Maintenance, Safety, Risk, Management and Life-Cycle Performance of Bridges –
Powers, Frangopol, Al-Mahaidi & Caprani (Eds)
© 2018 Taylor & Francis Group, London, ISBN 978-1-138-73045-8
Use of stochastic optimization in the analysis of weigh-in-motion data
F. El Hajj Chehade, R. Younes, H. Mroueh & F. Hage Chehade
Laboratory of Civil Engineering and geo-Environment (LGCgE) Lille 1 University, France
Modeling Center Doctoral School of Science and Technology, Lebanon
Lebanese University Faculty of Engineering, Lebanon
ABSTRACT: The calibration of traffic simulation models for bridge design or assessment is normally based
on weigh-in-motion or WIM data, this implies the estimation of a large number of parameters related to load
distribution and vehicle weights. In a similar stochastic problem, the use of genetic algorithm in the optimization process can greatly simplify the work by allowing the calculation of a very large number of parameters
even with a small amount of data and conditions. This method is illustrated by an application example of genetic optimization to evaluate the correlation between the gross vehicle weight and its distribution on single
vehicle axles.
1 INTRODUCTION
1.1 Background
The weigh-in-motion or WIM process is widely used
all over the world, it consists in the estimation of
total weight for a moving road vehicle and of the
proportion carried per each axle by measuring the
dynamic tire forces using specific sensors. Therefore
the distribution of all axle weights can be provided
directly from WIM data. Figure 1 present a redraw
from Burnos & Rys (2017) for a pavement
embedded WIM sensor.
Figure 1. Schema of pavement embedded WIM sensor
The recorded WIM data on a multi-lane carriageway can then be used for the design or the assessment of highway bridges(O’Brien et al. 2006).
Bridge weigh-in-motion or BWIM systems constitute now an alternative way to the pavement-based
weigh-in-motion systems and are considered more
convenient and cost effective(Yu et al. 2016), these
methods rely on measuring strains in some bridge
components to estimate the vehicle weights; consequently, some algorithms have been developed to
deduce axle loads especially Moses’ algorithm.
1.2 Approach
The analysis of the recorded data from the real time
traffic starts by fitting the histograms of measured
data such as Gross Vehicle Weight (GVW), axle
spacing, distribution of GVW, to a known probabilistic distribution which provides the basic parameters in the traffic simulation process.
The distribution of GVW between axles is used to
illustrate the problem as it has a strong influence on
the load effects especially on short to medium span
bridges, from 10 to 35 m (Enright & O’Brien 2013)
Normally, the distribution of GVW should be
performed for each axle in each vehicle class; it requires having the histograms of the distributions for
each axle in all the vehicle classes, so a lot of data.
In this paper we will try to do the same work based
only on the global distribution of all axles.
The genetic algorithm is often used to interpret
and analyze stochastic data from real world problem
which cannot be assimilated to a well-defined mathematical function; weigh-in-motion data resulting
from vehicle monitoring on highway have the same
stochastic status. Genetic algorithm will be the basic
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ones to form a new offspring while keeping
the same initial population size or number of
individuals;
4. Replace the current population with the transformed one;
5. Repeat steps 2, 3 and 4 until reaching the
number of generations fixed at the beginning
according to the problem circumstances.
This process is summarized in Figure 2.
tool in this work.
2 GENETIC OPTIMIZATION
The most common type of stochastic optimization is
represented by genetic algorithm (Holland, 1975)
which constitutes a robust optimization tool mimicking the same process of natural selection (biological
evolution). This algorithm is recommended when we
have little analytical knowledge about the search
domain and very large number of parameters, unlike
the FMINS functions in Matlab which require the
determination of an initial point that affects the optimization success, this random-search optimization
algorithm is oriented by some genetic operators like
selection, crossover and mutation which make its
population more diverse and thus more immune to
be trapped in a local optima and guarantee a global
optima solution. The selection of an individual is
based on how fit it is after a fitness evaluation
process through a fitness function, also called objective function and the fittest ones have more chance
to be inherited into the next generation, the strong
tend to adapt and survive and the weak tend to die
out, genetic algorithm uses the fitness function as
Black Box and doesn’t require any information
about its structure or its mathematical model which
isn’t available in many real world problems, for example, axle weights data given from real time monitoring where it seems difficult to obtain explicit
formula for the objective function with all constraints especially with variable and stochastic traffic
loading , the general formulation of the fitness function to be minimized used in this optimization problem is mentioned in Equation 1:
 (y
1
 y2 )2
(1)
where y1 is the data resulting from real time monitoring and y2 the corresponding value resulting from
genetic optimization, this function will be minimized
by varying in each iteration the parameters described
in Figure 5, no matter if we have underweight or
overweight, the goal is to reach a very small
difference.
