Environmental Engineering Reference
In-Depth Information
1. 17
Range of negative operating time values.
t
2
∫
ftdt Pt T t Ft Ft Qt Qt
()
= << =
(
)
( )
()
−
( )
=
().
−
1
2
2
1
2
1
t
1
After transition from the random variable
T
= {
t
} to the value
X
= {
x
}:
x
2
= (
t
2
-
T
0
) /
S
;
x
1
= (
t
1
-
T
0
) /
S
,
this leads to
2
t
∫
ftdt Qt Qt
( )
=
(
)
−
(
)
=
0.5
+Φ
(
x
)
−
0.5
−Φ
(
x
)
=Φ
(
x
)
−Φ
(
x
).
2
1
2
1
2
1
t
1
and, therefore, the normalisation factor
c
is equal to:
1
c
=
Φ
.
( )
x
−Φ
()
x
2
1
Since [Φ(
x
)(
x
2
)-Φ(
x
)(
x
1
)] < 1, then
c
> 1, so that
f
(
t
) >
f
(
t
). Curve
f
(
t
) is higher than
f
(
t
), since the areas under the curves
f
(
t
) and
f
(
t
) are
the same and equal to 1 (Fig. 1.18):
TS
+
3
t
0
2
∫
∫
f t dt
()
=
f t dt
()
TS
−
3
t
0
1
Reliability indices for the TND in the range (
t
1
,
t
2
):
f t cf t cf x S
()
=
()
( )/ ;
=
∞
∞
[
]
∫
∫
P t
()
=
cf t dt c f t dt c
()
=
()
0.5
( );
=
x
−Φ
t
t
[
]
Qt
( )
= −
1
c
0.5
+Φ
( ) ;
x
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