Elastic settlement of shallow foundations on granular soil: a
critical review
Braja M. Das
Dean Emeritus, California State University, Sacramento
Henderson, Nevada, U.S.A.
ABSTRACT: Developments in major procedures available in the literature relating to elastic
settlement of shallow foundations supported by granular soil are presented and compared. The
discrepancies between the observed and the predicted settlement are primarily due to the inability
to estimate the modulus of elasticity of soil using the results of the standard penetration tests and/or
cone penetration tests. Based on the procedures available at this time, recommendations have been
made for the best estimation of settlement of foundations
KEY WORDS: Cone penetration test, elastic settlement, granular soil, shallow foundation,
standard penetration test
1 INTRODUCTION
The estimation of settlement of shallow foundations is an important topic in the design and
construction of buildings and other related structures. In general, settlement of a foundation
consists of two major components—elastic settlement (Se) and consolidation settlement (Sc). In
turn, the consolidation settlement of a submerged clay layer has two parts; that is, the contribution
of primary consolidation settlement (Sp) and that due to secondary consolidation (Ss). For a
foundation supported by granular soil within the zone of influence of stress distribution, the elastic
settlement is the only component that needs consideration. This paper is a general overview of
various aspects of the elastic settlement of shallow foundations supported by granular soil deposits.
During the last fifty years or so, a number of procedures have been developed to predict elastic
settlement; however, there is a lack of a reliable standardized procedure.
2 ELASTIC SETTLEMENT CALCULATION PROCEDURES—GENERAL
Various methods to calculate the elastic settlement available at the present time can be divided into
three general categories. They are as follows:
1. Methods Based on Observed Settlement of Structures and Full Scale Prototypes. These
methods are empirical in nature and are correlated with the results of the standard in situ tests
such as the standard penetration test (SPT) and the cone penetration test (CPT). They include
1
procedures developed by Terzaghi and Peck (1948, 1967), Meyerhof (1956, 1965), DeBeer and
Martens (1957), Hough (1969), Peck and Bazaraa (1969), and Burland and Burbidge (1985).
2. Semi-Empirical Methods. These methods are based on a combination of field observations and
some theoretical studies. They include, for example, the procedures outlined by Schmertmann
(1970), Schmertmann et al. (1978), Briaud (2007), and Akbas and Kulhawy (2009).
3. Methods Based on Theoretical Relationships Derived from the Theory of Elasticity. The
relationships for settlement calculation available in this category contain the term modulus of
elasticity (Es).
The general outlines for some of these methods are given in the following sections.
METHODS BASED ON OBSERVED SETTLEMENT
3 TERZAGHI AND PECK’S METHOD
Terzaghi and Peck (1948) proposed the following empirical relationship between the settlement
(Se) of a prototype foundation measuring B×B in plan and the settlement of a test plate [Se(1)]
measuring B1×B1 loaded to the same intensity
Se
4
=
S e (1)
⎡ ⎛ B1
⎢1 + ⎜
⎣ ⎝ B
⎞⎤
⎟⎥
⎠⎦
(1)
2
Although a full-sized footing can be used for a load test, the normal practice is to employ a plate of
the order of 0.3 m to 1 m. Bjerrum and Eggestad (1963) provided the results of 14 sets of load
settlement tests. This is shown in Figure 1 along with the plot of Eq. (1). For these tests, B1 was
0.35 m for circular plates and 0.32 m for square plates. It is obvious from Figure 1 that, although
the general trend is correct, Eq. (1) represents approximately the lower limit of the field test results.
Bazaraa (1967) also provided several field test results. Figure 2 shows the plot of Se/Se(1) versus
B/B1 for all tests results provide by Bjerrum and Eggestad (1963) and Bazaraa (1967) as compiled
by D’Appolonia et al. (1970). The overall results with the expanded data base are similar to those
in Figure 1 as they relate to Eq. (1).
Terzaghi and Peck (1948, 1967) proposed a correlation for the allowable bearing capacity,
standard penetration number (N60), and the width of the foundation (B) corresponding to a 25 -mm
settlement based on the observation given by Eq. (1). This correlation is shown in Figure 3. The
curves shown in Figure 3 can be approximated by the relation
3q ⎛ B ⎞
Se (mm) =
⎜
⎟
N 60 ⎝ B + 0.3 ⎠
2
(2)
where q = bearing pressure in kN/m2
B = width of foundation (m)
If corrections for ground water table location and depth of embedment are included, then Eq. (2)
takes the form
3q ⎛ B ⎞
Se = CW CD
⎜
⎟
N 60 ⎝ B + 0.3 ⎠
2
(3)
where CW = ground water table correction
CD = correction for depth of embedment = 1 – (Df /4B)
Df = depth of embedment
2
Figure 1 Variation of Se/Se(1) versus B/B1 from the load settlement results of Bjerrum and Eggestad (1963)
(Note: B1 = 0.36 m for circular plates and 0.32 m for square plates).
Figure 2 Variation of Se/Se(1) versus B/B1 based on the data of Bjerrum and Eggestad (1963) and Bazaraa
(1967) (adapted from D’Appolonia et al., 1970).
The magnitude of CW is equal to 1.0 if the depth of water table is greater than or equal to 2B below
the foundation, and it is equal to 2.0 if the depth of water table is less than or equal to B below the
foundation. The N60 value that is to be used in Eqs. (2) and (3) should be the average value of N60
up to a depth of about 3B to 4B measured from the bottom of the foundation.
3
Figure 3. Terzaghi and Peck’s (1948, 1967) recommendation for allowable bearing capacity for 25-mm
settlement variation with B and N60.
Jayapalan and Boehm (1986) and Papadopoulos (1992) summarized the case histories of 79
foundations. Sivakugan et al (1998) used those case histories to compare with the settlement
predicted by the Terzaghi and Peck method. This comparison is shown in Figure 4. It can be seen
from this figure that, in general, the predicted settlements were significantly higher than those
observed. The average value of Se(predicted)/Se(observed) ≈ 2.18.
Similar observations were also made by Bazaraa (1967). With B1 = 0.3 m, Eq. (1) can be
rewritten as
Se
⎛ B ⎞
= 4⎜
⎟
Se (1)
⎝ B + 0.3 ⎠
2
or
2
⎛ B ⎞ 1 ⎛⎜ Se ⎞⎟
⎜
⎟ = ⎜
⎝ B + 0.3 ⎠ 4 ⎝ Se (1) ⎟⎠
(4)
Combining Eqs. (2) and (4)
⎛ 3q ⎞ 1 ⎛ Se ⎞
⎟
⎟⎟ ⎜
Se = ⎜⎜
⎜
⎟
⎝ N 60 ⎠ 4 ⎝ Se (1) ⎠
or
q
Se (1)
=
N 60
0.75
(5)
4
Figure 4. Sivakugan et al.’s (1998) comparison of predicted with observed settlement for 79 foundations—
predicted settlement based on Terzaghi and Peck method (1948, 1967).
Figure 5. Bazaraa’s plate load test results—plot of q/Se(1) versus N60.
Bazaraa (1967) plotted a large number of plate load test results (B1 = 0.3 m) in the form of q/Se(1)
versus N60 as shown in Figure 5. It can be seen that the relationship given by Eq. (5) is very
conservative. In fact, q/Se(1) versus N60/0.5 will more closely represent the lower limiting condition.
5
4 MEYERHOF’S METHOD
In 1956, Meyerhof proposed relationships for the elastic settlement of foundations on granular soil
similar to Eq. (2). In 1965 he compared the predicted (by the relationships proposed in 1956) and
observed settlements of eight structures and suggested that the allowable pressure (q) for a desired
magnitude of Se can be increased by 50% compared to what he recommended in 1956. The revised
relationships including the correction factors for water table location (CW) and depth of embedment
(CD) can be expressed as
Se = CW C D
1.25q
N 60
Se = CW CD
2q ⎛ B ⎞
⎜
⎟
N 60 ⎝ B + 0.3 ⎠
(for B ≤ 1.22 m)
(6)
and
2
(for B > 1.22 m)
(7)
CW = 1.0
(8)
and
CD = 1.0 −
Df
(9)
4B
If these equations are used to predict the settlement of the 79 foundations shown in Figure 4, then
we will obtain Se(predicted)/Se(observed) ≈ 1.46. Hence, the predicted settlements will overestimate the
observed values by about 50% on the average.
