Environmental Engineering Reference
In-Depth Information
Table 5.4
Prior and posterior of the mean and COV of the soil properties and probability of failure
given information on a survived loading condition
Prior
Posterior
Parameter
Mean
COV
Mean
COV
Unit weight
γ
(kN/m
3
)
19.0
5.0%
19.04
4.8%
37.40
15.5%
38.22
14.4%
Mean friction angle
μ
φ
(°)
1.43
114.6%
1.85
97.5%
Mean uplift coefficient
μ
K
Probability of failure
1.70
×
10
−
3
-
3.58
×
10
−
4
-
To obtain samples from the posterior distribution of γ, μ
K
, and μ
φ
, it suffices to sample
their prior distribution conditional on {
h
(γ, μ
K
, μ
φ
) ≤ 0}. The samples are obtained through
subset simulation. We also compute the updated failure probability by application of
Equation 5.41
.
The numerator in
Equation 5.41
is again computed with subset simulation.
The prior and posterior statistics of the random variables and probability of failure are
It is shown that the means of μ
φ
and μ
K
are increased compared to their priors, while their
COVs are decreased implying reduction of uncertainty. This is reflected in the decrease of
the probability of failure. The corresponding posterior reliability index is β″ = 3.38.
5.6 aPPlICatIon: FInIte-eleMent-baSeD uPDatIng
oF SoIl ParaMeterS anD relIabIlItY
This example is based on the previous stochastic model and reliability updating presented
in (Papaioannou and Straub 2012; Straub and Papaioannou 2014). We update the mate-
rial properties of the soil surrounding a geotechnical site based on a deformation mea-
surement performed in situ. Also, deformation measurements made at an intermediate
excavation depth are utilized to update the reliability of the construction site at the stage of
full excavation.
The site consists of a 5.0 m deep trench with cantilever sheet piles in a homogeneous soil
layer of dense cohesionless sand with uncertain spatially varying mechanical properties (see
Figure 5.13
)
. The soil is modeled in two dimensions (2D) with plane-strain finite elements.
For simplicity, neither groundwater nor external loading is considered. Additionally, we
u
x
5 m
12.5 m
10 m
Dense cohesionless sand
Figure 5.13
Sheet pile wall in sand.
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