Effect of spatial variability on the bearing capacity of
cement-treated ground
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Citation
Kasama, Kiyonobu, Andrew J. Whittle, and Kouki Zen. “Effect of
Spatial Variability on the Bearing Capacity of Cement-Treated
Ground.” Soils and Foundations 52, no. 4 (August 2012):
600–619. © 2012 The Japanese Geotechnical Society
As Published
http://dx.doi.org/10.1016/j.sandf.2012.07.003
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The Japanese Geotechnical Society
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Fri May 27 02:09:30 EDT 2016
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http://hdl.handle.net/1721.1/91637
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Soils and Foundations 2012;52(4):600–619
The Japanese Geotechnical Society
Soils and Foundations
www.sciencedirect.com
journal homepage: www.elsevier.com/locate/sandf
Effect of spatial variability on the bearing capacity of
cement-treated ground
Kiyonobu Kasamaa,n, Andrew J. Whittleb, Kouki Zena
a
Division of Civil and Structural Engineering, Faculty of Engineering, Kyushu University, Fukuoka, Japan
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
b
Received 15 September 2011; received in revised form 10 April 2012; accepted 1 May 2012
Available online 7 September 2012
Abstract
This paper presents a reliability assessment for the undrained bearing capacity of a surface strip foundation based on the results of a
probabilistic study in which the shear strength and unit weight of cement-treated ground are represented as random fields in Monte Carlo
simulations of undrained stability using numerical limit analyses. The results show how the bearing capacity is related to the coefficient of
variation and correlation length scale in both shear strength and unit weight. Based on the results, the authors propose an overdesign factor,
tolerable percentage of defective core specimens, and resistance factors for LRFD ultimate limit state of surface footings on cement-treated
ground in order to achieve a target reliability index and probability of failure. The proposed method is illustrated through example calculations
based on the spatial variation of unconfined compressive strength measured using a variety of cement-mixing methods from projects in Japan.
& 2012 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Cement stabilization; Bearing capacity; Limit analysis; Monte Carlo method; Probabilistic methods; Cohesive soils
1. Introduction
Cement-mixing techniques such as deep-mixing (DMM;
Terashi and Tanaka, 1981) and pre-mixing (Zen et al., 1992)
methods are becoming widely established for stabilizing soft
soils in applications ranging from the strengthening of weak
foundation soils to the mitigation of liquefaction. Although
there have been significant advances in the equipment and
methods used for cement-mixing, there remains a high degree
of spatial variability in the physical and mechanical properties
of the treated ground (i.e., unit weight, shear strength, etc.).
n
Corresponding author.
E-mail address: kasama@civil.kyushu-u.ac.jp (K. Kasama).
Peer review under responsibility of The Japanese Geotechnical Society.
This spatial variability introduces uncertainties in the design
of foundations on cement-treated ground.
Miyake et al. (1991), Kitazume et al. (2000) and
Bouassida and Porbaha (2004) have presented results of
physical (centrifuge) model tests to measure the bearing
capacity of strip footings founded on arrays of clay-cement
columns with area replacement ratios ranging from 18 to
80%. The papers include details of the interpreted failure
mechanisms and finite element simulations of the loaddeformation response. Broms (2004) has proposed a semiempirical method for interpreting the bearing capacity
based on the unconfined compressive strength of the
deep-mixing columns and local shear failure of the adjacent soft clay. A more rigorous theoretical approach is
presented by Bouassida and Porbaha (2004) based on
upper and lower bound plasticity solutions (based on the
earlier yield design theory developed by Bouassida et al.,
1995). These analyses treat the cement-treated ground as
a homogeneous mass. None of these prior studies has
0038-0806 & 2012 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.sandf.2012.07.003
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
Nomenclture
B
COVc
COVNc
COVqu
COVg
ci
cu
Fo
G
Gi
K
Nc
NcDet
NcL
NcU
NcLi
NcUi
Ncl99%
n
ne
PD
Pf
P[y]
Q
QDet
QL
QU
qu
R
RNc
RNc90%
RNc95%
RNc99%
ri
width of foundation
coefficient of variation of undrained shear
strength
coefficient of variation of bearing capacity
factor
coefficient of variation of unconfined
compressive strength
coefficient of variation of unit weight
undrained shear strength of ith element
undrained shear strength of cement-treated soil
overdesign factor
correlated standard Gaussian field with zero
mean unit variance
local value of standard Gaussian field with zero
mean unit variance for ith element;
correlation matrix
bearing capacity factor
bearing capacity factor of analytical Prandtl
solution
equivalent bearing capacity factor obtained by
LB-NLA
equivalent bearing capacity factor obtained by
UB-NLA
bearing capacity factor for ith realization for
LB-NLA
bearing capacity factor for ith realization for
UB-NLA
bearing capacity factor with 99% lower
confidence level
number of Monte Carlo iterations
total number of elements in the mesh
percent defective
probabilities of failure
probability
the ultimate bearing capacity of cement-treated
ground
ultimate bearing capacity
lower bound estimates of the collapse load
upper bound estimates of the collapse load
unconfined compressive strength
standard Gaussian field with zero mean unit
variance
mNcU/NcDet, reduced mean bearing capacity
ratio
reduced bearing capacity ratio with 90% lower
confidence level
reduced bearing capacity ratio with 95% lower
confidence level
Ncl99%/NcDet, reduced bearing capacity ratio
with 99% lower confidence level
standard Gaussian field with zero mean unit
variance for ith element
S
ST
SR
TPD
xi
aR
bT
g
gi
mc
mln c
mln Nc
mln g
mNc
mNcL
mNcU
mqu
mg
Yln c
y
yv
yh
yr
yo
yln c
yln g
rij
r(xij)
sc
sln c
sln Nc
sln g
sNc
sNcL
sNcU
squ
sg
fR
F(y)
601
upper triangular forms of matrix K
lower triangular forms of matrix K
strength ratio of unconfined compressive
strength between in situ specimen and
laboratory specimen
the tolerance of percent defective
position vector at center of ith element
sensitivity factor variabilities of load component and resistance for a strip foundation
target reliability index
unit weight
unit weight of ith element
mean undrained shear strength
mean log undrained shear strength
mean of log bearing capacity factor
mean log unit weight
mean of bearing capacity factor
mean of bearing capacity factor for LB-NLA
through Monte Carlo simulations
mean of bearing capacity factor for UB-NLA
through Monte Carlo simulations
mean unconfined compressive strength
mean unit weight
yln c/B ¼ yln g/B, normalized correlation length
spatial correlation length
vertical correlation length
horizontal correlation length
radial correlation length
orthogonal correlation length
spatial correlation length for undrained shear
strength
spatial correlation length for unit weight
correlation coefficient between element i and j
correlation coefficient between two random
values separated by a distance xij ¼ 9xi xj9
standard deviation of undrained shear strength
standard deviation of log undrained shear
strength
standard deviation of log bearing capacity
factor
standard deviation of log unit weight
standard deviation of bearing capacity factor
standard deviation of bearing capacity factor
for LB-NLA through Monte Carlo simulations
standard deviation of bearing capacity factor
for UB-NLA through Monte Carlo simulations
standard deviation of unconfined compressive
strength
standard deviation of unit weight
resistance factor for the ultimate limit state of a
strip foundation on the cement-treated ground
Cumulative normal function
602
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
considered the role of spatial variability of the cementtreated ground on the bearing capacity or ultimate limit
state failure modes. Omine et al. (2005) have proposed a
model for estimating the shear strength of cement-treated
soil that combines a statistical weakest link model (for the
brittle ‘soilcrete’) with a bundle model to represent the
level of mixing achieved at different scales. Tang et al.
(2001) have also evaluated the variation of unconfined
compression strength of in situ cement-treated soil due to
the change in water content and cement amount.
In order to evaluate the effects of the spatial variability
of soil parameters on the stability of shallow foundations,
Griffiths and Fenton (2001), Griffiths et al. (2002) and
Popescu et al. (2005) have analyzed the undrained bearing
capacity of strip footings on cohesive soils using displacement-based finite element analyses. In their studies,
undrained shear strength is treated as a random field
characterized by a lognormal distribution and a spatial
correlation length using methods of local area subdivision
(LAS; Fenton and Vanmarcke, 1990) and mid-point
discretization (Baecher and Christian, 2003). Namikawa
and Koseki (2009) investigated the effects of the spatial
correlation of the shear strength on the behavior of fullscale, cement-treated columns using similar random finite
element analyses. Tokunaga et al. (2009) evaluated the
reliability index and safety factor for cement-treated soils
with the estimated failure modes based on the results of a
series of reliability analyses.
