Environmental Engineering Reference
In-Depth Information
and the normalizing constant, α, is
−1
∞
∫
n
m
m
(
nm
−
)
−≥
(14.10)
α
=
px
(
≥
x
)
1
p xx
(
)
f
′
()
µ µ
d
T
µ
T
µ
−∞
Accordingly, the updated distribution of the mean of the bearing capacity ratio is
n
m
m
(
nm
−
)
−≥
f
′′
()
µα
=
px
(
≥
x
)
1
p xx
(
)
f
′
()
µ
(14.11)
T
µ
T
µ
Equation 14.11
has two special cases. When all tests pass (i.e.,
m
=
n
),
Equation 14.11
reduces to
Equation 14.4
.
If none of the tests passes (i.e.,
m
= 0), a dispersive posterior dis-
tribution will result. The updated distribution of the bearing capacity ratio can be obtained
using
Equation 14.8
.
14.3.3 Proof load tests conducted to failure
Let us now consider the proof load tests carried out to failure, that is, cases the pile capacity
values are known. To start with, the log-normally distributed pile capacity assumed previously
is now transformed into a normal variate. If the test outcome is a set of
n
observed values
representing a rando
m
sample following a normal distribution with a known standard devia-
tion σ and a mean of
x
, then the distribution of the pile capacity can be updated directly using
Bayesian sampling theory. Given a known prior normal distribution,
N
μ
(μ
′
, σ
′
), the posterior
density function of the bearing capacity ratio,
fx
X
″
()
is also normal (e.g., Ang and Tang 2007):
fx
X
′′
()
=
N
µσ
(,
′′
′′
)
(14.12)
X
XX
where
x
()
σµσ
σ
′
2
+ ′
(
2
/
n
)
µ
′′ =
(14.13)
X
()
2
+
(
σ
2
/
n
)
′
σσ
σ
σ
(
′
2
/
n
)
(14.14)
′′ =
1
+
X
()
2
+
(
σ
2
/
n
)
′
14.3.4 Multiple types of tests
In many cases, multiple test methods are used for construction quality assurance at a single
site. The construction control of driven H-piles in Hong Kong (Zhang et al. 2006b) and the
seven-step construction control process of Bell et al. (2002) are two examples. In the Hong
Kong practice, both dynamic formula and static analysis are adopted for determining the pile
length. Early test piles or preliminary test piles are required at the early stage of construction
to verify the construction workmanship and design parameters. In addition, not fewer than
10% of working piles may be tested using high-strain dynamic tests, for example, using a pile
driving analyzer (PDA). One-half of these PDA tests should be analyzed using a wave equa-
tion analysis such as the CAse Pile Wave Analysis Program (CAPWAP). Finally, at the end
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