Table 1 shows the comparison of the maximum observed settlements of mat foundations
considered by Meyerhof (1965) and the settlements predicted by Eq. (7). The ratios of the predicted
to observed settlements are generally in the range of 0.8 to 2. This is also what Meyerhof concluded
in his 1965 paper.
Table 1. Comparison of observed maximum settlements provided by Meyerhof (1965) for eight mat
foundations with those predicted by Eq. (7)
Structure
T. Edison, Sao Paulo
Banco do Brasil, Sao Paulo
Iparanga, Sao Paulo
C.B.I. Esplanada, Sao Paulo
Riscala, Sao Paulo
Thyssen, Dusseldorf
Ministry, Dusseldorf
Chimney, Cologne
B
(m)
18.3
22.9
9.15
14.6
3.96
22.6
15.9
20.4
Average
N60
15
18
9
22
20
25
20
10
q
(kN/m2)
229.8
239.4
220.2
383.0
229.8
239.4
220.4
172.4
Maximum
Se(observed)
(mm)
15.24
27.94
35.56
27.94
12.70
24.13
21.59
10.16
Se(predicted)
S e ( predicted)
by Eq. (7)
S e ( observed )
(mm)
29.66
1.95
25.74
0.99
45.88
1.29
33.43
1.20
19.86
1.56
18.65
0.77
21.23
0.98
33.49
3.30
Average ≈1.5
5 DE BEER AND MARTEN’S METHOD
DeBeer and Martens (1957) and DeBeer (1965) proposed the following relationship to estimate the
elastic settlement of a foundation
6
Se =
⎛ σ ′ + Δσ ⎞
2.3
⎟⎟ H
log10 ⎜⎜ o
C
⎝ σo′ ⎠
(10)
where C = a constant of proportionality
σ o′ = effective overburden pressure at the depth considered
Δσ = increase in pressure at that depth due to foundation loading
H = thickness of the layer considered
The value of C can be approximated as
C ≈ 1.5
qc
σ o′
(11)
where qc = cone penetration resistance.
Equation (10) is essentially in the form of the relationship for estimating the consolidation
settlement of normally consolidated clay. We can rewrite Eq. (10) as
Se =
where
⎛ σ ′ + Δσ ⎞
Cc
⎟⎟
H log10 ⎜⎜ o
1 + eo
⎝ σo′ ⎠
(12)
⎛ σ′ ⎞
Cc
= 1.5⎜⎜ o ⎟⎟
1 + eo
⎝ qc ⎠
(13)
Cc = compression index
eo = in situ void ratio
For the field cases considered by DeBeer and Martens (1957), the average ratio of predicted to
observed settlement was about 1.9. DeBeer (1965) further observed that the above stated method
only applies to normally consolidated sands. For overconsolidated sand, a reduction factor needs to
be applied which can be obtained from cyclic loading tests carried out in an oedometer. Hough
(1969) expressed Cc in Eq. (12) as
Cc = a (eo − b)
(14)
Approximate values of a and b are given in Table 2.
Table 2. Values of a and b from Eq. (14) (based on Hough, 1969)
Value of constant
Type of soil
a
b*
Uniform cohesionless material (uniformity coefficient Cu ≤ 2)
Clean gravel
0.05
0.50
Coarse sand
0.06
0.50
Medium sand
0.07
0.50
Fine sand
0.08
0.50
Inorganic silt
0.10
0.50
Well-graded cohesionless soil
Silty sand and gravel
0.09
0.20
Clean, coarse to fine sand
0.12
0.35
Coarse to fine silty sand
0.15
0.25
Sandy silt (inorganic)
0.18
0.25
*
The value of the constant b should be taken as emin whenever the latter is known or
can conveniently be determined. Otherwise, use tablulated values as a rough
approximation.
7
6 THE METHOD OF PECK AND BAZARAA
Peck and Bazaraa (1969) recognized that the original Terzaghi and Peck method in Section 3 was
overly conservative and revised Eq. (3) to the following form
Se = CW CD
2q ⎛ B ⎞
⎜
⎟
( N1 )60 ⎝ B + 0.3 ⎠
2
(15)
where Se is in mm, q is in kN/m2, and B is in m
(N1)60 = corrected standard penetration number
CW =
σ o at 0.5 B below the bottom of the foundation
σ o′ at 0.5 B below the bottom of the foundation
(16)
σo = total overburden pressure
σ o′ = effective overburden pressure
⎛ γD
CD = 1.0 − 0.4⎜⎜ f
⎝ q
⎞
⎟⎟
⎠
0 .5
(17)
γ = unit weight of soil
The relationships for (N1)60 are as follow:
( N 1 ) 60 =
4 N 60
1 + 0.04σ o′
( N 1 ) 60 =
4 N 60
3.25 + 0.01σ o′
(for σ o′ ≤ 75 kN/m 2 )
(18)
and
(for σ o′ > 75 kN/m2 )
(19)
where σ o′ is the effective overburden pressure (kN/m2)
D’Appolonia et al. (1970) compared the observed settlement of several shallow foundations
from several structures in Indiana (USA) with those estimated using the Peck and Bazaraa method,
and this is shown in Figure 6. It can be seen from this figure that the calculated settlement from
theory greatly overestimates the observed settlement. It appears that this solution will provide
nearly the level of settlement that was obtained from Meyerhof’s revised relationships (Section 5).
Figure 6 Plot of measured versus predicted settlement based on Peck and Bazaraa’s method (adapted from
D’Appolonia et al., 1970).
8
7 METHOD OF BURLAND AND BURBIDGE (1985)
Burland and Burbidge (1985) proposed a method for calculating the elastic settlement of sandy soil
using the field standard penetration number N60. The method can be summarized as follows:
7.1
Determination of Variation of Standard Penetration Number with Depth
Obtain the field penetration numbers (N60) with depth at the location of the foundation. The
following adjustments of N60 may be necessary, depending on the field conditions:
For gravel or sandy gravel,
N 60(a) ≈ 1.25N 60
(20)
For fine sand or silty sand below the ground water table and N60 > 15,
N 60(a) ≈ 15 + 0.5( N 60 − 15)
(21)
where N60(a) = adjusted N60 value
7.2
Determination of Depth of Stress Influence (z′)
In determining the depth of stress influence, the following three cases may arise:
Case I. If N60 [or N60(a)] is approximately constant with depth, calculate z' from
⎛ B
z′
= 1.4⎜⎜
BR
⎝ BR
⎞
⎟⎟
⎠
0 .75
(22)
where BR = reference width = 0.3 m
B = width of the actual foundation (m)
Case II. If N60 [or N60(a)] is increasing with depth, use Eq. (41) to calculate z'.
Case III. If N60 [or N60(a)] is decreasing with depth, calculate z' = 2B and z' = distance from the
bottom of the foundation to the bottom of the soft soil layer (= z"). Use z' = 2B or z' = z" (whichever
is smaller).