This paper presents a reliability assessment for the
bearing capacity of a surface strip foundation on cementtreated ground based on the results of a probabilistic study
in which the shear strength and unit weight of the cementtreated ground are represented as random fields in Monte
Carlo simulations of undrained stability for a surface strip
foundation using numerical limit analyses. The numerical
limit analysis used in this study offer a convenient method
for analyzing undrained stability problems and can readily
be adapted to simulate the effects of spatial variability of
soil parameters of cement-treated ground and natural soil
layers. The originality of the proposed analytical method is
to combine the numerical limit analysis with the random
field theory, which can offer a more convenient and
computationally efficient approach for evaluating effects
of variability in soil strength properties in geotechnical
stability calculations. The results show how the bearing
capacity is related to the coefficient of variation and
correlation length scale in the shear strength and unit
weight of cement-treated ground. Based on the results of
these analyses, we propose a systematic procedure for
selecting the overdesign factor, the tolerable percentage
of defective core specimens and the resistance factor in
LRFD for the bearing capacity of cement-treated ground
to obtain a target reliability index. Finally, the proposed
method is illustrated using spatial variability data from a
range of cement-mixing methods in order to discuss the
assessment of stability, design and quality control from the
view point of the reliability-based design.
2. Spatial variability of cement-treated ground
The main factors influencing the shear strength of the
cement-treated ground include the types and amounts of
binder/cement (e.g., Clough et al., 1981, Kamon and
Katsumi, 1999), physico-chemical properties of the in situ
soil (e.g., Chew et al., 2004), curing conditions (e.g., Consoli
et al., 2000) and effectiveness of the mixing process (e.g.,
Larsson, 2001; Omine et al., 1998). Since there are a number
of influential factors on shear strength, as mentioned above,
the in situ shear strength of cement-treated ground shows a
large degree of spatial variability.
For example, Table 1 summarizes the data of the
unconfined compressive strength from a series of construction projects of cement-treated ground in Japan. In each
case, measurements of unconfined compressive strength,
qu, were obtained from core samples cured in the field.
The mean values of qu range from 100 to 7500 kPa with
coefficients of variation, COVqu ¼ squ/mqu ¼ 0.14–0.99.
These results are consistent with the findings of a recent
review of US deep mixing projects by Navin and Filz
(2005) who report COVqu ¼ 0.17–0.67. This level of
variability is much higher than that expected for the
undrained shear strength of natural clays (e.g., Phoon
and Kulhawy, 1999; Matsuo and Asaoka, 1977).
Although there is quite extensive data for estimating
the coefficient of variation in the unconfined compressive
strength, there is much less information available to
understand the underlying spatial correlation structure.
Table 2 summarizes values of the correlation length, y, (in
both the vertical and horizontal directions) reported in five
separate studies in the literature. Two of these are based on
qu data from installed DMM columns, while two others
use cone penetration data in dredged fills. The study by
Larsson et al. (2005) uses a miniature penetrometer to
evaluate the spatial mixing structure within individual,
exhumed lime-cement columns. The results show that the
vertical correlation length can range from 0.2 to 4.0 m (this
is similar to the range of fluctuation scales quoted for
natural clay deposits). Navin and Filz (2005) find that the
horizontal correlation length is much larger for wet mix
DMM columns (12 m) than for dry-mix (o 3 m), while
Larsson et al. (2005) find a radial correlation length,
yr o 0.15 m within 0.6 m diameter columns. Overall, these
data suggest that the horizontal correlation length for
cement-treated ground is much smaller than for natural
sedimentary soil layers although some studies have found
that the horizontal scale of fluctuation can be an order of
magnitude greater than the vertical scale (e.g., James Bay
marine clay deposits; DeGroot and Baecher, 1993).
In order to examine the difference in shear strength
between in situ cement-treatment and laboratory cement–
soil mixtures, Fig. 1 summarizes the strength ratio (SR) of qu
between in situ specimen obtained from core samples in
cement-treated ground (using the pre-mixing method) and
laboratory specimens prepared with a similar target cement
content. The figure also shows the percentage of defective core
Table 1
Project conditions and unconfined compressive strength data for cement-treated soil from construction projects in Japan.
Depth (m)
Cement type
Cement amount
(kg/m3)
Curing period
(day)
Water content
w (%)
Sample
number
Mean qu (kPa)
COVqu
Reference
NP
NP
BP
NP
BP
Cement milk
140
150
14
135
150
15
28
28, 49
28
60
28–52
–
95–135
110–150
55–110
100–130
90–100
–
176
222
182
26
29
30
2140
3920
3760
3370
3770
790
0.358
0.353
0.440
0.331
0.485
0.290
CDIT (1999a)
Deep mixing method (offshore)
8.0 19.0
9.0 22.0
12.5 36.0
9.0 39.0
1.0 8.0
–
2.0 8.0
0.0 7.0
0.0 10.0
0.0 4.0
1.5 28.2
0.0 36.0
4.0 9.0
0.8 10.0
1.7 9.5
4.0 11.0
0.0 20.5
NP
–
Special cement
NP
NP and BP
NP
BP
BP
BP
BP
74
150
250
–
70–220
100–200
250
150
150
175
105–200
28
28
28
–
–
28, 56
28
28
28
28
110–140
–
100–200
–
–
–
–
–
–
–
54
47
493
36
100
116–157
27–
27
54
36
230
2360
3660
2820
1073–1712
754–1248
7321
7155
4127
4578
954–2631
0.480
0.420
0.140
0.423
0.27–0.39
0.26–0.28
0.24
0.3
0.18
0.26
0.12–0.58
CDIT (1999a)
Kohinata et al. (1995)
Noto et al. (1983)
Tamura et al. (1995)
Peng and But (2009)
Suzuki and Kawamura (2009)
Masuda et al. (2009)
–
–
–
BP
BP
BP
7.5%
3%
4%
91
28
28
6.5
9.4
11.8
32
13
25
661
360
120
0.470
0.990
0.750
CDIT (1999b)
Premixing method
Pneumatic flow mixing
0.0 10.0
–
–
–
–
50
7, 28
28
105
71–180
343
–
296
–
0.37
0.3–0.55
Oota et al. (2009)
Tang et al. (2001)
Deep mixing method (onshore)
Abe et al. (1997)
Hioki et al. (2009)
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
Cement mixing method
Notes: NP, Portland cement; BP, blast furnace slag cement; CDIT, Costal Development Institute of Technology.
603
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
604
Table 2
Summary of data on spatial correlation lengths for cement-treated ground.
Type of ground
Reference strength
COV
yh (m)
yv (m)
Reference
DMM columns
qu
–
0.4–4.0
Honjo (1982)
Cement-mixed dredged fill
Air-transported stabilized dredged fill
DMM columns
CPT
CPT
qu
0.21–0.36 (clay)
0.32–0.40 (sand)
0.114–0.194
–
0.34–0.74 (mean: 0.55)
0.5
0.22–0.74
–
Tang et al. (2001)
Porbaha et al. (1999)
Navin and Filz (2005)
Lime-cement columns
Hand-operated
penetrometer
2.0
–
12.0 (wet mixing)
o3.0 (dry mixing)
yr o 0.15
yo o 0.35
–
Larsson et al. (2005)
o0.6
Notes: yr, correlation length in radial direction; yo, correlation length in orthogonal direction.
Strength ratio SR(=qu in-situ /quo)
100
PDF of SR
0 0.25 0.50
0.25 0.50
0.25 0.50
0.25 0.5
10
1
32.8%
0.1
0.5
1
19.0%
PD=2.3%
4.4%
1.5
2
Overdesign factor Fo
2.5
3
Fig. 1. Strength ratio and observed percentage of defective core specimens as function of the overdesign factor used in pre-mixing construction
projects.
specimens (PD) with qu less than the reference laboratory
strength. The data are reported as functions of an ‘overdesign
factor’, Fo. The value of Fo is used in a conventional design
process to select the cement-mix proportions for achieving a
target value of SR for the cement-treated ground while
overcoming uncertainties associated with the high variability
of the in situ strength (i.e., percent defective). As PD¼ 32.8%
at Fo ¼ 1.0 (no overdesign) in Fig. 1, the bearing capacity
bearing capacity of the foundation is likely to be less than the
target design due to local ground failure and the existence of
weak areas of soilcrete. Conventional practice in soil improvement (e.g., Costal Development Institute of Technology,
1999a,b) uses an overdesign factor Fo = 1.5–2.0 (with a
corresponding reduction in PD, Fig. 1) in order to guarantee
the target bearing capacity even when there is a high degree of
spatial variability in cement-treated ground.
