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    NPTEL- Advanced Geotechnical Engineering

    Dept. of Civil Engg. Indian Institute of Technology, Kanpur 1

    Module 5

    CONSOLIDATION (Lectures 27 to 34)

    Topics

    1.1 FUNDAMENTS OF CONSOLIDATION1.1.1 General Concepts of One-dimensional Consolidation

    1.1.2 Theory of One-Dimensional Consolidation

    1.1.3 Relations of and for Other Forms of Initial Excess Pore WaterPressure Distribution

    1.1.4 Numerical Solution for One-Dimensional Consolidation

    Consolidation in a layered soil

    1.1.5 Degree of Consolidation under Time-Dependent Loading

    1.1.6 Standard One-Dimensional 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 one-dimensional consolidation settlement

    1.1.9 Calculation of coefficient of consolidation from laboratory test results

    Logarithm-of-time method

    Square-root-of-time method

    Sus maximum slope method

    Sivaram and Swamees computational method

    1.1.10Secondary Consolidation

    1.1.11Constant Rate-of-Strain consolidation Tests

    Coefficient of consolidation

    Interpretation of experimental results

    1.1.12Constant-Gradient Consolidating Test

    Interpretation of experimental results

    1.1.13One-Dimensional Consolidation with Visoelastic Models

    1.2 CONSOLIDATON BY SAND DRAINS

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    Dept. of Civil Engg. Indian Institute of Technology, Kanpur 2

    1.2.1 Sand Drains

    1.2.2 Free-Strain Consolidation with no Smear

    1.2.3 Equal-Strain 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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    NPTEL- Advanced Geotechnical Engineering

    Dept. of Civil Engg. Indian Institute of Technology, Kanpur 3

    Module 5

    Lecture 27

    Consolidation-1

    Topics

    1.1 FUNDAMENTS OF CONSOLIDATION1.1.1 General Concepts of One-dimensional Consolidation

    1.1.2 Theory of One-Dimensional 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 soils-which have a low coefficient

    of permeability-are 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 One-dimens 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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    NPTEL- Advanced Geotechnical Engineering

    Dept. of Civil Engg. Indian Institute of Technology, Kanpur 4

    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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    NPTEL- Advanced Geotechnical Engineering

    Dept. of Civil Engg. Indian Institute of Technology, Kanpur 5

    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 time-dependent and is referred to as the process ofconsolidation.

    1.1.2 Theory of One-Dimens ional Cons ol idat ion

    The theory for the time rate of one-dimensional 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 one-dimensional 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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    NPTEL- Advanced Geotechnical Engineering

    Dept. of Civil Engg. Indian Institute of Technology, Kanpur 6

    (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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    NPTEL- Advanced Geotechnical Engineering

    Dept. of Civil Engg. Indian Institute of Technology, Kanpur 7

    And

    (18)

    From equations (14), (17), and (18),

    or

    (19)

    The right-hand side of equation (19) is a function ofzonly and is independent oft; the left-hand 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 right-hand 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 two-way 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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    NPTEL- Advanced Geotechnical Engineering

    Dept. of Civil Engg. Indian Institute of Technology, Kanpur 8

    Where is the nondimensional time factor and is equal to

    To satisfy the first boundary co