7.3
Determination of Depth of Stress Influence Correction Factor α
The correction factor α is given as
α=
H⎛
H⎞
⎜2 − ⎟ ≤ 1
z′ ⎝
z′ ⎠
(23)
where H = thickness of the compressible layer
7.4
Calculation of Elastic Settlement
The elastic settlement of the foundation Se can be calculated as:
9
A. For normally consolidated soil
⎧⎪
Se
1.71
= 0.14α ⎨
BR
⎪⎩ N 60 orN 60(a)
[
⎡
⎛ L⎞ ⎤
⎫⎪⎢ 1.25⎜ B ⎟ ⎥
⎝ ⎠ ⎥
⎢
1.4 ⎬
⎪⎭⎢ 0.25 + ⎛⎜ L ⎞⎟ ⎥
⎢
⎝ B ⎠ ⎦⎥
⎣
]
2
⎛ B
⎜⎜
⎝ BR
⎞
⎟⎟
⎠
0 .7
⎛ q ⎞
⎜⎜ ⎟⎟
⎝ pa ⎠
(24)
where L = length of the foundation
pa = atmospheric pressure (≈ 100 kN/m2)
B. For overconsolidated soil (q ≤ σ c′ ; where σ c′ = overconsolidation pressure)
⎧⎪
Se
0.57
= 0.047 α ⎨
BR
⎩⎪ N 60 or N 60 (a)
[
⎡
⎛L⎞ ⎤
⎫⎪⎢ 1.25⎜ B ⎟ ⎥
⎝ ⎠ ⎥
⎢
1.4 ⎬
⎪⎭⎢ 0.25 + ⎛⎜ L ⎞⎟ ⎥
⎢
⎝ B ⎠ ⎥⎦
⎣
]
2
⎛ B
⎜⎜
⎝ BR
⎞
⎟⎟
⎠
0 .7
⎛ q ⎞
⎜⎜ ⎟⎟
⎝ pa ⎠
(25)
C. For overconsolidated soil (q > σ c′ )
⎧⎪
Se
0.57
= 0.14α ⎨
BR
⎪⎩ N 60 or N 60 (a)
[
⎡
⎛L⎞ ⎤
⎫⎪⎢ 1.25⎜ B ⎟ ⎥
⎝ ⎠ ⎥
⎢
1.4 ⎬
⎪⎭⎢ 0.25 + ⎛⎜ L ⎞⎟ ⎥
⎢
⎝ B ⎠ ⎥⎦
⎣
2
]
⎛ B
⎜⎜
⎝ BR
⎞
⎟⎟
⎠
0 .7
⎛ q − 0.67σ c′ ⎞
⎜⎜
⎟⎟
pa
⎝
⎠
(26)
The procedure works reasonably well. However it may be difficult under normal working
conditions to obtain the overconsolidation pressure in the field.
SEMI-EMPIRICAL METHODS
8 STRAIN INFLUENCE FACTOR METHOD
Based on the theory of elasticity, the equation for vertical strain ε z at a depth below the center of a
flexible circular load of diameter B, can be given as
εz=
q(1 + μs )
[(1 − 2 μs ) A′ + B′]
Es
Iz =
ε z Es
= (1 + μs )[(1 − 2 μs ) A′ + B′]
q
or
(27)
where A' and B' = f (z/B)
q = load per unit area
Es = modulus of elasticity
μ s = Poisson’s ratio
Iz = strain influence factor
10
Figure 7 Theoretical and experimental distribution of vertical strain influence factor below the center of a
circular loaded area (based on Schmertmann, 1970).
Figure 7 shows the variation of Iz with depth based on Eq. (27) for μ s = 0.4 and 0.5. The
experimental results of Eggestad (1963) for variation of Iz are also given in this figure. Considering
both the theoretical and experimental results cited in Figure 7, Schmertmann (1970) proposed a
simplified distribution of Iz with depth that is generally referred to as 2B–0.6Iz distribution and it is
also shown in Figure 7. According to the simplified method,
2B
S e = C1C2 q ∑
o
Iz
Δz
Es
(28)
q = net effective pressure applied at the level of the foundation
qo = effective overburden pressure at the level of the foundation
q
C1 = correction factor for embedment of foundation = 1− 0.5 o
(29)
q
⎛ t ⎞
(30)
C2 = correction factor to account for creep in soil = 1 + 0.2 log⎜ ⎟
⎝ 0.1 ⎠
t = time, in years
For use in Eq. (30) and the strain influence factor shown in Figure 7, it was recommended that
where
ES = 2qc
(31)
where qc = cone penetration resistance
Sivakugan et al. (1998) used the case histories of the 79 foundations given in Figure 4 and
compared those with the settlements obtained using the strain influence factor shown in Figure 7
and Eqs. (28) and (31), and this is shown in Figure 8. From this figure, it can be seen that
se(predicted)/Se(observed) ≈ 3.39.
Sivakugan and Johnson (2004) used a probabilistic approach to compare the predicted
settlements obtained by the methods of Terzaghi and Peck (1948, 1967), Schmertmann et al. (1970)
and Burland and Burbidge (1985). Table 3 gives a summary of their study; that is, predicted
settlement versus the probability of exceeding 25 mm settlement in the field. This shows that the
method of Burland and Burbidge (1985), although conservative, is an improved technique to
estimate elastic settlement. It also shows that the strain influence factor method of Schmertmann
(1970) is not substantially superior to that of Terzaghi and Peck (1948, 1967).
11
Figure 8 Sivakugan et al.’s comparison (1998) of predicted and observed settlements from 79 foundations—
predicted settlement based on 2B−0.6Iz procedure.
Table 3. Probability of exceeding 25 mm settlement in the field
Probability of exceeding 25 mm settlement in field
Predicted
Terzaghi and Peck
Schmertmann et al.
Burland and
settlement
(1948, 1967)
(1970)
Burbidge (1985)
(mm)
1
0.00
0.00
0.00
5
0.00
0.00
0.03
10
0.00
0.02
0.15
15
0.09
0.13
0.25
20
0.20
0.20
0.34
25
0.26
0.27
0.42
30
0.31
0.32
0.49
35
0.35
0.37
0.55
40
0.387
0.42
0.61
Compiled from Sivakugan and Johnson (2004)
Schmertmann et al. (1978) modified the strain influence factor variation (2B–0.6Iz) shown in
Figure 7. The revised distribution is shown in Figure 9 for use in Eqs. (28)–(30). According to this,
For square or circular foundation:
Iz = 0.1 at z = 0
Iz(peak) at z = zp = 0.5B
Iz = 0 at z = zo = 2B
For foundation with L/B ≥ 10:
Iz = 0.2 at z = 0
Iz(peak) at z = zp = B
Iz = 0 at z = zo = 4B
where L = length of foundation. For L/B between 1 and 10, interpolation can be done. Also
I z ( peak )
⎛ q ⎞
= 0.5 + 0.1⎜⎜ ⎟⎟
⎝ σ o′ ⎠
0.5
(32)
12
Figure 9 Revised strain influence factor diagram suggested by Schmertmann et al. (1978).
The value of σ ′o in Eq. (32) is the effective overburden pressure at a depth where Iz(peak) occurs.
Salgado (2008) gave the following interpolation for Iz at z = 0, zp, and zo (for L/B = 1 to L/B ≥ 10.
⎛L⎞
I z ( at z =0 ) = 0.1 + 0.0111⎜ ⎟ ≤ 0.2
⎝B⎠
(33)
zp
⎛L ⎞
= 0.5 + 0.0555⎜ − 1⎟ ≤ 1
B
⎝B ⎠
(34)
zo
⎛L ⎞
= 2 + 0.222⎜ − 1⎟ ≤ 4
B
⎝B ⎠
(35)
Noting that stiffness is about 40% larger for plane strain compared to axisymmetric loading,
Schmertmann et al. (1978) recommended that.
Es = 2.5qc (for square and circular foundations)
(36)
E s = 3.5qc (for strip foundation)
(37)
and
With the modified strain-influence factor diagram,
z = zo
S e = C1 C2 q ∑
z =0
Iz
Δz
Es
(38)
The modified strain influence factor and Eqs. (36) and (37) will definitely reduce the average ratio
of predicted to observed settlement. However, it may still overestimate the actual elastic settlement
in the field.
13
9 RECENT MODIFICATIONS IN STRAIN-INFLUENCE FACTOR DIAGRAMS
More recently some modifications have been proposed to the strain-influence factor diagram
suggested by Schmertmann et al. (1978). Two of these suggestions are discussed below.
9.1
Modification Suggested by Terzaghi, Peck and Mesri (1996)
The modification suggested by Terzaghi et al. (1996) is shown in Figure 10. For this case, for
surface foundation condition (that is, Df/B = 0)
Iz = 0.2 at z = 0
Iz = Iz(peak) = 0.6 at z = zp = 0.5B
Iz = 0 at z = zo
Figure 10 Strain influence diagram suggested by Terzaghi et al. (1996).
⎡
⎛ L ⎞⎤
zo = 2 ⎢1 + log⎜ ⎟⎥ ≤ 4
⎝ B ⎠⎦
⎣
(39)
For Df/B > 0, Iz should be modified to I z′ . Figure 11 shows the variation of I z′ / I z with Df/B.