3. Random field numerical limit analyses
3.1. Numerical limit analyses
The numerical limit analyses (NLA) used in this study
are based on 2-D, plane strain linear programming
formulations of the upper bound (UB) and lower bound
(LB) theorems for rigid, perfectly plastic materials presented by Sloan and Kleeman (1995) and Sloan (1988a).
The lower bound analyses (Sloan, 1988a) assume a
linear variation of the unknown stresses (sx, sy, txy) within
each triangular finite element. The formulation differs
from conventional displacement-based finite-element formulations by assigning each node uniquely within an
element, such that the unknown stresses are discontinuous
along adjacent edges between elements. Statically admissible stress fields are generated by satisfying: (i) a set of
linear equality constraints, enforcing static equilibrium
with triangular elements and along stress discontinuities
between the elements and (ii) inequality constraints that
ensure no violation of the linearized material failure
criterion. Tresca yield criterion is used to represent both
the undrained shear strength of the clay and the cohesive
strength of the cement-treated ground. The lower-bound
estimate of the collapse load is then obtained through an
objective function that maximizes the resultant force, Q,
acting on the footing. The linear programming problem is
solved efficiently using a steepest edge active set algorithm
(Sloan, 1988b).
The upper-bound formulation assumes linear variations
in the unknown velocities (ux, uy) within each triangular
finite element. Nodes are unique to each element and
hence, the edges between elements represent planes of
velocity discontinuities. Plastic volume change and shear
distortion can occur within each element as well as along
velocity discontinuities. The kinematic constraints are
defined by the compatibility equations and the condition
of associated flow (based on an appropriate linearization
of the Tresca criterion) within each element and along the
velocity discontinuities between elements. The external
applied load can be expressed as a function of unknown
nodal velocities and plastic multiplier rates. The upperbound on the collapse load can then be formulated as a
linear programming problem, which seeks to minimize the
external applied load using an active set algorithm (after
Sloan and Kleeman, 1995).
Recent numerical formulations of upper and lower
bound limit analyses for rigid perfectly plastic materials,
using finite element discretization and linear or non-linear
programming methods, provide a practical, efficient and
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
accurate method for performing geotechnical stability
calculations. For example, Ukritchon et al. (1998) proposed a solution to the undrained stability of surface
footings on non-homogeneous and layered clay deposits
under the combined effects of vertical, horizontal and
moment loading to a numerical accuracy of 7 5%. One of
the principal advantages of numerical limit analyses is that
the true collapse load is always bracketed by results from
the upper and lower bound calculations. Moreover, the
only parameter used in these NLA is the undrained shear
strength (which can vary linearly within a given soil layer).
Hence, NLA provides a more convenient method of
analyzing stability problems than conventional displacement-based finite element methods which also require the
specification of (elastic) stiffness parameters and simulation of the complete non-linear load-deformation response
up to collapse.
Fig. 2a and b shows typical finite element meshes used in
the current UB and LB analyses respectively for surface
foundations on cement-treated ground. The model considers a soil layer with depth z/B ¼ 2.0, where B is the
width of the surface strip foundation under vertical
loading. The dimension of square mesh divided into
four quarter elements is 0.125B. Previous studies (e.g.,
Ukritchon et al., 1998) have used a high element density
close to the stress singularities at the edges of the footing in
order to achieve more accuracy in lower bound analyses.
The current study computes lower bounds using an uniform mesh in order to ensure comparable accuracy in the
representation of spatially variable soil properties (i.e.,
y/B
0
-1
c u / μc
SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
-2
0
1
2
3
4
5
x/B
y/B
0
-1
c u / μc
SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
0
1
3
2
4
same element size used in LB and UB analyses). Extension
elements are introduced in the LB mesh to ensure that
lower-bound conditions are rigorously satisfied in the far
field. The soil is underlain by a rigid base, while far-field
lateral boundaries of the mesh extend beyond the zone
of all potential failure mechanisms. The analyses assume
full improvement of soils with cement-mixing around the
footing such that the zone of cement-treated ground
extends to the boundary. The current simulations also
assume that both shear strength and unit soil weight of the
cement-treated ground are spatially variable parameters.
The sliding resistance at the soil–foundation interface is
controlled by the shear strength of the cement-treated
ground. Therefore, sliding between soil and foundation
occurs when the shear stress on the soil–foundation interface
is more than the shear strength of cement-treated ground.
The collapse loads for UB and LB meshes as shown in Fig. 2
are represented by an equivalent bearing capacity factors,
NcU ¼ QU/(Bcu) and NcL ¼ QL/(Bcu) respectively, where QU
and QL are the upper or lower bound estimates of the
collapse load. Analyses for uniform clay produce values of
NcU ¼ 5.23 and NcL ¼ 5.00, such the analytical Prandtl
solution (NcDet ¼ 2þ p) is well bounded (i.e., 5.00rNcDet
r5.23) with errors of 72.25%.
3.2. Representation of spatial variability of soil parameter
The effects of inherent spatial variability of soil property
are represented in the analyses by modeling the undrained
shear strength, cu, and unit soil weight, g, of the cementedtreated soil as a homogeneous random field (Vanmarcke,
1984). The undrained shear strength and unit weight are
assumed to have an underlying log-normal distributions
with mean, mc and mg, and standard deviations, sc and sg,
and an isotropic scale of fluctuation (also referred to as the
correlation length), yln c and yln g. The current simulations
assume that correlation length for unit weight, yln g is
similar to that for undrained shear strength,yln c. The use
of the log-normal distribution is predicated by the fact that
cu and g are always positive quantities. Following Griffiths
et al. (2002) the current analyses present results based on
assumed values of the ratio of the correlation length to
footing width, Yln c ¼ yln c/B, which is referred to as a
‘‘normalized correlation length’’ in this paper.
The mean and standard deviation of log cu and log g are
readily derived from sc and mc and sg and mg as follows
(e.g., Baecher and Christian, 2003):
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1Þ
slnc ¼ lnð1þ COVc2 Þ; slng ¼ lnð1 þ COVg2 Þ
mlnc ¼ lnmc 12 s2lnc ;
-2
5
x/B
Fig. 2. Typical finite element mesh used in UB and LB numerical limit
analyses (Yln c ¼ 1.0, COVc ¼ 0.2). (a) FE mesh for upper bound
numerical limit analysis. (b) FE mesh for lower bound numerical limit
analysis.
605
mlng ¼ lnmg 12s2lng
ð2Þ
Spatial variability is incorporated within the numerical
limit analyses (both UB and LB meshes) by assigning the
undrained shear strength and unit weight corresponding to
the ith element
ci ¼ expðmlnc þ slnc UGi Þ
ð3aÞ
gi ¼ expðmlng þ slng UGi Þ
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
ð3bÞ
where Gi is a random variable that is linked to the spatial
correlation length, yln c.
It is assumed that unit weight of ith element, gi is
perfectly correlated with ci, based on experimental findings
that show strong correlation between undrained shear
strength and unit weight of cement-treated soils as presented by Tsuchida et al. (2007) and Kasama et al. (2007).
Noted that the auto-correction function is used to realize
the spatial variability of the soil properties in both
horizontal and vertical directions because the cross correlation between the soil properties in the vertical and
horizontal directions has been not fully clarified in prior
studies.
Values of Gi are obtained using a Cholesky Decomposition technique (CD, e.g., Matthies et al., 1997; Baecher
and Christian, 2003; Kasama et al., 2006; Kasama and
Whittle, 2011) using an isotropic Markov function which
assumes that the correlation decreases exponentially with
distance between two points i, j
2xij
rðxij Þ ¼ exp ð4Þ
ylnc
where r is the correlation coefficient between two random
values of cu and g at any points separated by a distance
xij = 9xi xj9 where xi is the position vector of i (located
at the centroid of element i in the finite element mesh).
Noted that an exponential autocorrelation function is used
to express the covariance structure of cement-treated
ground as experimentally shown by Honjo (1982), Navin
and Filz (2005) and Larsson et al. (2005) although the
influence of autocorrelation function on the variability of
the bearing capacity should be clarified in future study. It
is emphasized that the coordinate at the centroid of the
element is used to represent the spatial variability of soil
properties in this study. This coefficient can be used to
generate a correlation matrix, K, which represents the
correlation coefficient between each of the elements used in
the NLA finite element meshes
3
2
1
r12 r1ne
6r
1
r2ne 7
7
6 12
7
K ¼6
ð5Þ
7
6 ^
^
&
5
4
r1ne r2ne 1
where rij is the correlation coefficient between element i
and j, and ne is the total number of elements in the
mesh.
