Engineering Geology 90 (2007) 71 – 88
www.elsevier.com/locate/enggeo
Simplified models for assessing annual liquefaction probability — A
case study of the Yuanlin area, Taiwan
Ya-Fen Lee a , Yun-Yao Chi b,⁎, Der-Her Lee a,c , Charng Hsein Juang d , Jian-Hong Wu a,c
b
a
Department of Civil Engineering, National Cheng Kung University, Tainan 701, Taiwan
Department of Land Management and Development, Chang Jung Christian University, Kway Jen, Tainan 711, Taiwan
c
Sustainable Environment Research Center, National Cheng Kung University, Tainan, Taiwan
d
Department of Civil Engineering, Clemson University, Clemson, SC 29634, USA
Received 14 January 2006; received in revised form 25 October 2006; accepted 18 December 2006
Available online 22 December 2006
Abstract
For management and mitigation of liquefaction hazards on a regional basis, it is generally desirable to evaluate liquefaction
hazards in terms of annual probability of liquefaction (APL). In this study, an approach that combines the knowledge-based,
clustered partitioning technique with the Hasofer–Lind reliability method is developed for estimating the annual liquefaction
probability. Because it is generally difficult to validate the computed annual liquefaction probability, the results obtained from a
modified version of the Davis and Berrill's energy dissipation model are used as a reference. The two models are examined with
borehole data in the Yuanlin, Taiwan area that were investigated shortly after the 1999 Chi-Chi Earthquake. Results of the analysis
reveal that annual probabilities of liquefaction estimated by the two models are consistent with each other and both deemed
reasonable.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Annual probability of liquefaction; Reliability index; Knowledge-based clustered partitioning; Standard penetration test; Energy
dissipation model
1. Introduction
Earthquake-induced soil liquefaction in loose sand
layers often causes settlement and tilting of buildings.
Structural failure caused by liquefaction has been observed in many earthquakes (e.g., the 1964 Niigata,
Japan earthquake, the 1971 Los Angeles, California
earthquake, the 1995 Hyogoken-Nambu, Japan earth-
⁎ Corresponding author. Tel.: +886 6 2785123x2317; fax: +886 6
2785902.
E-mail address: [email protected] (Y.-Y. Chi).
0013-7952/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.enggeo.2006.12.003
quake, the 1999 Kocaeli, Turkey earthquake, and the
1999 Chi-Chi, Taiwan earthquake) (Hamada et al.,
1987; Ishihara, 1993; Japanese Geotechnical Society,
1996; Earthquake Engineering Research Institute, 2000;
Stewart, 2001; Ku et al., 2004). Located in the western
part of the circum-Pacific earthquake belt, Taiwan is the
site of convergence between the Philippine Sea plate and
the Eurasian plate that results in frequent earthquakes of
all magnitudes (Jeng et al., 2002). Furthermore, the
heavily-populated western coastal plain of Taiwan is
underlain by Quaternary alluvium composed of structurally weak silt, sands, and fine gravels that are susceptible to liquefaction. According to field investigation
72
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
reports (Stewart et al., 2000), the areas surrounding the
cities of Miaoli, Taichung, Changhua, Nantou, Yuanlin,
and Chiayi experienced various types of ground failures
as a result of the Chi-Chi earthquake, including sand
boiling, surface rupture, tilting and displacement of
structures, damage to pipelines and destruction of
underground facilities. The damage caused by liquefaction was most serious in the towns of Yuanlin, Nantou,
and Wufen in the 1999 Chi-Chi earthquake.
Many empirical methods for assessing liquefaction
hazard have been developed based on analysis of case
histories of liquefaction and non-liquefaction around the
world. These methods can be divided into three
categories. The first category predicts whether or not
liquefaction will occur with a safety factor calculated
for a given set of earthquake loading and soil properties.
The second category expresses liquefaction potential as
a probability. The third category simulates liquefaction
conditions based on dynamic behavior of the soil. Each
category of methods contributes to the assessment of
liquefaction hazards. However, significant uncertainties
exist in the estimated earthquake loading and soil properties obtained from subsurface exploration. It is therefore desirable to develop a probability-based method for
assessing liquefaction hazards that can: (1) account for
uncertainties in the liquefaction loading and resistance
parameters, (2) assess the reliability of an engineering
design decision, (3) evaluate the degree of liquefaction
hazards in a specific region, (4) facilitate the development of hazard mitigation strategies, and (5) provide a
basis for regulating hazard insurance policy.
Several approaches are available for evaluating liquefaction probability. These approaches include: (1) statistical regression equations (Christian and Swiger, 1975;
Liao et al., 1988; Lay et al., 1990; Cetin et al., 2004),
(2) reliability-based methods (Haldar and Tang, 1979;
Juang et al., 2000), and (3) energy dissipation-based
methods (Davis and Berrill, 1982). In many applications
such as fare calculation in a hazard insurance policy,
annual probability of liquefaction is needed. A realistic
assessment of the annual probability of liquefaction
requires a model that can account for (1) regional
earthquake probability, (2) liquefaction mechanism (as
opposed to an empirical equation obtained solely from
regression analysis of field data), and (3) variation in
earthquake occurrence and soil characterization.
In this study two models are developed for calculating the annual probability of liquefaction for use in
the seismic hazard analysis. The first one, referred to
herein as CL1 model, is based on Hasofer–Lind reliability method (Ditlevsen, 1981). In the Hasofer–Lind
method, generally known as the advanced first order
second moment (AFOSM) method, the reliability index
β is defined as the minimum distance from mean values
of variables to the boundary of the failure region, in the
vector direction of directional standard deviations (Low,
1997). The AFOSM relies on an optimization scheme to
determine the reliability index. Several methods including Lagrange's multiplier (Shinozuka, 1983), polynomial technique (Chowdhury and Xu, 1995) and EXCEL
solver tool (Low, 1997; Low and Tang, 1997) are
available for determining the reliability index. However,
these methods are not well suited for reliability analysis
with a complex limit state function and the need to
process a large amount of data efficiently. In this study a
knowledge-based clustered partitioning technique (Shi
et al., 1999) is used to optimize the objective function
(i.e., to determine reliability index).
The second model, referred to herein as CL2 model,
which is different from the reliability-based model
(CL1) described previously. The CL2 model is based on
energy dissipation models (Davis and Berrill, 1982);
however, it incorporates a limit state search technique
developed by Juang and Chen (2000). Details on these
two models including their formulation are presented
later.
Because of the influence of long recurrence interval
for strong earthquakes, the probability of annual
liquefaction is usually low (about 10− 2 to 10− 4) and
generally difficult to verify. Davis and Berrill (1982)
estimated the return period of liquefaction in the South
Market site due to re-rupture of the San Andreas Fault at
123 years and indicated that their estimation was
supported fairly by historical records, including lack
of liquefaction record in the 1868 earthquake and the
presence of liquefaction in the 1906 earthquake.
However, their approach is not suitable for the study
area, the town of Yuanlin, Taiwan, because of the
absence of sufficiently long historical records.
It should be noted that the annual probability of
liquefaction calculated at a given site is the “prediction”
for the future events, and the case histories in the
available database are observations of liquefaction or
non-liquefaction in the past earthquakes where the
actual probabilities were either 1 or 0. Thus, direct
validation of the computed annual probability of liquefaction is not possible using case histories. Nevertheless,
indirect verification of the computed annual probability
of liquefaction might be achieved by observing the
consistency and reasonableness of the results. This
approach of indirect verification is adopted for examining the two models, CL1 and CL2, based on the data
from 26 boreholes in Yuanlin, Taiwan that were
investigated shortly after the 1999 Chi-Chi earthquake.
