601
Technical Committee 102 /
Comité technique 102
whose period is about 10 m along the horizontal axis, is found,
the term sin{(x/5-1/2)π} was added to Equation (5). The
determined mean function is
m
1.98
0.816sin
x
5
1
2
0.157
z
(8)
The covariance function is determined by
C
ij
0.75
2
exp
x
i
x
j
6.14
z
i
z
j
0.63
i
j
C
ij
1.24
2
i
j
(9)
The horizontal correlation length is identified to be
approximately ten times of the vertical one. Since this rate is
similar to the values published previously (e.g. Soulie et. al.
1990), the correlation lengths identified here are judged to be
appropriate. The boundary between the base ground and the
embankment is determined based on the SWS results.
The
N
-distribution predicted based on the determined
statistical models with aid of the indicator simulation method
(Deutsch and Journel 1990), which is one of the geo-statistical
methods, and interpolates the point-estimated
N
-values, is
exhibited in Figure 4. The horizontal periodicity of the
N
-
values is presented according to the figure.
4 RELIAIBILITY-BASED DESIGN OF A FILL-
EMBANKMENT
4.1
Statistical model of an embankment
A stability analysis is conducted and the risk is evaluated for an
earth-fill dam at Site H to analyze the transversal section, the
mean of the equation. As a mean function, Equation (12) is
proposed by averaging Equation (8) along the
x
–axis, while the
covariance function is defined as Equation (13), in which
coordinate
x
is replaced by
y
of Equation (9), and depth
z
is
replaced by elevation
h
. This assumption is based on the reason
why the embankments are compacted horizontally in the
construction, and the correlation structure at the same elevation
is homogeneous.
m
1.89
0.157
z
(12)
C
ij
0.75
exp
y
i
y
j
2
6.14
h
i
h
j
0.63
i
j
C
ij
1.24
2
i
j
(13)
The analytical sections of the original embankment, and the
improved and restored embankment are exhibited in Figure 5.
The embankment is improved by constructing an inclined core,
and by covering the original embankment with the additional
soil for reinforcement. The material properties are given in
Table 1. The soil parameters are determined from the SPT
N
-
values and the laboratory soil tests. The Bs means the
embankment material; it is determined from the
N
-values based
on the SWS results to consider the spatial distribution. The
effective internal friction angle
'=
d
, is obtained from the
conversion, namely, Equation (14) (Hatanaka and Uchida
1996). In the equation, 5.3
f
is the conversion error, in which
f
is an
N
(0,1) type normal random variable, and the ratio of 5.3 is
the standard deviation.
Figure 3. Spatial distribution of NSWS and statistical model.
15
13
11
9
7
5
3
1
0 5 10 15 20 25 30 35 40
Depth (m)
Horizontal Coordinate x (m)
0
5
10
15
20
25
Figure 4. Predicted spatial distribution of N-value.
Bs
Ac
As
Gr
(a) Original embankment.
Bs
Ac
As
Gr
Core
Block
Rigid soil
(b) Restored embankment.
Figure 5. Cross sections and critical slip surfaces of embankments.
'
20
N
1
0.5
20
5.3
f
(14)
N
1
N
SPT
v
' /98
0.5
(15)
in which
v
' is the effective vertical stress.
4.2
Reliability analysis
In the stability analysis, the pore water pressure is required; it is
calculated with a saturated-unsaturated seepage finite element
analysis (e.g., Nishigaki 2000). In the restored embankment, the
water table level is dramatically reduced by the existence of the
impermeable zone. Consequently, this reduction can make the
embankment stable.
The circular slip surface method is employed as the stability
analysis in this study. For uncertain factors, the random
numbers are assigned, and the stability of the embankments is
evaluated as the probability of failure with the use of the Monte
Carlo method. For the reliability analysis, Equation (16) is
defined as a performance function, in which the internal friction
angle is a probabilistic parameter. As the load of the
earthquake, the design earthquake intensity of 0.15 is
considered.
g
fi
si
l
i
i
1
n
(16)
