Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 246

While this rejection-sampling algorithm is easy to implement, it is generally inefficient,
since the acceptance rate is small. In the example shown in Figure 5.10 , the acceptance rate
is 35%, but this rate decreases with increasing dimension (number of random variables)
and with an increasing number of measurements. The BUS approach overcomes this prob-
lem by resorting to structural reliability methods for representing the observation domain
h ( x , p ) ≤ 0}. In Straub (2011), importance sampling is applied for this purpose. Straub and
Papaioannou (2014) demonstrated how subset simulation is effective for solving this prob-
lem. As also demonstrated in Betz et al. (2014a), the algorithm is particularly efficient in
high dimensions, making it suitable for Bayesian updating of models involving random
fields. This is shown in the application presented in Section 5.6.

5.5 aPPlICatIon: FounDatIon oF tranSMISSIon
toWerS unDer tenSIle loaDIng

In this example, we examine a shallow square foundation of a transmission tower embedded
in a sandy soil. We consider two different information sets: cone penetration tests (CPTs) at
the site of the foundation, and the observation that the system survived an extreme loading
condition. We utilize the information to update the distribution of the material parameters
of the soil as well as to update the reliability of the foundation subjected to tensile loading.
The base width of the footing and the depth of the foundation are taken as B = D = 2 m
(see Figure 5.11 ) ; both the base thickness and pedestal width equal 0.6 m. Transmission
towers apply tensile loads to their foundations, mainly due to transient wind loads. The LSF
expressing the performance of the foundation under tensile force can be expressed as

gQ F

=−,

(5.43)

where F is the applied tensile loading and Q u is the uplift capacity of the foundation. It is
noted that for the design of transmission tower foundations, other limit states must also be
checked, but Equation 5.43 is often the critical one (Pacheco et al. 2008). The tensile loading
is modeled as F = kV 2 , where V is a random variable modeling the maximum annual wind
speed and k is a deterministic coefficient incorporating several factors, such as geometry of
the structure and geographical terrain. For simplicity here, we neglect the uncertainty in this
coefficient, it is k = 0.2 tn/m. Considering drained conditions, the uplift capacity of a spread
foundation can be expressed as (Kulhawy et al. 1983)

QWQ
u

=+, (5.44)

su

F

Q su

D

W

B

Figure 5.11 Spread foundation of a transmission tower subjected to the uplift load.