NPTEL – ADVANCED FOUNDATION ENGINEERING-I
Module 4
Lecture 14
SHALLOW FOUNDATIONS: ALLOWABLE BEARING CAPACITY AND
SETTLEMENT
Topics
1.1 SETTLEMENT CALCULATION
1.2 ELASTIC SETTLEMENT BASED ON THE THEORY OF
ELASTICITY
1.3 ELASTIC
SETTLEMENT
OF
FOUNDATION
ON
SATURATED CLAY
1.4 SETTLEMENT OF SANDY SOIL: USE OF STRAIN
INFLUENCE FACTOR
1.5 RANGE OF MATERIAL PARAMETERS FOR COMPUTING
ELASTIC SETTLEMENT
1.6 CONSOLIDATION SETTLEMENT
1.7 SKEMPTON-BJERRUM MODIFICATION FOR CONSOLIDATION
SETTLEMENT
1.8 CONSOLIDATION SETTLEMENT-GENERAL COMMENTS
AND A CASE HISTORY
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SETTLEMENT CALCULATION
ELASTIC SETTLEMENT BASED ON THE THEORY OF ELASTICITY
The elastic settlement of a shallow foundation can be estimated by using the theory of
elasticity. Referring to figure 4.16 and using Hooke’s law,
Figure 4.16 Elastic settlement of shallow foundation
H
1
H
Se = ∫0 εz dz = E ∫0 �∆pz − μs Δpx − μs Δpy �dz
Where
s
[4.27]
Se = elastic settlement
Es = modulus of elasticity of soil
H = thickness of the soil layer
μs = Poisson′ s ratio of the soil
Δpx , Δpy , Δpz =
stress increase due to the net applied foundation load in the x, y, and z, directions, respectively
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Theoretically, if the depth of foundation Df = 0, H = ∞, and the foundation is perfectly
flexible, according to Harr (1966) the settlement may be expressed as (figure 4.17)
Figure 4.17 Elastic settlement of flexible and rigid foundations
Se =
Se =
Bq o
Es
Bq o
Es
Where
(1 − μ2s )
α
2
(1 − μ2s )α
1
�1+m 21 +m 1
α = π �In �
m1 = L/B
�1+m 21 −m 1
(corner of the flexible foundation)
(center of the flexible foundation)
�1+m 21 +1
� + m In �
�1+m 21 −1
��
[4.28]
[4.29]
[4.30]
[4.31]
B = width of foundation
L = length of foundation
The values of α for various length-to-width (L/B) ratios are shown in figure 4.18. The
average immediate settlement for a flexible foundation also may be expressed as
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Se =
Bq o
Es
Figure 4.18 Values of α, αav , and αr −equations (28, 29, 32 and 32a)
(1 − μ2s )αav = average for flexible foundation)
[4.32]
Figure 18 also shows the values of αav for various L/B ratios of foundation.
However, if the foundation shown in figure 17 is rigid, the immediate settlement will be
different and may be expressed as
Se =
Bq o
Es
(1 − μ2s )αr = (rigid foundation)
[4.32a]
The value of αr for various L/B ratios of foundation are shown in figure 4.18.
If Df = 0 and H < ∞ due to the presence of a rigid (incompressible) layer as shown in
figure 17,
Se =
Bq o
Se =
Bq o
Es
And
Es
(1 − μ2s )
[�1−μ 2s �F 1 +�1−μ s −2μ 2s �F 2 ]
2
(corner of flexible foundation)
(1 − μ2s )[(1 − μ2s )F1 + (1 − μs − 2μ2s )F2 ](corner of flexible foundation)
[4.33a]
[4.33b]
The variation of F1 and F2 with H/B are given in figures 4.19 and 4.20, respectively
(Steinbrenner, 1934).
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Figure 4.19 Variation of F1 with H/B (based on Steinbrener, 1934
Figure 4.20 Variation of F2 with H/B (based on Steinbrenner, 1934)
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It is also important to realize that the preceding relationships for Se assume that the depth
of the foundation is equal to zero. ForDf > 0, the magnitude of Se will decrease.
Example 5
A foundation is 1m × 2m in plan and carries a net load per unit area, q o = 150 kN/m2 .
