Environmental Engineering Reference
In-Depth Information
C:Cost ($1000):
Design OK
20-0.4
x
*+0+0.1
n
P
(
X
≥
x
*)
x
*
E
(
C
|
x
*) =20-0.4
x
*
+
P
(
X
<
x
*)×100
+
0.1
n
No QA/QC tests
Design fails
x
*: Design
strength
20-0.4
x
*+0+0.1
n
P
(
X
<
x
*)
Des ign OK
20-0.4
x
*+0+0.1
n
P
(
X
<
x
* |
n
,
x
)
x
*
E
(
C
|
x
*,
n
,
x
)=20-0.4x*
+
P
(
X
<
x
*|
n
,
x
)×100
+
0.1
n
X
=
x
Design fails
x
*: Design
strength
Run n QA/QC
tests
n
20-0.4
x
*+ 100+0.1
n
X
: Sample
mean
P
(
X
<
x
*|
n
,
x
)
n
: number
of tests
0.10
Prior total
distribution of X
X
~
N
(μ′
X
T
, σ′
X
T
)
0.10
Updated total
distribution of X
|
x
X
|μ
X
~
N
(μ
X
, σ
X
)
σ
X
= σ
X
/
÷n
0.08
0.08
x=40 kPa, n=10
μ
X
= μ
X
~
N
(μ′
μ
, σ′
μ
)
0.06
μ′′
T
X
= 41.5
kPa
σ′′
T
X
= 10.4
kPa
0.06
μ′
X
T
= μ′
μ
= 8.7 kpa
σ′
X
T
=
÷
ææææ
Prior total
distribution of X
n
=10
0.04
0.04
σ
X
2
+σ′
μ
2
μ′
T
X
= 50 kPa
σ′
T
X
= 12.5 kPa
σ
X
= 4.5 kpa
σ′
X
T
= 8.7 kpa
0.02
0.02
0.00
0.00
0
20
40
60
80
100
0
20
40
60
80
100
x
, μ
x
(kPa)
x
(kPa)
Figure 13.26
Preposterior decision tree to design QA/QC program for testing compacted fill.
The preposterior decision tree to design the QA/QC program for the compacted fill is
shown in
Figure 13.26
.
The expected cost for a given value of n can then be obtained as
follows:
X
=+∞
∫
ECn
(
|
)
=
E Cn x
(|,)
×
f
()
xdx
(13.25)
Min
X
X
=−∞
where
100
204
2
EC nx
(|,)
=−×
[
20
04
.
(
xnx
*
|
,
)]
+ ×−
×
100
Φ
ln
−
ln(
σσ
µ
2
+
"
2
)
+
01
.
n
Min
Opt
X
π
.
2
(13.26)
and
(
)
×
100
204
2
−
xnx
Opt
∗
|,
=− +
µ
″
σ
2
σ
″2
ln
ln(
σσ
2
+
″2
)
(13.27)
X
X
π
×
.
2
µ
µ
µ
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