Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 249

Assuming that the measurement errors ε m of the CPT recordings, the transformation
uncertainties ε t , and the friction angle φ at different locations are pairwise-independent
random variables, we can obtain the likelihood describing the combined information from
all five recordings, as follows:

5

∏ 1

L () µ = L ( . µ (5.53)

ϕ i ϕ

i =

Note that μ K and the CPT recordings are conditionally independent given μ φ , which
implies that f = for all Q tni . Therefore, we can update the joint

distribution of μ K and μ φ with the likelihood of Equation 5.53 , which gives

( Q | µµ , ) f ( Q | µ )

Qtni ϕ K Q tni ϕ

tni tni

f ′′ (, µµ µ ) = aLf ( ) ′ ( µ µ , ,

(5.54)

µµ K ϕ ϕ µµ K ϕ

K ϕ K ϕ

where a is the proportionality constant.
We perform the updating by application of the BUS approach with subset simulation (see
Section 5.4.3). We also update the reliability of the foundation by application of Equation
5.41 . It is noted that the denominator in Equation 5.41 is computed as a by-product of
the Bayesian updating algorithm. Therefore, for updating the reliability, only the numera-
tor has to be computed additionally. The latter is computed by application of the subset
simulation.
The updated mean and COV of μ φ and μ K as well as the updated probability of failure
are displayed in Table 5.3 together with the corresponding priors. Figure 5.12 shows the
prior joint PDF of μ φ and μ K and their posterior joint PDF conditional on the CPT measure-
ments. It is shown that the means of the two parameters decrease compared to the priors.
Moreover, their COVs also decrease, which demonstrate the effect of the CPT test on the
reduction of the uncertainty of both μ φ and μ K . The reduction is larger for μ φ than for μ K .
This is due to the fact that the tip resistance of the CPT test correlates with the friction
angle, while μ K is influenced indirectly due to its dependence with μ φ . The posterior prob-
ability of failure decreases slightly compared to the prior due to the combined effect of the
reduction of the mean and of the uncertainty of the random variables. The posterior reli-
ability index is β″ = 3.01.

5.5.4 updating with survived loading conditions
We now assume a different scenario: the foundation survived an extreme loading condition
caused by a wind speed of V m = 40 m/s, which is significantly higher than the design 50-year
return period wind speed. We can use this information to update the distribution of the

Table 5.3 Prior and posterior of the mean and COV of the soil properties and probability of
failure given CPT measurement outcomes

Prior Posterior

Parameter Mean COV Mean COV

Mean friction angle μ φ (°) 37.40 15.5% 35.41 5.4%

1.43 114.6% 1.32 78.6%

Mean uplift coefficient μ K

Probability of failure - -

1.70 × 10 − 3 1.31 × 10 − 3