The end of construction settlement can be estimated as
z = zo
Se = q ∑
z =0
I z′
Δz
Es
(40)
The settlement due to creep can be calculated as
⎛ 0.1 ⎞
⎛ t
⎞
Screep = ⎜⎜ ⎟⎟ zo log⎜⎜ days ⎟⎟
⎝ 1 day ⎠
⎝ qc ⎠
(41)
14
Figure 11 Variation of I z′ / I z with Df /B (after Terzaghi et al. 1996).
where qc = weighted mean value of measured qc values of sublayers between z = 0 and z = zo
(MN/m2)
It has also been suggested that
Es ( L / B )
E s ( L / B =1)
⎛L⎞
= 1 + 0.4 log⎜ ⎟ ≤ 1.4
⎝B⎠
(42)
where Es ( L / B =1) = 3.5qc
(43)
Figure 12 shows the plot of Es versus qc from 81 foundations and 92 plate load tests on which
Eq. (43) has been established. The magnitude of Es recommended by Eq. (43) is about 40% higher
than that obtained from Eq. (36). Figure 13 shows a comparison of the end-of-construction
predicted [using Eqs. (40), (42) and (43)] and measured settlement of foundations on sand and
gravelly soils (Terzaghi et al., 1996).
9.2
Modification Suggested by Lee et al. (2008)
Based on finite element analysis, Lee et al. (2008) suggested the following modifications to the
strain influence factor diagram suggested by Schmertmann et al. (1978). This assumes that Iz(peak)
and Iz at z = 0 is the same as given by Eqs. (32) and (33). However Eqs. (34) and (35) are modified
as
zp
⎡⎛ L ⎞ ⎤
L
= 0.5 + 0.11⎢⎜ ⎟ − 1⎥ ≤ with a maximum of 1 at = 6
B
B
⎣⎝ B ⎠ ⎦
(44)
zo
⎧⎡ π L
⎤ ⎫
L
=6
= 0.95 cos ⎨ ⎢ ⎛⎜ − 1⎞⎟⎥ − π ⎬ + 3 ≤ with a maximum of 4 at
5
B
B
B
⎝
⎠
⎦ ⎭
⎩⎣
(45)
With these modifications, the elastic settlement can be calculated using Eq. (21).
15
Figure 12 Correlation between Es and qc for square and circularly loaded areas [adapted from Terzaghi et al.
(1996)].
Figure 13 Comparison of end of construction predicted and measured Se of foundations on sand and gravelly
soils based on Eqs. (40), (42) and (43) [adapted from Terzaghi et al. (1996)].
10 LOAD-SETTLEMENT CURVE APPROACH BASED ON PRESSUREMETER TESTS
(PMT)
Briaud (2007) presented a method based on field Pressuremeter tests to develop a load-settlement
curve for a given foundation from which the elastic settlement at a given load intensity can be
estimated. This takes into account the foundation load eccentricity, load inclination, and the
location of the foundation on a slope (Figure 14). Following is a step-by-step procedure of the
procedure suggested by Briaud (2007).
16
Figure 14 Pressuremeter test to obtain load-settlement curve.
Figure 15 Adjustment of field Pressuremeter test plot of pp versus ΔR/Ro.
1. Conduct several Pressuremeter tests at the site at various depths.
2. Plot the PMT curves as pressure pp on the cavity wall versus relative increase in cavity radius
ΔR/Ro. Extend the straight line part of the PMT curve to zero pressure and shift the vertical axis
to the value of ΔR/Ro where that strain line portion intersects the horizontal axis (Figure 15).
3. Plot the strain influence factor diagram proposed by Schmertmann et al. (1978) for the
foundation. Based on the pp versus ΔR/Ro diagrams (Step 2) and the location of the depth of the
tests, develop a mean plot of pp versus ΔR/Ro as shown in Figure 16.
The mean pp for a given ΔR/Ro can be given as
p p ( mean ) =
A1
A
A
p p (1) + 2 p p ( 2 ) + 3 p p ( 3) + . . .
A
A
A
(46)
where A1, A2, A3, . . . are the areas tributary to each test under the influence diagram
A = total area of the strain-influence factor diagram
4. Convert the plot of pp(mean) versus ΔR/Ro plot to q versus Se/B plot using the following
equations.
17
Figure 16 Development of the mean pp versus ΔR/Ro plot.
Figure 17 Variation of Г function.
q = ( Γ )( f L / B f e f δ fβ,d ) p p ( mean )
(47)
Se
ΔR
= 0.24
B
Ro
(48)
where Г = Gamma function linking q and pp(mean) (see Figure 17)
⎛B⎞
f B / L = shape factor = 0.8 + 0.2⎜ ⎟
⎝L⎠
(49)
⎛e⎞
f e = load eccentrici ty factor = 1 − 0.33⎜ ⎟ (center)
⎝B⎠
(50)
⎛e⎞
fe = 1 − ⎜ ⎟
⎝B⎠
0.5
(51)
(edge)
⎡ δ (degrees) ⎤
f δ = inclination factor = 1 − ⎢
⎥⎦ (center)
90
⎣
18
(52)
⎡ δ (degrees) ⎤
fδ = 1 − ⎢
⎥⎦
360
⎣
0.5
(53)
(edge)
0.1
f β ,d
f β ,d
⎛ d⎞
= slope factor = 0.8⎜1 + ⎟ (3 : 1 slope)
⎝ B⎠
0.15
⎛ d⎞
= 0.7⎜1 + ⎟
(2 : 1 slope)
⎝ B⎠
(54)
(55)
5. Based on the load-settlement diagram developed in Step 4, obtain the actual Se(maximum) which
corresponds to the actual intensity of load q to which the foundation will be subjected.
6. To account for creep over the life-span of the structure,
⎛t⎞
Se (t ) ≈ Se (maximum) ⎜⎜ ⎟⎟
⎝ t1 ⎠
0.3
(56)
where Se(t) = settlement after time t
Se(maximum) = settlement obtained from Step 5
t = time, in minutes
t1 = reference time = 1 minute
10 FIELD TESTS ON LOAD-SETTLEMENT BEHAVIOR
Akbas And Kulhawy (2009) evaluated 167 load-settlement relationships obtained from field tests.
Figure 18 shows a generalized relationship of load (Q) versus settlement (Se) from these field tests
which they referred to as the L1–L2 method. From this figure, note that: (a) QL1 is the load at
settlement level Se(L1); (b) QT is the load at settlement level Se(T); and (c) QL2 is the load at settlement
level Se(L2), which is the ultimate load (≈Qu).
The field test results yielded the mean value of Se(L1) to be 0.23% of the width of the foundation,
B. Similarly, the mean value of Se(L1) was 5.39% of B. The final analysis showed a nondimensional
load-settlement relationship as given in Figure 19. The mean plot can be expressed as
Q
=
QL 2
Se
B
⎛ Se ⎞
0.69⎜
⎟ + 1.68
⎝ B ⎠
(57)
Figure 18 General nature of the load versus settlement plot obtained from the field (L1–L2 method)
19
Figure 19 Nondimensional plot of Q/QL2 versus Se/B (after Akbas and Kulhawy, 2009).
In order to find Q for a given settlement level, one needs to know that QL2 = Qu for which the
following is recommended.
1. For B > 1m:
QL 2 = Quγ + Quq =
1
2 γ B Nγ Fγs Fγd Fγc + q Nq Fqs Fqd Fqc
Qγ
u
(58)
Qq
u
where
Nγ, Nq = bearing capacity factors
Fγs, Fqs = shape factors
Fγd, Fqd = depth factors
Fγc, Fqc = compressibility factors
(see Vesic, 1973)
2. For B ≤ 1m:
QL 2 =
Quγ
+ Quq
B
(59)
Note that Fγc and Fγc are functions of the modulus of elasticity of soil (Es). This procedure will
give a fairly good estimate of Se for a given value of Q or vice versa; however, a good estimate of
the modulus of elasticity (Es) and soil friction angle (φ′) will be required.
SETTLEMENT CALCULATION BASED ON THEORY OF ELASTICITY
12 STEINBRENNER’S (1934) AND FOX’S (1948) THEORY
Based on the observations made on elastic settlement calculation using empirical correlations and
the wide range in the predictions obtained, it is desirable to consider alternative solutions based on
the theory of elasticity. With that in mind, Figure 20 shows a schematic diagram of the elastic
settlement profile for a flexible and rigid foundation. The shallow foundation measures B×L in plan
and is located at a depth Df below the ground surface. A rock layer (or a rigid layer) is located at a
depth H below the bottom of the foundation.
