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The maintenance of highway infrastructure requires major expenditure in many countries. By minimizing the repair or replacement of highway bridges in particular, this cost can be reduced significantly. Of the two bridge assessment components, loading

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Bridge and Infrastructure Research in Ireland: Symposium 2006
Bridge Traffic Loading: The Implications of Some Recent Findings
Colin C. Caprani Paraic H. Rattigan
Abstract
The maintenance of highway infrastructure requires major expenditure in many countries. By minimizing the repair or replacement of highway bridges in particular, this cost can be reduced significantly. Of the two bridge assessment components, loading is more difficult to estimate than strength, due to its more variable nature. Consequently, bridge traffic loading has been an area of intensive research in recent years. Recent research has focused on assumptions inherent in previous work and the results are presented and discussed. In this paper, the latest statistical analyses adapted for use in the bridge traffic loading problem are reported. Comparisons to the previous state of the art are made and it is shown that a revised approach reflects the underlying phenomenon of bridge traffic loading more accurately. A method which is shown to reduce the variability of the statistical extrapolation process is also presented. Of more significance, a statistical approach which joins the dynamic and static effects of traffic loading is presented. An assumption inherent in much previous research in this area is that free-flowing traffic with coincident dynamic effects is more critical than congested traffic (which has practically no dynamic effect) for short- to medium-length bridges. Given that about 90% of bridge stock is of this length, this assumption has critical implications for the expenditure on bridge rehabilitation. A sample application of the proposed statistical method is presented and the results are shown to be of great significance. It is shown that the level of dynamic interaction is not sufficient for free-flowing traffic to govern and that it is congested traffic that may govern the vast majority of bridges. The implications of the cumulative effect of these various findings are discussed with reference to the future direction of research into bridge traffic loading and current practice in bridge assessment for traffic loading.
Keywords:
Bridges, Loading, Statistics, Simulation, Traffic, Trucks
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Colin Caprani studied at DIT Bolton St. and graduated first in the class of 1999 with a First Class Honours Degree in Structural Engineering. After over 2 years in building structures consultancy, he commenced his PhD research into bridge traffic loading at UCD under Prof. Eugene OBrien. He obtained his PhD in 2006 and has published 18 scholarly articles to-date. Presently he lectures in both UCD and DIT Bolton St, and also works with Roughan & O’Donovan Consulting Engineers in the design of bridges. Paraic Rattigan has a Bachelors degree in Civil Engineering from University College Dublin 2003. He is currently pursuing a PhD in the Bridge and Transportation Infrastructure Research Group of UCD, with the aid of an Irish Research Council Scholarship, and under the supervision of Prof. Eugene OBrien and Dr. Arturo González. Has presented research at recent international conferences such as IABMAS06, Oporto, EURODYN 05, Paris, and MSO 04, Hawaii. Represented Ireland at the young researchers seminar, YRS 05, in The Hague.
INTRODUCTION Background
The maintenance costs associated with ageing bridge stocks across the world represent an increasing proportion of total road infrastructure expenditure. The EU expenditure on the repair, rehabilitation and maintenance of bridge structures is estimated to be €4–6 bn annually [1]. As only the 15 member states up to May 2004 are included in this estimate, in the recently-enlarged EU, bridge maintenance expenditure is likely to be more than €6 bn annually. As a result of the high maintenance cost, research into the assessment of existing infrastructure has received great focus recently. This is so, as significant savings are possible, through more accurate modelling of both the physical and statistical phenomena associated with the problem. In particular, given that bridge traffic loading is significantly more variable than bridge capacity, it is in this area that much progress towards reducing maintenance expenditure may be made. The focus of this paper is to present some recent findings in the statistical analysis of bridge traffic loading, and to discuss their implications. These recent findings are briefly compared to relevant reports in the literature. The implications of these findings to ongoing research and the practical assessment of existing bridges are also discussed.
