Verification of geotechnical reliability using load tests and integrity tests - Risk and Reliability in Geotechnical Engineering - page 530

in which q and r are the parameters of beta distribution and B () is the beta function. The
beta distribution is a conjugate distribution of the binomial distribution. That is, if the prior
p d is a beta distribution, the updated p d is still a beta distribution (Raiffa and Schlaifer
2000). In this case, the updated PDF of p d is a beta distribution with parameters q ″ = q + m
and r ″ = r + n − m .
If no information on p d is available, then a diffuse prior for f ′( p d ) may be used. In this
case, one can obtain

p

d m

( 1 − p )

nm −

d

fp ′′ () =

d

(14.21)

1

∫

p

d m

( 1 − p )

nm −

d p

d d

0

14.4.3 updating mean thickness of toe debris
The thickness of toe debris, x , is also assumed to follow an exponential distribution with the
mean thickness t as a parameter

1 x

t

fxt (|) = exp − , x ≥ 0 (14.22)

t

For mathematical convenience and simplicity, in the absence of any information, the
inverted gamma distribution, which is the conjugate distribution of the exponential distri-
bution, may be taken as the prior distribution of t (Raiffa and Schlaifer 2000)

κ + 1

ν
κ

k

1 ν

′

(14.23)

ft () = exp − , t ≥ 0

Γ () t t

in which Г() is the gamma function and κ and ν are the parameters of the distribution. When
results from on-site interface coring are available, the PDF of t can be updated using the
Bayesian approach. The posterior PDF of t is also an inverted gamma distribution with param-
eters κ ″ = κ + m and ν ″ + Σ i m x 1 (Raiffa and Schlaifer 2000) in which m is the number of

piles found to be defective and x i is the observed toe debris thickness in the i ith defective pile.

= ii

14.4.4 Cases of test outcome
Three cases of test outcome are considered in this chapter: (1) no toe debris is detected, (2)
toe debris is detected without repair, and (3) toe debris is detected and repaired.
Suppose n piles are selected randomly from a site for interface coring. If the construction
quality at the site is excellent and no toe debris is found in the sampled piles, then no actions
need to be taken after the tests. In this case, the occurrence probability of toe debris will
be updated after the tests, and the updated occurrence probability will be smaller than the
estimated prior value. Since no information on the thickness of toe debris is obtained, the
distribution of thickness of toe debris cannot be updated. According to Equation 14.15 ,
the updated probability of unsatisfactory performance of the pile will be smaller than that
before the integrity tests. The reliability of the piles will be improved although no remedial
actions are taken.
If n piles are selected randomly from a site for interface coring and m out of these are
found to contain toe debris, then a decision on immediate repair or replacement must be