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Title Monte Carlo simulation of extreme traffic loading on short and medium span bridges
Authors(s) Enright, Bernard; O'Brien, Eugene J.
Publication date 2013-12
Publication information Structure and Infrastructure Engineering, 9 (12): 1267-1282
Publisher Informa UK (Taylor & Francis)
Item record/more information http://hdl.handle.net/10197/4868
Publisher's statement This is an electronic version of an article published in Structure and Infrastructure
Engineering (2013) 9(12): 1267-1282. Structure and Infrastructure Engineering is available
online at: www.tandfonline.com, DOI: http://dx.doi/org/10.1080/15732479.2012.688753.
Publisher's version (DOI) 10.1080/15732479.2012.688753
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Monte Carlo Simulation of extreme traffic loading on short and medium span bridges
Bernard Enright
1
and Eugene J. OBrien
2
1
Dublin Institute of Technology, Ireland
2
University College Dublin, Ireland
ABSTRACT
The accurate estimation of site-specific lifetime extreme traffic load effects is an important element
in the cost-effective assessment of bridges. A common approach is to use statistical distributions
derived from weigh-in-motion (WIM) measurements as the basis for Monte Carlo simulation of
traffic loading over a number of years, and to estimate lifetime bridge load effects by extrapolation
from the results of this simulation. However, results are sensitive to the assumptions made, not just
with regard to vehicle weights but also to number of axles, inter-axle spacings and gaps between
vehicles. This paper carries out a critical review of the assumptions involved in the process. It
presents a comprehensive model for Monte Carlo simulation of bridge loading for free-flowing
traffic that can be applied to different sites, and shows how the model matches results obtained from
extensive sets of WIM measurements for highway sites in five European countries. The model
allows for the simulation of vehicles which are heavier and have more axles than those recorded in
the WIM data, and uses techniques for modeling axle configuration that can be applied to any type
of vehicle. The model presented in this paper has been optimized to allow the simulation of 1000 or
more years of traffic and this greatly reduces the variance in the process of calculating estimates for
lifetime loading from the simulation model. Using this approach, it is possible to analyze the type of
loading scenarios that cause the maximum lifetime load effects. Conclusions can be drawn about
the type of vehicles likely to be involved in maximum lifetime loading scenarios, and the results
highlight the importance of special vehicles in bridge loading. The approach described here does
not remove the uncertainty inherent in estimating lifetime maximum loading from data collected
over time periods which are much shorter than the bridge lifetime.
Keywords: Bridge, assessment, weigh-in-motion, Monte Carlo simulation, traffic loading, special
vehicles
2
1. INTRODUCTION
In recent years, the improved quality and increasing use of weigh-in-motion (WIM) technology [1]
has meant that more accurate measurements of vehicle weights are now available for periods
covering many months or even years of traffic at selected locations. These extensive measurements
can be used to refine probabilistic bridge loading models for the assessment of existing bridges, and
to monitor the implications for bridge design of trends in vehicle weights and types. Codes of
practice for the design of bridges such as the Eurocode [2] must be sufficiently general to be
applicable to many different bridge types with widely varying traffic loading conditions [3,4].
Assessment codes are also general, and in many cases may be conservative [5] despite the fact that
bridge maintenance is expensive, and bridge owners need to allocate limited resources efficiently.
Site-specific assessment, based on measured traffic, can lead to very significant cost reductions for
maintenance [6], and the application of site-specific models for bridge assessment has been widely
studied [5,7-9].
European and North American codes are based on relatively small amounts of data collected some
years ago. The U.S. and Canadian codes are based on data collected in Ontario in 1975 for 9250
trucks [10,11]. The Eurocode [2] was initially based on a number of weeks of data from Auxerre in
the 1980s [3,12], and was confirmed using data from a number of French sites in 1997 [13].
Changing truck weights [14], composition of traffic, and vehicle sizes all have implications for
bridge loading, and codes need to be periodically re-calibrated based on current traffic. This study is
based on WIM data collected between 2005 and 2008 for 2.7 million trucks at five European sites –
in the Netherlands, Slovakia, the Czech Republic, Slovenia and Poland. Codes segregate normal
legal traffic (with some allowance for illegal overloading) from special vehicles which require
permits [2,15]. Special vehicles are very important for bridge loading [5,16], and the model
developed for this study seeks to include all vehicles, normal and special, that are likely to cross a
bridge at full highway speed during its lifetime.
