Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 247

where W is the weight of the foundation and enclosed soil, and Q su is the side resistance (see
Figure 5.7 ) . Q su is evaluated applying the effective stress model (Kulhawy et al. 1983) by

D

0

QpKz = () σ ′ ( )tan z ϕ () , z dz (5.45)

su b

where p b = 4 B is the perimeter of the foundation, φ( z ) is the drained friction angle, σ′( z ) = z γ
is the effective stress, γ is the unit weight of the soil, and K ( z ) is the operative horizontal
stress coefficient. The friction angle and horizontal stress coefficient are spatially variable
quantities. We adopt the common (but possibly unconservative) approach discussed in
Section 5.3.1 and instead of explicitly modeling the quantities φ and K as random fields,
we model their mean values μ φ and μ K as random variables. The spatial variability of the
unit weight γ is neglected, that is, γ is assumed to vary uniformly within the soil profile.
Equation 5.45 can then be written as

γµ µ ϕ

Q = 2 D tan. (5.46)

su K

5.5.1 Prior probabilistic model
The wind speed V is modeled by the Gumbel distribution with a COV of 30% and a
mean value of 16.88 m/s corresponding to a 50-year return period wind speed of 30 m/s.
In sandy soils, the mean friction angle μ φ is expected to vary between 30° and 45°. These
values are taken as the 10 and 90% quantiles of μ φ . We model μ φ with the β-distribution
with bounds 10° and 80°. The parameters of the beta-distribution are evaluated by match-
ing the 10 and 90% quantiles to the aforementioned values, for which we obtain a mean
value of 37.40° and a standard deviation of 5.80°. The mean horizontal stress coefficient
μ K typically depends on the mean friction angle and can vary between K 0 = 1 − sinμ φ and
(2/3) K p = (2/3)tan 2 (45 + μ φ /2), where K 0 is the in situ coefficient of horizontal stress and K p
is the coefficient of passive soil stress (Kulhawy et al. 1991). K 0 and (2/3) K p are taken as
the 10 and 90% quantiles of the distribution of μ K conditional on μ φ and are used to fit the
parameters of the lognormal distribution. Hence, the conditional distribution of μ K given
μ φ is obtained as

2

1 1

2

ln µλ

ζ

−

K

f ( µµ | ) = exp − , (5.47)

µ K ϕ

K

µζ π 2

K

wherein

µ

2
3

ϕ

2

λ = 05 .lntan 45 + ( 1 − sin µ ), (5.48)

ϕ

2

tan(

2

45
1

+ ( µ

µ

/ 2 ))

2
3

ϕ
ϕ

ζ = 039 . ln . (5.49)

− sin