Given, for the soil, Es = 10,000 kN/m2 ; μs = 0.3. Assuming the foundation to be
flexible, estimate the elastic settlement at the center of the foundation for the following
conditions:
a. Df = 0; H = ∞
b. Df = 0; H = 5 m
Solution
Part a
From equation (29)
Se =
Bq o
Se =
(1)(150)
Es
(1 − μ2s )α
For L/B = 2/1 = 2, from figure 18, α ≈ 1.53, so
10,000
Part b
(1 − 0. 32 )(1.53) = 0.0209 m = 20.9 mm
From equation (33b)
Se =
Bq o
Se =
(1)(150)
Es
(1 − μ2s )[(1 − μ2s )F1 + (1 − μs − 2μ2s )F2 ]
For L/B = 2 and H/B = 5, from figures 19 and 20, F1 ≈ 0.525 and F2 ≈ 0.06
10,000
(1 − 0. 32 )[(1 − 0. 32 )(0.525) + (1 − 0.3 − 2 × 0. 32 )(0.06)] =
0.007 m = 7.0 mm
ELASTIC SETTLEMENT OF FOUNDATION ON SATURATED CLAY
Janbu et al. (1956) proposed an equation for evaluating the average settlement of flexible
foundations on saturated clay soils (Poisson’s ratio, μs = 0.5). For the notation used in
figure 4.21, this equation is
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Figure 4.21 Values of A1 and A2 for elastic settlement calculation-equation (34) (after
Christian and Carrier, 1978)
Se = A1 A2
qo B
Es
[4.34]
Where A1 is a function of H/B and L/B and A2 is a function of Df /B
Christian and Carrier (1978) modified the values of A1 and A2 to some extent, as
presented in figure 21.
SETTLEMENT OF SANDY SOIL: USE OF STRAIN INFLUENCE FACTOR
Settlement of granular soils can be evaluated by use of a semi-empirical strain influence
factor (figure 4.22) proposed by Schmertmann and Hartman (1978). According to this
method, the settlement is
NPTEL – ADVANCED FOUNDATION ENGINEERING-I
z I
Se = C1 C2 (q� − q) ∑02 Ez ∆z
s
[4.35]
Figure 4.22 Elastic settlement calculation by using strain influence factor
Where
Iz = strain influence factor
C1 = a 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡ℎ𝑒𝑒 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑ℎ 𝑜𝑜𝑜𝑜 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 1 −
0.5[𝑞𝑞/(𝑞𝑞� − 𝑞𝑞)]
𝐶𝐶2 = 𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐r𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 =
1 + 0.2 𝑙𝑙𝑙𝑙𝑙𝑙(𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦/0.1)
𝑞𝑞� = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑡𝑡ℎ𝑒𝑒 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
𝑞𝑞 = 𝛾𝛾𝐷𝐷𝑓𝑓
The variation of the strain influence factor with depth below the foundation is shown in
figure 22a. Note that, for square or circular foundations,
𝐼𝐼𝑧𝑧 = 0.1 𝑎𝑎𝑎𝑎 𝑧𝑧 = 0
𝐼𝐼𝑧𝑧 = 0.5 𝑎𝑎𝑎𝑎 𝑧𝑧 = 𝑧𝑧1 = 0.5 𝐵𝐵
𝐼𝐼𝑧𝑧 = 0 𝑎𝑎𝑎𝑎 𝑧𝑧 = 𝑧𝑧2 = 2𝐵𝐵
Similarly, for foundation with 𝐿𝐿/𝐵𝐵 ≥ 10,
𝐼𝐼𝑧𝑧 = 0.2 𝑎𝑎𝑎𝑎 𝑧𝑧 = 0
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𝐼𝐼𝑧𝑧 = 0.5 𝑎𝑎𝑎𝑎 𝑧𝑧 = 𝑧𝑧1 = 𝐵𝐵
𝐼𝐼𝑧𝑧 = 0 𝑎𝑎𝑎𝑎 𝑧𝑧 = 𝑧𝑧2 = 4𝐵𝐵
Where
𝐵𝐵 = 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝐿𝐿 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙ℎ 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
For values of 𝐿𝐿/𝐵𝐵 between 1 and 10, necessary interpolations can be made.