Basis of Research
Modelling bridge traffic load effect requires the input of actual highway traffic data, obtained from suitable installations. Weigh-In-Motion (WIM) technology is frequently used for this purpose. The work reported here is based on measured traffic from the Paris to Lyon A6 motorway near Auxerre, France. The site has 4 lanes of traffic (2 in each direction) but only the traffic recorded in the slow lanes was used (it is acknowledged that this results in conservative loading for a 2-lane bridge, for example). In total 17 756 and 18 617 trucks were measured in the north and south slow lanes respectively, giving an average daily truck flow of 6744 trucks. It is important to note that though the particular results may be influenced by the quantity of data available, the methodologies presented in this work are general. Monte Carlo simulation of traffic streams is based upon the measured traffic from a given site. The bridge loading induced by such a traffic stream is then obtained by using influence lines (whether theoretical, site-measured, or obtained from finite-element modelling of the bridge) for the load effect of interest. Detailed information on the simulation process used to obtain the results herein is described by Caprani [2] and OBrien and Caprani [3]. Of relevance to this
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paper, as only hourly variations in traffic flow and composition are modelled, the statistical models are only stationary once inference is based on data of at least 24 hours’ duration. In addition, it is taken that the ‘economic year’ is equivalent to about 50 weeks of weekday traffic and consequently 250 ‘simulation days’ are taken to represent a calendar year.
STATIC TRAFFIC LOAD EFFECT Statistical Methods in the Literature
Overview There have been many different methods used to predict the lifetime bridge load effect from measured or simulated load effect data. Caprani [2] presents a review of these methods, which are briefly summarised here. In a series of studies by Nowak and others (see [4] –[6] for example), straight or curved lines
are superimposed on the tails of the distributions and extrapolated to determine the characteristic load effect values. As part of the background studies to the Eurocode for bridge loading [7], Bruls et al [8] and Flint and Jacob [9] consider and compare several methods of
extrapolation of the basic histogram of load effect, based on measured traffic samples. A weighted least-squares approach is used by Grave et al [10] to fit Weibull distributions to load effect values. Castillo’s [11] recommendation to use upper 2
√
n data points is adopted by Grave et al [10]. Bailey and Bez [12, 13] determine that the Weibull distribution is most
appropriate to model the tails of the load effect distributions and used maximum likelihood estimation, whilst Cooper [14, 15] uses measured truck loading events to determine the
distribution of load effect. Cooper raises this distribution to a power to establish the distribution of the maximum load effect from 4.5 days of traffic. This is fit with a Gumbel distribution which is used to extrapolate to a 2400 year return period. Of a more statistically advanced nature is the work of Crespo-Minguillón and Casas [16] who adopt a Peaks-Over-Threshold approach and use the Generalized Pareto Distribution to model the exceedances of weekly maximum traffic effects over a certain threshold. Problems The attributes required of a robust statistical extrapolation procedure are described by Caprani [2] and summarised here. Of most importance is that a model should not be subjective: different results obtained as a result of different processing decisions do not induce confidence in any of the results. Other requirements are as follows:
•
Choice of Population:
The population upon which the analysis is based must be in keeping with the limitations of the statistical model to be applied. In many cases the stationarity assumption of many statistical models is violated.
•
Distribution of Extreme Load Effects:
Often decisions about which extreme value distribution to use are made. This is unnecessary given that the Generalized Extreme Value distribution incorporates all three.
•
Estimation:
The means by which the model parameter estimation is done is often graphical or least-squares-based when more accurate methods, such as maximum likelihood estimation exist.
•
Choice of Thresholds:
Many authors make decisions regarding the data which is to be kept as a basis for the analysis – the ‘tail’ data problem. This is unnecessary if the correct model is being applied to the correct population using good estimation procedures. Save for that of [16] other means of extrapolation generally fail to meet the minimum requirements of a statistical model. Also, variability of the characteristic load effect is not generally assessed. Extrapolations are carried out to the return period, rather than to find the actual characteristic value (which for the Eurocode for bridge loading [7] is 10% probability of exceedance in 100 years).
