Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 248

The prior joint distribution of μ K and μ φ can then be obtained as

f ′ (, µµ µµ µ ) = f ( | )(). f ′

(5.50)

µµ K ϕ µ K ϕµ ϕ

K ϕ K ϕ

The unit weight of the soil γ follows the normal distribution with mean value 19 kN/m 3
and a COV of 5%. The unit weight of concrete that enters the evaluation of the self-weight
of the foundation in Equation 5.46 is assumed deterministic and taken as γ c = 23 kN/m 3 .

5.5.2 reliability analysis based on the prior model
The reliability analysis is performed by application of the subset simulation (Au and Beck
2001). The method is implemented in the standard normal space. The Rosenblatt trans-
formation (Hohenbichler and Rackwitz 1981) is applied to transform the samples of the
independent standard normal distribution into samples of the joint distribution of V, γ , μ K
and μ φ . The computed prior failure probability is Pr( F ) = 1.70 × 10 −3 and the corresponding
reliability index is β′ = 2.93.

5.5.3 updating with CPt test outcomes
We assume that a CPT is performed at the site of the foundation. Typically, data from elec-
tronic CPT tests are recorded at 5-cm depth intervals (Mayne 2007). Here, we assume for
simplicity that five recordings of the tip resistance q t are obtained along the depth of the
foundation. The recordings after normalization read: Q tn 1 = 38.2, Q tn 2 = 43.5, Q tn 3 = 45.8,
Q tn 4 = 35.4, Q tn 5 = 41.1, with Q tn = ( q t / p a )( p a /σ′) 0.5 ; p a is the atmospheric pressure and σ′ is
the effective stress. Each measurement outcome is associated with an additive measurement
error ε m modeled by a zero mean normal random variable with a COV of 5%, which lies
within the typical range of the COV of the measurement error of CPT tests (Orchant et al.
1988; Phoon and Kulhawy 1999). The recordings of the CPT test can be used to update the
joint distribution of μ K and μ φ employing the following correlation between the normalized
tip resistance and the friction angle (Kulhawy and Mayne 1990):

ϕ= 17 6 10 . + . log Q tn +

ε

. (5.51)

10

The zero mean random variable ε t expresses the transformation uncertainty of the corre-
lation model and has a standard deviation of 2.8° (Kulhawy and Mayne 1990). We further
assume that ε t follows the normal distribution. Equation 5.51 describes a point-wise cor-
relation between the friction angle and the normalized tip resistance Q tn . To construct the
likelihood of the measurement of Q tn for learning μ φ , we model the conditional distribution
of φ given μ φ at each point by a beta-distribution with mean μ φ , COV 10%, and bounds 0°
and 90°. The chosen COV agrees with the typical COV of inherent variability of the friction
angle of sandy soils (Phoon and Kulhawy 1999). The likelihood of each measurement Q tni
for learning μ φ can then be expressed as

L () µ = f ( Q | µ )

i ϕ Q tni ϕ

tni

∞ ∞

∫ ∫

= f ( QQf Qf | ) ( | ϕ ) ( ϕµ ϕ | ) dd Q

Qtni tn Qtn ϕ ϕ tn

tni tn

(5.52)

0 − ∞

∞ ∞

∫ ∫

= fQ ( − Qf ) ( ϕ − 17 6 10 . − . og Qf ) ( ϕµ ϕ | ) dd Q tn .

∈ tni tn ∈ 10 tn ϕ ϕ

m t

0 − ∞