To use equation (35) first requires evaluation of the approximate variation of he modulus
of elasticity with depth (figure 4.22). This evaluation can be made by using the standard
penetration numbers or cone penetration resistances. The soil layer can be divided into
several layers to a depth of 𝑧𝑧 = 𝑧𝑧2 , and the elastic settlement of each layer can be
estimated. The sum of the settlement of all layers equals 𝑆𝑆𝑒𝑒 , Schmertmann (1970)
provided a case history of a rectangular foundation (Belgian bridge pier) having L = 23 m
and B = 2.6 m and being supported by a granular soil deposit. For this foundation we may
assume that 𝐿𝐿/𝐵𝐵 ≈ 10 for plotting the strain influence factor diagram. Figure 4.23 shows
the details of the foundation along with the approximate variation of the cone penetration
resistance, 𝑞𝑞𝑜𝑜 , with depth. For this foundation [equation (35)], note that
Figure 4.23 Variation of 𝐼𝐼𝑧𝑧 𝑎𝑎𝑎𝑎𝑎𝑎 𝑞𝑞𝑟𝑟 below the foundation
NPTEL – ADVANCED FOUNDATION ENGINEERING-I
�
= 178.54 𝑘𝑘𝑘𝑘/𝑚𝑚2
𝑞𝑞
𝑞𝑞 = 31.39 𝑘𝑘𝑘𝑘/𝑚𝑚2
𝑞𝑞
31.39
𝐶𝐶1 = 1 − 0.5 𝑞𝑞�−𝑞𝑞 = 1 − (0.5) �178.54−31.39� = 0.893
𝑡𝑡 𝑦𝑦𝑦𝑦
𝐶𝐶2 = 1 + 0.2 𝑙𝑙𝑙𝑙𝑙𝑙 � 0.1 �
For 𝑡𝑡 = 5 𝑦𝑦𝑦𝑦
5
𝐶𝐶2 = 1 + 0.2 𝑙𝑙𝑙𝑙𝑙𝑙 �0.1� = 1.34
𝑧𝑧
The following table shows the calculation of ∑02 (𝐼𝐼𝑧𝑧 /𝐸𝐸𝑠𝑠 ) ∆𝑧𝑧 in conjunction with figure
4.23.
∆𝑧𝑧(𝑚𝑚)
Layer
𝑞𝑞𝑐𝑐 (𝑘𝑘𝑘𝑘/
𝑚𝑚2 )
𝐸𝐸𝑠𝑠𝑎𝑎 (𝑘𝑘𝑘𝑘/
𝑚𝑚2 )
𝑧𝑧 to the 𝐼𝐼𝑧𝑧 at the (𝐼𝐼𝑧𝑧 /
center of center of 𝐸𝐸𝑠𝑠 ) ∆𝑧𝑧(𝑚𝑚2 /
the layer the layer 𝑘𝑘𝑘𝑘)
(m)
1
1
2,450
8,575
0.5
0.258
2
1.6
3,430
12,005
1.8
0.408
3
0.4
3,430
12,005
2.8
0.487
4
0.5
6,870
24,045
3.25
0.458
5
1.0
2,950
10,325
4.0
0.410
6
0.5
8,340
29,190
4.75
0.362
3.00
× 10−5
5.43
× 10−5
1.62
× 10−5
0.95
× 10−5
3.97
× 10−5
0.62
× 10−5
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7
1.5
14,000
49,000
5.75
0.298
8
1
6,000
21,000
7.0
0.247
9
1
10,000
35,000
8.0
0.154
10
1.9
4,000
14,000
9.45
0.062
a
𝛴𝛴 10.4 𝑚𝑚
= 4 𝐵𝐵
𝐸𝐸𝑠𝑠 ≈ 3.5𝑞𝑞𝑜𝑜 [𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 (40)]
0.91
× 10−5
1.17
× 10−5
0.44
× 10−5
0.84
× 10−5
𝛴𝛴 18.95
× 10−5
Hence the immediate settlement is calculated as
𝐼𝐼
𝑆𝑆𝑒𝑒 = 𝐶𝐶1 𝐶𝐶2 (𝑞𝑞� − 𝑞𝑞)𝛴𝛴 𝐸𝐸𝑧𝑧 𝛥𝛥𝑧𝑧
𝑠𝑠
= (0.893)(1.34)(178.54 − 31.39)(18.95 × 10−5 )
= 0.03336 ≈ 33 𝑚𝑚𝑚𝑚
After five years, the actual maximum settlement observed for the foundation was about
39 mm.
RANGE OF MATERIAL PARAMETERS FOR COMPUTING ELASTIC
SETTLEMENT
Section 8-10 presented the equations for calculating immediate settlement of foundations.