The main steps of genetic optimization can be
summarized as follow:
1. Generate randomly initial population of individuals, each individual is assimilated to a
chromosome and represents a set of the parameters which can be a feasible solution of
the optimization problem;
2. Using the fitness function, calculate the fitness of each individual in the population;
3. Select the more fit individuals, those that give
the minimum gap with the reference data,
then cross over and mutate the remaining
Figure 2. Schema of a simple genetic algorithm
3 MODELING DISTRIBUTION OF GVW
BETWEEN AXLES
Normally vehicles can be simply classified according to their number of axles and axle layout, so the
proportion of GVW carried per each individual axle
is modeled for each vehicle class separately, this
means that a very large number of parameters is to
be estimated, for example Enright & O’Brien (2013)
proposed a bimodal normal distribution for the proportion carried per each axle for each vehicle class,
thus for the 6-axle vehicle class we should consider
the histogram of the percentage of GVW for each
axle separately in order to estimate the 6 parameters
of the bimodal normal distribution, in total we need
to estimate 36 parameters for the modeling of GVW
distribution for only one vehicle class based on a
complex analysis of the measured data in order to
get the 6 histograms needed for the fitting
process.Simon F. & Rolf(1999) proceeded in a similar way by considering bimodal beta distributions for
modeling axle group loads for each vehicle class
separately. In other cases, the proportion of GVW
carried per axle is sometimes simply assumed as deterministic (GRAVE et al. 2000). Correlation be-
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tween axle loads is also performed for each vehicle
class in order to get the distribution of GVW among
axles (Centre de recherche: ICOM EPF-Lausanne et
al. 2006), for this purpose a complex work is established to deduce the correlation between axle loads
for each axle on each vehicle class based on the correlation for every individual vehicle in this class.
The latter case is chosen to prove the effectiveness
of genetic optimization in the following paragraphs.
load of the GVW and so on, after that we can select
the linear trend line that can describe the correlation
function, an illustrative schema is used to explain the
described procedure of calculating correlation coefficients (Figure 4), the general matrix of population
used in the optimization process is presented in Figure 5.
4 DESCRIPTION OF WIM DATABASE
The Swiss Federal Roads Office (FEDRO) has
equipped the national road network in Switzerland
with eight main weigh-in-motion monitoring stations(Treacy & Brühwiler 2011); the traffic data
from the Mattstetten A1 motorway station are chosen to illustrate the proposed methodology in WIM
analysis since they give a good representation of local or national traffic, these data are adopted as presented in the research work published by the FEDRO in 2006 (Centre de recherche: ICOM EPFLausanne et al. 2006) , all needed data are perfectly
stripped in this report.
Figure 3. Distribution of GVW for all vehicles
4.1 Vehicle modeling
According to the number of axles and axle layout,
vehicles are classified in 12 classes with a known
range of GVW for each vehicle class, the distributions of GVW for each vehicle class are also given
and then fitted to bimodal beta distributions, since
the proportion of each vehicle class from the total
traffic is taken similar to that in the measured traffic,
all GVW can be simulated using inverse-transform
method as shown in Figure 3. Each class is designated by a series of characters which refer to the type
and position of the axle, for example in case of 123a
class, all vehicles have one single axle, followed by
a tandem then a tridem axle , the letter “a” means
that vehicles in this class are articulated.The letter
“r” refers to the vehicles with trailer. All vehicle
class labels are summarized in Table 1.
