Environmental Engineering Reference
In-Depth Information
Table 4.6
Results of the maximum likelihood analyses with various assumptions
Distribution type
for z
Model
probability
Model form
BIC
μ
δ
Normal(Q
1
)
0.905
0.273
107.16
0.13
F
µ
µδ
−
⋅
S
P
1
Φ
=−
L
Lognormal(Q
2
)
0.914
0.281
104.94
0.40
(
)
ln()ln
F
1
2
−
µ
+
δ
S
P
1
=−
Φ
L
ln(
1
2
)
+
δ
Minimum
Gumbel(Q
3
)
1.028
0.337
112.38
0.01
‡
Ž
(
F
−
µπ
)(
−
0 5772
.
6
⋅
δ
)
‰
‰
‰
‰
S
P
exp xp
=
−
L
6
⋅⋅
µδ
Maximum
Gumbel(Q
4
)
0.800
0.282
104.71
0.45
‡
Ž
−− −
(
F
µπ
)(
0 5772
.
6
⋅
δ
)
‰
‰
‰
‰
S
P
1
exp xp
=− −
L
6
⋅⋅
µδ
As analyzed in the previous section, the adjusting weights for the calibration database are
With Solver, the optimal values of θ are θ* = {μ*, δ*} = {0.914, 0.281}.
Based on
Equation 4.27
and the knowledge of model error (bias), which is character-
ized with θ* = {μ*, δ*} = {0.914, 0.281}, the following
P
L
−
F
S
mapping function can be
established:
0.128
+
ln()
F
S
P
=1
−
Φ
(4.31)
L
0 276
.
Thus, the probability of liquefaction can be computed once the nominal factor of safety is
calculated using the Robertson and Wride method.
The calibrated results and model probabilities of the four models are also shown in Table
4.6. Among the four models considered, the lognormal model and maximum Gumbel model
are the most supported by the adopted database, whereas the normal model and minimum
Gumbel model are much less supported by the database. Thus, either the lognormal model
or the maximum Gumbel model may be used for future applications. It should be noted
that this particular conclusion differs somewhat from that obtained by Ku et al. (2012), as
they employed a different set of adjusting weights (
w
L
= 0.660 and
w
NL
= 2.063). The recom-
study is practically identical, however, to that obtained and recommended by Ku et al. (2012).
4.5 eStIMatIon oF lIqueFaCtIon-InDuCeD SettleMent
4.5.1 Probabilistic model for predicting
liquefaction-induced settlement
Liquefaction-induced settlement has been the subject of multiple investigations, from which
many semiempirical methods have been developed based on results of both laboratory testing
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