20
Figure 20 Settlement profile for shallow flexible and rigid foundation.
Theoretically, if the foundation is perfectly flexible (Figure 20), the settlement may be expressed as
(see Bowles, 1987)
S e = q(α ′B ′)
1 − μs2
IsI f
Es
(60)
where q = net applied pressure on the foundation
μ s = Poisson’s ratio of soil
Es = average modulus of elasticity of the soil under the foundation, measured from z =
0 to about z = 4B
B' = B/2 for center of foundation (= B for corner of foundation)
Is = shape factor (Steinbrenner, 1934) = F1 +
1 − 2 μs
F2
1 − μs
(61)
F1 =
1
( A0 + A1 )
π
(62)
F2 =
n
tan −1 A2
2π
(63)
A0 = m ln
A1 = ln
A2 =
(1 +
(
)
m2 + 1 m2 + n2
m 1 + m2 + n2 + 1
(64)
)
)
(m +
m2 + 1 1 + n2
(65)
m + m2 + n2 + 1
m
(66)
2
n + m + n2 +1
⎛ Df
L⎞
I f = depth factor (Fox, 1948) = f ⎜⎜
, μ s , and ⎟⎟
B⎠
⎝ B
α' = a factor that depends on the location below the foundation where settlement is being
calculated
21
(67)
To calculate settlement at the center of the foundation, we use
α′ = 4
(68)
L
B
(69)
m=
and
n=
H
⎛B⎞
⎜ ⎟
⎝2⎠
(70)
To calculate settlement at a corner of the foundation,
α′ =1
m=
L
B
n=
H
B
(71)
and
The variations of F1 and F2 with m and n are given Tables 4 and 5. Based on the works of Fox
(1948), the variations of depth factor If for μ s = 0.3 and 0.4 and L/B have been determined by
Bowles (1987) and are given in Table 6. Note that If is not a function of H/B.
Due to the non-homogeneous nature of a soil deposit, the magnitude of Es may vary with depth.
For that reason, Bowles (1987) recommended
Es =
∑ Es ( i ) Δz
z
(72)
where Es(i) = soil modulus within the depth Δz
z = 5B or H (if H < 5B)
Bowles (1987) also recommended that
Es = 500( N 60 + 15) kN/m2
(73)
The elastic settlement of a rigid foundation can be estimated as
S e( rigid ) ≈ 0.93S e(flexible, center)
(74)
Bowles (1987) compared this theory with 12 case histories that provided reasonable good results.
22
Table 4. Variation of F1 with m and n
n
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75
10.00
20.00
50.00
100.00
1.0
0.014
0.049
0.095
0.142
0.186
0.224
0.257
0.285
0.309
0.330
0.348
0.363
0.376
0.388
0.399
0.408
0.417
0.424
0.431
0.437
0.443
0.448
0.453
0.457
0.461
0.465
0.468
0.471
0.474
0.477
0.480
0.482
0.485
0.487
0.489
0.491
0.493
0.495
0.496
0.498
0.529
0.548
0.555
1.2
0.013
0.046
0.090
0.138
0.183
0.224
0.259
0.290
0.317
0.341
0.361
0.379
0.394
0.408
0.420
0.431
0.440
0.450
0.458
0.465
0.472
0.478
0.483
0.489
0.493
0.498
0.502
0.506
0.509
0.513
0.516
0.519
0.522
0.524
0.527
0.529
0.531
0.533
0.536
0.537
0.575
0.598
0.605
1.4
0.012
0.044
0.087
0.134
0.179
0.222
0.259
0.292
0.321
0.347
0.369
0.389
0.406
0.422
0.436
0.448
0.458
0.469
0.478
0.487
0.494
0.501
0.508
0.514
0.519
0.524
0.529
0.533
0.538
0.541
0.545
0.549
0.552
0.555
0.558
0.560
0.563
0.565
0.568
0.570
0.614
0.640
0.649
1.6
0.011
0.042
0.084
0.130
0.176
0.219
0.258
0.292
0.323
0.350
0.374
0.396
0.415
0.431
0.447
0.460
0.472
0.484
0.494
0.503
0.512
0.520
0.527
0.534
0.540
0.546
0.551
0.556
0.561
0.565
0.569
0.573
0.577
0.580
0.583
0.587
0.589
0.592
0.595
0.597
0.647
0.678
0.688
1.8
0.011
0.041
0.082
0.127
0.173
0.216
0.255
0.291
0.323
0.351
0.377
0.400
0.420
0.438
0.454
0.469
0.481
0.495
0.506
0.516
0.526
0.534
0.542
0.550
0.557
0.563
0.569
0.575
0.580
0.585
0.589
0.594
0.598
0.601
0.605
0.609
0.612
0.615
0.618
0.621
0.677
0.711
0.722
23
m
2.0
0.011
0.040
0.080
0.125
0.170
0.213
0.253
0.289
0.322
0.351
0.378
0.402
0.423
0.442
0.460
0.476
0.484
0.503
0.515
0.526
0.537
0.546
0.555
0.563
0.570
0.577
0.584
0.590
0.596
0.601
0.606
0.611
0.615
0.619
0.623
0.627
0.631
0.634
0.638
0.641
0.702
0.740
0.753
2.5
0.010
0.038
0.077
0.121
0.165
0.207
0.247
0.284
0.317
0.348
0.377
0.402
0.426
0.447
0.467
0.484
0.495
0.516
0.530
0.543
0.555
0.566
0.576
0.585
0.594
0.603
0.610
0.618
0.625
0.631
0.637
0.643
0.648
0.653
0.658
0.663
0.667
0.671
0.675
0.679
0.756
0.803
0.819
3.0
0.010
0.038
0.076
0.118
0.161
0.203
0.242
0.279
0.313
0.344
0.373
0.400
0.424
0.447
0.458
0.487
0.514
0.521
0.536
0.551
0.564
0.576
0.588
0.598
0.609
0.618
0.627
0.635
0.643
0.650
0.658
0.664
0.670
0.676
0.682
0.687
0.693
0.697
0.702
0.707
0.797
0.853
0.872
3.5
0.010
0.037
0.074
0.116
0.158
0.199
0.238
0.275
0.308
0.340
0.369
0.396
0.421
0.444
0.466
0.486
0.515
0.522
0539
0.554
0.568
0.581
0.594
0.606
0.617
0.627
0.637
0.646
0.655
0.663
0.671
0.678
0.685
0.692
0.698
0.705
0.710
0.716
0.721
0.726
0.830
0.895
0.918
4.0
0.010
0.037
0.074
0.115
0.157
0.197
0.235
0.271
0.305
0.336
0.365
0.392
0.418
0.441
0.464
0.484
0.515
0.522
0.539
0.554
0.569
0.584
0.597
0.609
0.621
0.632
0.643
0.653
0.662
0.671
0.680
0.688
0.695
0.703
0.710
0.716
0.723
0.719
0.735
0.740
0.858
0.931
0.956
Table 4. (Continued)
m
n
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75
10.00
20.00
50.00
100.00
4.5
0.010
0.036
0.073
0.114
0.155
0.195
0.233
0.269
0.302
0.333
0.362
0.389
0.415
0.438
0.461
0.482
0.516
0.520
0.537
0.554
0.569
0.584
0.597
0.611
0.623
0.635
0.646
0.656
0.666
0.676
0.685
0.694
0.702
0.710
0.717
0.725
0.731
0.738
0.744
0.750
0.878
0.962
0.990
5.0
0.010
0.036
0.073
0.113
0.154
0.194
0.232
0.267
0.300
0.331
0.359
0.386
0.412
0.435
0.458
0.479
0.496
0.517
0.535
0.552
0.568
0.583
0.597
0.610
0.623
0.635
0.647
0.658
0.669
0.679
0.688
0.697
0.706
0.714
0.722
0.730
0.737
0.744
0.751
