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U.S.-Taiwan Workshop on Soil Liquefaction
A Practical Reliability-Based Method for
Assessing Soil Liquefaction Potential
Jin-Hung Hwang
National Central University, Taiwan
Outline
Previous studies
Reliability model
Probability density function of CSR
Probability density function of CRR
Liquefaction probability and safety factor
Summary and discussion
Previous Studies
Haldar
and Tang (1975),
Fardis and Veneziano (1982),
Chameau and Clough (1983),
Liao et al. (1988),
Youd and Nobel (1997),
Toprak et al. (1999) ,
Juang et al. (2000a,2000b)
Some
comments
Soil
parameters and data should be updated.
Probabilistic cyclic strength curves without the
statistics.
Juang’s work is a notable advancement, however
ANN is a little unfamiliar to engineers.
Reliability Model
Based
on Seed’85 method
Assume CSR and CRR are normal distribution

R  s
 
2
R
2
S
Pf  1.0   (  )
Probability Density
τL
τR
fL(L)
fR(R)
S, R
Z < 0 , liquefy

Z > 0 , non-liquefy
βσz
fz(z)
R  s
 R2   S2
Pf  1.0   (  )
liquefaction
probability , Pf
σz
μZ
σz
Z
Fig.1 Probability density distribution for the liquefaction performance function.
Assume
CSR and CRR are log-normal distributions
 Z  ln R   ln S


2
2
Z
 ln   ln S
Pf  1.0  (  )
    1 
R

 

ln
  S    1  

1/ 2
2
2
ln(  R  1)( S  1)

2
S
2
R
1/ 2

Flow
chart of calculation
Geological data
Earthquake data
Earthquake magnitude M and
hypocentral distance R
Attenuation formula
to compute Amax
SPT
N60
Effective
overburden stress
v (kg / cm2 )
Magnitude
scaling factor
M
MSF  ( ) 1.11
7.5
N1 60 
1
 v'
Fines content
KS  f (FC)
If FC  10
K S  1 .0
 N 60
If
FC  10
K S   0 .00009 FC 2  0 .0168 FC  0 .841
CSR statistics
CSR7.5  0.65 
Amax  v
 rd / MSF
g  v
CRR statistics
2
 CRR  exp[2.63  0.06008( N1 ) 60  0.000507( N1 ) 60
]
 CRR  0.604
 CSR  0.581
Reliability index


  ln CSR
Z
 ln CRR
2
Z
 ln CRR   ln2 CSR
    2  1 1 / 2 
 
ln  CRR  CSR
2
  CSR   CRR  1  

1/ 2
2
2
ln( CRR
 1)( CSR
 1)

Liquefaction probability
Pf  1   (  )

Information
required
Mean
values and variance coefficients of
CSR and CRR
Table 2 Mean values and variance coefficients of CSR and CRR
Mean value
CSR
CRR
0.65 
 v Amax

 rd  MSF ( M )
'
v g
2
exp[ 2.63  0.06008( N1 )60  0.000507( N1 )60
]
Variance coefficient
0.581
0.604
PDF of CSR
CSR  0.65
 v Amax
rd ( z ) / MSF ( M )  a  Amax
v g
 1 ln( CSR)   ln(CSR ) 2 
f CSR (CSR) 
exp  (
) 
2

2  ln(CSR )  CSR


ln(CSR )
1
5.0
depth = 10m
G.W.T. = 5.3m
2
σ v = 20.3 t/m
Probability Density
4.0
2
σ ' v = 15.3 t/m
r d = 0.899
PGA = 0.28g
μ ln(CSR) = -1.757
σ ln(CSR) = 0.677
3.0
2.0
1.0
0.0
0
0.2
0.4
0.6
0.8
1
Cyclic Stress Ratio (CSR )
Fig.2 Calculated probability density function of a soil at a depth of 10 m.
PDF of CRR
2
  ln( 1 / PL  1)   0  1 ( N1 ) 60cs   2 ( N1 ) 60

cs
CSR  exp 

3


1.0
0.7
P L = 0.99
Table 1 Parameters in the logistic model
Parameter
β0
β1
β2
β3
Regressed result
10.4
-0.2283
-0.001927
3.8
Cyclic Resistance Ratio (CRR)
0.8
0.9
0.3
0.1
0.5
0.01
0.6
0.4
0.2
0.0
0
10
20
30
40
50
Corrected Blow Count , (N 1)60
Fig.3 Probabilistic cyclic resistance curves
regressed by the logistic model.
PDF of CRR
ab(CRR )b 1
f (CRR )  
(1  a(CRR )b ) 2
12
10
Probability Density
(N 1)60 = 5
8
The greater (N 1)60 , the greater δ
6
4
CRR
(N 1)60 = 30
2
0
0.0
0.2
0.4
0.6
0.8
1.0
Cyclic Resistance Ratio, CRR
Fig.4 Probability density function of the soil cyclic resistance ratio.

