Assessing the value of information to design site investigation and construction quality assurance programs - Risk and Reliability in Geotechnical Engineering - page 493

(a) 0.0040

θ = 1.0 kPa/m

0.0035

θ = 1.3 kPa/m

0.0030

0.0025

θ = 1.6 kPa/m

0.0020

0.0015

0.0010

0.0005

0

0 500 1000 1500 2000

Pile capacity (kN)

Pile length = 30 m

(b) 0.0030

θ = 1.0 kPa/m

0.0025

θ = 1.3 kPa/m

0.0020

θ = 1.6 kPa/m

0.0015

0.0010

0.0005

0

0 500 1000 1500 2000

Pile capacity (kN)

Pile length = 35 m

Figure 13.14 Conditional prior probability distributions for spatial variability in pile capacity given different
model parameter values for alternative pile lengths in pile foundation design.

where φ(.) is the standard normal probability density function and d ε i is a negligibly small
interval about the measurement result (note that the magnitude of this interval is not sig-
nificant since it does not affect the relative likelihood). This likelihood function is shown
in Figure 13.16 for an example test result. Note that since it is the relative likelihood that
controls the “informativeness” of the function via filtering, the likelihood values are shown
relative to the maximum likelihood. The resulting updated probability distribution for the
model parameter obtained from Equation 13.9 is shown in Figure 13.17 , and the resulting
updated probability distributions for the consequences obtained from Equations 13.10 and
13.11 are shown in Figure 13.18 . The preferred pile length based on this load test result is
now the shorter pile ( Figure 13.18 ).
Next, consider the possibility of performing multiple pile load tests. If n tests are per-
formed and the results are statistically independent between piles, then the likelihood func-
tion is given by

ε − µ

∏∏

i R θ

P (, , ε

1 …= ==

ε θ ) P r ( ε ) φ d ε (13.16)

n i i

σ

R θ

all ε all ε

i i