0.758
0.896
0.989
1.020
6.0
0.010
0.036
0.072
0.112
0.153
0.192
0.229
0.264
0.296
0.327
0.355
0.382
0.407
0.430
0.453
0.474
0.484
0.513
0.530
0.548
0.564
0.579
0.594
0.608
0.621
0.634
0.646
0.658
0.669
0.680
0.690
0.700
0.710
0.719
0.727
0.736
0.744
0.752
0.759
0.766
0.925
1.034
1.072
7.8
0.010
0.036
0.072
0.112
0.152
0.191
0.228
0.262
0.294
0.324
0.352
0.378
0.403
0.427
0.449
0.470
0.473
0.508
0.526
0.543
0.560
0.575
0.590
0.604
0.618
0.631
0.644
0.656
0.668
0.679
0.689
0.700
0.710
0.719
0.728
0.737
0.746
0.754
0.762
0.770
0.945
1.070
1.114
8.0
0.010
0.036
0.072
0.112
0.152
0.190
0.227
0.261
0.293
0.322
0.350
0.376
0.401
0.424
0.446
0.466
0.471
0.505
0.523
0.540
0.556
0.571
0.586
0.601
0.615
0.628
0.641
0.653
0.665
0.676
0.687
0.698
0.708
0.718
0.727
0.736
0.745
0.754
0.762
0.770
0.959
1.100
1.150
24
9.0
0.010
0.036
0.072
0.111
0.151
0.190
0.226
0.260
0.291
0.321
0.348
0.374
0.399
0.421
0.443
0.464
0.471
0.502
0.519
0.536
0.553
0.568
0.583
0.598
0.611
0.625
0.637
0.650
0.662
0.673
0.684
0.695
0.705
0.715
0.725
0.735
0.744
0.753
0.761
0.770
0.969
1.125
1.182
10.0
0.010
0.036
0.071
0.111
0.151
0.189
0.225
0.259
0.291
0.320
0.347
0.373
0.397
0.420
0.441
0.462
0.470
0.499
0.517
0.534
0.550
0.585
0.580
0.595
0.608
0.622
0.634
0.647
0.659
0.670
0.681
0.692
0.703
0.713
0.723
0.732
0.742
0.751
0.759
0.768
0.977
1.146
1.209
25.0
0.010
0.036
0.071
0.110
0.150
0.188
0.223
0.257
0.287
0.316
0.343
0.368
0.391
0.413
0.433
0.453
0.468
0.489
0.506
0.522
0.537
0.551
0.565
0.579
0.592
0.605
0.617
0.628
0.640
0.651
0.661
0.672
0.682
0.692
0.701
0.710
0.719
0.728
0.737
0.745
0.982
1.265
1.408
50.0
0.010
0.036
0.071
0.110
0.150
0.188
0.223
0.256
0.287
0.315
0.342
0.367
0.390
0.412
0.432
0.451
0.462
0.487
0.504
0.519
0.534
0.549
0.583
0.576
0.589
0.601
0.613
0.624
0.635
0.646
0.656
0.666
0.676
0.686
0.695
0.704
0.713
0.721
0.729
0.738
0.965
1.279
1.489
100.0
0.010
0.036
0.071
0.110
0.150
0.188
0.223
0.256
0.287
0.315
0.342
0.367
0.390
0.411
0.432
0.451
0.460
0.487
0.503
0.519
0.534
0.548
0.562
0.575
0.588
0.600
0.612
0.623
0.634
0.645
0.655
0.665
0.675
0.684
0.693
0.702
0.711
0.719
0.727
0.735
0.957
1.261
1.499
Table 5. Variation of F2 with m and n
m
n
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75
10.00
20.00
50.00
100.00
1.0
0.049
0.074
0.083
0.083
0.080
0.075
0.069
0.064
0.059
0.055
0.051
0.048
0.045
0.042
0.040
0.037
0.036
0.034
0.032
0.031
0.029
0.028
0.027
0.026
0.025
0.024
0.023
0.022
0.022
0.021
0.020
0.020
0.019
0.018
0.018
0.017
0.017
0.017
0.016
0.016
0.008
0.003
0.002
1.2
0.050
0.077
0.089
0.091
0.089
0.084
0.079
0.074
0.069
0.064
0.060
0.056
0.053
0.050
0.047
0.044
0.042
0.040
0.038
0.036
0.035
0.033
0.032
0.031
0.030
0.029
0.028
0.027
0.026
0.025
0.024
0.023
0.023
0.022
0.021
0.021
0.020
0.020
0.019
0.019
0.010
0.004
0.002
1.4
0.051
0.080
0.093
0.098
0.096
0.093
0.088
0.083
0.077
0.073
0.068
0.064
0.060
0.057
0.054
0.051
0.049
0.046
0.044
0.042
0.040
0.039
0.037
0.036
0.034
0.033
0.032
0.031
0.030
0.029
0.028
0.027
0.026
0.026
0.025
0.024
0.024
0.023
0.023
0.022
0.011
0.004
0.002
1.6
0.051
0.081
0.097
0.102
0.102
0.099
0.095
0.090
0.085
0.080
0.076
0.071
0.067
0.068
0.060
0.057
0.055
0.052
0.050
0.048
0.046
0.044
0.042
0.040
0.039
0.038
0.036
0.035
0.034
0.033
0.032
0.031
0.030
0.029
0.028
0.028
0.027
0.026
0.026
0.025
0.013
0.005
0.003
1.8
0.051
0.083
0.099
0.106
0.107
0.105
0.101
0.097
0.092
0.087
0.082
0.078
0.074
0.070
0.067
0.063
0.061
0.058
0.055
0.053
0.051
0.049
0.047
0.045
0.044
0.042
0.041
0.039
0.038
0.037
0.036
0.035
0.034
0.033
0.032
0.031
0.030
0.029
0.029
0.028
0.014
0.006
0.003
25
2.0
0.052
0.084
0.101
0.109
0.111
0.110
0.107
0.102
0.098
0.093
0.089
0.084
0.080
0.076
0.073
0.069
0.066
0.063
0.061
0.058
0.056
0.054
0.052
0.050
0.048
0.046
0.045
0.043
0.042
0.041
0.039
0.038
0.037
0.036
0.035
0.034
0.033
0.033
0.032
0.031
0.016
0.006
0.003
2.5
0.052
0.086
0.104
0.114
0.118
0.118
0.117
0.114
0.110
0.106
0.102
0.097
0.093
0.089
0.086
0.082
0.079
0.076
0.073
0.070
0.067
0.065
0.063
0.060
0.058
0.056
0.055
0.053
0.051
0.050
0.048
0.047
0.046
0.045
0.043
0.042
0.041
0.040
0.039
0.038
0.020
0.008
0.004
3.0
0.052
0.086
0.106
0.117
0.122
0.124
0.123
0.121
0.119
0.115
0.111
0.108
0.104
0.100
0.096
0.093
0.090
0.086
0.083
0.080
0.078
0.075
0.073
0.070
0.068
0.066
0.064
0.062
0.060
0.059
0.057
0.055
0.054
0.053
0.051
0.050
0.049
0.048
0.047
0.046
0.024
0.010
0.005
3.5
0.052
0.087
0.107
0.119
0.125
0.128
0.128
0.127
0.125
0.122
0.119
0.116
0.112
0.109
0.105
0.102
0.099
0.096
0.093
0.090
0.087
0.084
0.082
0.079
0.077
0.075
0.073
0.071
0.069
0.067
0.065
0.063
0.062
0.060
0.059
0.057
0.056
0.055
0.054
0.052
0.027
0.011
0.006
4.0
0.052
0.087
0.108
0.120
0.127
0.130
0.131
0.131
0.130
0.127
0.125
0.122
0.119
0.116
0.113
0.110
0.107
0.104
0.101
0.098
0.095
0.092
0.090
0.087
0.085
0.083
0.080
0.078
0.076
0.074
0.072
0.071
0.069
0.067
0.066
0.064
0.063
0.061
0.060
0.059
0.031
0.013
0.006
Table 5. (continued)
m
n
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75
10.00
20.00
50.00
100.00
4.5
0.053
0.087
0.109
0.121
0.128
0.132
0.134
0.134
0.133
0.132
0.130
0.127
0.125
0.122
0.119
0.116
0.113
0.110
0.107
0.105
0.102
0.099
0.097
0.094
0.092
0.090
0.087
0.085
0.083
0.081
0.079
0.077
0.076
0.074
0.072
0.071
0.069
0.068
0.066
0.065
0.035
0.014
0.007
5.0
0.053