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
2. An overview of elements related to proposed models
2.1. Annual probability of liquefaction
Liquefaction or cyclic mobility occurs when excess
pore water pressure caused by an earthquake is equal in
magnitude to the effective overburden stress. Based on
the assumption suggested by Nasser and Shokooh
(1979) that pore water pressure is directly related to
the amount of seismic energy dissipated in the soil,
Davis and Berrill (1982) developed a seismic energy
dissipation model to calculate the annual liquefaction
probability. By incorporating the limit state search
technique developed by Juang and Chen (2000), a
modified energy-based model for annual liquefaction
probability is developed in this paper with additional
data from the 1999 Chi-Chi, Taiwan earthquake.
2.2. Variability of seismic and soil parameters
Variability in the parameters of geotechnical materials
has long been studied (e.g. Harr, 1987; Phoon and
Kulhawy, 1999; Baecher and Christian, 2003). Sources of
variability include inherent variability in the materials and
testing error. In addition, variability in earthquake loading
is very high in most areas of interest, as there are too few
events in the historical record. Ground accelerations are
influenced by such factors as earthquake magnitude,
distance from the epicenter, and local geology. Generally
speaking, variability of earthquake parameters is greater
than that of soil properties for the evaluation of annual
liquefaction probability of a site or an area. Variability in
parameters may be determined through statistical analysis
and characterized in terms of standard deviation (σ) or
coefficient of variation (Ω). The variability in parameters
is local in nature; in other words, the same parameter in
different areas could have different degrees of variability.
Therefore, in order to correctly characterize the liquefaction hazard in a given area, liquefaction probability should
be calculated based on the local variability in the input
parameters. However, because variability data could be
lacking in the area of interest, the coefficients of variation
of the parameters from other areas reported in the
literature may be employed in a preliminary analysis,
and the calculated probability could be “updated” if
additional local information becomes available.
73
reliability index. In this paper, a knowledge-based
clustered partitioning technique (Shi et al., 1999) is
employed for the optimization task. This technique is
very efficient although it is conceptually more complex
than other optimization techniques. First, the most
promising region for a given set of restricted conditions
is determined and separated into sub-regions based on
desired features of the study or the characteristics of the
objective function. Second, random sampling is carried
out in each of the sub-regions, with each point in the
sub-region having a positive probability of being
selected. Third, an estimate of the promising index for
each sub-region is calculated and compared to each
other. The most promising index within sub-regions can
then be obtained. Finally, if more than one region is
equally promising, the sub-region with the highest
promising index value is selected as the most promising
region in the next iteration. On the other hand, if the
highest promising index falls within another sub-region,
the program backtracks to the preceding level to resample, calculate, and compare indices.
2.4. Liquefaction limit state
In a reliability analysis of soil liquefaction potential,
it is necessary to define a limit state that separates
liquefaction from non-liquefaction. Davis and Berrill
(1982) defined a limit state based on statistical analysis
of 57 case histories of liquefaction and non-liquefaction. Juang and Chen (2000) used an artificial neural
network to search for the liquefaction limit state
(Fig. 1). In this paper, for the CL1 model, the boundary
curve in the Standard Penetration Test (SPT)-based
simplified method recommended by Youd et al. (2001)
is adopted as the limit state. For the CL2 model, which
is a modified energy-based model, the limit state is
2.3. Knowledge-based clustered partitioning technique
for optimization
The AFOSM requires an optimization process to find
the shortest distance to the limit state surface or the
Fig. 1. Conceptual model for searching for points on limit state surface
(Juang and Chen, 2000).
74
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
defined by the search technique developed by Juang
and Chen (2000).
2.5. Hasofer–Lind reliability index
The Hasofer–Lind second moment reliability index
β is defined as (Hasofer and Lind, 1974; Ditlevsen,
1981):
b ¼ min
X aF
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðX −mÞT H −1 ðX −mÞ
ð1Þ
where X = the vector representing a set of random
variables Xi; m = the mean values matrix of Xi; H = the
covariance matrix of Xi; F = the failure region (corresponding to the domain in which factor of safety,
FS ≤ 1).
Using the scenario where the limit state is defined
with two random variables as an example, the Hasofer–
Lind reliability index can be illustrated with Fig. 2. Low
(1997) interpreted the Hasofer–Lind reliability index β
as the ratio of the size of the dispersion ellipsoid that
touches the failure surface to the size of 1-σ dispersion
ellipsoid. Low (1997) and Low and Tang (1997)
demonstrated this solution approach using the Excel
Solver. As noted previously, several other algorithms
such as Lagrange's multiplier (Shinozuka, 1983) and
polynomial technique (Chowdhury and Xu, 1995) are
also available. In this paper, the knowledge-based
clustered partitioning technique, a very efficient but
less well-known technique, is adopted. This technique is
well suited for processing a large number of cases at
once (as opposed to case-by-case solution using the
Excel Solver). The computer code that implements this
optimization technique for calculating the reliability
index and conditional probability of liquefaction is
available to the interested reader upon request.
2.6. Seismic energy dissipation theory
Seismic loading applied to soils is a matter of energy
release, including the distance from the point of energy
release and the amount of energy released. In addition, it
is affected by inherent soil conditions and geology of the
site. Davis and Berrill (1982) developed the relationship
between the increase in excess pore pressure and energy
dissipation as follows:
Du ¼
CðN1 Þ
pffiffiffiffiffiffi 101:5M
R2 r0V
ð2Þ
where Δu = excess pore pressure (kPa); M = earthquake
magnitude; R = distance from the epicenter (km); σ0′ =
effective overburden stress (kPa); N1 = corrected standard penetration blow count as per Liao and Whitman
(1986), which is commonly denoted as (N1)60; and C(N1)
is a function of N1. Based on statistical regression of
liquefaction case histories, Davis and Berrill (1982)
expressed the function C(N1) as:
CðN1 Þ ¼
450
N12
ð3Þ
A general equation for the developed excess pore water
pressure may be expressed as:
Du
Ad101:5M
¼ f ðM; R; N1 ; r0VÞ ¼ B C VD
r0V
R N1 r0
ð4Þ
where f (.) is a dimensionless term (Law et al., 1990).
By definition, when the excess pore pressure is equal
to the effective overburden stress (i.e.,Δu / σ0′ = 1), the
liquefaction will occur. This condition (or state) is
essentially the initiation of liquefaction. Eq. (4) can be
rewritten as:
1:5M ¼ logð1=AÞ þ B log R þ C log N1 þ D logðr0VÞ
ð5Þ
Fig. 2. Illustration of the reliability index β in two-dimensional setting
(Low, 1997).
where A, B, C, and D are unknown regression coefficients that may be calibrated using case histories of
liquefaction where the surface manifestation was
observed.
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
75
According to Gutenberg and Richter (1954), the relationship between the total annual number of earthquakes
(T ) and the number (n) exceeding a specific earthquake
magnitude (m) can be represented by:
3. Development of the models
logðnÞ ¼ logðT Þ−bm;
For development of the proposed models, 40 case
histories compiled by the writers from the 1999 Chi-Chi
earthquake (Table 1) and 90 cases reported by
Tokimatsu and Yoshimi (1983) are used. The data
taken from Tokimatsu and Yoshimi (1983) are not
repeated here. The depths at which the cases were
reported range from 1.3 m to 16 m. The corrected
standard penetration blow count N1 ranges from 1 to
about 27, and the fines content in percent ranges from 0
to 99. The vertical effective and total stresses in kPa are
in the ranges of 17 to 322, and 24 to 329, respectively.
The peak ground surface acceleration amax ranges from
0.1 g to 0.5 g, and the earthquake magnitude ranges
from 6.0 to 7.6.