It is necessary to estimate as accurately as possible the probable maximum bridge load effects
(bending moments, shears) over a selected lifetime. For assessment, this can be 5 to 10 years [17],
whereas for design the U.S. AASHTO code is based on the distribution of the 75-year maximum
loading [10]. The Eurocode [2] for the design of new bridges is based on the distribution of the 50-
year maximum, and the characteristic load is calculated as the value with a 5% probability of being
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exceeded in the 50 year lifetime, which is approximately equivalent to the value with a return
period of 1000 years. Even with the relatively large amounts of truck data gathered for this study, it
is still necessary to extrapolate from the measured data to calculate estimates of lifetime maximum
bridge loading. This is true regardless of the particular method adopted. One approach is to fit a
statistical distribution to the calculated load effects for the measured traffic, and to use these
distributions to estimate characteristic lifetime maximum effects [10,18,19]. An alternative
approach adopted by many authors is to use Monte Carlo (MC) simulation [20-22], and this is the
approach adopted here.
In the development of the U.S. AASHTO code for bridge design [10], load effects were calculated
for the measured trucks on different bridge spans and plotted on Normal probability paper. The
curves were extrapolated to give estimates for the mean 75-year load effect, and the coefficient of
variation was estimated by raising the distributions to a power based on typical truck volumes. This
process requires a significant degree of engineering judgment and subjectivity, as noted by Kulicki
[4], Miao and Chan [19] and by Gindy and Nassif [23] who report variations in estimated lifetime
maxima of up to 33%. In the development of the Eurocode [2], traffic measurements were collected
over some weeks at different times, and a number of different extrapolation techniques were
applied. Multimodal Normal and Gumbel distributions were fitted to measured load effects for
individual loading events, and the Gumbel extreme value distribution was fitted to periodic maxima
calculated from simulation. Dawe [12] reports variations of up to 20% between results from the
different approaches used in the development of the Eurocode.
In the Monte Carlo simulation approach, statistical distributions for vehicle weights, inter-vehicle
gaps and other characteristics are derived from the measurements, and are used as the basis for the
simulation of traffic, typically for some number of years. It is thus possible to simulate vehicles and
combinations of vehicles that have not been observed during the period of measurement. Lifetime
maximum load effects have usually been estimated by extrapolating from the results of the
simulation. Cooper [24] uses the Gumbel extreme value distribution for extrapolation, whereas the
Generalized Extreme Value (GEV) distribution is applied by Caprani et al. [25] for simulations of
up to five years of traffic, and by James [26] who notes its sensitivity to changes in the shape
parameter. The GEV distribution incorporates the Gumbel, Fréchet (unbounded) and Weibull
(bounded) distributions. Fitting a distribution to the full data set of periodic maxima can give
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excessive emphasis to loading scenarios which do not make any significant contribution to the
characteristic value. Castillo [27] recommends fitting the Gumbel distribution to the upper tail of
the data (by selecting the top
n2
data points), but this risks placing excessive emphasis on a small
number of extreme cases. Crespo-Minguillón and Casas [28] and James [26] use the peaks over
threshold extreme value approach, while Cremona [29] adopts the Rice level-crossing technique. In
this paper, the MC model has been optimized to make it practical to simulate thousands of years on
a conventional desktop computer, and if the simulation is run for a sufficiently long time, the
lifetime maximum load effects can be found directly from the results of the simulation. Running the
simulation for 1000 years and fitting a distribution for smoothing purposes is more efficient than
simulating many thousands of years, and is found to give sufficiently good estimates. Using long-
run simulations avoids the problems of extrapolating from short simulation runs, and gives much
more consistent results compared with existing MC simulation approaches.
In order to simplify the simulation process, various restrictions are often placed on the traffic model
used – some authors specify a maximum value for vehicle weights, and many use a limited set of
vehicle classes with a fixed maximum number of axles [20,30-32]. Some employ limited modeling
of inter-vehicle gaps [10,18,30]. Vehicle models are typically based on existing vehicle types only,
without attempting to extrapolate for vehicle types other than those recorded [24]. The approach
used here is to build a detailed MC model, without any restrictive assumptions, and to calibrate it
against extensive WIM data collected at five European sites. The model is designed to extrapolate
both vehicle weights and types (axle configurations), and while this extrapolation is based on
assumptions which will influence the results, it is considered to give a more realistic estimate of
lifetime loading than previous MC models.
Estimating lifetime loading from short periods of measured or simulated data does not give a clear
idea of what types of trucks are likely to be involved in lifetime maximum loading events. Long-run
simulations provide examples of the types and combinations of vehicles that might be expected to
feature in extreme bridge loading. This is useful in identifying the relative importance of factors
such as gross vehicle weight (GVW), the weights of individual axles and of groups of axles,
wheelbase, and axle layout. This in turn may help in identifying useful legal restrictions on truck
types. Simulating 1000 years or more of traffic also makes it possible to model extremely rare
events such as one exceptionally heavy truck or a number of extremely heavy trucks meeting on a