These equations contain the elastic parameters, such as 𝐸𝐸𝑠𝑠 𝑎𝑎𝑎𝑎𝑎𝑎 𝜇𝜇𝑠𝑠 . If the laboratory test
results for these parameters are not available, certain realistic assumptions have to be
made. Table 5 shows the approximate range of the elastic parameters for various soils.
Several investigators have correlated the values of the modulus of elasticity, 𝐸𝐸𝑠𝑠 , with the
field standard penetration number, 𝑁𝑁𝐹𝐹 and the cone penetration resistance, 𝑞𝑞𝑐𝑐 , Mitchell
and Gardner (1975) compiled a list of these correlations, Schmertmann (1970) indicated
that the modulus of elasticity of sand may be given by
𝐸𝐸𝑠𝑠 (𝑘𝑘𝑘𝑘/𝑚𝑚2 ) = 766𝑁𝑁𝐹𝐹
[4.36]
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Where
𝑁𝑁𝐹𝐹 = 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠a𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛
In English units
𝐸𝐸 (𝑈𝑈. 𝑆𝑆. 𝑡𝑡𝑡𝑡𝑡𝑡/𝑓𝑓𝑓𝑓 2 ) = 8𝑁𝑁𝐹𝐹
[4.37]
Table 5 Elastic Parameters of Various Soils
Modulus of elasticity, 𝐸𝐸𝑠𝑠
Loose sand
𝑙𝑙𝑙𝑙/𝑖𝑖𝑖𝑖2
1,500-3,500
𝑀𝑀𝑀𝑀/𝑚𝑚2
10.35-24.15
Poisson’s ratio, 𝜇𝜇𝑠𝑠
Medium dense sand
2,500-4,000
17.25-27.60
0.25-0.40
Dense sand
5,000-8000
34.50-55.20
0.30-0.45
Silty sand
1,500-2,500
10.35-17.25
0.20-0.40
Sand and gravel
10,000-25,000
69.00-172.50
0.15-0.35
Soft clay
600-3,000
4.1-20.7
Medium clay
3,000-6,000
20.7-41.4
Stiff clay
6,000-14,000
41.4-96.6
Type of soil
0.20-0.40
0.20-0.50
Similarly,
𝐸𝐸𝑠𝑠 = 2𝑞𝑞𝑐𝑐
[4.38]
Where
𝑄𝑄𝑐𝑐 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
Schmertmann and Hartman (1978) further suggested that the following correlations may
be used with the strain influence factors described in section 10:
𝐸𝐸𝑠𝑠 = 2.5𝑞𝑞𝑐𝑐
[4.39]
And
(𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓)
𝐸𝐸𝑠𝑠 = 3.5𝑞𝑞𝑐𝑐
(𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓)
[4.40]
Note: Any consistent set of units may be used in equation (38)-(40).
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The modulus of elasticity of normally consolidated clays may be estimated as
𝐸𝐸𝑠𝑠 = 250𝑐𝑐 𝑡𝑡𝑡𝑡 500 𝑐𝑐
[4.41]
𝐸𝐸𝑠𝑠 = 750𝑐𝑐 𝑡𝑡𝑡𝑡 1000 𝑐𝑐
[4.42]
And for overconsolidated clays as
Where
𝑐𝑐 = 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
CONSOLIDATION SETTLEMENT
As mentioned before, consolidation settlement occurs over time, and it occurs in
saturated clayey soils when they are subjected to increased load caused by foundation
construction (figure 4.24). Based on the one-dimensional consolidation settlement