ð6Þ
where log(T ) = a, which means that T = 10a. Both parameters a and b are specific to local seismicity and may be
determined from records of past earthquakes. The probability of all magnitudes greater than a specific earthquake magnitude (m) can be defined by:
n
P½M zm ¼ ¼ 10−bm
ð7Þ
T
Substituting Eqs. (4) into (7), the annual liquefaction
probability can be obtained as follows:
P t Du=r0Vz1b ¼ P t M zð2=3ÞlogðRB N1C r0VD =AÞb
¼ ½RB N1C r0VD =A−3b
2
ð8Þ
3.2. Development of the CL1 model
According to the hypothesis of cumulative liquefaction
probability (Davis and Berrill, 1982), the probability of
earthquake occurrence at every location along the length
of the fault is uniform and the cumulative annual
liquefaction probability of every fault can be calculated
as follows:
−ð2b
3Þ
p
C
VD
p½ðDu=r0VÞz1 ¼ ðN1 Þðr0 Þ=A
L
CðgÞ
;
2
½
ð2Rmid
3.1. Description of database of case histories
gþ1
Þ C
2
ð9Þ
g
where γ = (2/ 3)Bb − 1 and Γ(γ) is a gamma function. L =
fault length (km); Rmid = distance from the middle of the
fault on the surface to a specific site. If the earthquake
occurrences along a fault have a random Poisson distribution, the annual liquefaction probability in soils can be
expressed by:
Pf ¼ 1−expð−T P t Du=r0Vz1b Þ;
ð10Þ
Substituting Eqs. (9) into (10), the following equation is
obtained:
( )
T C VD −23b
CðgÞ
Pf ¼1−exp −k
N1 r0 =A d
2 ;
L
ð2Rmid Þg C gþ1
2
ð11Þ
where T / L = number of earthquake occurrences per unit
fault length.
The proposed CL1 model combines the HasoferLind second moment reliability index with the
knowledge-based clustered partitioning technique to
calculate the conditional probability of liquefaction.
With the Hasofer–Lind approach, the evaluation of the
second moment reliability index may be treated as a
problem of linear programming; every parameter or
random variable Xi can vary within a given range:
Xi ¼ mi þ Ki ri ;
i ¼ 0 to 5
ð12Þ
where mi and σi are the mean and standard deviation of
Xi respectively; Ki is a coefficient and can vary
randomly within the probable range of Xi, from + ∞
to − ∞. Six variables, including earthquake magnitude
(M ), peak ground acceleration (amax), depth to
groundwater table (HWT), SPT blow count (N), fines
content (FC) and saturated soil unit weight (γt), are
generally recognized as the major variables that
influence liquefaction potential. In this paper, they are
treated as random variables in the analysis of
liquefaction probability. Thus, the vector X in Eq.
(12) consists of six random variables, M, amax, HWT, N,
FC and γt. Table 2 shows the typical coefficients of
variation (Ω) for these variables. It should be noted that
the variability in these parameters is local in nature; in
other words, different areas have different degrees of
variability. Therefore, use of local parameter variability
is preferred in the reliability analysis for liquefaction
probability. Because of the lack of variability data,
however, the coefficient of variation of each input
76
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
Table 1
Case histories of liquefaction in Yuanlin, Taiwan in the 1999 Chi-Chi earthquake
Number
Hole
number
Sample
number
Borehole
depth (m)
Groundwater
depth (m)
SPT-N
Soil unit weight
(t/m3)
Fines content
(%)
Liquefied?
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Bh-14
Bh-18
Bh-18
Bh-18
Bh-26
Bh-26
Bh-26
Bh-26
Bh-26
Bh-26
Bh-26
Bh-26
Bh-27
Bh-27
Bh-27
Bh-27
Bh-27
Bh-27
Bh-27
Bh-27
Bh-28
Bh-29
Bh-29
Bh-29
Bh-29
Bh-29
Bh-29
Bh-30
Bh-30
Bh-30
Bh-30
Bh-35
Bh-35
Bh-39
Bh-39
Bh-45
Bh-47
Bh-47
Bh-47
Bh-47
S-2
S-1
S-2
S-3
S-3
S-4
S-5
S-6
S-7
S-8
S-9
S-10
S-3
S-4
S-5
S-6
S-7
S-8
S-9
S-10
S-4
S-1
S-2
S-3
S-7
S-8
S-9
S-1
S-2
S-3
S-4
S-2
S-3
S-1
S-2
S-4
S-3
S-4
S-5
S-6
2.78
1.28
2.78
4.28
4.28
5.78
7.28
8.78
10.28
11.78
13.48
14.78
4.28
5.78
7.28
8.78
10.28
11.78
13.48
14.78
6.00
2.28
4.28
5.78
11.78
13.28
14.78
2.28
3.78
5.28
6.78
2.78
4.28
1.38
4.78
5.78
4.28
6.78
8.78
10.28
1.60
0.60
0.60
0.60
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
2.30
2.00
2.00
2.00
2.00
2.00
2.00
1.10
1.10
1.10
1.10
2.30
2.30
0.20
0.20
1.30
2.30
2.30
2.30
2.30
3
3.5
3
3
6
5
9
9
9
10
10
8
3
3
5
12
13
14
14
12
4.5
6
3
4
21
24
26
4
6
6
7
6
9
13
4
4
3
4.5
5.5
14.5
2.02
1.86
1.93
1.87
1.95
1.97
2.02
2
1.95
1.85
1.89
1.99
2.02
1.98
1.89
2.01
2.22
1.89
1.98
1.99
1.89
2.14
1.9
1.95
2.09
2.22
2.23
1.92
1.99
2.04
2.17
2.03
2.08
2.52
2.31
1.85
1.84
1.9
1.82
2.07
27
54
47
98
78
82
60
65
98
94
80
44
76
47
93
20
9
15
5
3
46
15
99
54
18
7
20
59
17
61
15
90
26
16
35
86
84
99
94
23
N
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
N
N
N
N
N
N
N
N
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Table 2
Coefficients of variation for input variables in the liquefaction analysis
Parameters
Ω
References
Adopted value
Peak ground acceleration (amax)
Depth to groundwater (HWT)
Mean grain size (D50)
SPT N-value
0.51–0.84
0.2
0.12
0.26
0.06–0.5
0.15–0.45
0.05
–
0.03
Haldar and Tang (1979), Haldar and Miller (1984)
0.3⁎
0.2
0.12
0.2
Magnitude (Mw)
Fines content (FC)
Soil unit weight (γt)
Harr (1987)
Elton and Tarik (1990)
Duncan (2000)
Juang et al. (1999)
Same as D50
Harr (1987)
0.05
0.12
0.03
⁎ Uncertainty in the peak ground acceleration (amax) at a site in a future event considering uncertain seismic source and return period would be much
higher. However, the uncertainty of amax for a case history where amax was derived from a calibrated local attenuation relationship could be smaller.
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
variable in each case in the database is assumed based
on those reported in the literature. Thus, the coefficients
of variation of the six input variables,ΩM, Ωamax, ΩHWT,
ΩN, ΩFC, and Ωγt are assumed to be 0.05, 0.3, 0.2, 0.2,
0.12, and 0.03, respectively, as shown in Table 2. This
set of coefficients is used herein as an example to
illustrate the model development.
The six coefficients, K0 through K5, in Eq. (12) are
continuous random variables. Optimization with six
continuous random variables is very time-consuming
with traditional algorithms. In this paper, this sixdimensional space is “transformed” into a polar
coordinate system with one length L and three angles
θ1, θ2, and θ3 as follows:
u0 ¼ L sinðh1 Þ sinðh2 Þ sinðh3 Þ
u1 ¼ L sinðh1 Þ sinðh2 Þ cosðh3 Þ
u2 ¼ L sinðh1 Þ cosðh2 Þ sinðh3 Þ
u3 ¼ L sinðh1 Þ cosðh2 Þ cosðh3 Þ
u4 ¼ L cosðh1 Þ sinðh2 Þ
u5 ¼ L cosðh1 Þ cosðh2 Þ
ð13Þ
where ui (i = 0, 5) are the Ki values in the transformed
space. The angles (θ1, θ2, and θ3) and the length (L) are
the unknowns that are to be determined. Thus, the
optimization that is supposed to involve the six variables
in the original space now becomes one that involves four
basic variables (θ1, θ2, θ3, and L). If all the “points” ui
(i = 0, 5) on the limit state surface are found and analyzed,
the shortest distance to the origin, L, is the reliability
index that is to be determined.
