Civil Engineering Reference
In-Depth Information
case is also shown in Fig. 15.3. The results show clearly the importance of
considering damage accumulation. Note that in this particular case, the
probability of failure at any time increases with respect to the case of no
damage accumulation. This implies a large underestimation of failure prob-
ability if damage accumulation is not considered.
In the experiments, it was observed that the effect of COV on the damage
accumulation models is not signifi cant compared with the fi rst case. Although
smaller values of COV result in larger probability of failure, differences are
not signifi cant. For example at
t
50, the probability of failure
P
f
(
t
) is 0.93,
0.92, 0.91 and 0.88 for COVs of 25%, 35%, 50% and 75%, respectively.
For the case of damage accumulation, MTTF takes the values of 23.6,
24.3, 25.1 and 26.6 years for COV
=
25%, 35%, 50% and 75%. It can be
noticed that there is a substantial reduction in the structural lifetime when
damage accumulation is considered. Note also that the structural lifetime
increases with COV. However, in all cases considered, the differences are
not as signifi cant compared with the infl uence of other parameters such as
the occurrence rate of the earthquakes and the parameters of the shock
size distribution.
=
15.3.5 Successive reconstruction
General life-cycle models describe the performance (i.e., deterioration) of
a system or a component throughout its lifetime. In a life-cycle model, once
a structure is put in service damage starts accumulating as a result of pro-
gressive deterioration (e.g., corrosion, fatigue) or shocks (i.e., earthquakes)
until the structure fails; it is then repaired or reconstructed and the process
restarts. A sample path describing the performance of structural system
throughout its lifetime is depicted in Fig. 15.4.
Shock-based damage accumulation
The lifetime of repairable systems consists of successive independent and
stochastically identical cycles. A cycle starts with the system in a condition
'as good as new' and it deteriorates as a result of shocks; once the system
fails, immediate reconstruction (or maintenance) is carried out and a new
cycle starts. Let's defi ne the counting process {
0} as the number of
interventions (i.e., early or full replacement) by time
t
. Then, a cycle is
defi ned as the time between any two interventions; and
Z
(
t
)
denotes the
(
t
),
t
≥
(
t
)-th replacement time (initiation of a cycle). Then, the total amount of
shocks in the
(
t
)-th cycle can be computed as (Sánchez-Silva
et al.
, 2011):
()
Nt
⎧
⎪
⎪
∑
∑∑
⎫
⎪
⎪
()
=
Y
t
1
j
j
=
1
()
()
=
Qt
t
[15.13]
(
)
()
Nt
NZ
t
()
−
()
>
1
Y
−
Y
t
1
j
j
j
=
1
j
=
1
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