equations given in chapter 1, we write
Figure 4.24 Consolidation settlement calculation
𝑆𝑆𝑐𝑐 = ∫ 𝜀𝜀𝑧𝑧 𝑑𝑑𝑑𝑑
Where
𝜀𝜀𝑧𝑧 = 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
𝛥𝛥𝑒𝑒
= 1+𝑒𝑒
𝑜𝑜
NPTEL – ADVANCED FOUNDATION ENGINEERING-I
𝛥𝛥𝑒𝑒 = 𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑜𝑜𝑜𝑜 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
= 𝑓𝑓(𝑝𝑝𝑜𝑜 , 𝑝𝑝𝑐𝑐 , 𝑎𝑎𝑎𝑎𝑎𝑎 𝛥𝛥𝑝𝑝)
So
𝑐𝑐 𝑐𝑐
𝑆𝑆𝑐𝑐 = 1+𝑒𝑒
𝑙𝑙𝑙𝑙𝑙𝑙
𝐶𝐶 𝐻𝐻
𝑝𝑝 𝑜𝑜 +𝛥𝛥𝑝𝑝 𝑎𝑎𝑎𝑎
𝐶𝐶 𝐻𝐻
𝑝𝑝 𝑜𝑜 +𝛥𝛥𝑝𝑝 𝑎𝑎𝑎𝑎
𝑜𝑜
(𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)
𝑝𝑝 𝑜𝑜
(From chapter 1)
𝑠𝑠 𝑐𝑐
𝑆𝑆𝑐𝑐 = 1+𝑒𝑒
𝑙𝑙𝑙𝑙𝑙𝑙
𝑜𝑜
[64]
(𝑓𝑓𝑓𝑓𝑓𝑓 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)
𝑝𝑝 𝑜𝑜
(From chapter 1)
𝐶𝐶 𝐻𝐻
𝑝𝑝
[66]
𝐶𝐶 𝐻𝐻
𝑠𝑠 𝑐𝑐
𝑐𝑐 𝑐𝑐
𝑆𝑆𝑐𝑐 = 1+𝑒𝑒
𝑙𝑙𝑙𝑙𝑙𝑙 𝑝𝑝 𝑐𝑐 + 1+𝑒𝑒
𝑙𝑙𝑙𝑙𝑙𝑙
𝑜𝑜
𝑜𝑜
𝑝𝑝𝑐𝑐 < 𝑝𝑝𝑜𝑜 + 𝛥𝛥𝑝𝑝𝑎𝑎𝑎𝑎 )
𝑜𝑜
𝑝𝑝 𝑜𝑜 +𝛥𝛥𝑝𝑝 𝑎𝑎𝑎𝑎
𝑝𝑝 𝑜𝑜
(From chapter 1)
(𝑓𝑓𝑓𝑓𝑓𝑓 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑤𝑤𝑖𝑖𝑖𝑖ℎ 𝑝𝑝𝑜𝑜 <
[66]
Where
𝑝𝑝a =
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
𝛥𝛥𝑝𝑝𝑎𝑎𝑎𝑎 =
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑜𝑜𝑜𝑜 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑏𝑏𝑏𝑏 𝑡𝑡ℎ𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑝𝑝𝑐𝑐 = 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑒𝑒𝑜𝑜 = 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
𝐶𝐶𝑐𝑐 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝐶𝐶𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖d𝑒𝑒𝑒𝑒
𝐻𝐻𝑐𝑐 = 𝑡𝑡ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
The procedures for determining the compression and swelling indexes were discussed in
chapter 1.
Note that the increase of pressure, 𝛥𝛥𝑝𝑝, on the clay layer is not constant with depth. The
magnitude of 𝛥𝛥𝑝𝑝 will decrease with the increase of depth measured from the bottom of
the foundation. However, the average increase of pressure may be approximated by
𝛥𝛥𝑝𝑝𝑎𝑎𝑣𝑣 = 16(𝛥𝛥𝑝𝑝𝑡𝑡 + 4𝛥𝛥𝑝𝑝𝑚𝑚 + 𝛥𝛥𝑝𝑝𝑏𝑏 )
[4.43]
NPTEL – ADVANCED FOUNDATION ENGINEERING-I
where 𝛥𝛥𝑝𝑝𝑡𝑡 , 𝛥𝛥𝑝𝑝𝑚𝑚 , 𝑎𝑎𝑎𝑎𝑎𝑎 𝛥𝛥𝑝𝑝𝑏𝑏 are the pressure increases at the top, middle, and bottom of the
clay layer that are caused by the foundation construction.
The method of determining the pressure increase caused by various types of foundation
load is discussed in sections 2-7, 𝛥𝛥𝑝𝑝𝑎𝑎𝑎𝑎 can also be directly obtained from the method
presented in section 5.
Example 6
A foundation 1 𝑚𝑚 × 2𝑚𝑚 in plan is shown in figure 4.25. Estimate the consolidation
settlement of the foundation.