The matrix K is positive definite and hence, the standard
Cholesky Decomposition algorithm can be used to factor
the matrix into upper and lower triangular forms, S and
ST, respectively:
ð6Þ
ST S ¼ K
The components of ST are specific to a given finite
element mesh (for either UB or LB) and selected value of
the correlation length, yln c.
The vector of correlated random variables, G (i.e., {G1,
G2, y, Gne}, where Gi specifies the random component of
the undrained shear strength and unit weight in element i,
Eq. (3) can then be obtained from the product
G ¼ ST R
ð7Þ
where R is a vector of statistically independent, random
numbers {r1, r2, y, rne} with a standard normal distribution (i.e., with zero mean and unit standard deviation).
The current implementation implicitly uses the distance
between the centroids to define the correlations between
undrained shear strengths and unit weights in adjacent
elements. This is an approximation of the random field,
which involves the integral of the correlation function over
the areas of the two elements. Fig. 3 compares the exact
correlation function (Eq. (4)) with results using a typical
realization obtained using the proposed CD technique for
the FE mesh shown in Fig. 2. The data show good
agreement with the correlation function for intervals as
small as 0.05B, corresponding to the minimum distance
between the centroids of adjacent elements. The results
suggest that the current mesh can provide an adequate
representation for correlation lengths, Yln c Z 0.25.
Values of the random variable vector R are re-generated
for each realization in a set of Monte Carlo simulations.
Fig. 2 illustrates the spatial distribution of shear strength
in the cement-treated ground for one example simulation
with input parameters cu ¼ 100 kPa, COVc ¼ (sc/mc)¼ 0.2
and Yln c ¼ 1.0. The lighter shaded regions indicate areas of
higher shear strength.
Based on the literature review of the variability and
correlation lengths for cement-treated ground (Tables 1
and 2), a parametric study has been performed using the
ranges listed in Table 3. It is noted that input coefficient of
variability of undrained shear strength, COVc, ranges from
0.2 to 1.0 while the input coefficient of variability of unit
weight, COVg, is fixed at 0.1 because the spatial variability
of unit weight of cement-treated ground is less than that of
1.2
: Cholesky Decomposition
: Exact Correlation
1
Correlation coefficient
606
0.8
Θ =2.0
lnc
0.6
1.0
0.4
0.5
0.25
0.2
0
Random
-0.2
0
0.2
0.4
0.6
distance/B
0.8
1
Fig. 3. Comparison of exact correlation relation and realized correlation
obtained by proposed Cholesky Decomposition technique.
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
Table 3
Input parameters for current study.
Parameter
Selected values
Mean undrained shear strength mc
COVc of undrained shear strength
Mean unit weight mg
COVg of unit weight
Ratio of horizontal and vertical correlation lengths
Normalized correlation length Yln c ¼ yln c/B
100 kPa
0.2, 0.4, 0.6, 0.8, 1.0
15.0 kN/m3
0.1
1.0
Random, 0.25, 0.5,
1.0, 2.0, 4.0
1000
Monte Carlo iteration
shear strength reported by Kitazume et al. (2004) and
Kitazume and Takahashi (2008). The horizontal correlation length is identical to the vertical correlation length in
this study. For each set of parameters, a series of 1000
Monte Carlo simulations have been performed.
4. Stochastic bearing capacity
4.1. Failure mechanism against vertical loading
Fig. 4a and b shows the UB failure situations against
vertical loading for the initial UB mesh of Fig. 2a. Each
figure shows the deformed mesh, vectors of the UB
velocity field, zone of plastic shear distortion (dark zones
within the velocity field). It can be seen that the computed
failure mechanisms are not symmetric (with associated
rotation of the footing) and find paths of least resistance,
passing through weaker regions of the cement-treated
ground with active passive rigid body wedges under the
foundation. It can be seen that the failure mechanisms do
not extend below the depth z ¼ B (this applies for all of the
simulations) which suggests that it is critical to examine the
shear strength distribution of the shallow foundation soils
(z r B).
4.2. Stochastic bearing capacity factor
In order to evaluate the stochastic bearing capacity of
cement-treated ground with spatial variability in shear
strength and unit weight, the computed bearing capacity
factor can then be reported for each realization, i, of the
random field, NcUi for UB-NLA and NcLi for LB-NLA.
Hence, the mean, mNcU and mNcL, and standard deviation,
sNcU and sNcL, of the bearing capacity factor are recorded
through each set of Monte Carlo simulations, as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
n
1X
1 X
mNcU ¼
ð8aÞ
NcUi ; sNcU ¼
ðNcUi mNcU Þ2
ni¼1
n1 i ¼ 1
mNcL
n
1X
¼
NcLi ;
ni¼1
sNcL
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
1 X
¼
ðNcLi mNcL Þ2
n1 i ¼ 1
ð8bÞ
Fig. 5 illustrates one set of results for the case with
n ¼ 1000, Yln c ¼ 1.0 and COVc ¼ 0.2, 0.6 and 1.0. The
607
results confirm that the collapse load for any given
realization is well bounded by the UB and LB calculations
and furthermore the mean and standard deviation of NcU
by UB-NLA is always more than those of NcL by LB-NLA
for a given COVc and Yln c. Moreover, the mean and
standard deviation of Nc both become stable within 1000
simulations and hence, reliable statistical interpretation of
the data can be obtained from this set of simulations.
Several studies (e.g., Griffiths et al. 2002; Phoon, 2008)
have performed to determine an appropriate number of
Monte Carlo iteration combining reasonable accuracy of
the results in terms manageable computational efforts for a
large parametric study.
Table 4 summarizes the statistical data for the bearing
capacity factor for all combinations of the input parameters. In all cases the results show mNcU 4 mNcL, while
difference (mNcU mNcL) increases with increasing COVc
and decreasing Yln c. In all cases, except where Yln c is
random, the collapse load is bounded within 7 2.5–20%
showing acceptable accuracy from the numerical limit
analyses. The data also show sNcU 4 sNcL. This latter
result may reflect differences in the upper bound and lower
bound limit analyses. However, it is notable that the
numerical limit analyses generate much smaller coefficients
of variation in bearing capacity than were reported by
Griffiths et al. (2002) from displacement-based finite
element simulations (with LAS representations of the
random fields).
Fig. 6 shows a 25-bin histogram of the bearing capacity
factor from one complete series of Monte Carlo simulations
with n ¼ 1000, Yln c ¼ 1.0 and COVc ¼ 0.4 and 0.8. It is
seen that most of Nci are less than NcDet, suggesting that
weak soil elements have a reducing effect for the bearing
capacity of spatially variable ground, as shown in Fig. 4. In
order to obtain the distribution function of the bearing
capacity factor based on w2 goodness-of-fit tests, Table 4
summarizes w2 statistics for all of the simulations and
confirms that normal or log-normal distribution functions
can be used to characterize the bearing capacity at a 5%
significance level (with acceptance level, w2252122 ½0:05 ¼
33:92).
4.3. Reduction of bearing capacity factor
The role of spatial variability in reducing the expected
bearing capacity can be more conveniently seen in Fig. 7,
which reports the reduced mean bearing capacity ratio
RNc ¼ mNcU/NcDet and mNcL/NcDet, where NcDet is the bearing capacity factor of the analytical Prandtl solution
(NcDet ¼ 5.14). Fig. 7a and b is the results of UB-NLA
and LB-NLA respectively. There are large reductions in
RNc as COVc increases for a given normalized correlation
length, Yln c, while the reduction rate of RNc increases with
decreasing Yln c. It can be characterized that the expected
mean bearing capacity of cement-treated ground for typical
coefficients of variation (COVc ¼ 0.4–0.8, c.f. Table 1) is
50–80% of the deterministic value.
