Civil Engineering Reference
In-Depth Information
ity can be computed by replacing
k*
by
s*
in equation 15.7. Equation 15.7
depends on the number of shocks that have occurred by time
t
, which is a
random variable. Therefore, removing the condition on the number of
shocks, the instantaneous failure rate can be rewritten as:
∞
∞
⎡
⎣
⎤
⎦
∑
()
=
∫
()
( )
(
)
d
Pt
λ
tGxPN ntt
d
=
,
d
[15.8]
(
)
Vts n
,,
*
n
=
0
n
,
t
) is the probability of having
n
shocks by time
t
. The lower limit of the integral in equation 15.8 is a func-
tion of the initial remaining life and the shock history; then,
where
N
is the number of shocks and
P
(
N
=
vs
−
*
(
)
=
∫
0
(
)
( )
()
n
[15.9]
Vts n
,
*
v
−
s
*
−
x G
d
x
0
0
where
dG
(
n
)
is the
n
th Stieltjes convolution of
G
(
x
) with itself. Then, the
probability that an intervention is required before or at time
t
is given by:
t
(
)
=
()
=
∫
()
P
intervention before
t
Λ
t
d
P t
[15.10]
0
Distribution of time to failure
Assuming independence between failures and disturbances, the distribution
of the time to
n
th shock can be computed in terms of the time required for
the remaining life to fall below
s*
; this is:
∞
∑
[
()
≤
]
=
(
)
(
)
PV t
s
*
Pv
−
s
*
≤
D Pnt
,
[15.11]
0
s
n
=
0
∞
∑
0
vs
−
*
⎡
⎣
⎤
⎦
0
∫
()
()
(
)
=
1
−
d
GxPn
n
,
t
0
n
=
where
P
(
n
,
t
)
n
,
t
) is the probability of having
n
shocks by time
t
(Sánchez-Silva
et al.
, 2011).
=
P
(
N
=
Illustrative example
Consider a structure that is put into service at time
t
=
0 with an initial
25. The structure is located in a seismic region where earthquakes occur
according to a Poisson process with rate
capacity
ν
0
=
100 (capacity units); the minimum acceptable capacity is
k*
=
λ
=
0.1 and with
damaging intensity
that follows a lognormal distribution with
40. For comparison purposes,
several coeffi cients of variation (i.e., COV) were selected: 25%, 35%, 50%
and 75%.
μ
=
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