Environmental Engineering Reference
In-Depth Information
1.0
0.8
0.6
0.4
D
=
1.2 m
L
=
25 m
L
=
50 m
L
=
75 m
0.2
0.0
0
50
100
150
200
250
300
350
ickness of toe debris (mm)
Figure 14.9
The relationships between pile capacity reduction factor and toe debris thickness.
mobilized toe resistance only consists of a small fraction of the pile capacity. Therefore, the
pile capacity is less affected by the presence of toe debris.
After obtaining the reduction factor, the resulting bias factor λ
RD
for a pile with toe debris
can be calculated using Equation 14.28. Accordingly, the reliability index of the defective
pile can be calculated using
Equation 14.24
and the probability of unsatisfactory perfor-
mance of the pile with a given toe debris,
P
(
F
|
x
), can be calculated based on the calculated
reliability index.
Figure 14.10
shows the relationships between the reliability index and the
toe debris thickness for three pile lengths. The reliability index decreases significantly due
to the effect of toe debris when the bedrock levels are not extremely deep (say, smaller than
50 m). For instance, for a 1.2 m diameter, 25 m long pile, the reliability index decreases from
3.5 when no toe debris is present to 2.11 when a 100-mm-thick toe debris is present. The
reliability index at zero toe debris thickness, 3.50, and the corresponding probability of fail-
ure,
P
(
F
|
Ē
) = 2.33 × 10
−4
, account for other types of imperfections besides toe debris as well
as other sources of uncertainty. The conditional probability of unsatisfactory performance
given the presence of toe debris,
P
(
F
|
E
), can be calculated using
Equation 14.17
.
To do so,
three functions,
P
(
F|x
),
f
(
x|t
), and
f
(
t
) must be available.
P
(
F|x
) can be obtained by curve
fitting based on the data in
Figure 14.10
;
f
(
x|t
) is expressed by
Equation 14.22
;
and
f
(
t
) is
expressed by the inverted gamma distribution shown in
Figure 14.8
and
Equation 14.23
.
The
P
(
F|E
) values are 0.077 for
L
= 25 m, 0.039 for
L
= 50 m, and 0.00088 for
L
= 75 m, and the
updated values are 0.07 for
L
= 25 m, 0.034 for
L
= 50 m, and 0.00087 for
L
= 75 m.
Having determined
P
(
F
|
E
),
P
(
F
|
Ē
), and the Bayesian estimators of
p
d
, the reliability of the
piles at a site before and after the tests can be calculated by
Equation 14.15
for each of the
three cases of test outcome and follow-up actions described earlier. Let us examine the sce-
nario of a diffuse prior
p
d
first. The updated PDFs of the occurrence probability of toe debris
have been shown in
Figure 14.7
.
Figure 14.11
shows the reliability of the 1.2-m diameter
piles of different lengths before and after tests. The reliability after the tests is significantly
higher than that before the tests, particularly for relatively short piles. The effectiveness of
the tests decreases with the pile length. For the 75-m long piles in case 2, the reliability index
only increases from 3.26 before the tests to 3.47 after the tests. The outcome of the tests
also affects the updated reliability as expected. For example, for the 25-m long piles, the
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