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
608
0
y/B
y/B
0
-1
-1
SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
-2
0
1
2
3
4
SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
-2
5
0
3
4
x/B
Initial mesh and strength distribution
0
1
2
0
1
2
0
1
x/B
Initial mesh and strength distribution
y/B
-1
0
1
2
3
4
(kPa*B): 0.005 0.01 0.015 0.02 0.025 0.03
-2
5
x/B
Displacement vector and dissipated energy
3
4
x/B
Displacement vector and dissipated energy
-1
-1
SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
0
1
2
3
4
SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
-2
5
x/B
Deformed mesh
3
x/B
Deformed mesh
-1
0
1
2
3
4
SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
-2
5
x/B
Initial mesh and strength distribution
3
4
x/B
Initial mesh and strength distribution
-1
5
-1
(kPa*B): 0.005 0.01 0.015 0.02 0.025 0.03
0
1
2
3
4
(kPa*B): 0.005 0.01 0.015 0.02 0.025 0.03
-2
5
0
x/B
Displacement vector and dissipated energy
0
1
2
3
4
x/B
Displacement vector and dissipated energy
5
0
y/B
y/B
2
0
y/B
y/B
0
-1
-1
SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
-2
5
-1
SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
-2
4
0
y/B
y/B
0
-2
5
0
y/B
y/B
0
-2
5
-1
(kPa*B): 0.005 0.01 0.015 0.02 0.025 0.03
-2
2
0
0
y/B
1
0
1
2
3
x/B
Deformed mesh
4
SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
5
-2
0
1
2
3
x/B
Deformed mesh
4
5
Fig. 4. Deformed mesh and plastic failure zone at failure against vertical loading. (a) Yln c ¼ 0.25, COVc ¼ 0.2; (b) Yln c ¼ 0.5, COVc ¼ 0.2; (c) Yln c ¼
1.0, COVc ¼ 0.2; and (d) Yln c ¼ 2.0, COVc ¼ 0.2.
7
6
COVc =0.2
5
0.6
4
3
1.0
2
: Upper bound limit analysis
: Lower bound limit analysis
1
0
609
3
Θ lnc =1.0
Standard Deviation of Bearing
Capacity Factor, Nc
Mean Bearing Capacity Factor, Nc
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
1
10
100
Number of Simulations
Θ lnc =1.0
2.5
: Upper bound limit analysis
: Lower bound limit analysis
2
1.5
1.0
1
0.6
0.5
COV c =0.2
0
1000
1
10
100
Number of Simulations
1000
Fig. 5. Summary statistics of bearing capacity factor as functions of the number of Monte Carlo simulations. (a) Mean bearing capacity factor.
(b) Standard deviation of bearing capacity factor.
Table 4
Bearing capacity factor statistics and goodness of fit results for normal and log-normal distributions.
Yln c
COVc
LB-NLA
UB-NLA
mNc
sNc
w2
mlnNc
slnNc
w2
mNc
sNc
w2
mlnNc
slnNc
w2
Random
0.2
0.4
0.6
0.8
1.0
4.23
3.40
2.71
2.16
1.75
0.058
0.090
0.103
0.104
0.099
1.90
2.75
2.63
2.29
2.11
1.44
1.22
0.99
0.77
0.56
0.014
0.027
0.038
0.048
0.057
1.82
3.38
2.93
2.63
2.72
4.78
4.08
3.40
2.83
2.37
0.074
0.126
0.150
0.157
0.159
1.86
2.15
4.06
3.24
3.14
1.56
1.41
1.22
1.04
0.86
0.016
0.031
0.044
0.055
0.067
4.14
3.15
6.58
5.47
5.05
0.25
0.2
0.4
0.6
0.8
1.0
4.44
3.73
3.08
2.55
2.12
0.154
0.245
0.284
0.293
0.285
1.98
1.53
1.88
2.27
1.58
1.49
1.31
1.12
0.93
0.74
0.035
0.065
0.092
0.115
0.134
2.03
3.20
6.30
8.01
8.96
4.81
4.18
3.54
2.98
2.53
0.191
0.332
0.411
0.442
0.447
7.99
8.78
6.71
7.06
2.04
1.57
1.43
1.26
1.08
0.91
0.040
0.079
0.116
0.147
0.175
7.55
7.68
7.72
7.47
4.89
0.5
0.2
0.4
0.6
0.8
1.0
4.54
3.89
3.28
2.76
2.33
0.249
0.396
0.468
0.491
0.487
2.92
1.46
1.81
1.46
1.75
1.51
1.35
1.18
1.00
0.83
0.055
0.102
0.142
0.177
0.206
3.09
4.60
8.14
15.31
18.77
4.80
4.19
3.58
3.05
2.61
0.294
0.511
0.635
0.686
0.692
2.42
4.15
6.50
4.16
4.40
1.57
1.43
1.26
1.09
0.92
0.061
0.121
0.176
0.222
0.261
2.29
3.52
5.04
5.63
8.43
1.0
0.2
0.4
0.6
0.8
1.0
4.63
4.05
3.48
2.99
2.59
0.340
0.547
0.661
0.710
0.721
2.28
3.21
3.10
3.75
4.60
1.53
1.39
1.23
1.07
0.92
0.073
0.134
0.188
0.234
0.273
2.52
6.98
11.95
27.29
31.87
4.82
4.26
3.70
3.21
2.80
0.394
0.684
0.855
0.932
0.951
2.75
2.72
3.43
3.25
4.11
1.57
1.44
1.28
1.13
0.97
0.082
0.160
0.228
0.285
0.331
2.10
2.79
5.28
9.09
13.01
2.0
0.2
0.4
0.6
0.8
1.0
4.72
4.23
3.74
3.31
2.95
0.375
0.618
0.774
0.863
0.905
2.77
1.98
1.92
3.36
3.38
1.55
1.43
1.30
1.16
1.04
0.079
0.145
0.205
0.256
0.300
3.58
11.01
22.53
25.68
21.77
4.86
4.38
3.91
3.48
3.12
0.429
0.757
0.971
1.090
1.148
1.59
2.98
2.68
4.38
4.97
1.58
1.46
1.33
1.20
1.07
0.088
0.171
0.245
0.306
0.356
2.64
3.25
6.00
10.05
16.06
4.0
0.2
0.4
0.6
0.8
1.0
4.81
4.43
4.04
3.68
3.38
0.346
0.586
0.758
0.876
0.952
2.03
2.30
1.92
3.06
3.42
1.57
1.48
1.38
1.28
1.18
0.072
0.132
0.186
0.235
0.276
5.26
16.33
15.20
26.39
18.37
4.94
4.55
4.17
3.83
3.53
0.389
0.700
0.924
1.073
1.168
1.44
1.30
1.83
2.51
4.25
1.59
1.50
1.40
1.30
1.21
0.079
0.153
0.219
0.275
0.323
2.45
5.88
7.71
15.61
28.34
Note: acceptance criterion w2252122 ½0:05r 33:92.
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
610
0.020
Θlnc=1.0
COVc =0.4
NcDet
PDF of Nc
0.015
UB-NLA
μNc =4.26
COVNc =0.684
LB-NLA
μNc=4.05
COVNc=0.547
0.010
0.005
0.000
Θ lnc=1.0
COV c =0.8
UB-NLA
μNc=3.21
COVNc=0.932
0.015
PDF of Nc
0.020
LB-NLA
μNc=2.99
COVNc =0.710
0.010
0.005
2.3
2.8
3.3 3.7 4.2 4.7 5.2 5.6
Bearing capacity factor Nc
0.000
6.1
NcDet
1
1.7
2.3 3 3.7 4.3 5 5.7
Bearing capacity factor Nc
6.3
100
Reduced bearing capacity ratio,
RNc = Nc
NcDet
Reduced bearing capacity ratio,
RNc = Nc
NcDet
Fig. 6. Histogram of bearing capacity factor. (a) Yln c ¼ 1.0, COVc ¼ 0.4 and (b) Yln c ¼ 1.0, COVc ¼ 0.8.
Mean bearing capacity factor
80
60
Θ lnc
Random
0.25
0.5
1.0
2.0
4.0
40
20
0
0
0.2
0.4
0.6
0.8
Coefficient of variation of cu, COVc
100
1
Mean bearing capacity factor
80
60
Θ lnc
Random
0.25
0.5
1.0
2.0
4.0
40
20
0
0
0.2
0.4
0.6
0.8
Coefficient of variation of cu, COVc
1
100
99% lower confidence bound
80
60
Θ
40
Random
0.25
0.5
1.0
2.0
4.0
20
0
0
R
(μ σ )
0.2
0.4
0.6
0.8
Coefficient of variation of cu, COVc
1
Reduced bearing capacity ratio,
Ncl99%
RNc 99% = N
cDet
Reduced bearing capacity ratio,
Ncl99%
RNc99%= N
cDet
Fig. 7. Reduced bearing capacity ratio against COVc (mNc). (a) UB-NLA and (b) LB-NLA.
