Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 255

of the measurement. Moreover, the results show the influence of the autocorrelation of the
prior distribution. The change of the posterior means is steeper in the vertical than in the
horizontal direction, which is due to the fact that the prior correlation length in the hori-
zontal direction is larger than in the vertical direction. The low values of Young's modulus
observed in the bottom right of Figure 5.15 cannot be explained by the measurement and
hence are attributed to sampling error.

5.6.3 updating the reliability with deformation measurements
We now consider that deformation measurements are taken at an intermediate excavation
step and utilize the measurements to update the reliability at the final excavation state. The
failure event F is defined as the event of the horizontal displacement at the top of the trench
u x ( X ) exceeding a threshold of u x,t = 100 mm at full excavation. Mathematically, this is
expressed through the following LSF:

g () x = u − u ( , x (5.58)

xt , x

This is a serviceability limit state, reflecting the assumed serviceability design require-
ments. The reliability analysis is performed by means of subset simulation. Without mea-
surements, the computed failure probability is Pr( F ) = 1.36 × 10 −2 with a corresponding
reliability index β′ = 2.21.
We assume that a measurement of the displacement u x ,2.5m is made at an intermediate
excavation step of 2.5 m depth. The measurement is subjected to an additive error ε m , which
is described by a normal PDF f ∈ with zero mean and standard deviation σ ∈ m . This informa-
tion is expressed by an event Z , described by the following likelihood function:

L () x = f [ u − u (). x ]

(5.59) ∈ xm , x , . 25m

m

For the estimation of the updated failure probability conditional on the measurement
event Z , we apply the BUS approach combined with subset simulation. The constant c is
again selected as c = σ , which satisfies the condition cL ( x ) ≤ 1. The reliability updating

was performed for different measurement outcomes u x,m , and different values of the stan-
dard deviation σ ∈ m of the measurement error. The results are summarized in Table 5.6 and
the computed reliability indices are plotted in Figure 5.17 . For comparison, the (a priori)
expected value of the measurement outcome u x,m is computed as 2.6 mm.
Not surprisingly, for measurements significantly higher than the expected value, the
updated failure probability is higher than the prior probability. This difference is more pro-
nounced when the measurement device is more accurate, that is, when σ ∈ m is smaller. For
measurements lower than the expected value, the updated failure probability is lower than

m

Table 5.6 Updated probability of failure (prior probability of failure Pr( F ) = 1.36 × 10 − 2 )

σ ε m = 2mm σ ε m = 1mm

Measurement

Pr(F|Z)

β″

Pr(F|Z)

β″

u x,m = 10 mm 2.18 × 10 − 1 0.78 3.31 × 10 − 1 0.44

2.04 1.80

u x,m = 5 mm 2.09 × 10 − 2 3.59 × 10 − 2

2.47 2.90

u x,m = 2 mm 6.74 × 10 − 3 1.84 × 10 − 3