PDF of CRR
2
a  exp   0  1 ( N1 ) 60cs   2 ( N1 ) 60
cs
b  3
1.0
Mean value
0.8
Cyclic Resistance Ratio (CRR)

P L =0.6
0.6
0.4
0.2
Median value (P L =0.5)
0.0
0
10
20
30
40
50
Corrected Blow Count , (N 1)60
Fig.5 Mean and median curves compared with the probabilistic curve of PL=0.6.
Liquefaction Probability and Safety Factor
    2  1 1/ 2 
ln  R  S2  
  S   R  1  
ln( FS )

 0.013 
1
/
2
0.7758
ln(  R2  1)( S2  1)


Pf  1.0  (  )
1.0
assume δ
δ = 0.0
CSR
=δ
CRR
Liquefaction Probability , PL
0.8
0.6
0.4
δ = 1.0
0.2
0.0
0
1
2
3
4
5
6
Safety Factor , FS
Fig.7 Relations of liquefaction probability with the
safety factor for different variance coefficients.
Compared
with the safety factor defined by the
Seed’85 method
1.0
P L = 0.6 0.5 0.2
Cyclic Resistance Ratio ( CRR)
0.8
(N 1)60=30, PL =0.57, Cr =1.03
0.6
(N 1)60=29, PL =0.30, Cr =1.38
Seed'85 Method
0.4
(N 1)60=8, PL =0.32, Cr =1.35
(N 1)60=28, PL =0.22, Cr =1.55
0.2
(N 1)60=20, PL =0.35, Cr =1.31
(N 1)60=14, PL =0.44, Cr =1.18
0.0
0
10
20
30
40
50
Corrected Blow Count , (N 1)60
Fig.8 Comparison of the probabilistic CRR curves with the
empirical curve proposed by Seed’85 method.
Compared
with Juang’s result
1.0
Juang et al. (2002)
Liquefaction Probability , PL
0.8
0.6
Cr = 1.18
Cr = 1.30
0.4
Cr = 1.55
0.2
0.0
0
1
2
3
4
5
Safety Factor , FS Seed
Fig.9 Relation of liquefaction probability with the
safety factor calculated by Seed’85 method.
6
Parameter Study
Influences
of ( N1 )60 , Fines Content FC(%), and
the ground water table on the liquefaction
probability Pf
100%
Depth = 8m
G.W.T. = 2m
FC = 5%
Probability Liquefaction
80%
60%
40%
20%
0%
0
10
20
30
40
Corrected Blow Count , (N 1)60
Fig.10(a) Variation of liquefaction probability with (N1)60.
Parameter Study
Influences
of ( N1 )60 , Fines Content FC(%), and
the ground water table on the liquefaction
probability Pf
100%
Depth = 8m
G.W.T. = 2 m
FC = 5~35%
Probability Liquefaction
80%
60%
40%
FC = 5%
20%
FC = 35%
0%
0
10
20
30
40
Corrected Blow Count , (N 1)60
Fig.10(b) Influence of fines content on liquefaction probability.
Parameter Study
Influences
of ( N1 )60 , Fines Content FC(%), and
the ground water table on the liquefaction
probability Pf
100%
Depth = 8m
G.W.T. = 0~6m
FC = 5%
Probability Liquefaction
80%
60%
40%
G.W.T. = 0 m
20%
G.W.T. = 6 m
0%
0
10
20
30
40
Corrected Blow Count, (N 1)60
Fig.10(c) Influence of ground water table on liquefaction probability.
Application Example
Active
Hsinhwa fault (12km rupture)
1946 Tainan earthquake
Caused extensive liquefaction
Design earthquake M  6.8, PGA  0.28g
L
Result of liquefaction analysis
Application Example
PL
Table 3 Result of liquefaction analysis for the site near the Hsinhwa fault
depth
(m)
Unit weight
(t/m3)
SPT-N
FC
(%)
Soil classification
F.S.
(Seed)
PL
(%)
1.3
1.97
3
73
CL-ML
-
-
2.8
2.02
6
69
CL-ML
-
-
4.3
2.00
7
75
CL-ML
-
-
5.8
1.89
15
82
ML
-
-
7.3
1.93
6
99
ML
-
-
8.8
2.01
6
91
CL-ML
-
-
10.3
1.98
17
33
SM
1.2
35%
11.8
1.95
23
29
SM
1.4
19%
13.3
1.87
18
33
SM
1.2
35%
14.8
1.96
13
14
SM
0.8
62%
16.3
1.95
9
99
CL
-
-
18.8
2.04
33
25
SM
2.0
6%
19.3
2.19
33
20
SM
1.9
9%
Application Example
Simplified profile
0
0
10
20
0
30
50
0
100
1
2
3
0
0
0
0
Liquefaction probability , P f
Safety factor , FS
FC (%)
SPT-N
0
PGA = 0.28g
ML = 6.8
Seed85 method
CL
PGA = 0.28g
ML = 6.8
5
5
5
5
0.5
5
10
10
depth (m)
10
depth (m)
10
depth (m)
depth (m)
depth(m)
ML
10
SM
15
15
15
15
15
20
20
20
20
CL
SM
20
Fig.11 Result of liquefaction analysis for the site near the Hsinhwa fault.
1
Summary and Discussion
A simple
and practical reliability method for
liquefaction analysis was proposed
The liquefaction probability is just a
probability under a given earthquake event
It needs to combine the probability of
earthquake occurrence