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be selected arbitrarily. However, it is suggested that
E
[
B
i
]
k
*)/
e
g
(
T
)
;
where
e
g
(
T
)
is the number of shocks by time
T
(i.e., time-window for the
study) according to the trend function
g
(
t
).
=
(
v
0
−
15.4.5 Comparison of graceful deterioration models
Consider a structure that deteriorates continuously with time; for example
a concrete sewage pipe subjected to bacterial attack. The deterioration
process can be approximated by a discrete shock-based approach according
to the models presented in previous sections. Consider that the pipe is put
into service at time
t
=
0 with an initial capacity
v
0
=
100 (capacity units);
the minimum acceptable capacity is
k
*
25. Three models for graceful
deterioration that follow a Gamma process are presented in Fig. 15.5. The
cases shown were constructed for a scale parameter and the following time-
dependent shape parameter functions:
v
(
t
)
=
0.125
t
2
and
v
(
t
)
=
5.25
t
,
v
(
t
)
=
=
exp(0.11
t
).
Consider now the case in which shocks are
iid
and occur at fi xed time
according to a function
g
. In Fig. 15.6 three sample paths of the deteriora-
tion history process are shown. Every process corresponds to a different
trend
function
g
. It can be clearly observed that the times at which the
random increments are evaluated depend on the trend function. Thus, they
80
Δ
t
70
60
Fix time intervals
50
Δ
t
Δ
t
40
30
v
(
t
) = 5.25
t
v
(
t
) = 0.125
t
2
20
10
v
(
t
) = exp
0.11
t
0
0
5 0 520
25
Time
30
35
40
45
50
15.5
Gamma model for graceful deterioration (adapted from Sánchez-
Silva and Riascos-Ochoa, 2012a,b).
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