Civil Engineering Reference
In-Depth Information
eters
(
t
) and
u
. When using the Gamma process for modeling deteriora-
tion, it is assumed that the scale parameter remains constant throughout
the process while the shape parameter is a right continuous, non-decreasing,
real-valued function (Noortwijk, 2009).
ν
15.4.4 Trend-dependent
In this approach, graceful deterioration is modeled by a stochastic process
in which shocks are assumed to be independent and equally distributed;
and the time between shocks is deterministic but not necessarily equally
spaced (Sánchez-Silva
et al.
, 2011). Let's defi ne
B
i
as the loss of capacity
caused by a 'progressive deterioration shock'
i
, and assume that these
shocks are independent and identically distributed with
W
B
(
x
)
=
P
(
B
≤
x
).
At a given time
t
and in a cycle
(
t
), the accumulated damage
S
(
t
) can be
computed as:
()
∑
et
()
=
St
B
⋅
1
,
i
=
23
, ,...,
[15.18]
{
}
j
Zt
≤<
Z
()
()
+
j
=
0
t
t
1
where
e
(
t
) is the number of fi ctitious shocks, simulating progressive deterio-
ration, that have occurred by time
t
−
Z
(
t
)
. Note that
e
(
t
) is not a random
variable but a known value; i.e.,
e
(
t
)
∈
. Then, the structural capacity in
cycle
(
t
) by time
t
becomes:
(
)
=
()
Vtk
,*
v
−
k
*
−
St
[15.19]
()
−
t
1
which results by subtracting, from the structural capacity available (i.e.,
v
(
t
)−1
k
*), the deterioration caused by progressive deterioration in the
form of small shocks occurring at fi xed times, i.e.,
S
(
t
).
Because shocks that simulate progressive deterioration are evaluated at
fi xed times, it is possible to simulate possible trends of the process (e.g.,
linear or exponential decay). Then, if the occurrence times are governed by
any deterministic function
g
(
t
), equation 15.19 becomes
−
(
)
=
()
Vtk
,*
u
−
k
*
−
S t
[15.20]
()
−
t
1
g
where
S
g
(
t
) is the total damage caused by progressive deterioration as a
result of jumps that will occur by time
t
, according to function
g
(
t
). The
number of jumps
e
g
(
t
) is computed based on the selection of the time
between shocks, i.e.,
Δ
t
, which, in turn, is computed by solving:
(
)
−
()
=>
gt
+
Δ
t
gt
ct
;
0
[15.21]
where
c
is a constant. For example, for
g
(
t
)
=
t
, jumps are equally spaced,
while for
g
(
t
)
(
t
) gets smaller as time increases. In this approach,
shocks are assumed to be
iid
and, in principle, the shock distribution can
=
exp(
α
t
),
Δ
Search WWH ::
Custom Search