus, the settlement. This phenomenon is the
subject of discussion of this chapter.
1.1 FUNDAMENTS OF CONSOLIDATION1.1.1 General Conc epts of Onedimens ional Cons ol idat ion
To understand the basic concepts of consolidation, consider a clay layer of thickness located below thegroundwater level and between two highly permeable sand layers as shown in Figure 5.1. If a surcharge of
intensity is applied at the ground surface over a very large area, the pore water pressure in the clay layerwill increase. For a surcharge ofinfinite extent, the immediate increase of the pore water pressure, , at alldepths of the clay layer will be equal to the increase of the total stress, . Thus, immediately after theapplication of the surcharge.
Figure 5.1

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Since the total stress is equal to the sum of the effective stress and the pore water pressure at all depth soft
the clay layer the increase of effective stress due to the surcharge (immediately after application) will be
equal to zero (i.e., where is the increase of the effective stress). In other words, at time t= 0,the entire stress increase at all depths of the clay is taken by the pore water pressure and none b y the soil
skeleton. This is shown in Figure 5.2a. (It must be pointed out that, for loads applied over a limited area, it
may to be true that the increase of the pore water pressure is equal to the increase of vertical stress at anydepth at time t = 0.
After application of the surcharge (i.e., at time ), the water in the void spaces of the clay layer will besqueezed out and will flow toward both the highly permeable sand layers, thereby reducing the excess porewater pressure. This, in turn, will increase the effective stress by an amount since . Thus, attime ,
And
This fact is shown in Figure 5.2b.
Theoretically, at time the excess pore water pressure at all depths of the clay layer will be dissipatedby gradual drainage. Thus, at time ,
Figure 5.2 Change of pore water pressure and effective stress in the clay layer shown
in Figure 5. 1 due to the surcharge

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And
This shown in Figure 5.2c.
This gradual process of increase of effective stress in the clay layer due to the surcharge will result in a
settlement which is timedependent and is referred to as the process ofconsolidation.
1.1.2 Theory of OneDimens ional Cons ol idat ion
The theory for the time rate of onedimensional consolidation was first proposed by Terzaghi (1925). The
underlying assumption in the derivation of the mathematical equations are as follows:
1. The clay layer is homogeneous.
2. The clay layer is saturated.
3. The compression of the soil layer is due to the change in volume only, which, in turn, is due to the
squeezing out of water from the void spaces.
4. Darcys law valid.
5. Deformation of soil occurs only in the direction of the load application.
6. The coefficient of consolidation [equation (15)] is constant during the consolidation.
With the above assumptions, let us consider a clay layer of thickness as shown in Figure 5.3. The layeris located between two highly permeable sand layers. In this case of onedimensional consolidation, the flow
of water into and out of the soil element is in one direction only, i.e., in the zdirection. This means that
are equal to zero, and thus the rate of low into and out of the soil element can be givenby:
(1)
Where (2)
we obtain
Figure 5.3 Clay layer undergoing consolidation

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(3)
Where is the coefficient of permeability [k=]. However,
(4)
where is the unit weight of water. Substitution of equation (4) and (3) and rearranging gives
(5)
During consolidation the rate of change of volume is equal to the rate of change of the void volume. So,
(6)
Where is the volume of voids in the soil element. But
(7)
Where is the volume of soil solids in the element, which is constant, and is the void ratio. So,
(8)
Substituting the above relation into equation (5), we get
(9)
The change in void ratio, , is due to the increase of effective stress; assuming that these are linearlyrelated, then
(10)
Combining equations (9) and (11),
(12)
Where
(13)
Or
(14)
Where (15)
Equation (14) is the basic differential equation of Terzaghis consolidation theory and can be solved with
proper boundary conditions. To solve the equation, assume u to be the product of two functions, i.e., the
product of a function ofzand a function oft, or
(16)So,
(17)

