module 5 - nptelnptel.ac.in/courses/105104132/module5/lecture27.pdf · module 5 lecture 27...

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 CONSOLIDATION

1.1.1 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 Water

Pressure 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 cirve

1.1.8 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

Su’s maximum slope method

Sivaram and Swamee’s computational method

1.1.10 Secondary Consolidation

1.1.11 Constant Rate-of-Strain consolidation Tests

Coefficient of consolidation

Interpretation of experimental results

1.1.12 Constant-Gradient Consolidating Test

Interpretation of experimental results

1.1.13 One-Dimensional Consolidation with Visoelastic Models

1.2 CONSOLIDATON BY SAND DRAINS



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



Module 5

Lecture 27

Consolidation-1

Topics




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 will

immediately 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.



To understand the basic concepts of consolidation, consider a clay layer of thickness located below the

groundwater 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 layer

will increase. For a surcharge of infinite extent, the immediate increase of the pore water pressure, , at all

depths of the clay layer will be equal to the increase of the total stress, . Thus, immediately after the

application of the surcharge.

Figure 5.1



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 any

depth at time t = 0.

After application of the surcharge (i.e., at time ), the water in the void spaces of the clay layer will be

squeezed out and will flow toward both the highly permeable sand layers, thereby reducing the excess pore

water pressure. This, in turn, will increase the effective stress by an amount since . Thus, at

time ,

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 dissipated

by 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



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 of consolidation.


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. Darcy’s 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 layer

is 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 z direction. This means that

are equal to zero, and thus the rate of low into and out of the soil element can be given

by:

(1)

Where (2)

we obtain

Figure 5.3 Clay layer undergoing consolidation



(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 linearly

related, then

(10)

Combining equations (9) and (11),

(12)

Where

(13)

Or

(14)

Where

(15)

Equation (14) is the basic differential equation of Terzaghi’s 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 of z and a function of t, or

(16)

So,

(17)



And

(18)

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

or

(19)

The right-hand side of equation (19) is a function of z only and is independent of t; the left-hand side of the

equation is a function of t only and is independent of z. 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 that H is the length of the longest drainage path. In this case, which is two-way drainage condition (top

and bottom of the clay layer), H is 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)



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)


(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 depth – i.e., if (Figure 5.4) – referring to equation

(28),

So,

(29)

Figure 5.4 Initial excess pore water pressure-constant with depth (double drainage)



Note that the term in the above equation is zero for cases when n is even; therefore, u is also

zero. 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 depth z is defined as

(31)

Where is the increase of effective stress at a depth z due to consolidation. From equations (30) and (31),

(32)

Figure 5.5 shows the variation of with depth for various values of the non-dimensional 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 maximum

consolidation settlement. Note, in this case, that .


Figure 5.5 Variation of with and



(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% for

90% .

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)



It must be pointed out that, if we have a situation of one-way 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 distribution-one 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 distribution-two-way drainage



Figure 5.9 Sinusoidal initial excess pore water pressure distribution-two-way 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.

module 5 - nptelnptel.ac.in/courses/105104132/module5/lecture27.pdf · module 5 lecture 27...

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