Environmental Engineering Reference
In-Depth Information
and 0.59, respectively. The actual measured value of Y 5 is −0.727 (
σ pa
′
/P
=
0
.
483
(Rochelle
et al. 1988).
The updated version of Y 6 is an SU distribution with a X = 2.039/0.671 = 3.038,
b X = (−0.517-0.028)/0.671 = −0.799, a Y = 1.427, and b Y = −1.461. The unconditioned mean
and standard deviation of Y 6 are −1.05 and 0.82, respectively. The conditioned mean and
standard deviation of Y 6 are −1.06 and 0.51, respectively. The actual measured value of Y 6 is
−1.196 (
′ σ 00 (Rochelle et al. 1988). The unconditioned and conditioned distribu-
tions for Y 5 and Y 6 are shown in Figure 1.35 .
It is also possible to plot the bivariate distribution of (Y 5 , Y 6 ). Note that (X 5 , X 6 ) are
bivariate normal with mean equal to μ [1] and covariance matrix equal to C [11] . The bivariate
normal PDF of (X 5 , X 6 ) is
s uv
/
.
3 2
)
T
x
x
x
1
1
2
ˆ
Š
'
'
ˆ
Š
'
'
(
) −
1
(
) =
5
5
[]
1
[]
11
µ []
1
f X
xx
,
×
exp
−
−
µ
C
−
(1.118)
56
x 6
2
π
C
[]
11
6
According to Equation 1.95 , we have
(
)
(
)
−
1
−
1
(1.119)
x
=
ba
+
×
sinh
yb a
−
x
=
b
+
a
×
sinh
y
−
ba
5
X
5
X
5
5
Y
5
Y
5
6
X
6
X
6
6
Y
6
Y6
because both (Y 5 , Y 6 ) are Johnson SU. The derivative of x with respect to y is
dx
dy
a
dx
dy
a
5
5
X
5
6
6
X
6
=
=
(1.120)
a
1
+−
[(
y
ba
)
]
2
a
1
+−
[(
y
ba
)
]
2
Y
5
5
Y
5
Y
5
Y
6
6
Y
6
Y
6
(a)
(b)
1.2
0
Unconditional Y 6
Conditional Y 6 using (Y 3 ,Y 4 ,Y 8 ,Y 10 )
Unconditional Y 6
Conditional Y 6 using (Y 3 ,Y 4 ,Y 8 ,Y 10 )
0.9
1
0.8
0.7
0.8
0.6
0.6
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0
-4
-4
-2
0
2
4
6
8
-2
0
2
4
Y 5 = ln(σ p / P a )
Y 6 = ln( S u /σ v )
Figure 1.35 Conditional probability density distribution based on Y 3 = 1.793, Y 4 = − 0.912 , Y 8 = 0.504, and
Y 10 = 1.265 for (a) normalized preconsolidation stress (Y 5 ) and (b) normalized undrained shear
strength (Y 6 ). (The arrow indicates actual measured value reported by Rochelle, P.L., Zebdi,
M., Leroueil, S., Tavenas, F., and Virely, D. 1988. Proceedings of the 1st International Symposium on
Penetration Testing , Orlando, 2, 831-841.)
 
 
Previous Page
Next Page
Risk and Reliability in Geotechnical Engineering
Search WWH ::




Custom Search
Home