0.087
0.109
0.122
0.130
0.134
0.136
0.136
0.136
0.135
0.133
0.131
0.129
0.126
0.124
0.121
0.119
0.116
0.113
0.111
0.108
0.106
0.103
0.101
0.098
0.096
0.094
0.092
0.090
0.088
0.086
0.084
0.082
0.080
0.078
0.077
0.075
0.074
0.072
0.071
0.039
0.016
0.008
6.0
0.053
0.088
0.109
0.123
0.131
0.136
0.138
0.139
0.140
0.139
0.138
0.137
0.135
0.133
0.131
0.129
0.127
0.125
0.123
0.120
0.118
0.116
0.113
0.111
0.109
0.107
0.105
0.103
0.101
0.099
0.097
0.095
0.093
0.091
0.089
0.888
0.086
0.085
0.083
0.082
0.046
0.019
0.010
7.0
0.053
0.088
0.110
0.123
0.132
0.137
0.140
0.141
0.142
0.142
0.142
0.141
0.140
0.138
0.137
0.135
0.133
0.131
0.130
0.128
0.126
0.124
0.122
0.120
0.118
0.116
0.114
0.112
0.110
0.108
0.106
0.104
0.102
0.101
0.099
0.097
0.096
0.094
0.092
0.091
0.053
0.022
0.011
8.0
0.053
0.088
0.110
0.124
0.132
0.138
0.141
0.143
0.144
0.144
0.144
0.144
0.143
0.142
0.141
0.139
0.138
0.136
0.135
0.133
0.131
0.130
0.128
0.126
0.124
0.122
0.121
0.119
0.117
0.115
0.114
0.112
0.110
0.108
0.107
0.105
0.104
0.102
0.100
0.099
0.059
0.025
0.013
26
9.0
0.053
0.088
0.110
0.124
0.133
0.138
0.142
0.144
0.145
0.146
0.146
0.145
0.145
0.144
0.143
0.142
0.141
0.140
0.139
0.137
0.136
0.134
0.133
0.131
0.129
0.128
0.126
0.125
0.123
0.121
0.120
0.118
0.117
0.115
0.114
0.112
0.110
0.109
0.107
0.106
0.065
0.028
0.014
10.0
0.053
0.088
0.110
0.124
0.133
0.139
0.142
0.145
0.146
0.147
0.147
0.147
0.147
0.146
0.145
0.145
0.144
0.143
0.142
0.140
0.139
0.138
0.136
0.135
0.134
0.132
0.131
0.129
0.128
0.126
0.125
0.124
0.122
0.121
0.119
0.118
0.116
0.115
0.113
0.112
0.071
0.031
0.016
25.0
0.053
0.088
0.111
0.125
0.134
0.140
0.144
0.147
0.149
0.151
0.152
0.152
0.153
0.153
0.154
0.154
0.154
0.154
0.154
0.154
0.154
0.154
0.154
0.153
0.153
0.153
0.153
0.152
0.152
0.152
0.151
0.151
0.150
0.150
0.150
0.149
0.149
0.148
0.148
0.147
0.124
0.071
0.039
50.0
0.053
0.088
0.111
0.125
0.134
0.140
0.144
0.147
0.150
0.151
0.152
0.153
0.154
0.155
0.155
0.155
0.156
0.156
0.156
0.156
0.156
0.156
0.157
0.157
0.157
0.157
0.157
0.157
0.157
0.156
0.156
0.156
0.156
0.156
0.156
0.156
0.156
0.156
0.156
0.156
0.148
0.113
0.071
100.0
0.053
0.088
0.111
0.125
0.134
0.140
0.145
0.148
0.150
0.151
0.153
0.154
0.154
0.155
0.155
0.156
0.156
0.156
0.157
0.157
0.157
0.157
0.157
0.157
0.158
0.158
0.158
0.158
0.158
0.158
0.158
0.158
0.158
0.158
0.158
0.158
0.158
0.158
0.158
0.158
0.156
0.142
0.113
Table 6. Variation of If (Fox, 1948)*
L/B
1.4
1.6
1.8
Poisson’s ratio μs = 0.30
0.05 0.979 0.981 0.982 0.983 0.984
0.10 0.954 0.958 0.962 0.964 0.966
0.20 0.902 0.911 0.917 0.923 0.927
0.40 0.808 0.823 0.834 0.843 0.851
0.60 0.738 0.754 0.767 0.778 0.788
0.80 0.687 0.703 0.716 0.728 0.738
1.00 0.650 0.665 0.678 0.689 0.700
2.00 0.562 0.571 0.580 0.588 0.596
Poisson’s ratio μs = 0.40
0.05 0.989 0.990 0.991 0.992 0.992
0.10 0.973 0.976 0.978 0.980 0.981
0.20 0.932 0.940 0.945 0.949 0.952
0.40 0.848 0.862 0.872 0.881 0.887
0.60 0.779 0.795 0.808 0.819 0.828
0.80 0.727 0.743 0.757 0.769 0.779
1.00 0.689 0.704 0.718 0.730 0.740
2.00 0.596 0.606 0.615 0.624 0.632
*
Adapted from Bowles (1987)
Df/B
1.0
1.2
2.0
5.0
0.985
0.968
0.930
0.857
0.796
0.747
0.709
0.603
0.990
0.977
0.951
0.899
0.852
0.813
0.780
0.675
0.993
0.982
0.955
0.893
0.836
0.788
0.749
0.640
0.995
0.988
0.970
0.927
0.886
0.849
0.818
0.714
13 ANALYSIS OF MAYNE AND POULOS BASED ON THEORY OF ELASTICITY
Mayne and Poulos (1999) presented an improved formula for calculating the elastic settlement of
foundations. The formula takes into account the rigidity of the foundation, the depth of embedment
of the foundation, the increase in the modulus of elasticity of the soil with depth, and the location
of rigid layers at a limited depth. To use the equation of Mayne and Poulos, one needs to determine
the equivalent diameter Be of a rectangular foundation, or
Be =
4 BL
π
(75)
For circular foundations,
Be = B
(76)
where B = diameter of foundation
Figure 21 shows a foundation with an equivalent diameter Be located at a depth Df below the
ground surface. Let the thickness of the foundation be t and the modulus of elasticity of the
foundation material be Ef. A rigid layer is located at a depth H below the bottom of the foundation.
Figure 21 Mayne and Poulos’ procedure (1999) for settlement calculation.
27
The modulus of elasticity of the compressible soil layer can be given as
E s = Eo + kz
(77)
where k = rate of increase in Es with depth (kN/m2/m)
With the preceding parameters defined, the elastic settlement below the center of the foundation
is
Se =
qBe I G I R I E
(1 − μs2 )
Eo
(78)
⎛
E
H ⎞
⎟
I G = influence factor for the variation of Es with depth = f ⎜⎜ β = o ,
kBe Be ⎟⎠
⎝
IR = foundation rigidity correction factor
IE = foundation embedment correction factor
Figure 22 shows the variation of IG with β = Eo/kBe and H/Be. The foundation rigidity correction
factor can be expressed as
where
IR =
π
+
4
1
⎛
⎜
Ef
4.6 + 10⎜
⎜⎜ E + Be
o
2
⎝
(79)
⎞
⎟ ⎛ 2t ⎞ 3
⎟⎜⎜ ⎟⎟
k ⎟⎟⎝ Be ⎠
⎠
Similarly, the embedment correction factor is
IE = 1−
1
(80)
⎞
⎛B
3.5 exp (1.22 μs − 0.4)⎜ e + 1.6 ⎟
⎟
⎜D
⎠
⎝ f
Figures 23 and 24 show the variation of IR with IE as a function of the terms expressed in Eqs.
(79) and (80).
Figure 22 Variation of IG with β.
28
Figure 23 Variation of IR with KF.
Figure 24 Variation of IE with μs and Df/Be.