100
99% lower confidence bound
80
60
Θ
40
Random
0.25
0.5
1.0
2.0
4.0
20
0
0
R
(μ σ )
0.2
0.4
0.6
0.8
Coefficient of variation of cu, COVc
1
Fig. 8. Reduced bearing capacity ratio against COVc (99% lower confidence bound of N(c). (a) UB-NLA and (b) LB-NLA.
It is well known in most building codes that the nominal
value of the material strength (resistance) is usually defined
as the 5% or 10% fractile of the material strength. In order
to examine nominal values for the bearing capacity, Fig. 8
shows the reduced bearing capacity ratio at the 99% lower
confidence level, RNc99% ¼ Ncl99%/NcDet, where Ncl99% is
estimated by assuming a log-normal distribution with mNc
and sNc (from Table 4). RNc99% shows a large reduction in
small range of COVc irrespective of the normalized
correlation length, Yln c. In the design of a surface strip
foundation, the bearing capacity can be estimated approximately by multiplying NcDet with the characteristic value of
soil strength (which is generally assumed to be a conservative value of the strength given by mc a sc) to
consider the reduction of bearing capacity due to the
spatial variability in shear strength. Fig. 8 shows the
reduced bearing capacity ratio for the case where the
characteristic soil strength is mc sc (i.e. a ¼ 1), RNca ¼ 1.
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
100
80
60
40
Mean bearing capacity factor
20
0
: UB-NLA
: LB-NLA
0
0.5
100
Reduced bearing capacity ratio (%)
Reduced bearing capacity ratio (%)
COV =0.2
0.4
0.6
0.8
1.0
1
1.5
2
2.5
3
3.5
Normalized correlation length lnc
95% lower confidence bound
COV =0.2
80
0.4
60
0.6
0.8
1.0
40
20
: UB-NLA
: LB-NLA
0
0
0.5
1
1.5
2
2.5
3
3.5
Normalized correlation length lnc
90% lower confidence bound
COV =0.2
80
0.4
60
0.6
0.8
1.0
40
20
0
4
: UB-NLA
: LB-NLA
0
0.5
1
1.5
2
2.5
3
3.5
Normalized correlation length lnc
4
100
Reduced bearing capacity ratio (%)
Reduced bearing capacity ratio (%)
100
611
: UB-NLA
: LB-NLA
80
COV =0.2
60
0.4
40
0.6
0.8
1.0
20
99% lower confidence bound
0
4
0
0.5
1
1.5
2
2.5
3
3.5
Normalized correlation length lnc
4
Fig. 9. Reduced bearing capacity ratio against Yln c. (a) Mean bearing capacity factor, mNc; (b) 90% lower bound of Nc; (c) 95% lower bound of Nc; and
(d) 99% lower bound of Nc.
7
7
Target probability of failure,P = 10
6
Overdesign factor Fo
Overdesign factor Fo
6
Θ =1.0
5
2.0
0.5
4.0
0.25
4
3
Random
2
1
Target probability of failure,P = 10
2.0
Θ =1.0
0.5
4.0
5
4
0.25
3
Random
2
0
0.2
0.4
0.6
0.8
1
Coefficient of variation of cu, COVc
1.2
1
0
0.2
0.4
0.6
0.8
1
Coefficient of variation of cu, COVc
1.2
Fig. 11. Overdesign factor Fo and COVc. (a) Target probability of failure, Pf ¼ 10 2 and (b) target probability of failure, Pf ¼ 10 3.
The results show that for COVc r 0.6, RNc99% r RNca ¼ 1
and hence, a 4 1.0 is needed for a conservative estimate of
the nominal bearing capacity.
Fig. 9a–d shows the relationships between the reduced
mean bearing capacity ratio, RNc, and the 90%, 95% and
99% lower confidence bounds (RNc90%, RNc95%, RNc99%,
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
612
100
10-1
10-2
10-3
COVc =1.0
10-4
0.2 0.4
10-5
1
0.6
2
3
4
Overdesign factor Fo
100
0.4
1
0.6
UB-NLA
LB-NLA
0.8
2
3
4
Overdesign factor Fo
100
P[Q < QDet]
P[Q < QDet]
COVc =1.0
5
Θlnc=1.0
10-1
COVc =1.0
10-3
0.8
10-4
0.2
1
0.4
2
3
4
Overdesign factor Fo
10-2
COVc =1.0
10-3
0.2
10-5
5
0.8
0.6
10-4
UB-NLA
LB-NLA
0.6
100
1
UB-NLA
LB-NLA
0.4
2
3
4
Overdesign factor Fo
100
Θlnc =2.0
5
Θlnc =4.0
10-1
P [Q<QDet]
10-1
P[Q<QDet]
10-3
10-5
5
Θlnc=0.5
10-2
10-2
COVc =1.0
10-3
0.6
10-4
0.2
10-5
10-2
0.2
10-1
10-5
Θ lnc=0.25
10-4
UB-NLA
LB-NLA
0.8
Pf by Baecher & Christian
Phoon et al.
10-1
Θlnc=Random
P[Q < QDet]
P[Q<QDet]
100
Pf by Baecher & Christian
Phoon et al.
1
0.8
0.4
2
3
4
Overdesign factor Fo
10-2
COVc =1.0
10-3
0.8
0.6
-4
10
UB-NLA
LB-NLA
0.2
5
10-5
1
0.4
2
3
4
Overdesign factor Fo
UB-NLA
LB-NLA
5
Fig. 10. P[Q o QDet] and overdesign factor Fo. (a) Yln c ¼ random; (b) Yln c ¼ 0.25; (c) Yln c ¼ 0.5; (d) Yln c ¼ 1.0; (e) Yln c ¼ 2.0; and (f) Yln c ¼ 4.0.
respectively) of Nc as functions of Yln c and COVc. The results
for spatially random shear strength are plotted at Yln c ¼ 0
for simplicity. There is a very small difference between the
upper and lower bound solutions for Yln c 4 0.5, and hence
the reductions in bearing capacity are well defined in this
range of correlation lengths. The widening gap between the
solutions for Yln c o 1.0 (i.e., loss of accuracy) reflects the
underlying problem of stochastic discretization that requires
elements to be smaller than the spatial correlation length
(Matthies et al., 1997). Overall, the results show minimum
values of RNc90%, RNc95% and RNc99% at Yln c ¼ 1.0 irrespective of COVc while RNc increases with increasing Yln c.
Qualitatively similar results have been presented by Griffiths
et al. (2002) who also report a local minimum in RNc for
Yln c E 0.5, which is not seen in the current numerical limit
analyses (Fig. 9a). As a result, the current analyses do not
converge to the theoretical limits for the case of spatially
random shear strength.
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
35
Target probability of failure,P = 10
30
25
20
COV
0.2
0.4
0.6
0.8
1.0
Allowable percent defective (%)
Allowable percent defective (%)
35
15
10
5
0
1
2
3
Normalized correlation length, lnc
4
Target probability of failure,P = 10
30
COV
0.2
0.4
0.6
0.8
1.0
25
20
15
10
5
0
0
613
0
1
2
3
Normalized correlation length, lnc
4
Fig. 12. Allowable percent defective and normalized correlation length. (a) Target probability of failure, Pf ¼ 10 2 and (b) target probability of failure,
Pf ¼ 10 3.
5. Reliability assessment
5.1. Overdesign factor and tolerance of percent defective
In the conventional design of a surface strip foundation
with width, B, on cohesive soil, the ultimate bearing capacity
QDet ¼ B mc NcDet, in which the mean undrained shear
strength mc and the bearing capacity factor NcDet are based
on the simplified failure mechanism obtained for a homogeneous soil mass. If the bearing capacity factor Nc for
cement-treated ground is a probabilistic parameter and the
mean undrained shear strength mc is augmented (using the
overdesign factor, Fo) to account for the increase in mean
undrained shear strength due to cement-mixing. The ultimate
bearing capacity Q can then be found from Q¼ B Fo mc Nc. In the current calculations, considering the reduction
in the bearing capacity of cement-treated ground due to the
spatial variability of soil parameter and the increase in mean
undrained shear strength due to cement-mixing, the probability that the bearing capacity of cement-treated ground Q is less
than QDet can be obtained by assuming that Nc of cementtreated ground is described by a log-normal distribution. The
probability P[Q o QDet] is given by
lnð½2þ p=Fo ÞmlnNc
P½Q o QDet ¼ P½Fo Nc o NcDet ¼ F
slnNc
ð9Þ
where F(y) is the cumulative normal function, mln Nc and
sln Nc are mean and standard deviation of ln Nc obtained by
following equations using COVNc ¼ sNc/mNc:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð10Þ
slnNc ¼ lnð1þ COVN2 c Þ
mlnNc ¼ lnmNc 0:5Us2lnNc
ð11Þ
Fig. 10 summarizes P[Q o QDet] for a given Yln c as a
functions of Fo and COVc. The target probabilities of failure
considered in LRFD codes for shallow foundations are
reported in the range, Pf ¼ 10 2–10 3 (Baecher and
Christian, 2003; Phoon et al., 2000). It is noted that the
difference of P[QoQDet] between UB-NLA and LB-NLA
increases as COVc increases. It can be seen that P[QoQDet]
decreases very markedly with increasing Fo irrespective of
COVc, but also increases with increasing COVc for a given
Fo suggesting that an overdesign factor should be properly
determined (depending on COVc) to satisfy the required
probability of failure. In addition, P[QoQDet] depends on
values of Yln c, however, the probability for a given overdesign
factor shows maximum around Yln c=1.0 in current analysis.
