1.1.3 relations of and for other forms of initial excess...

NPTEL- Advanced Geotechnical Engineering

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

Module 5

Lecture 28

Consolidation-2

Topics

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

1.1.3 Relations of and for Other Forms of Initial Excess Pore Water

Pressure Distribution

Using the basic equation for excess pore water pressure and with proper boundary conditions, relations for

and for various other types of initial excess pore water pressure distribution can be obtained. Figure

5.10 and 5.11 present some of these cases.

Figure 5.10 Some forms of initial excess pore water pressure distribution



Example 1 Consider the case of initial excess hydrostatic pore water that is constant with depth, i.e.,

(Figure 5.12). For , determine the degree of consolidation at a depth H/3 measured from

the top of the layer.

Figure 5.12

Solution From equation (32), for constant pore water pressure increase,

Figure 5.11 Variation of for initial excess pore water pressure diagrams shown in Figure 5. 10



. We can now make a table to calculate 1.

2. 0.3 0.3 0.3

3. 0 1 2

4.

5.

6. 1.273 0.4244 0.2546

7. 0.4770 0.00128

8. 0.5 1.0 0.5

9.

0.3036 0.00005

Using the value of 0.3041 calculated in step 9, the degree of consolidation at depth H/3 is

Note that in the above table we need not go beyond , since the expression in step 9 is negligible for

.

Example 2 Due to certain loading conditions, the excess pore water pressure in a clay layer (drained at top

and bottom) increased in the manner shown in Figure 5.13. For a time factor , calculate the average

degree of consolidation.

Figure 5.13

Solution The excess pore water pressure diagram shown in Figure 5.13 can be expressed as the difference

of two diagrams, as shown in Figure 5.14b and c. the excess pore water pressure diagrams in Figure 5.

14b shows a case where varies linearly with depth. Figure 5.14c can be approximated as a sinusoidal

variation.



Figure 5.14

The area of the diagram in Figure 5.14b is

The area of the diagram in Figure 5.14c is

The average degree of consolidation can now be calculated as follows:

Form Figure 5. 6, for . So

Example 3uniform surcharge of is applied on the ground surface as shown in Figure 5.

15a.

(a) Determine the initial excess pore water pressure distribution in the clay layer.

(b) Plot the distribution of the excess pore water pressure with depth in the clay layer at a time for which

.



Figure 5.15

Solution Part (a): the initial excess pore water pressure will be and will be the same

throughout the clay layer (Figure 5.15b; refer to Prob. 1 in chapter 4).

Part (b): From equation (31), . For can be

obtained from the top half of Figure 5. 5 as shown in Figure 5.16a.

0 0 0.63 740

0.2 2 0.65 700

0.4 4 0.71 580

0.6 6 0.78 440

0.8 8 0.89 220

1.0 10 1 0

Figure 5.16

Figure 5.16b shows the variation of excess pore water pressure with depth.



Example 4 A clay layer is shown in Figure 5.17. Due to a certain loading condition, the initial excess pore

water pressure in the clay layer is of a sinusoidal nature, given by the equation . Calculate the excess pore water pressure at a the midheight of the clay layer for

.

Figure 5.17

Solution From equation (28),

Let us evaluate the term A:

Or

Note that the above integral is zero if so the only nonzero term is obtained when .

Therefore,

Since only for is A not zero,

At the midheight of the clay layer, and so

The values of the excess pore water pressure are tabulated below:



0.2 30.52

0.4 18.64

0.6 11.38

0.8 6.95

1.1.3 Numerical Solution for One-Dimensional Consolidation

In this section, we will consider the finite-difference solution for one-dimensional consolidation, starting

from the basic differential equation of Terzaghi’s consolidation theory:

(14)

Let by any arbitrary reference excess pore water pressure, time, and distance, respectively.

From these we can define the following nondimensional terms:

Nondimensional excess pore water pressure:

(43)

Nondimensional time:

(44)

Nondimensional depth:

(45)

From equations (43), (44), and the left-hand side of equation (14),

(46)

Similarly, from equations (43), (45), and the right-hand side of equation (14),

(47)

From equations (46), and (47),

Or

(48)

If we adopt the reference time in such a way that , then equation (48) will be of the form

(49)

The left-hand side of equation (49) can be written as

(50)



Where are the nondimensional pore water pressure at point O (Figure 5.18a) at nondimensional

times .