To convert into the polar coordinate system, the
term Ki in Eq. (12) may be replaced by any ui but
each ui can only be selected once. Thus, the first
coefficient K0 can take its value from six possibilities.
For the second through sixth coefficients (K1 through
K5), the number of possibilities reduces to five, four,
three, two and one, respectively. Therefore, a total of
720 (i.e., 6!) combinations are possible, and repeating
the analysis 720 times would be a time consuming
process. In this study, the knowledge-based cluster
partitioning technique (Shi et al., 1999) is utilized to
reduce the total number of combinations through four
main steps, including (1) partitioning, (2) random
sampling, (3) calculation of the length, and (4)
backtracking. These four steps are followed by two
additional steps for the determination of the annual
77
liquefaction probability. A brief description of each
step follows:
3.2.1. Step 1 — partitioning
To implement Eq. (13), ui may be divided into three
partitions as shown in Table 3. Among them, u0 and u1
may be clustered in the same partition because of the
similarity of the function. Similarly, u2 and u3, and u4
and u5, may be clustered into the same partition. Within
a partition, the order of ui is not important; for example,
[u0, u1] and [u1, u0] are considered the same alternative.
3.2.2. Step 2 — random sampling
Since Ki in Eq. (12) is to be replaced by ui, the six
coefficients Ki (i= 0, 5) may be separated into three partitions, similar to the partitioning of the six coefficients ui.
Thus, in each sampling, two ui values are chosen at a time,
and are then assigned to the corresponding Ki values.
As shown in Fig. 3, the sampling process is carried
out in three stages similar to Table 3 with three partitions.
Sampling starts with K0 and K1, and the corresponding
ui are chosen according to the partitioning principle
established in Section 3.2.1: Step 1. Thus, there are 15
possible combinations (C26 = 15) to assign K0 and K1.
Next, sampling with K2 and K3 is carried out and the
corresponding ui can be determined, which has 6
possible combinations. Finally, sampling with K4 and
K5 is carried out and there is only one alternative. The
number of all possible combinations in assigning the
values of the six variables through partition is 90,
resulting from 15 × 6 × 1 = 90. This number is far less than
720, the total number of combinations without partitioning. The result shows that the knowledge-based cluster
partition is an effective and systematic means to reduce
Table 3
List of three partitions of six dimensions
Partition
Coefficients
Remarks
1
u0 = L × sin(θ1) ×
sin(θ2) × sin(θ3)
0° ≤ θ1 ≤ 180°;
0° ≤ θ2 ≤ 180°;
0° ≤ θ3 b 360°;
2
3
u1 = L × sin(θ1) ×
sin(θ2) × cos(θ3)
u2 = L × sin(θ1) ×
cos(θ2) × sin(θ3)
u3 = L × sin(θ1) ×
cos(θ2) × cos(θ3)
u4 = L × cos(θ1) ×
sin(θ2)
u5 = L × cos(θ1) ×
cos(θ2)
L represents the length(equivalent to
reliability index β); ui are random
coefficients, i = 0 to 5.
78
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
Fig. 3. An architecture of knowledge-based clustered partitioning.
the computational time as the number of combinations is
greatly reduced.
3.2.3. Step 3 — calculation of the length L (reliability
index)
As shown in Fig. 4, a searching procedure that involves 3 levels of search for the length L and 2 levels of
search for the angles is adopted. The starting values of
the angles θ1, θ2, and θ3 are all setpto
ffiffiffi 0. The starting
value of the length L is set to L ¼ n, where
pffiffinffi is the
number of variables (thus, in this case, L ¼ 6). This
starting L is adequate for the problem at hand, although
other values may be used. In the search, described later,
the initial increment in angles θ1, θ2, and θ3 is set to be
45° and the change in length L is based on the bisection
method. After each sampling, the coefficients K i
(determined from Eq. (13)) should be checked to ensure
the corresponding Xi values (from Eq. (12)) are within
the pre-set bounds. The sampling process is repeated
until a set of L, θ1, θ2, and θ3 is produced that yields a
satisfactory Xi.
The ranges for the earthquake magnitude, peak
ground acceleration, and SPT N-value are assumed to
be 4 to 9, 0 to 1.5 g and 0 to 30, respectively. The depth to
groundwater table is search within the range of 0 to 20 m
since the analysis of liquefaction potential proceeds only
to a depth of 20 m. The range of FC (fines content) is set
to be 0 to 100% since FC of up to 95% has been recorded
for liquefied soils (Ishihara et al., 1993). Soil unit
weights are searched in the range of 9.8 to 28.4 kN/m3,
which is based on the actual cases in the database.
The objective in the search (optimization) is to find the
minimum L value, denoted as Lmin, which satisfies all
restriction conditions. As shown in Fig. 4, the factor of
safety (FS) against the occurrence of liquefaction for a
satisfactory sample is calculated using the SPT-based
simplified procedure (Youd et al., 2001). Depending on
the calculated FS, the length Lmin will be increased or
decreased. Initially, the range of Lmin is set to be [a, b] =
[0,L]. If FS ≤ 1, it suggests that Lmin falls in the range [0,
L / 2], and in this situation, set b = L / 2, and repeat the
process. Otherwise, Lmin falls in the range [L / 2, L];
accordingly, set a = L / 2, and repeat the process until |a −
b| b ε1 = 0.0001. In the end, Lmin = (a + b) / 2.
After Lmin is determined for a given set of angles (θ1,
θ2, θ3), the process is repeated for all other sets of (θ1,
θ2, θ3). This process is carried out by changing one
angle at a time (changing θ1 first, then θ2 and finally
θ3), as illustrated in Fig. 4. In each angle change, the
increment is set to be half of the previous angle
increment. With each new set θ1, θ2, and θ3, the process
of determining Lmin through the bisection method based
on the calculated FS is repeated. In the end, a new Lmin
is determined for the new set of angles. This process is
repeated until the angle increment is less than ε2 = 1° in
all the three angles. The minimum value of all the Lmin
values obtained for all possible sets of angles is the
optimum solution of the minimum length at this stage. It
should be noted that in the search process, another
stopping criterion of L N 5, θ1 N180°, θ2 N 180°, and
θ3 N 360° is implemented. These restrictions on the three
angles are placed at their upper bounds, and the
restriction of L N 5 is selected because the probability
is approaching zero if the reliability index is greater than
5. The entire process described previously is repeated
for all possible Ki (or ui) combinations (Fig. 3).
3.2.4. Step 4 — backtracking
If the random variable Xi does not fall in the pre-set
range in a certain sampling, a new combination of ui has
to be created based on new partition, and the above steps
need to be repeated.
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
Fig. 4. Flowchart of the procedure in solving for reliability index.
79
80
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
3.2.5. Step 5 — determination of minimum L and
probability
The smallest value of all the minimum L values
obtained for all combinations of ui is taken as the
Hasofer–Lind reliability index. If the central factor of
safety, the one calculated with “mean” parameter
values, is greater than one, the center (m1, m2) of the
ellipsoid in Fig. 2 lies in the safe region and the
conditional probability of liquefaction is PCL = 1 − ϕ(β ).
If the central factor of safety is less than one, the center
(m1, m2) of the ellipsoid lies in the unsafe region and
the conditional probability of liquefaction is calculated
as: PCL = ϕ(β ).