14 BERARDI AND LANCELLOTTA’S METHOD
Berardi and Lancellotta (1991) proposed a method to estimate the elastic settlement that takes into
account the variation of the modulus of elasticity of soil with the strain level. This method is also
described by Berardi et al. (1991). According to this procedure,
Se = I s
qB
Es
(81)
where Is = influence factor for a rigid foundation (Tsytovich, 1951)
Es = modulus of elasticity of soil
The variation of Is (Tsytovich, 1951) with Poisson’s ratio μs = 0.15 is given in Table 7.
Table 7. Variation of Is
L/B
1
2
3
5
10
Depth of influence HI /B
0.5
1.0
1.5
2.0
0.35
0.56
0.63
0.69
0.39
0.65
0.76
0.88
0.40
0.67
0.81
0.96
0.41
0.68
0.84
0.99
0.42
0.71
0.89
1.06
29
Using analytical and numberical evaluations, Berardi and Lancellotta (1991) have shown that,
for a circular foundation,
H 25 = (0.8 to 1.3) B
(82)
For plane strain condition (that is, L/B ≥ 10)
H 25 = (1.5 to 1.7 ) H 25(circle)
(83)
where H25 = depth from the bottom of the foundation below which the residual settlement is 25%
of the total settlement
The above implies that H25 ≤ 2.5B for practically all foundations. Thus the depth of influence HI
can be taken to be H25. The modulus of elasticity Es in Eq. (81) can be evaluated as (Janbu, 1963)
⎛ σ ′ + 0.5Δσ ′ ⎞
⎟⎟
Es = K E pa ⎜⎜ o
pa
⎝
⎠
0 .5
(84)
where pa = atmospheric pressure
σ o′ and Δσ' = effective overburden pressure and net effective stress increase due to the
foundation loading, respectively, at a depth B/2 below the foundation
KE = dimensionless modulus number
After reanalyzing the performance of 130 structures foundations on predominantly silica sand as
reported by Burland and Burbidge (1985), Berardi and Lancellotta (1991) obtained the variation of
KE with the relative density Dr at Se/B = 0.1% and KE at varying strain levels. Figures 25 and 26
show the average variation of KE with Dr at Se/B = 0.1% and K E ( Se /B ) / K E ( Se /B =0.1%) with Se/B.
In order to estimate the elastic settlement of the foundation, an iterative procedure is suggested
which can be described as follows:
Figure 25 Variation of KE with Dr and N60 (adapted from Berardi and Lancellotta, 1991).
30
Figure 26 Plot of K E ( Se /B ) / K E ( Se /B=0.1%) with Se/B (adapted from Berardi and Lancellotta, 1991).
1. Determine the variation of the blow count N60 from standard penetration tests within the zone
of influence, that is H25.
2. Determine the corrected blow count (N1)60 as
⎞
⎛
2
⎟⎟
( N1 )60 = N 60 ⎜⎜
⎝ 1 + 0.01σo′ ⎠
(85)
where σ o′ = vertical effective stress in kN/m2
3. Determine the average corrected blow count from standard penetration tests (N1 )60 and hence
the average relative density as
⎛N ⎞
Dr = ⎜⎜ 1 ⎟⎟
⎝ 60 ⎠
0.5
(86)
4. With a known value of Dr, determine K E ( Se /B =0.1%) from Figure 23 and hence Es from Eq. (84)
for Se/B = 0.1%.
5. With the known value of Es (Step 4), the magnitude of Se can be calculated from Eq. (81).
6. If the calculated Se/B is not the same as the assumed value, then use the calculated value of Se/B
from Step 5 and Figure 26 to estimate a revised K E ( Se /B ) . This value can now be used in Eqs.
(84) and (81) to obtain a revised Se. The iterative procedures can be continued until the
assumed and calculated values are the same.
Based on a probabilistic study conducted by Sivakugan and Johnson (2004), the probability of
exceeding 25 mm settlement in the field for various predicted settlement levels using the iteration
procedure of Berardi and Lancellotta (1991) is shown in Table 8. When compared with Table 3,
this shows a promise of improved prediction in elastic settlement.
31
Table 8. Probability of exceeding 25 mm settlement in the field
—procedure of Berardi and Lancellotta (1991)
(based on Sivakugan and Johnson, 2004)
Predicted
settlement
(mm)
1
5
10
15
20
25
30
35
40
Probability of exceeding
25 mm in the field
(%)
6
19
32
43
52
60
66
72
77
15 GENERAL COMMENTS AND CONCLUSIONS
A general review of the major developments over the last sixty years for estimating elastic
settlement of shallow foundations on granular soil is presented. Based on the above review, the
following general observations can be made.
1. Meyerhof’s relationship (1965) is fairly simple to use. It will probably yield predicted
settlements that are 50% higher on the average than those observed in the field. Peck and
Bazaraa’s method (1969) provides results that are almost similar to those obtained from
Meyerhof’s method (1965).
2. Burland and Burbidge’s solution (1985) will provide more reasonable estimations of Se than
those obtained from the solution of Meyerhof (1965). However it will be difficult to determine
the overconsolidation ratio and the preconsolidation pressure for granular soils from field
exploration.
3. The modified strain influence factor diagrams presented by Schmertmann et al. (1978),
Terzaghi et al. (1996), and Lee et al. (2008) will all provide reasonable estimations of the
elastic settlement provided a more realistic value of Es is assumed in the calculation. The author
feels that the empirical relationships for Es provided by Eqs. (42) and (43) are more reasonable.
4. The relationships for Es provided by Eqs. (42) and (43) are based on the field cone penetration
resistance. These equations can be converted to expressions in terms of N60 and D50 (mean
grain size). Figure 27 shows some of the relationships available in the literature. Based on the
data of Burland and Burbidge et al. (1985)
⎛ qc ⎞
⎜⎜ ⎟⎟
⎝ pa ⎠ = 8D 0.305
50
N 60
(87)
Based on the data of Robertson and Campanella (1983) and Seed and DeAlba (1986)
⎛ qc ⎞
⎜⎜ ⎟⎟
⎝ pa ⎠ = 6D 0.228
50
N 60
(88)
Based on the data of Anagnostopoulos et al. (2003)
32
Figure 27 Variation of (qc/pa)/N60 with D50. (a) Adapted from Terzaghi et al. (1996); (b) Adapted from
Anagnostopoulos, 2003).
⎛ qc ⎞
⎜⎜ ⎟⎟
⎝ pa ⎠ = 7.6429 D 0.26
50
N 60
(89)
where pa = atmospheric pressure (same unit as qc)
D50 = mean grain size, in mm.
5. The procedure for developing the load-settlement plot based on pressuremeter tests is a
versatile technique; however, the cost effectiveness should be taken into account.
6. The load-settlement plot based on the L1–L2 method should give good results provided the
values of Es and φ′ are properly chosen.
7. Relationships for elastic settlement using the theory of elasticity will be equally as good as the
other methods, provided a realistic value of Es is adopted. This can be accomplished using the
iteration method suggested by Berardi and Lancellotta (1991). In lieu of that, the Es relationship
given by Terzaghi et al. (1996) can be used.
In his landmark paper in 1927 entitled “The Science of Foundations,” Karl Terzaghi wrote
“Foundation problems, throughout, are of such character that a strictly theoretical mathematical
treatment will always be impossible. The only way to handle them efficiently consists of finding
33
out, first, what has happened on preceding jobs of a similar character; next, the kind of soil on
which the operations were performed; and, finally, why the operations have lead to certain results.
By systematically accumulating such knowledge, the empirical data being well defined by the
results of adequate soil investigations, foundation engineering could be developed into a semiempirical science, . . . .”
What is presented in this paper is a systematic accumulation of knowledge and data over the
past sixty years. In summary, the parameters for comparing settlement prediction methods are
accuracy and reliability. Reliability is the probability that the actual settlement would be less than
that computed by a specific method. In choosing a method for design, it all comes down to keeping
a critical balance between reliability and accuracy which can be difficult at times knowing the nonhomogeneous nature of soil in general. We cannot be over-conservative but, at the same time, not
be accurate. We need to keep in mind what Karl Terzaghi said in the 45th James Forrest Lecture at
the Institute of Civil Engineers in London: “Foundation failures that occur are not longer ‘an act of
God’.”
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