Civil Engineering Reference
In-Depth Information
t
()
=
()
∫
()
Vt
v
−
Q
t
−
1
δτ τ
d
[15.22]
()
t
()
−
t
1
Z
()
−
t
Progressive deterioration increases the probability of failure by reducing
the shock size required to exceed the failure or intervention threshold
value. The lower limit of the integral in equation 15.17 changes to:
−−
()
vsgt
d
*
0
(
)
=
∫
[
()
−
]
()
()
n
[15.23]
Vtns
,,*
v
−
s
*
−
g t
x G
d
x
0
d
0
Typical forms of
g
d
(
t
) are linear, cuadratic or exponential (Mori and
Ellingwood, 1994; Pandey, 1998). More elaborate models have been pro-
posed to include the randomness of progressive deterioration (Streicher
et al.
, 2008; Klutke and Yang, 2002). In general these can be classifi ed into
two broad categories, the random variable (RV) model and the stochastic
process model. In the RV model, the probabilistic nature is included as
random parameters of the deterministic function of deterioration
g
d
(
t
). In
the other hand, the stochastic process models, that include Markov chains,
continuous-time Markov chains or the Gamma process model, incorporates
the temporal uncertainty of the deterioration process and do not have a
specifi c sample path associated (Pandey
et al.
, 2009).
15.5.2 Illustrative example
Consider a structure that is put into service at time
t
=
0 with an initial
capacity
v
0
0. It will be
assumed that strutcural deterioration results from the combined action of
shocks and progressive degradation. Shocks occur according to a Poisson
process with
=
100 and minimum acceptable capacity of
k
*
=
0.1 and with
damaging intensity
that follows a lognormal
distribution with
λ
=
μ
=
10 and
σ
=
5. If damage accumulates due to shocks
only, the structure's MTTF
106 years with standard deviation of 36 years.
On the other hand, three deterministic models were used to represent
graceful deterioration:
g
p
1
(
t
)
=
0.0008
t
3
. All three
models have in common that their time to failure is 50 years. In addition to
these three models, a RV linear deterioration model
g
d
(
t
)
=
2
t
,
g
p
2
(
t
)
=
0.04
t
2
and
g
p
3
(
t
)
=
Mt
was also
considered in the analysis. In this model
M
is a random variable that refl ects
the inherent variability of the process.
M
was assumed to be gamma dis-
tributed with shape and scale parameters:
=
η
and
δ
respectively. These
parameters were chosen so that MTTF
50 years; furthermore, three COV
(0.3, 0.6 and 0.9) of the time to failure were also considered (Pandey
et al.
,
2009) to evaluate the sensitivity of the results to the variability in
M
. The
parameters of the Gamma distribution for the three cases mentioned above
are shown in Table 15.1.
=
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