Recent Advances
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The Statistical Nature of Bridge Traffic Load Effect Recent work [17] has concluded that bridge traffic load effect is not a single statistical generating mechanism. In essence and as is intuitively reasonable, the distribution of load effects caused by a 2-truck event (two trucks concurrently present on the bridge) differs to that of a 3-truck event. When each loading event-type is isolated, it is found [2] that the Generalized Extreme Value Distribution (GEV) is appropriate to model the daily maximum load effects that result. Therefore a composite distribution of daily maximum load effect,
S
, is required as a basis for extrapolation. An appropriate generalized extreme value composite distribution model,
( )
C
G
⋅
, is described by Caprani et al [17] to be:
[ ]
( ) ( )
1
N C j j
P S s G s G s
=
≤ = =
∏
(1) where
( )
j
G
⋅
is the GEV distribution of the
j
th of
N
possible event-types. This model has been shown to exhibit greater fidelity in fitting distributions of load effect, and meets minimum requirements for a good extrapolation model [2]. Predicting the Lifetime Load Effect Extrapolations to a return period result in a single value of load effect. This is statistically flawed: repeating the process would generally yield a different result. Thus it may be thought that there exists a distribution of characteristic values, but this is not necessary. Instead, the definition of characteristic value should be unequivocal, as is that, for example, of the Eurocode [7]. Therefore focus should be centred on the estimation of the lifetime distribution of load effect, from which the characteristic value is then derived. Of significant further value would be a means by which allowances for modelling uncertainties, such as parameter confidence intervals, could be included. Predictive likelihood is a method for estimation which allows both for sampling and modelling uncertainties. Caprani and OBrien [18] have applied this method to the bridge loading problem and shows that the traditional return period approach yields different results to the direct estimate of the characteristic value from the lifetime distribution of load effect [19]. The method has also been shown [20] to be effective in predicting extreme vehicle weights.
Comparison
The net effect of the application of the two advances described, in comparison to a statistical model which represents the best of the models in the literature, is shown in Fig. 1 for three load effects:
•
Load Effect 1: Bending moment at the mid-span of a simply supported bridge;
•
Load Effect 2: Left support shear in a simply-supported bridge;
•
Load Effect 3: Bending moment at central support of a two-span continuous bridge. From Fig. 1 it can be seen that the differences are generally small, with notable exceptions for spans of about 40 m. In particular, Load Effect 2 is sensitive to spans around 40 m due to the shape of its influence line, and the inter-vehicle gaps. That the differences are not excessive, despite the advances in analysis, shows a certain degree of robustness amongst the better statistical extrapolation methods. Caprani [2] describes these differences in more detail. A difference in load effect of up to 14% is substantial when existing bridges fail assessments by only a few percent. As a result substantial savings are possible, not only in materials and labour, but more importantly through a reduction in traffic disruption.
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-10-261401020304050Bridge Length (m)
D i f f e r e n c e ( % )
60Load Effect 1Load Effect 2Load Effect 3
Fig. 1
Change in Predicted Load Effect Due to Recent Advances TRAFFIC LOAD EFFECT ALLOWING FOR DYNAMIC INTERACTION Incorporation of Dynamic Effects into Bridge Traffic Load Models
The dynamic amplification factor (DAF) is defined as the ratio of total to static load effect, where total load effect results from the truck and bridge interacting dynamically, in addition to the static load effect. Allowances for dynamic interaction are made in bridge loading codes, based on the notion of the DAF. Usually however, the worst possible DAF is applied to the critical static load effect. This does not take into account the reduced likelihood of these events coinciding. Indeed it is intuitively reasonable that grossly overloaded vehicles are not as dynamically lively as unloaded vehicles, for example. The conservative loading that results from the application of a critical DAF to a critical static load effect has been an area of active research in University College Dublin in recent years.
Recent Advances in Dynamic Interaction Modelling
Advances in knowledge and technology have led to increased accuracy in the assessment of the dynamic bridge-vehicle interaction that occurs for given bridge loading scenarios. A number of experimental studies on the dynamic loading of beam and slab (girder) bridges have been carried out previously [21, 22]. These studies reveal that a reasonable level of
accuracy may be achieved using simplified planar beam, or grillage models. However, more complex three-dimensional finite element approaches yield more accurate validations, and allow for the investigation of numerous varying bridge load effects [22, 23, 24].
Also of importance are computationally-efficient methods for the estimation of bridge-traffic dynamic interaction. Significant advances have been made with such models in recent years. In particular, numerically efficient solutions have been developed to determine the influence of vehicle speed and road irregularities at particular points on the road and have been shown to be reasonably accurate [25].
Predicting the Level of Dynamic Interaction for the Lifetime Load Effect
Statistical Background Total and static load effects are related through the DAF, which is not constant but there remains a degree of correlation between these statistical variables. The recent statistical theories of multivariate extreme values has been applied to this problem to extrapolate these correlated variables to their design lifetime values. Their ratio at this level is therefore the level of dynamic interaction applicable for the bridge design lifetime. This has been termed the assessment dynamic ratio (ADR) by Caprani et al [26] in recognition that it does not arise from any one single loading event.
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