Fig. 11a and b illustrates the optimized values of the
overdesign factor, Fo, for mix designs of cemented-treated
grounds (with specified parameters, COVc and Yln c) in
order to satisfy the target probability of bearing failure for
a surface strip footing with Pf ¼ 10 2 and 10 3. These
results use the higher values of Fo from UB and LB
computations shown in Fig. 10. It can be seen that the
largest values of Fo occur for Yln c ¼ 1.0–2.0 and that Fo
always increases with COVc. Typical that cement-treated
ground with a large degree of variability (COVc Z 0.6 and
Yln c ¼ 1.0) will require Fo 4 3.0 to satisfy Pf ¼ 10 2 while
Fo ¼ 1.5–2.5 is appropriate for cement-treated ground
with small spatial variability (COVc ¼ 0.2–0.4 similar to
naturally deposited soils).
The results in Fig. 11 can be re-arranged to focus on the
quality of the in situ shear strength of the cement-treated
ground. Fig. 12a and b shows the Tolerated Percentage of
Defective core specimens TPD needed to achieve Pf ¼ 10 2
and 10 3 as a functions of Yln c and COVc. Noted that
TPD is directly related to Fo as shown in Eq. (9). It can be
seen that Yln c has much more influence on TPD compared
to COVc. The minimum tolerance level, for Pf ¼ 10 2 occurs
at Yln c ¼ 0.5 with TPD E 10%, while results for Pf ¼ 10 3
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
614
1
COVc =0.2
0.8
Resistance factor R
Resistance factor R
1
0.4
0.6
0.6
0.8
0.4
1.0
0.2
0
αR = 0.5
Θlnc= Random
1
1.5
2
2.5
3
3.5
Target reliability index T
1.0
0.2
0
4
Resistance factor R
Resistance factor R
0.8
0.4
αR = 0.5
Θlnc = 0.25
COVc =0.2
0.4
0.6
0.6
0.4
0.8
1.0
0.2
α R= 0.5
1
1.5
1
1.5
0.8
: UB-NLA
: LB-NLA
2
2.5
3
3.5
Target reliability index T
4
COVc =0.2
0.4
0.6
0.6
0.4
0.8
1.0
0.2
αR = 0.5
: UB-NLA
: LB-NLA
Θlnc = 0.5
2
2.5
3
3.5
Target reliability index T
Θlnc = 1.0
0
4
1
1
1.5
: UB-NLA
: LB-NLA
2
2.5
3
3.5
Target reliability index T
4
1
0.8
COVc =0.2
Resistance factor R
Resistance factor R
0.6
1
0.8
0.4
0.6
0.6
0.4
0.8
1.0
0.2
0
0.4
0.6
: UB-NLA
: LB-NLA
1
0
COVc =0.2
0.8
α R= 0.5
1.5
0.4
0.6
0.6
0.8
1.0
0.4
0.2
αR = 0.5
: UB-NLA
: LB-NLA
Θlnc = 2.0
1
COVc =0.2
0.8
2
2.5
3
3.5
Target reliability index T
Θlnc = 4.0
4
0
1
1.5
: UB-NLA
: LB-NLA
2
2.5
3
3.5
Target reliability index T
4
Fig. 13. Resistance factor and target reliability index (aR ¼ 0.5). (a) Yln c ¼ random; (b) Yln c ¼ 0.25; (c) Yln c ¼ 0.5; (d) Yln c ¼ 1.0; (e) Yln c ¼ 2.0; and
(f) Yln c ¼ 4.0.
suggest minimum TPD values in the range 5–6% for
Yln c ¼ 0.5–2.0.
5.2. Resistance factor in LRFD
In order to extend the numerical results in the current
analysis to LRFD code for cement-treated ground, a
resistance factor fR for the ultimate limit state of a surface
strip foundation on cement-treated ground is calculated
assuming that the load component to a strip foundation on
the cement-treated ground is modeled by log-normal
distribution as follows (JGS, 2006):
mNc
1
fR ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðaR UbT UslnNc Þ
N
1þ COVNc
cDet
ð12Þ
where aR is a sensitivity factor to represent the ratio
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
1
COVc =0.2
0.8
0.4
0.6
0.6
0.8
0.4
1.0
0.2
0
α R= 0.8
1.5
2
2.5
3
3.5
Target reliability index T
Resistance factor R
Resistance factor R
0.8
1.0
0.2
0
4
COVc =0.2
0.6
0.4
0.4
0.6
0.2
0.8
1.0
: UB-NLA
: LB-NLA
αR = 0.8
Θlnc = 0.5
αR = 0.8
1
1.5
: UB-NLA
: LB-NLA
2
2.5
3
3.5
Target reliability index T
4
1
1.5
2
2.5
3
3.5
Target reliability index T
0.8
COVc =0.2
0.6
0.4
0.4
0.6
0.2
αR = 0.8
Θlnc = 1.0
0
4
1
1
1.5
0.8
1.0
: UB-NLA
: LB-NLA
2
2.5
3
3.5
Target reliability index T
4
1
0.8
Resistance factor R
Resistance factor R
0.6
0.4
1
0.8
COVc =0.2
0.6
0.4
0.4
0.6
0.2
0.8
1.0
0
0.4
0.6
Θlnc = 0.25
1
0
COVc =0.2
0.8
: UB-NLA
: LB-NLA
Θlnc= Random
1
Resistance factor R
Resistance factor R
1
615
αR = 0.8
Θlnc = 2.0
1
1.5
0.8
COVc =0.2
0.6
0.4
0.6
0.4
0.8
1.0
0.2
αR = 0.8
: UB-NLA
: LB-NLA
2
2.5
3
3.5
Target reliability index T
4
0
Θlnc = 4.0
1
1.5
: UB-NLA
: LB-NLA
2
2.5
3
3.5
Target reliability index T
4
Fig. 14. Resistance factor and target reliability index (aR ¼ 0.8). (a) Yln c ¼ random; (b) Yln c ¼ 0.25; (c) Yln c ¼ 0.5; (d) Yln c ¼ 1.0; (e) Yln c ¼ 2.0; and
(f) Yln c ¼ 4.0.
between variabilities of the load component and resistance,
and bT is a target reliability index. Noted that the nominal
bearing capacity (resistance) in Eq. (12) is assumed to be
equal to NcDet. Figs. 13 and 14 show the resistance factor
fR for aR = 0.5 and 0.8 as functions of the target
reliability index for selected values of COVc and Yln c. It
is seen that the resistance factor fR for Yln c Z 0.25
decreases with increasing target reliability index bT and
the decrease rate of fR increases with increasing Yln c
although fR for Yln c = random is almost constant irrespective of COVc. In addition, the difference of UB-NLA
and LB-NLA decreases as Yln c increases. JGS (2006)
proposed that a resistance factor fR for the ultimate limit
state of a strip foundation on a clayey ground with
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
COVc ¼ 0.3 for bT ¼ 1.5–3.0 ranges 0.77–0.62 and 0.67–
0.47 for aR ¼ 0.5 and 0.8 respectively, which is less than
those obtained from this study.
6. Case study
This section illustrates how spatial variability affects
the ultimate limit state design of shallow foundations
on cement-treated ground. The numerical limit analyses
assume a normalized correlation length Yln c ¼ 0.25,
based on empirical data on spatial correlation lengths
for a range of ground treatment methods (Table 2) and
shallow foundations with widths ranging from 5.0 to
10.0 m.