Figure 4.25
Solution
The clay is normally consolidated. Thus
𝐶𝐶 𝐻𝐻
𝑐𝑐
𝑆𝑆𝑐𝑐 = 1+𝑒𝑒
𝑙𝑙𝑙𝑙𝑙𝑙
𝑜𝑜
𝑝𝑝 𝑜𝑜 +𝛥𝛥𝑝𝑝 𝑎𝑎𝑎𝑎
𝑝𝑝 𝑜𝑜
𝑝𝑝𝑜𝑜 = (2.5)(16.5) + (0.5)(17.5 − 9.81) + (1.25)(16 − 9.81) = 41.25 + 3.85 + 7.74 =
52.84𝑘𝑘𝑘𝑘/𝑚𝑚2
From equation (43),
𝛥𝛥𝑝𝑝𝑎𝑎𝑎𝑎 = 16(𝛥𝛥𝑝𝑝𝑡𝑡 + 4𝛥𝛥𝑝𝑝𝑚𝑚 + 𝛥𝛥𝑝𝑝𝑏𝑏 )
Now the following table can be prepared (note: 𝐿𝐿 = 2 𝑚𝑚; 𝐵𝐵 = 1 𝑚𝑚):
2
2
𝑚𝑚̇1 = 𝐿𝐿/𝐵𝐵
2
𝑧𝑧(𝑚𝑚)
2+2.5/2 = 3.25
𝑧𝑧/(𝐵𝐵/2) = 𝑛𝑛1
4
6.5
0.190
𝐼𝐼𝑐𝑐
≈ 0.085
∆𝑝𝑝 = 𝑞𝑞𝑜𝑜 𝐼𝐼𝑐𝑐
28.5 = ∆𝑝𝑝𝑡𝑡
12.75 = ∆𝑝𝑝𝑚𝑚
NPTEL – ADVANCED FOUNDATION ENGINEERING-I
2
2+2.5 = 4.5
9
0.045
Table 3
6.75 = ∆𝑝𝑝𝑏𝑏
Equation (10)
𝛥𝛥𝑝𝑝𝑎𝑎𝑎𝑎 = 16(28.5 + 4 × 12.75 + 6.75) = 14.38 𝑘𝑘𝑘𝑘/𝑚𝑚2
So
𝑆𝑆𝑐𝑐 =
(0.32)(2.5)
1+0.8
52.84+14.38
𝑙𝑙𝑙𝑙𝑙𝑙 �
52.84
SKEMPTON-BJERRUM
SETTLEMENT
� = 0.0465 = 46.5 𝑚𝑚𝑚𝑚
MODIFICATION
FOR
CONSOLIDATION
The consolidation settlement calculation presented in the preceding section is based on
equations (64, 66, and 68). These equations, as shown in chapter 1, are based on onedimensional laboratory consolidation tests. The underlying assumption for these
equations is that the increase of pore water pressure, ∆𝑢𝑢, immediately after the load
application equals the increase of stress, ∆𝑝𝑝, at any depth. For this case
∆𝑒𝑒
𝑆𝑆𝑐𝑐(𝑜𝑜e𝑑𝑑) = ∫ 1+𝑒𝑒 𝑑𝑑𝑑𝑑 = ∫ 𝑚𝑚𝑣𝑣 ∆𝑝𝑝(1) 𝑑𝑑𝑑𝑑
Where
𝑜𝑜
𝑆𝑆𝑐𝑐(𝑜𝑜𝑜𝑜𝑜𝑜 ) =
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 (64, 66, 𝑎𝑎𝑎𝑎𝑎𝑎 68 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 1)
∆𝑝𝑝(1) = 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑜𝑜𝑜𝑜 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ∆𝑝𝑝)
𝑚𝑚𝑣𝑣 = 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 (𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 1)
In the field, however, when load is applied over a limited area on the ground surface, this
assumption will not be correct. Consider the case of a circular foundation on a clay layer
as shown in figure 4.26. The vertical and the horizontal stress increases at a point in the
clay layer immediately below the center of the foundation are ∆𝑝𝑝(1) 𝑎𝑎𝑎𝑎𝑎𝑎 ∆𝑝𝑝(3) ,
respectively. For saturated clay, the pore water pressure increase at that depth (chapter 1)
is
∆𝑢𝑢 = ∆𝑝𝑝(3) + 𝐴𝐴[∆𝑝𝑝(1) − ∆𝑝𝑝(3) ]
[4.44]
NPTEL – ADVANCED FOUNDATION ENGINEERING-I
Figure 4.26 Circular foundation on a clay layer
Where
𝐴𝐴 = 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
For this case
S𝑐𝑐 = ∫ 𝑚𝑚𝑣𝑣 ∆𝑢𝑢 𝑑𝑑𝑑𝑑 = ∫(𝑚𝑚𝑣𝑣 )�∆𝑝𝑝(3) + 𝐴𝐴�∆𝑝𝑝(1) − ∆𝑝𝑝(3) ��𝑑𝑑𝑑𝑑
[4.45]
Thus we can write
𝐾𝐾𝑐𝑐𝑐𝑐𝑐𝑐 = 𝑆𝑆
Where
𝑆𝑆𝑐𝑐
𝑐𝑐(𝑜𝑜𝑜𝑜𝑜𝑜 )
=
𝐻𝐻
∫0 𝑐𝑐 𝑚𝑚 𝑣𝑣 ∆𝑢𝑢 𝑑𝑑z
H
∫0 c m v ∆p (1) dz
= A + (1 − A) �
H
∫0 c ∆p (3) dz
�
H
∫0 c ∆p (1) dz
K cir = settlement ratios for circular foundations
[4.46]
The settlement ratio for a continuous foundation (K str ) can be determined in a manner
similar to that or a circular foundation. The variation of K cir and K str with A and Hc /B is
given in figure 4.27. (Note: B = diameter of a circular foundation, and B = width of a
continuous foundation).