(51)

Equating the right sides of equations (50) and (51),

)

Or

(52)

For equation (52) to coverage, must be chosen such that is less than 0.5.

When solving for pore water pressure at the interface of a clay layer and an impervious layer, equation (52)

can be used. However, we need to take point 3 as the mirror image of point 1 (Figure 5.18b); thus . So equation (52) becomes

(53)

Consolidation in a layered soil

It is not always possible to develop a closed-form solution for consolidation in layered soils. There are

several variables involved, such as different coefficients of permeability, the thickness of layers, and

different values of coefficient of consolidation. Figure 5.19 shows the nature of the degree of consolidation

of a two-layered soil.

Figure 5.18



In view of the above, numerical solutions provide a better approach. If we are involved with the calculation

of excess pore water pressure at the interface of two different types (i.e., different values of of clayey

soils, equation (52) will have to be modified to some extent. Referring to Figure 5.20, this can be achieved

as follows (Scott, 1963): from equation (14),

Figure 5.19 Degree of consolidation in two-layered soil. (Figure 5.19b after U. Luscher,

Discussion. Soil Mech. Found. Div., ASCE, vol. 91, no. SM1, 1965)

Figure 5.20



Based on the derivations of equation (163a in chapter 2)

(54)

Where are the coefficients of permeability in layers 1 and 2, respectively. and are

the excess pore water pressures at time t for points 0, 1, and 3, respectively.

Also, the average volume change for the element at the boundary is

(55)

Where and are the excess pore water pressures at point 0 at times t and , respectively.

Equating the right-hand sides of equations (54) and (55), we get

Or

Or

(56)

Assuming and combining equations (43) to (45) and (56), we get

(57)

Example 5 A uniform surcharge of is applied at the ground surface of the soil profile

shown in Figure 5.21. Using the numerical method, determine the distribution of excess pore water pressure

for the clay layers after 10 days of load application.

Figure 5.21



Solution Since this is a uniform surcharge, the excess pore water pressure immediately after the load

application will be throughout the clay layers. However, due to the drainage conditions, the

excess pore water pressures at the top of layer 1 and bottom of layer 2 will immediately become zero. Now,

let . So and .

Figure 5.22 shows the distribution of at time ; note that . Now,

Let for both layers. So, for layer 1,

For layer 2,

.

(52)

At [note: this is the boundary of two layers, so we will use equation (57)],

Figure 5.22



Or

.

,

,

The variation of the nondimensional excess pore water pressure is shown in Figure 5.22. Knowing , we can plot the variation of u with depth.

1.1.3 Degree of Consolidation under Time-Dependent Loading

Olson (1977) presented a mathematical solution for one-dimensional consolidation due to a single ramp

load. Olson’s solution can be explained with the help of Figure 5.23, in which a clay layer is drained at the

top and at the bottom (H is the drainage distance). A uniformly distributed load q is applied at the ground

surface. Note that q is a function of time, as shown in Figure 5.23b.



Figure 5.25 One-dimensional consolidations due to single ramp load

The expression for the excess pore water pressure for the case where is given in equation (30) as

Where

As stated above, the applied load is a function of time:

(58)

Where is the time of application of any load.

For a differential load applied at time the instantaneous pore pressure increase will be . At

time t, the remaining excess pore water pressure at a depth z can be given by the expression.

(59)

The average degree of consolidation can be defined as

(60)

Where is the total load per unit area applied at the time of the analysis. The settlement at time is,

of course, the ultimate settlement. Note that the term in the denominator of equation (60) is equal to the

instantaneous excess pore water pressure ( ) that might have been generated throughout the clay layer

had the stress been applied instantaneously.

Proper integration of equations (59) and (60) gives the following:



For

(61)

And

(62)

For

(63)

And

(64)

Where

(65)

Figure 5.24 shows the plot of for various values of .

Figure 5.24 Plot of against time factor for single ramp load. (After Olsen 1977)

1.1.3 relations of and for other forms of initial excess...

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