3.2.6. Step 6 — determination of annual liquefaction
probability
The probability determined by the reliability index
obtained with the procedure illustrated in Fig. 4 is the
conditional probability for a specific earthquake. According to the concept of fault rupture mode that is used in the
evaluation of seismic hazard, every point on a fault may
be the focus of an earthquake. At every site, there is a
different peak acceleration that depends on the distance
from the epicenter. Thus, the effect of distance of a
specific site from the epicenter must be taken into consideration when calculating annual liquefaction probability (Kramer, 1996). This effect may be expressed as the
probability of exceeding a specified ground-motion
parameter:
ZZ
⁎
P½Y Ny ¼
P½Y Ny⁎jðm; rÞ fM ðmÞfR ðrÞdmdr ð14Þ
where Y is a ground-motion-induced parameter, y⁎ is a
specified value of Y (for example, Y could represent the
excess pore water pressure generated by the ground
motion and y⁎ would be the effective overburden stress),
m is a given earthquake magnitude, r is a given distance to
the fault, fM (m) and fR(r) are the probability density
functions of earthquake magnitude (M) and distance to the
fault (R), respectively. The term P[Y N y⁎|(m,r)] is the
conditional probability of liquefaction PCL determined
from Section 3.2.5: Step 5. Furthermore, based on the
Gutenberg–Richter law (Eq. (7)), the cumulative probability density function and the probability density function
are (Kramer, 1996):
FM ðmÞ ¼ P½MbmjM Nm0 ¼ 1−e−cðm−m0 Þ
fM ðmÞ ¼
d
Fm ðm; Þ ¼ ce−cðm−m0 Þ
dm
ð15Þ
ð16Þ
where c = 2.303b (note: b is the seismicity parameter
defined in Eq. (6)) and m0 is the smallest earthquake
magnitude associated with liquefaction hazard. Because
liquefaction hazards are unlikely to be associated with
earthquakes with magnitude b 4, the smallest earthquake
magnitude is set at m0 = 4. With regard to earthquake
sources, in this study the average annual liquefaction
probability of the entire fault is estimated using an
approximation according to the Simpson's rule. Assuming that distances from a specific site to the two ends of a
fault and its center are k1, k2, and k3, respectively, and the
annual probability of liquefaction based on the distance of
ki (i = 1, 3) is:
P½Y Ny⁎jki ¼
Z
P½Y Ny⁎jðm; ki Þ fM ðmÞdm
ð17Þ
Then, the annual probability of liquefaction caused by the
entire fault can be approximated as:
1
P½Y Ny⁎ ¼ fP½Y Ny⁎jk1 þ 4P½Y Ny⁎jk3 6
þ P½Y Ny⁎jk2 g
ð18Þ
3.3. Development of the CL2 model
The CL2 model is developed using the functional
form (Eq. (5)) of the energy dissipation theory by Davis
and Berrill (1982). However, the limit state of liquefaction initiation is replaced with one extracted from the
database. It should be noted that the “condition” of each
of the 130 cases in the database is not necessarily at the
limit state of liquefaction initiation. In other words, a
case history represent a data point with a set of soil and
seismic parameters that were either in the safe region
(i.e., non-liquefied condition) or the failure region (i.e.,
liquefied condition), but not necessarily “right” at the
limit state surface (i.e., the boundary of the two regions).
In this study, a search technique by Juang and Chen
(2000), illustrated previously in Fig. 1, is used to search
the corresponding “point” on the limit state surface for
each case history. Using the parameter values of the
points on the limit state surface, the coefficients A, B, C,
and D in Eq. (5) can be determined. The procedure is
summarized in the following.
First, for each of the 130 cases, the magnitude (M) was
maintained at the same level and the peak ground acceleration (amax) was increased or decreased until the factor
of safety, calculated from the SPT-based simplified
method (Youd et al., 2001), reached one (FS = 1). Next,
the attenuation relationship by Campbell (1981) was used
to estimate the distance from the epicenter to the site based
on the magnitude and the “searched” peak acceleration.
For each case history, a data point that is on the unknown
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
81
limit state surface is “located.” Thus, a total of 130 limit
state data points were obtained. Similarly, for each case,
the search can be carried out by maintaining the peak
ground acceleration at the same level while changing the
earthquake magnitude. Each search resulted in a limit
state data point, and another group of 130 data points was
obtained. It should be noted that the FS calculated with the
Youd et al. (2001) method was used as a means to update
the magnitude or peak acceleration in the search process.
Finally, a nonlinear regression of the 260 data points is
conducted using the functional form of the energy
dissipation theory (i.e., Eq. (5)), which yields:
1:5M ¼ logð1=0:003945Þ þ 3:665 logR
þ 0:975 log N1 þ 0:610 logðr0VÞ
ð19Þ
In Eq. (19), all regressive coefficients and the model
as a whole pass the t- and F-statistics tests, and the
coefficient of determination R2 is 0.97. Fig. 5 shows the
searched limit state data points along with Eq. (19)
plotted in a two-dimensional X–Y graph, using X = log
(N1) and Y = 1.5 M − 3.665 log(R) − 0.611 log(σ0′). Fig. 6
shows a similar plot as in Fig. 5 except with the actual
cases, rather than the searched limit state data points. All
but one liquefied case is below the line (Eq. (19)),
indicating that the CL2 model can accurately predict
liquefied cases. This suggests that the CL2 model is quite
conservative and suitable for use in a design scenario.
Comparing Eqs. (5) and (19), the following set of
coefficients is obtained: A = 0.003945, B = 3.665,
C = 0.975 and D = 0.610. With these coefficients calibrated from the database of 130 case histories, Eq. (11)
Fig. 6. Plot of Eq. (19) with actual case histories of liquefaction and
non-liquefaction.
can be used to estimate the annual probability of
liquefaction. This is referred to herein as the CL2
model. Thus, the energy dissipation theory by Davis and
Berrill (1982) and the limit state search technique by
Juang and Chen (2000) have been employed here to
develop the CL2 model.
It should be noted that the CL2 model is based on the
energy dissipation theory by Davis and Berrill (1982)
and as such, it inherits the errors come with basic
assumptions of the theory. Likely sources of error
include: (1) Because geologic strata are not uniform, the
transmission of earthquake waves are not uniform in all
directions; (2) The exact location of the center of energy
release may not be known; (3) For intense earthquakes,
actual seismic energy may not be calculated from
earthquake magnitude; (4) The attenuation relationship
is different for nearby locations than it is for locations
farther from the epicenter; (5) The depth to the
groundwater table at the time of earthquake may not
be known; (6) The in situ test data are generally obtained
after the earthquake and may not represent the soil
conditions before the event; and (7) The influences of
inclined slopes and the constraints by the buildings are
ignored. The first four sources of errors are related to
seismological factors and the last three sources of errors
arise from geological and environmental factors.
4. Summary
Fig. 5. Plot of Eq. (19) with limit state points.
The two models, CL1 and CL2, have been developed
for computing the annual probability of liquefaction. The
82
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
CL1 model consists of the following components: (1) a
reliability analysis that treats the SPT-based deterministic method by Youd et al. (2001) as the limit state, and
considers the uncertainty in all basic input variables, (2) a
knowledge-based clustered partitioning technique for
efficient optimization in the reliability analysis, and 3)
use of the Simpson's rule to approximate the annual
probability of liquefaction caused by the entire fault.
Because the CL1 model was based on the well calibrated
SPT-based boundary curve (Youd et al., 2001) as the
limit state and the well established theory for reliability
analysis, there is little need to re-calibrate the model
using “time-independent” case histories from different
sites and events. What is critically needed is the
validation of the whole CL1 model for the annual
probability of liquefaction at individual sites with long
term field observations, which is challenging because of
lack of such data.