NPTEL – ADVANCED FOUNDATION ENGINEERING-I
Figure 4.27 Settlement ratios for circular (K cir ) and continuous (K str ) foundation
Following is the procedure for determining consolidation settlement according to
Skempton and Bjerrum (1957).
1. Determine the consolidation settlement, Sc(oed ) , using the procedure outlined in
section 12. (Note the change of notation from Sc .
2. Determine the pore water pressure parameter, A.
3. Determine Hc /B.
4. Obtain the settlement ratio-in this case, from figure 27.
5. Calculate the actual consolidation settlement:
Sc = Sc(oed ) × settlement ratio
[4.47]
This technique is generally referred to as the Skempton-Bjerrum modification for
consolidation settlement calculation.
CONSOLIDATION SETTLEMENT-GENERAL COMMENTS AND A CASE
HISTORY
In predicting the consolidation settlement and the time rate of settlement for actual field
conditions, an engineer has to make several simplifying assumptions. They include the
NPTEL – ADVANCED FOUNDATION ENGINEERING-I
compression index, coefficient of consolidation, preconsolidation pressure, drainage
conditions, and thickness of the clay layer. Soil layering is not always uniform with ideal
properties; hence field performance may deviate from the prediction, requiring
adjustments during construction. The following case history on consolidation, as reported
by Schnabel (1972), illustrates this reality.
Figure 4.28 shows the subsoil conditions for the construction of a school building in
Waldorf, Maryland. Upper Pleistocene sand and gravel soils are underlain by deposits of
very loose fine silty sand, soft silty clay, and clayey silt. The softer surface layers are
underlain by various layers of stiff to firm silty clay, clayey silt, and sandy silt to a depth
of 50 ft. before construction of the building began; a compacted fill having a thickness of
8-10 ft was placed on the ground surface. This fill initiated the consolidation settlement
in the soft silty clay and clayey silt.
Figure 4.28 Subsoil conditions for the construction of school building (note: SPT N value
are uncorrected, i.e., after Schnabel, 1972)
To predict the time rate of settlement, based on the laboratory test results, the engineers
made the following approximations:
NPTEL – ADVANCED FOUNDATION ENGINEERING-I
a. The preconsolidation pressure, pc was 1600 to 2800 lb/ft 2 in excess of the
existing overburden pressure.
b. The swell index, Cs , was 0.01 to 0.03.
c. For the more compressive layers Cv ≈ 0.36 ft 2 /day, and for the stiffer soil layers
Cv ≈ 3.1 ft 2 /day.
The total consolidation settlement was estimated to be about 3 in. under double drainage
conditions, 90% settlement was expected to occur in 114 days.
Figure 4.29 shows a comparison of measured and predicted settlement with time, with
indicates that
a.
S c (observed )
S c (estimated )
≈ 0.47
b. 90% of the settlement occurred in about 70 days; hence t 90(observed ) /
t 90(estimated ) ≈ 0.58.
The reality rapid settlement in the field is believed to be due to the presence of a fine sand
layer within the Miocene deposits.
Figure 4.29 Comparison of measured and predicted consolidation settlement with time
(after Schnabel, 1972)
NPTEL – ADVANCED FOUNDATION ENGINEERING-I