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 X ~ N 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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