On the other hand, in the CL2 model the limit state for
liquefaction initiation was derived based on a search
technique developed by Juang and Chen (2000) using 130
case histories. As such, it was essential to calibrate this
new limit state that is expressed in the form of Eq. (5)
based on the energy dissipation theory (Davis and Berrill,
1982). The results of the calibration with “timeindependent” case histories from different sites and
events, as shown in Figs. 5 and 6, were deemed satisfactory. The remaining components of the CL2 model are
basically an application of the approach developed by
Davis and Berrill (1982) for computing the annual
probability of liquefaction, which had already been validated (Davis and Berrill, 1982). However, re-validation of
the CL2 model is required since it is implemented with a
new limit state of liquefaction initiation.
The CL1 model is considered superior of the two
models for its strength in the well-calibrated limit state
and reliability methods, but the whole model for annual
Table 4
Differences in the approaches of models CL1 and CL2
With respect to
Limit state
Model CL1
SPT-based boundary
curve recommended by
Youd et al. (2001)
Mechanism of the
Related to shear stress
increase in excess generated by earthquake
pore pressure
shaking
Estimation of
Based on the Simpson's
probability of
rule using the values at
the entire fault
the two ends and the
center of the fault
Earthquake source
Distance to the two ends
and the middle of the fault
Model CL2
Eq. (19)
Related to seismic
energy dissipation
Based on Poisson's
distribution
Distance from the
epicenter
probability of liquefaction has yet to be validated. The
CL2 model is based on the energy dissipation theory and
the approach for annual probability of liquefaction has
previously been validated. However, the new limit state
that is included in the CL2 model has not been
extensively calibrated. Both models need to be examined and validated. Table 4 further compares the features
of the two models, and case study of the two models is
presented in Section 5.
5. Case study of Yuanlin area for annual probability
of liquefaction using the two models
5.1. Earthquake parameters
The town of Yuanlin is located in the Changhua
County, which is in central Taiwan. The eastern part of
the town is located on the hills of Ba-gua Mountain, and
the western part of the town is located on a plain
underlain by recent alluvium. The main fault in the
vicinity is the Chelungpu Fault that caused the 1999
Chi-Chi earthquake (Mw = 7.6). Extensive liquefaction
damage was observed in Yuanlin (Lee et al., 2003). The
Chelungpu Fault, a thrust fault with an N–S strike,
forms the eastern boundary of the Taichung basin and
extends from Fengyuan to Mingyuan. According to a
geologic map published by the Central Geological
Survey (Chang et al., 1998), the length of the fault is
86 km. Cheng et al. (1998) divided the earthquakes
along the Chelungpu Fault into 20 shallow-focus (0–
35 km) earthquakes and 6 deep-focus (35–200 km)
earthquakes. Yuanlin lies in the eastern region of the
shallow-foci earthquakes and the earthquake parameter
b in the Gutenberg–Richter law (Eq. (6)) equals 1.087,
and the parameter a equals 4.348 based on the
assumption that an earthquake with magnitude of 4 or
greater occurs once a year in Yuanlin. Thus, the total
number of earthquakes per unit length of the fault (T / L)
per year is approximately equal to 259/km.
5.2. Analysis of annual liquefaction probability in Yuanlin
In the area with long return period of strong
earthquakes, the annual probability of liquefaction
tends to be very low, which makes it difficult to verify
the calculated annual liquefaction probability. Because
of the lack of liquefaction records over a sufficiently long
period, the verification approach used by Davis and
Berrill (1982) is not suitable for case study of the
Yuanlin, Taiwan area. In this study, additional 168 cases
(Table 5) taken from sand layers at sites in the Yuanlin
area are analyzed for their annual liquefaction
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
83
Table 5
Data of critical layers in Yuanlin, Taiwan and the calculated annual probabilities
Number
Hole
number
Sample
number
Depth
(m)
Groundwater
depth (m)
SPT-N
Soil unit weight
(t/m3)
Fines content
(%)
CL1
CL2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
BH3
BH3
BH3
BH3
BH3
BH5
BH7
BH7
BH7
BH7
BH7
BH10
BH10
BH10
BH10
BH10
BH12
BH12
BH12
BH12
BH12
BH12
BH14
BH14
BH14
BH15
BH15
BH15
BH15
BH15
BH17
BH17
BH17
BH18
BH18
BH18
BH18
BH18
BH18
BH18
BH18
BH18
BH21
BH21
BH21
BH21
BH21
BH25
BH25
BH25
BH25
BH26
BH26
BH26
BH26
BH26
S-2
S-9
S-10
S-11
S-12
S-2
S-3
S-7
S-10
S-11
S-13
S-1
S-4
S-5
S-6
S-8
S-1
S-4
S-5
S-8
S-11
S-13
S-1
S-2
S-6
S-2
S-4
S-5
S-6
S-11
S-7
S-8
S-9
S-1
S-2
S-3
S-4
S-6
S-7
S-8
S-9
S-10
S-1
S-3
S-6
S-7
S-8
S-3
S-6
S-7
S-10
S-2
S-3
S-4
S-5
S-6
2.78
13.28
16.23
17.23
18.73
4.77
4.28
10.28
14.78
16.28
19.28
1.28
5.78
8.23
9.23
11.78
1.28
5.78
7.28
11.78
16.28
19.28
1.28
2.78
10.28
2.98
6.58
7.53
9.23
17.78
11.78
13.28
14.78
1.28
2.78
4.28
5.78
8.78
10.28
11.78
13.28
14.78
1.28
4.28
11.78
13.28
14.78
5.78
10.28
11.78
19.28
2.78
4.28
5.78
7.28
8.78
0.6
0.6
0.6
0.6
0.6
0.7
1.6
1.6
1.6
1.6
1.6
2.0
2.0
2.0
2.0
2.0
2.5
2.5
2.5
2.5
2.5
2.5
1.6
1.6
1.6
0.9
0.9
0.9
0.9
0.9
2.3
2.3
2.3
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.7
0.7
0.7
0.7
0.7
0.8
0.8
0.8
0.8
1.0
1.0
1.0
1.0
1.0
5
16
13
18
19
5
4
10
12
13
9
2
7
11
11
11
3
3
7
6
11
27
13
3
10
7
3
5
12
16
10
17
20
3.5
3
3
5
10
9
10
10
22
2
9
6
9
11
6
11
12
13
4
6
5
9
9
2.01
1.92
2.10
2.17
2.16
1.93
1.98
1.98
1.99
2.12
1.86
1.90
1.98
2.11
2.11
2.09
1.81
1.80
1.97
1.98
2.04
2.10
1.95
2.02
2.06
1.89
2.00
1.97
2.05
2.01
2.10
2.09
2.08
1.86
1.93
1.87
1.85
1.96
1.82
1.84
1.99
2.13
2.02
2.70
1.79
1.89
1.99
1.98
1.89
2.12
2.16
2.08
1.95
1.97
2.02
2.00
18
21
13
16
16
49
9
6
32
25
67
39
8
10
17
12
44
33
90
76
74
22
7
27
39
99
55
56
50
12
24
23
16
54
47
98
97
97
98
45
30
8
7
51
68
59
11
57
97
13
60
19
78
82
60
65
0.0031
0.0011
0.0017
0.0010
0.0009
0.0037
0.0060
0.0031
0.0012
0.0012
0.0014
0.0016
0.0029
0.0017
0.0013
0.0016
0.0017