Table 1. Labels of all vehicle classes
Vehicles
alone
11
12
22
Vehicles with
trailer
1111r
112r
1211r
1112r
Articulated
vehicles
112a
113a
122a
Mixed
vehicles
111
122
Figure 4. Calculation steps for correlation coefficients
4.2 Filtering of WIM data
The measured WIM data should be filtered to remove unreliable values, OBrien & Enright (2011)
have well detailed the cleaning process, only vehicles more than 3.5 tones are retained , in our case,
the following constraints will be considered to reject
unreliable data:
1. Wheelbase > Vehicle length
2. Distance between two consecutive axles > 20
m
3. Distance between two consecutive axles <0.4
m
4. Sum of all axle loads > GVW
5. Axle load > 40 t
6. Axle load > 85% of GVW
After simulation of GVW, it should be distributed
to the vehicle axles; as cited in the paragraph 3 , linear correlation between axle loads is performed for
every axle in each vehicle class, for example to calculate the correlation between two loaded axles in a
given class, we should plot for every measured vehicle, the load of the first axle as a function of the
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Population
Best
Crossover
Mutation
Figure 5. General matrix of population
Data cleaning constrains will be used to limit the
search domain in the optimization process.
5 CALCULATION OF CORRELATION
COEFFICIENTS
5.1 General framework
The proposed methodology will be summarized in
the following flowchart:
supported by the first tandem and the weight supported by the second tandem.
The correlation equations can be written as follow:
Q  Q  
2
3
tot
(2)
3
Q1  Qtot  Q2
(3)
The search domain for α and β can be limited according to the following constraints:
1. The slope of the correlation line should be less
than 1 (0 < α3 < 1);
2. Axle weight should be positive (Qi> 0) ;
3. Single axle load should be less than 40t;
4. Single axle load should be less than 85% of
GVW.
Q1
Q2
Therefore, the intervals of α and β will be:
● 0 3 1
0  Q  0.85Q

●  0   Q    0.85Q

   Q    (0.85   )Q
2
tot
3
3
Figure 6. Flowchart of the proposed framework
5.2 Illustrative example
To better present the proposed methodology, more
details will be explained for the vehicle class 22
(trucks with two tandem axles). Qtot, Q1 and Q2 are
respectively the total vehicle weight, the weight
3
tot
tot
3
tot
3
tot
This should be valid for all Qtot in the vehicle
class 22, so we can choose the minimum Qtot in this
class to reduce the search interval.
The same work is done for all the vehicle classes,
so we will have to calculate 48 parameters between
the identified maximum and minimum limits.
(Figure 5)
5.3 Results
After 15 generations of 30 populations of the 48 correlation coefficients, the best fitting coefficient val-
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ues are used to calculate the distribution of all vehicle axles for one million vehicles.
The distribution of all vehicle axles based on the
optimization results relative to the given distribution
is shown in Figure 7.
5.4 Discussion
The distribution of all vehicle axles based on the
calculated correlation coefficients using genetic optimization is very consistent with the given distribution of all vehicle classes based on monitoring data
(Figure 7). The distribution of every individual axle
in each vehicle class can also be deduced, Figure 7Figure 8 show the distributions of each axle for the
vehicle class 22, in this case the values of correlation
coefficients are also presented , α=0.38 and β=14.2,
the distributions appear to be bimodal; it should be
noted that, this is a sample from a set of 48 parameters calculated for 12 vehicle classes, this required a
total of 15⨯30=450 iterations while improving the
values of the searched coefficients in each new generation according to the minimum value of the objective function.
6 CONCLUSION
Figure 7. Given and simulated distributions of all vehicle axle
loads
The proposed methodology allows calculating a
large number of unknown parameters based only on
a little amount of data and some constraints to limit
the search domain.
In the same way, having the limitations for axle
distances in each vehicle class and the general distribution of the axle distance, the probabilistic distribution for each distance in a given class can be detected directly from the general histogram and all the
distribution parameters can be deduced using a global genetic optimization.
Genetic algorithm also constitutes a robust tool in
the optimization process in BWIM techniques, it can
be included in the Moses’ algorithm in order to
compute the GVW by minimizing the error between
measured and theoretical response.
Figure 8. Distribution of Q1 for vehicle class 22 based on calculated correlation coefficients
7 REFERENCES
Figure 9. Distribution of Q2 for vehicle class 22 based on calculated correlation coefficients; Q2=αQtot+β /α=0.38, β=14.2
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