r - The Hong Kong Polytechnic University

fatigue life corresponding to a target fatigue reliability of 1.65 is also presented in the table. As expected, the
predicted fatigue life is longer than that if we assume the road roughness is poor or very poor and shorter than
that if we assume the road roughness is good or very good. For the current modeling of the vehicle speed and
road surface deteriorations, the fatigue life of the bridge components is comparable with the case with a 60m/s
vehicle speed and an average road-roughness condition.
ESTIMATE THE EXTREME STRUCTURAL RESPONSE
This section is focus on the estimation of the extreme structural response based on monitoring data. At present,
the live load applied to calculate the reliability index was computed based on the live load models that were
developed by Nowak (1993) and used in the calibration of the early version of AASHTO LRFD Bridge Design
Specifications (AASHTO 1994). The models were derived from the available statistical data on 9,250 selected
truck surveys, and weigh-in-motion measurements. For a structure with a given capacity, its reliability index is
only related to the maximum load effect distribution corresponding to the structure’s service life. Assuming a
normal distribution for the individual truck load, the maximum live load effects (moment or shear) for various
time periods were determined by extrapolation as follows.
Let the live load effects following certain distribution Ω, and the number of trucks in the surveying interval
is . Then, the total number of trucks passing through the bridge in an expected service life ,
will be
(13)
The maximum live load effects
corresponding to any expected bridge service life is
where is the inverse of the standard distribution function. According to AASHTO specifications, the
expected service life for a new bridge is 75 years.
According to Orcesi and Frangopol (2010), the extreme value distribution of SHM data is assumed to approach
a Gumbel probability distribution (Gumbel 1958).
(15)
where is the cumulative distribution function of Gumbel probability distribution, both μ and σ are
constants to be determined from the measured data. Thus, the extreme values of the SHM data in a mean
recurrence interval, , ( is longer than the monitoring period), can be predicted as
(16)
Using these methods to estimate the extreme value of load effects in a mean recurrence interval, the information
of the number of the trucks running through the bridge must be available. However, sometimes it is not easy to
identify the number of trucks only by dealing with the recorded strain data. Even the number of the trucks is
known, some cases whose maximum structural response is induced by multiple presences of vehicles side by
side or one after another in a same span are still excluded in the reliability calculation.
The aim of this part is to develop a methodology that can establish the maximum live load effects distribution
for a mean recurrence interval with extreme value theories based on short-term monitoring data of structural
response.
Extreme Value Theory
Since only the maximum structure response is concerned, it is reasonable to use extreme theories to estimate the
long-term maximum response from the short-term records of structure response monitoring. In this study, the
Gunbel distribution is used to model the extreme values of long (finite) sequences of independent, identically
distributed random variables. Gumbel distributions also known as Type I extreme value distribution within the
extreme value theory. Let the variable be the maximum of independent random variables . Since
the inequality implies for all i , it follows that
The probabilities are referred to as the initial distributions of the variables . In the particular case in
(14)
(17)
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