Civil Engineering Reference
In-Depth Information
s
t
n
−1
g
(
t
)
where
Δ
α
=
α
t
n
−1
−
α
t
n
−1
(shock deterioration),
Δ
α
=
(d
α
t
/d
t
)|
t
≠
t
i
. This
implies that
Δ
α
g
(
t
) is the rate of change of
α
t
during gradual deterioration
s
t
0
(or
t
≠
t
i
). The value of
Δ
α
=
0 because there is no shock in the time
−
∫
Δ
t
1
()
=
interval [0,
t
−
]. Also we have,
αα
=−
α
g
tt
d
α
. Now combining
t
−
0
0
t
1
0
Eqs. (16.6) and (16.7),
D
t
n
is written as follows:
n
−
1
+
()
∑
DY
=++
α
Δ
α
s
Rt
−
[16.8]
t
t
0
α
n
t
n
n
i
i
=
1
t
()
=
∫
Δ
()
where
Rt
α
g
t t
d
.
α
0
16.3.2 Modeling the effect of deterioration on capacity
Adopting the previously used additive form, the effect of deterioration on
capacity of a system is written as follows:
−
=+ +
t
n
∫
()
CC
Δ
C
s
Δ
Ct
g
d
t
[16.9]
−
1
t
t
t
n
−
1
n
n
−
1
+
t
n
1
where
Δ
C
t
n
−1
=
C
t
n
−1
−
C
t
n
−1
(shock deterioration),
Δ
C
g
(
t
)
=
(d
C
t
/d
t
)|
t
≠
t
i
. This
C
g
(
t
) is the rate of change of
C
t
during gradual deterioration
implies that
Δ
(or
t
≠
t
i
). As explained with respect to
α
i
,
Δ
C
t
0
=
0 and
C
t
0
=
C
0
. The capacity
C
t
is then written in terms of the total change as follows:
(
)
−
∑
Nt
+
()
CC
=+
1
Δ
CRt
s
[16.10]
t
0
C
t
i
i
=
t
()
=
∫
Δ
()
where
Rt
Ct t
g
d
.
C
0
16.3.3 Formulation of failure
As discussed previously, for a given failure mode, failure can be of two
types: excessive demand or excessive deterioration. The failure of the system
due to excessive demand during
n
th
load occurs if [(
C
t
−
−
D
t
n
) < 0] where
(
C
t
−
−
D
t
n
) is given as follows:
n
−
1
(
)
+
[
()
−
()
]
∑
(
)
−+
CD C
−
−= −
α
Y
ΔΔ
C
s
−
α
s
Rt
−
Rt
−
[16.11]
t
0
0
t
Cn
α
n
t
t
t
n
n
i
i
n
i
=
1
Now normalizing with
u
0
=
(
C
0
−
α
0
) and assuming that (
C
0
−
α
0
) > 0 (this
assumption does not result in any loss of generality because
α
0
is an arbi-
trary constant), the failure event is written as follows:
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