Fig. 15 summarizes COVqu as functions of the mean
unconfined compressive strength (mqu) for the cementtreated soils presented in Table 1. Although there is a
large scatter in the data associated with different construction methods, the results show that the coefficient of
variability generally decreases with the level of the unconfined compressive strength. The behavior can be well
Coefficient of Variation of qu, COVqu
1
0.8
DMM(offshore)
DMM(onshore)
Premixing Method
Pneumatic Flow Mixing Method
0.6
COVqu =0.51*exp(-0.0022*μqu )+0.32
0.4
0.2
0
0
1000 2000 3000 4000 5000 6000 7000
Mean unconfined compressive strength qu (kPa)
8000
Reduced bearing capacity ratio, RNc
Fig. 15. The relationships between COVqu and mqu.
100
Open: UB-NLA
80
Solid: LB-NLA
60
: Eqn. 13
40
: DMM(offshore)
: DMM(onshore)
: Premixing Method
: Pneumatic Flow Mixing Method
20
0
Θ =0.25
0
1000 2000 3000 4000 5000 6000 7000 8000
Mean unconfined compressive strength qu (kPa)
approximated by an exponential decay regression function
as follows:
COVqu ¼ 0:51Uexpð0:0022Umqu Þþ 0:32
ð13Þ
This function suggests that COVqu is approximately
constant for mqu 4 2000 kPa. Fig. 16a and b compares the
reduced mean bearing capacity ratio (RNc) and 99% lower
confidence level (RNc99%) respectively, against mean unconfined compressive strength from numerical limit analyses
from Fig. 7 using values of COVqu directly from Table 1
and from the correlation given in Eq. (13). These results
confirm that the proposed correlation (Eq. (13)) provides a
good approximation for design. There is a large reduction
in bearing capacity for mqu o1000 kPa but is well bounded
(RNc ¼ 77–86% and RNc99% ¼ 68–73%) for mqu 42000 kPa.
This suggests that the design unconfined compressive
strength for cement-treated ground should be greater than
2000 kPa to prevent a large reduction in bearing capacity
due to the spatial variability.
Fig. 17a–d shows similar comparisons for the LRFD
resistance factor, fR for aR ¼ 0.5 and 0.8 and bT ¼ 2.0
and 3.0. It can be seen that resistance factor increases with
increasing mqu up to 2000 kPa and then remains constant
value for mqu 4 2000 kPa irrespective of aR and bT. The
resistance factor obtained by Eq. (13) for aR ¼ 0.5 ranges
0.72–0.80 for bT ¼ 2.0–3.0 while 0.68–0.77 for aR ¼ 0.8
which seems to be identical to those for clayey ground
proposed by Foye et al. (2006).
Finally, in order to guarantee a target bearing capacity
for cements-treated ground in terms of probability-based
design, Fig. 18 summarizes an overdesign factor Fo for the
design shear strength of cement-treated ground against
mean unconfined compressive strength mqu. It can be seen
that overdesign factor Fo decreases sharply when mqu
o 2000 kPa and then remains constant value around 1.5.
In addition, the difference of overdesign factor between the
target probability of failure, Pf ¼ 10 2 and 10 3 is very
small. An overdesign factor Fo ¼ 1.5 can be recommended
for the cement-treated ground with mean qu 4 2000 kPa,
Reduced bearing capacityratio, RNc99%
616
100
Open: UB-NLA
80
60
Solid: LB-NLA
: Eqn. 13
40
: DMM(offshore)
: DMM(onshore)
: Premixing Method
: Pneumatic Flow Mixing Method
20
0
Θ =0.25
0
1000 2000 3000 4000 5000 6000 7000 8000
Mean unconfined compressive strength qu (kPa)
Fig. 16. Reduced bearing capacity ratio against mean unconfined compressive strength mqu (Yln c ¼ 0.25). (a) Reduced mean bearing capacity ratio, RNc
and (b) 99% lower confidence level, RNc99%.
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
: Eqn. 13
β =2.0, α =0.5, Θ =0.25
0.4
: DMM(offshore)
: DMM(onshore)
: Premixing Method
: Pneumatic Flow Mixing Method
0.2
0
Resistance factor R
1
Resistance factor R
0.6
0
1
Solid: LB-NLA
Open: UB-NLA
0.4
: Eqn. 13
0.2
0.6
0.4
: Eqn. 13
0.2
0
1
Solid: LB-NLA
Open: UB-NLA
0.6
0.8
0
1000 2000 3000 4000 5000 6000 7000 8000
Mean unconfined compressive strength qu
0.8
Solid: LB-NLA
Open: UB-NLA
β =2.0, α =0.8, Θ =0.25
Resistance factor R
Resistance factor R
1
0
1000 2000 3000 4000 5000 6000 7000 8000
Mean unconfined compressive strength qu
1000 2000 3000 4000 5000 6000 7000 8000
Mean unconfined compressive strength qu
Solid: LB-NLA
Open: UB-NLA
0.8
0.6
0.4
: Eqn. 13
0.2
β =3.0, α =0.5, Θ =0.25
0
617
β =3.0, α =0.8, Θ =0.25
0
0
1000 2000 3000 4000 5000 6000 7000 8000
Mean unconfined compressive strength qu
Fig. 17. Resistance factor against mean unconfined compressive strength mqu (Yln c ¼ 0.25). (a) aR ¼ 0.5, bT ¼ 2.0, Yln c ¼ 0.25; (b) aR ¼ 0.8, bT ¼ 2.0,
Yln c ¼ 0.25; (c) aR ¼ 0.5, bT ¼ 3.0, Yln c ¼ 0.25; and (d) aR ¼ 0.8, bT ¼ 3.0, Yln c ¼ 0.25.
4
: DMM(offshore)
: DMM(onshore)
: Premixing Method
: Pneumatic Flow Mixing Method
Overdesign factor Fo
3.5
3
Solid: Target probability of failure, P = 10-3
f
Open: Target probability of failure,Pf = 10-2
2.5
Θlnc=0.25
2
1.5
1
0
1000
2000 3000 4000 5000 6000 7000
Mean unconfined compressive strength qu
8000
Fig. 18. Overdesign factor Fo against mean unconfined compressive strength
mqu (Yln c ¼ 0.25).
while higher values are needed for smaller values of mean
unconfined compressive strength.
7. Conclusions
This paper has presented a reliability assessment for
estimating the bearing capacity of a surface strip
foundation on cement-treated ground using numerical
limit analyses with random field theory and Monte Carlo
simulation. Using these results, we propose values of the
overdesign factors, tolerable percentage of defective core
specimens (TPD) and LRFD resistance factors for the
bearing capacity of cement-treated ground in order to
obtain a target reliability index and probability of failure.
The main conclusions are as follows:
(1) The bearing capacity factor of cement-treated ground
considering the spatial variability of shear strength and
unit soil weight can be characterized by both normal
and log-normal distribution functions with 5% significance level.
(2) The expected mean bearing capacity of cement-treated
ground for a typical coefficient of variation (COVc
¼ 0.4–0.8) is 50–80% of that estimated by assuming
an uniform strength ground. It can be characterized
that the bearing capacity with a lower confidence level
shows the minimum value at normalized correlation
length Yln c ¼ 1.0 irrespective of COVc while mean
bearing capacity increases with increasing Yln c.
(3) Cement-treated ground for COVc Z 0.6 and normalized correlation length Y ¼ 1.0 needs a large overdesign factor Fo over 3.0 although Fo ¼ 1.5–2.5 is
618
K. Kasama et al. / Soils and Foundations 52 (2012) 600–619
appropriate for COVc ¼ 0.2–0.4. The tolerable percentage of defective cores TPD ¼ 10% and 5% are needed
to obtain a probabilities of failure Pf ¼ 10 2 and
10 3, respectively, for a simplified quality management
in in situ strength of cement-treated ground.
(4) For cement-treated ground with mean unconfined compressive strength greater than 2000 kPa, the resistance
factor fR ¼ 0.73–0.76 can be used for surface strip
foundations (with target reliability indices, bT ¼ 2.0–3.0
and sensitivity factors, aT ¼ 0.5–0.8). An overdesign
factor, Fo ¼ 1.5 can be used for the same range of mqu.
Acknowledgment
A grateful acknowledgment is made to Professors Hidetoshi
Ochiai and Noriyuki Yasufuku, Associate Professors Guangqi
Chen and Kiyoshi Omine of Kyushu University for their
helpful advice and encouragement.
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