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And
(18)
From equations (14), (17), and (18),
or
(19)
The righthand side of equation (19) is a function ofzonly and is independent oft; the lefthand side of the
equation is a function oftonly and is independent ofz. therefore, they must be equal to a constant, say.So,
(20)
A solution to equation (20) can be given by
(21)Where and are constants.
Again, the righthand side of equation (19) may be written as
(22)
The solution to equation (22) is given by
(23)
Where is a constant. Combining equations (16), (21), and (23),
(24)
Where .
The constants in equation (24) can be evaluated from the boundary conditions, which are as follows:
1. At time (initial excess pore water pressure at any depth).2. .3. .
Note thatHis the length of the longest drainage path. In this case, which is twoway drainage condition (top
andbottom of the clay layer),His equal to half the total thickness of the clay layer, .
The second boundary condition dictates that , and from the third boundary condition we get
Where n is an integer. From the above, a general solution of equation (24) can be in given the form
(25)

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Where is the nondimensional time factor and is equal to
To satisfy the first boundary condition, we must have the coefficients of such that
(26)
Equation (26) is a Fourier sine series, and can be given by
(27)
Combining equations (25) and (27),
(28)
So far we have not made any assumptions regarding the variation of with the depth of the clay layer.Several possible types of variation for are considered below.
Constant with depth. if is constant with depthi.e., if (Figure 5.4)referring to equation(28),
So, (29)
Figure 5.4 Initial excess pore water pressureconstant with depth (double drainage)

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Note that the term in the above equation is zero for cases when n is even; therefore, u is alsozero. For the nonzero terms, it is convenient to substitute where m is an integer. So equation(29) will no read
(30)
Where . At a given time, the degree of consolidation at any depthzis defined as
(31)
Where is the increase of effective stress at a depthzdue to consolidation. From equations (30) and (31),
(32)
Figure 5.5 shows the variation of with depth for various values of the nondimensional time factor, ;these curves are called isocrones.
In most cases, however, we need to obtain the average degree of consolidation for the entire layer. This is
given by
(33)
The average degree of consolidation is also the ratio of consolidation settlement at any time to maximumconsolidation settlement. Note, in this case, that .
Combining equations (30) and (33),
Figure 5.5 Variation of with and

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(34)
Figure 5.6 gives the variation of (also see table 1)
Terzaghi suggested the following equations for to approximate the values obtained from equation (34):
For (35)
For (36)
Sivaram and Swamee (1977) gave the following equation for varying from 0 to 100%:
(37)
Or
(38)
Equations (37) and (38) give an error in of less than 1% for 0% and less than 3% for90% .
Table 1 Variation of [equation (34)
0 0 60 0.287
10 0.008 65 0.342
20 0.031 70 0.403
30 0.071 75 0.478
35 0.096 80 0.567
40 0.126 85 0.684
45 0.159 90 0.848
50 0.197 95 1.127
55 0.238 100
Figure 5.6 Variation of average degree of consolidation (for conditions given in figs. 4, 7, 8, and 9)

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It must be pointed out that, if we have a situation of oneway drainage as shown in Figure 5.7a and b,
equation (34) would still be valid. Note, however, that the length of the drainage path is equal to the total
thickness of the clay layer.
Figure 5.7 Initial excess pore pressure distributionone way drainage, constant with depth
Linear variation of. The linear variation of the initial excess pore water pressure, as shown in Figure 5.8, may be written as
(39)
Substitution of the above relation for into equation (28) yields
Figure 5.8 linearly varying initial excess pore water pressure distributiontwoway drainage

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Figure 5.9 Sinusoidal initial excess pore water pressure distributiontwoway drainage
(40)
The average degree of consolidation can be obtained by solving equations (40) and 33):
This is identical to equation (34), which was for the case where the excess pore water pressure is constant
with depth, and so the same curves as given in Figure 5.6 can be used.
Sinusoidal variation of. Sinusoidal variation (Figure 5.9) can be represented by the equation
(41)
The solution for the average degree of consolidation for this type of excess pore water pressure distribution
is of the form
(42)
The variation of for various values of is given in Figure 5.6.