0.0038
0.0021
0.0025
0.0013
0.0003
0.0008
0.0040
0.0020
0.0012
0.0036
0.0026
0.0011
0.0012
0.0013
0.0007
0.0007
0.0017
0.0030
0.0036
0.0026
0.0014
0.0016
0.0014
0.0014
0.0008
0.0074
0.0017
0.0031
0.0021
0.0027
0.0019
0.0011
0.0015
0.0008
0.0027
0.0017
0.0023
0.0013
0.0014
0.0030
0.0012
0.0013
0.0010
0.0010
0.0035
0.0033
0.0016
0.0014
0.0013
0.0017
0.0048
0.0018
0.0012
0.0012
0.0012
0.0047
0.0042
0.0023
0.0024
0.0015
0.0008
0.0019
0.0051
0.0020
0.0022
0.0038
0.0026
0.0014
0.0011
0.0014
0.0009
0.0008
0.0036
0.0038
0.0037
0.0025
0.0015
0.0016
0.0014
0.0014
0.0008
0.0065
0.0020
0.0026
0.0019
0.0016
0.0020
0.0013
0.0012
0.0010
0.0029
0.0021
0.0024
0.0015
0.0015
(continued on next page)
84
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
Table 5 (continued )
Number
Hole
number
Sample
number
Depth
(m)
Groundwater
depth (m)
SPT-N
Soil unit weight
(t/m3)
Fines content
(%)
CL1
CL2
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
BH26
BH26
BH26
BH26
BH26
BH27
BH27
BH27
BH27
BH27
BH27
BH27
BH27
BH27
BH27
BH27
BH28
BH28
BH28
BH28
BH28
BH28
BH28
BH28
BH29
BH29
BH29
BH29
BH29
BH29
BH29
BH29
BH29
BH29
BH29
BH30
BH30
BH30
BH30
BH30
BH30
BH30
BH30
BH30
BH30
BH31
BH31
BH31
BH31
BH31
BH31
BH31
BH32
BH32
BH32
BH32
BH35
BH35
S-7
S-8
S-9
S-10
S-11
S-2
S-3
S-4
S-5
S-6
S-7
S-8
S-9
S-10
S-11
S-12
S-3
S-4
S-5
S-7
S-8
S-9
S-10
S-11
S-1
S-2
S-3
S-4
S-6
S-7
S-8
S-9
S-10
S-11
S-12
S-1
S-2
S-3
S-4
S-5
S-7
S-8
S-9
S-10
S-11
S-3
S-4
S-5
S-6
S-10
S-11
S-13
S-5
S-6
S-8
S-10
S-1
S-2
10.28
11.78
13.48
14.78
16.28
2.78
4.28
5.78
7.28
8.78
10.28
11.78
13.48
14.78
16.28
17.78
4.28
6.00
7.28
10.28
11.78
15.05
16.28
17.78
2.28
4.28
5.78
7.28
10.28
11.78
13.28
14.78
16.28
17.78
19.28
2.28
3.78
5.28
6.78
8.28
11.28
12.78
14.33
15.78
17.28
4.28
5.78
7.28
8.78
14.78
16.28
19.28
7.28
8.78
11.78
14.78
1.28
2.78
1.0
1.0
1.0
1.0
1.0
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
4.2
4.2
4.2
4.2
4.2
4.2
4.2
2.0
2.0
2.0
2.0
2.3
2.3
9
10
10
8
10
3
3
3
5
12
13
14
14
12
14
15
3.5
4.5
3.5
5.5
6
18
18
17
6
3
4
11
13
21
24
26
18
23
11
4
6
6
7
12
17
22
18
24
22
5
2.5
4.5
5
10
4
8
12
19
18
14
7
6
1.95
1.85
1.89
1.99
1.99
1.95
2.02
1.98
1.89
2.01
2.22
1.89
1.98
1.99
2.03
2.06
2.03
1.89
1.89
2.02
2.02
2.12
2.12
2.11
2.14
1.90
1.95
2.09
1.93
2.09
2.22
2.23
2.18
2.17
2.10
1.92
1.99
2.04
2.17
2.19
2.09
2.08
2.06
2.27
2.30
1.97
1.82
1.96
1.83
1.93
1.87
1.97
2.12
2.10
2.11
2.04
2.03
2.03
98
94
80
44
30
38
76
47
93
20
9
15
5
3
4
4
31
46
59
31
70
9
9
8
15
99
54
9
34
18
7
20
10
6
29
59
17
61
15
15
20
7
18
18
10
83
41
15
50
23
90
38
18
17
20
30
28
90
0.0014
0.0013
0.0012
0.0015
0.0012
0.0034
0.0039
0.0042
0.0030
0.0015
0.0020
0.0014
0.0019
0.0024
0.0018
0.0016
0.0027
0.0025
0.0033
0.0025
0.0022
0.0012
0.0012
0.0013
0.0019
0.0032
0.0030
0.0024
0.0011
0.0006
0.0009
0.0004
0.0012
0.0009
0.0012
0.0023
0.0030
0.0024
0.0034
0.0019
0.0010
0.0010
0.0010
0.0006
0.0010
0.0011
0.0022
0.0027
0.0016
0.0011
0.0019
0.0010
0.0009
0.0005
0.0005
0.0007
0.0008
0.0013
0.0015
0.0014
0.0014
0.0016
0.0013
0.0040
0.0038
0.0038
0.0026
0.0014
0.0012
0.0012
0.0012
0.0013
0.0011
0.0011
0.0035
0.0029
0.0034
0.0024
0.0022
0.0010
0.0010
0.0010
0.0026
0.0041
0.0033
0.0016
0.0014
0.0009
0.0008
0.0008
0.0010
0.0009
0.0014
0.0036
0.0026
0.0025
0.0022
0.0015
0.0011
0.0009
0.0011
0.0009
0.0009
0.0022
0.0035
0.0023
0.0021
0.0012
0.0024
0.0014
0.0010
0.0007
0.0008
0.0009
0.0024
0.0025
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
85
Table 5 (continued )
Number
Hole
number
Sample
number
Depth
(m)
Groundwater
depth (m)
SPT-N
Soil unit weight
(t/m3)
Fines content
(%)
CL1
CL2
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
BH35
BH35
BH35
BH35
BH35
BH35
BH39
BH39
BH41
BH41
BH41
BH41
BH41
BH41
BH41
BH43
BH43
BH43
BH43
BH43
BH43
BH43
BH44
BH44
BH44
BH44
BH44
BH44
BH44
BH44
BH44
BH44
BH45
BH45
BH45
BH45
BH45
BH45
BH45
BH46
BH46
BH46
BH46
BH46
BH46
BH47
BH47
BH47
BH47
BH47
BH47
BH47
BH47
BH47
S-3
S-4
S-6
S-7
S-9
S-10
S-1
S-2
S-1
S-5
S-6
S-7
S-8
S-9
S-12
S-6
S-7
S-8
S-9
S-10
S-11
S-13
S-1
S-4
S-5
S-6
S-7
S-8
S-9
S-10
S-11
S-12
S-3
S-4
S-5
S-7
S-8
S-9
S-10
S-2
S-4
S-5
S-7
S-8
S-11
S-2
S-3
S-4
S-5
S-6
S-7
S-9
S-10
S-11
4.28
5.78
9.99
13.53
16.78
18.28
1.38
4.78
1.28
7.28
8.78
10.28
11.78
13.28
17.78
8.78
10.28
11.78
13.28
15.73
16.73
19.73
2.78
7.28
8.78
10.28
11.78
13.28
14.78
16.28
17.78
19.28
4.28
5.78
7.28
11.78
13.28
14.78
16.28
2.78
5.78
7.28
10.28
11.78
16.28
2.78
4.28
6.78
8.78
10.28
11.78
14.78
16.28
17.78
2.3
2.3
2.3
2.3
2.3
2.3
0.2
0.2
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.4
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.0
1.0
1.0
1.0
1.0
1.0
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.3
9
4
11
16
20
12
13
4
2
5
10
18
18
20
8
6
14
15
20
23
15
11
4
8
5
13
15
21
18
22
13
15
4
4
5
15
14
21
23
5
3
2
4
12
25
4
3
4.5
5.5
14.5
11
17
10.5
19
2.08
1.92
2.04
2.00
2.09
2.03
2.52
2.31
1.80
1.99
1.85
2.13
2.25
2.13
1.89
1.89
2.07
2.08
2.08
2.19
2.04
1.85
1.92
2.04
2.15
1.87
2.01
2.10
2.05
2.05
2.02
1.96
1.83
1.85
1.92
1.88
1.93
1.96
2.11
1.94
1.92
1.83
1.93
1.92
2.12
1.94
1.84
1.90
1.82
2.07
1.94
2.05
1.96
1.97
26
53
70
53
15
8
16
35
88
85
54
11
11
10
85
96
15
20
10
12
18
78
26
18
25
78
16
14
13
10
97
95
70
86
28
66
46
9
11
38
84
79
82
17
12
30
84
99
94
23
31
67
30
20
0.0013
0.0028
0.0013
0.0008
0.0009
0.0022
0.0004
0.0023
0.0028
0.0032
0.0017
0.0013
0.0013
0.0012
0.0019
0.0029
0.0016
0.0013
0.0011
0.0008
0.0013
0.0013
0.0028
0.0027
0.0036
0.0011
0.0014
0.0009
0.0012
0.0010
0.0011
0.0009
0.0031
0.0035
0.0034
0.0008
0.0011
0.0011
0.0009
0.0021
0.0044
0.0061
0.0038
0.0018
0.0007
0.0020
0.0031
0.0028
0.0026
0.0011
0.0014
0.0007
0.0014
0.0008
0.0018
0.0033
0.0015
0.0011
0.0010
0.0014
0.0011
0.0023
0.0062
0.0028
0.0017
0.0011
0.0011
0.0010
0.0019
0.0025
0.0013
0.0012
0.0010
0.0009
0.0012
0.0015
0.0036
0.0020
0.0028
0.0014
0.0013
0.0010
0.0011
0.0009
0.0013
0.0012
0.0035
0.0034
0.0028
0.0013
0.0013
0.0010
0.0009
0.0030
0.0041
0.0054
0.0032
0.0014
0.0008
0.0034
0.0041
0.0030
0.0026
0.0012
0.0015
0.0011
0.0015
0.0010
86
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
Fig. 7. Comparison of annual liquefaction probabilities of 168 cases in
Yuanlin calculated by the two models.
