Civil Engineering Reference
In-Depth Information
permissible value
w
a
. The value of
w
a
is such that if
W
t
>
w
a
then an insig-
nifi cantly small demand in the future will cause failure in mode
k
. The main
difference between the two types of failure is that the former can take place
only during the occurrence of a load whereas the latter does not require a
load. It must be noted that both the failure types correspond to the same
mode of failure. For example, a bridge column might fail due to excessive
deformation demand during an earthquake. The column might also be
considered as failed due to excessive deterioration in deformation capacity
due to corrosion.
For a particular failure mode, the probability distribution of the number
of shocks to failure
n
F
is given by the following equation:
n
⎡
⎢
⎤
⎥
(
)
[
]
=
∩
Pn
>
n
P
C
−
D
≥
0
[16.1]
F
−
t
t
n
n
i
=
1
where
P
[
A
] is the probability of
A
and the symbol
denotes the intersec-
tion of events. It should be noted here that the events [(
C
t
i
−
−
D
t
i
)
≥
0] are
not independent. Therefore,
(
)
(
)
(
)
[
]
≠
PnnPCD
>
⎣
−
≥
0
⎦
×
PCD
⎣
−
≥
0
⎦
××
...
PCD
⎣
−≥
0
⎦
F
−
t
−
t
−
t
t
1
t
2
t
n
1
2
n
[16.2]
The time to failure considering only failures due to excessive demand is
given by
t
n
F
. Defi ning
N
(
t
) as the number of loads or shocks in a timespan
t
, the distribution for
t
n
F
is given as follows:
[
]
=
[
]
()
Pt
>
t
Pn
>
Nt
[16.3]
n
F
F
While in many engineering fi elds,
N
(
t
)
is commonly modeled using a
Poisson process (Ang and Tang, 2007), the stochastic formulation used in
this chapter is applicable for any random counting process. The distribution
of time to failure
t
F
, considering excessive demand and excessive deteriora-
tion, is computed as follows:
[
]
=
{
()
}
∩
Pt
>
t
P n
⎣
>
Nt
W
≤
w
⎦
[16.4]
F
F
t
a
Now, for a system having
k
independent modes of failure, we can write,
k
=
∏
[
]
[
]
=
Pt
>
t
Pt
>
t
[16.5]
F
F
,
i
i
1
denotes algebraic product. In general, for Eq. (16.4), closed-form solutions
are not available and simulations are required. Often, such simulations are
conducted based on the basic 'hit and miss' methods in which the entire
where,
t
F,
i
represents
t
F
for
i
th mode of failure only and the symbol
Π
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