Assessing the value of information to design site investigation and construction quality assurance programs - Risk and Reliability in Geotechnical Engineering - page 492

1.0
0.9
0.8
0.7
0.6

0.5
0.4
0.3

0.33 0.33 0.33

0.2
0.1
0.0

0.9 1.3
Gradient of unit side shear versus depth, θ (kPa/m)

1.6

Figure 13.13 Prior probability distribution for model parameter in pile foundation design.

RAxial PileCapacity 02 where the mean capacity is a function

of an uncertain gradient of unit side shear versus depth, Μ

= ∼Μσ N ( , = . × Μ ),

RR R

/ where c is
the pile circumference of 1.2 m, l is the pile length, and Θ is the gradient of unit side shear
versus depth. Figure 13.13 shows the prior probability distribution for the gradient of unit
side shear versus depth, P (

= ( 12 × ×× cl

2

) Θ

R

Θ= and Figure 13.14 shows the possible conditional prob-
ability distributions for axial pile capacity given the possible values for the model parameter
(i.e., gradient of unit side shear vs. depth). The axial loads on the piles are also uncertain and
modeled with a normal distribution: S

),

= xial Pile Load = N ( µ = 400 kN , σ = 60 kN ).

S S

The expected consequence for a given pile length is obtained as a function of the cost of
construction for that length, plus the expected cost of failure:

[ ] ×>

EConsequencefor Pile Length l ( ) = ostl () + 10 × Cost l () PSRl ( ) (13.13)

where Cost ( l ) is the construction cost, the cost of failure is 10 times the construction cost,
and PS

( > Rl ) is the probability of failure (load exceeds capacity) for that pile length

µ − µ

∑ θ

R θ S

PS ( > Rl ) = Φ − P ( Θθ = ) (13.14)

σσ

2

+

2

all S

R θ

where Φ(.) is the standard normal cumulative distribution function, µ = ( 12 / × ×× cl

2

) θ ,

R θ

02. . The resulting prior probability distributions for the consequences
obtained from Equation 13.11 are shown in Figure 13.15 . For the prior decision, the lon-
ger pile is preferred with an expected cost of $6550 per pile versus the shorter pile with an
expected cost of $6870 per pile ( Figure 13.15 ).
A possible alternative is to perform pile load tests before deciding between the shorter or
longer pile lengths ( Figure 13.12 ). First, consider the possibility of a single-load test. The
probability of measuring a single capacity, ε 1 , is given by

and σ =× µ

R | θ R | θ

ε − µ

ε

1 R θ

P ( ε | θ ) === P r ( ε ) φ d (13.15)

1 1

σ

1

R θ