probabilities subjected to future re-rupture of the
Chelungpu Fault using both the CL1 model and the
CL2 model. In these analyses, the following assumptions
regarding the seismic conditions are made: (1) fault
rupture at earthquake focus is assumed and the same
probability of earthquake occurrence for a specific time
and point in space is assumed; (2) the attenuation
relationship by Campbell (1981) is employed; (3) only
earthquakes of magnitude 4 or greater are considered.
The annual liquefaction probabilities calculated by
the CL1 and CL2 models, respectively, for the 168 cases
are compared and shown in Fig. 7. The results obtained
from the two models are shown to agree well with each
other, as the coefficient of correlation of the annual
liquefaction probabilities calculated with the two
models is quite high (ρ = 0.81). Considering that the
two models, CL1 and CL2, are very different in their
principles and formulations, as discussed previously,
consistent results in the calculated annual liquefaction
probabilities suggest that more likely than not, both
models possess a certain degree of reliability. However,
no records are currently available to confirm the validity
of each model individually.
It should be noted that the annual liquefaction probability calculated by the model CL1 or CL2 is for a soil
element at a given depth subjected to a given seismic
loading. For the overall annual liquefaction probability
over the entire soil column, given the information of a
borehole or a subsurface soil profile, a weighted average
of the annual liquefaction probabilities of all depths
ranging from z = 0 to z = 20 m (based on the suggestion of
Iwasaki et al., 1982 that liquefaction at depths greater
than 20 m was rarely observed) may be calculated. This
may be accomplished by defining a weighted annual
probability of liquefaction (WAPL):
WAPL ¼
20
X
PðzÞd W ðzÞdz
ð20Þ
z¼0
Where WAPL is an index for weighted annual probability of liquefaction, z is depth of the soil element in
Fig. 8. Comparison of weighted annual probabilities of liquefaction at 26 sites in Yuanlin calculated by the two models.
Y.-F. Lee et al. / Engineering Geology 90 (2007) 71–88
meter (z = 0 to 20), P(z) is the annual liquefaction
probability at depth z, W(z) is the weighting function,
defined as W(z) = 10–0.5z. It should be noted that the
formulation of this weighted average was inspired by the
“Liquefaction Potential Index” defined by Iwasaki et al.
(1982). The annual liquefaction probability P(z) in Eq.
(18) may be calculated with the CL1 or CL2 model.
Thus, the index WAPL can be calculated using both the
CL1 and the CL2 models for each of the 26 sites that
were investigated by Moh and Associates (MAA, 2000)
through borehole sampling and SPTs in the post Chi-Chi
event investigation. Fig. 8 shows a comparison of the
WAPL values calculated with the two models. As with
the data shown in Fig. 7 previously, there is high correlation between the weighted averages of annual liquefaction probabilities obtained by the two models (the
coefficient of correlation ρ = 0.92). Again, this suggests
that both models possess a certain degree of reliability.
Since there is no evidence to favor use of one model over
the other, the annual probability of liquefaction may be
estimated by either one or by taking the average of the
probabilities obtained from both models.
6. Summary and concluding remarks
Two models, CL1 and CL2, for estimating the annual
liquefaction probability are developed in this study. The
unique features of the CL1 model include (1) use of the
start-of-the-art SPT-based boundary curve (Youd et al.,
2001) as the limit state for the reliability analysis, (2) use
of the knowledge-based clustered partitioning technique, a very efficient algorithm, to solve the optimization problem of finding the minimum reliability index
under the specified constraints, and (3) use of a
simplified approach, as expressed in Eqs. (14)–(18), to
formulate the annual probability of liquefaction. The
CL2 model is essentially a modified energy dissipation
method that was originally developed by Davis and
Berrill (1982). The new feature in the CL2 model is the
limit state that was developed using a searching
technique developed by Juang and Chen (2000) and
an expanded database of case histories. Additional
unique feature that applies to both models is the
implementation of the concept of weighted annual
probability of liquefaction (Eq. (20)). This feature
allows for an assessment of the annual probability of
liquefaction at a site based on the entire profile (up to
20 m) of SPT blow counts and other borehole data. The
feature greatly facilitates the task of mapping annual
probability of liquefaction in a city or county, which can
be used as a tool for building code development and/or
enforcement.
87
While the theoretical bases for these models appear
to be sound, it is generally difficult to verify the annual
probability of liquefaction because of the long recurrence interval of strong earthquakes (and thus, the very
low annual probability of liquefaction). Nevertheless,
the annual liquefaction probabilities calculated for
typical sand layers in Yuanlin area using the two
fundamentally different models agreed well with each
other, suggesting that both models possess a certainty
degree of reliability. The results of the analysis of the 26
borehole data from the Yuanlin area using the two
models, again, show that there is high correlation
(ρ = 0.92) between the weighted averages of annual
liquefaction probabilities obtained by the two models.
For a forward analysis in a future case, the annual
liquefaction probability may be estimated by either one
of the two models or by taking the average of the
probabilities obtained from both models.
Limitations of the developed models should be
noted. They include: (1) the assumptions made in the
model development, (2) uncertainties in earthquake and
geologic data, and (3) accuracy and limitation of in-situ
tests and field observations. Further study of the
developed model to ease these limitations is warranted.
Acknowledgments
The study on which this paper is based was supported
by the National Science Council (NSC), Taipei, Taiwan
through Grant No. 92-2211-E-309-004. This financial
support is greatly appreciated. The fourth author wishes
to acknowledge the support of National Science
Foundation for his participation in this study through
Grant No. CMS-0218365. Any opinions, findings, and
conclusions or recommendations expressed in this
material are those of the authors and do not necessarily
reflect the views of the National Science Council and the
National Science Foundation.
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