# Chapter 28: Consolidation -2 - Class Notes, Civil engineering Notes

## : Chapter 28: Consolidation -2 - Class Notes, Civil engineering Notes

``` Page 1

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

1.1.3 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
Page 2

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

1.1.3 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

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

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
Page 3

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

1.1.3 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

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

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

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

Page 4

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

1.1.3 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

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

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

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

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

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

.
Page 5

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

1.1.3 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

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

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

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

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

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

.

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

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.

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