Keywords: bridge, statistics, loading, simulation, predictive likelihood, traffic.

To minimise the repair or replacement of existing bridges, accurate knowledge of both the in-situ strength of the bridge and the loads to which it is subject is required. The assessment of the in-situ strength is relatively well understood compared with that of the in-situ loads. In recent years statistical methods have been increasingly employed to assist in the estimation of the extreme lifetime traffic loading to which a bridge is subject. This paper describes the recent statistical methods used, and their computational aspects, to determine the lifetime maximum traffic load effect [1].

Firstly, measured weigh-in-motion data is statistically modelled to characterise the traffic at the site of measurement. The period of measurement is usually quite short and so Monte Carlo simulation of the modelled traffic is used to synthetically extend the amount of traffic data available. Indeed, using object-oriented programming, up to four years of traffic has been simulated in a single run on an ordinary desktop computer. This simulated traffic is passed over bridge lengths and influence lines of interest, to determine the load effects that result. Theoretical, finite-element derived and "measured" influence lines may be used. The resulting load effect data forms a population upon which a statistical analysis is carried out.

The distribution of bridge traffic loading is a mixture of different types of loading events. For example, 1-truck presence events are very common whereas 4-truck presence events are rare, but significant for loading. A statistical model that takes account of this mixture is proposed and compared to theoretical examples. The fitting algorithm used is explained, as is the bootstrapping method applied to the examples. It is concluded that the proposed model better reflects the underlying mixture of the traffic loading phenomenon than other methods reported in the literature.

The conventional approach to the bridge loading problem is to use Monte Carlo techniques to simulate months or years of data. Maximum values for a specified period such as a day, are identified. The maximum-per-day data is then fitted to an extreme value statistical distribution and extrapolated to determine the design load effect. The Eurocode for traffic loading on bridges [2] defines the design value to be that value which has probability of exceedance of 10% in a 100 year design life. The associated return period is 1000 years, and it is usual to extrapolate to determine this singular value. In this paper an alternative approach, termed predictive likelihood [3], is used instead. In general, the two approaches give differing values of design load effect. However, as predictive likelihood returns more information from the sample, this is considered to be a more accurate result.

The predictive likelihood approach requires extensive numerical work. Indeed, allowing for the underlying mixture distribution of bridge traffic load effect results in significant complication. The problem is solved by using several optimisation techniques. In the main, the distribution of lifetime load effect is established by determining the likelihood of observing both the data and a value of lifetime load effect, for a range of alternative trial values for the characteristic maximum-in-lifetime effect. It therefore becomes possible to plot a distribution of the likelihood of each of the alternative maximum-in-lifetime values. Each postulated value requires optimisation of the parameter set of the mixed distribution. Constrained optimisation is used once to provide initial parameter values, and again to ensure that the model does not deviate from its corresponding data set. Numerical methods for the determination of the information matrices [4], and to ensure numerical stability, are also discussed. For a given data set, this procedure provides information not only on the most likely maximum-in-lifetime value but also the nature and shape of that distribution.

In conclusion, the latest statistical models to be applied to the bridge traffic load problem are described. These models are shown to result in an increase in information from the data, and computational methods are integral to this. Further, the numerical aspects of the problem are described and the solutions adopted discussed. It is concluded that great improvements in the accuracy of bridge traffic loading are obtained through the use of statistical computational methods.

1

C.C. Caprani, "Probabilistic Analysis of Highway Bridge Traffic Loading", Ph.D. Thesis, School of Architecture, Landscape, and Civil Engineering, University College Dublin, Ireland, 2005.

2

EC 1: "Basis of design and actions on structures, Part 3: Traffic loads on bridges", European Prestandard ENV 1991-3: European Committee for Standardisation, TC 250, Brussels, 1994.

3

J.F. Bjørnstad, "Predictive likelihood: a review", Statistical Science, 5(2), 242-254, 1990.

4

J.K. Lindsey, "Parametric Statistical Inference", Oxford: Oxford University Press, 1996.

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