### Abstract

**Introduction**

Many bridges of the world’s highway networks have been in service for decades and are subject to escalating volumes of traffic. Consequently, there is a growing need for the rehabilitation or replacement of bridges due to deterioration and increased loading. The assessment of the strength of an existing bridge is relatively well understood, whereas the traffic loading it is subject to, is not as well understood. Accurate assessment of the loading to which bridges may be subject, can result in significant savings for highway maintenance budgets internationally. In recent years, a general approach has emerged in the research literature: the characteristics of the traffic at a site are measured and used to investigate the load effects to which the bridge may be subject in its remaining lifetime.
**Objectives and scope**

This research has the broad objective of developing better methods of statistical analysis of highway bridge traffic loading. The work focuses on short- to medium-length (approximately 15 to 50 m), single- or two-span bridges with two opposing lanes of traffic. Dynamic interaction of the trucks on the bridge is generally not included.
**Outline of the Research**

Intuitively, it can be accepted that the gap between successive trucks has important implications for the amount of load that may be applied to any given bridge length. This work describes, in quantitative terms, the implications for various bridge lengths and load effects. A new method of modelling headway for this critical time-frame is presented.
When daily maximum load effects (for example) are considered as the basis for an extreme value statistical analysis of the simulation results, it is shown that although this data is independent, it is not identically distributed. Physically, this is manifest as the difference in load effect between 2- and 3-truck crossing events. A method termed composite distribution statistics is presented which accounts for the different distributions of load effect caused by different event types. Exact equations are derived, as well as asymptotic expressions which facilitate the application of the method.
Due to sampling variability, the estimate of lifetime load effect varies for each sample of load effect taken. In this work, the method of predictive likelihood is used to calculate the variability of the predicted extreme for a given sample. In this manner, sources of uncertainty can be taken into account and the resulting lifetime load effect is shown to be calculated with reasonable assurance.
To calculate the total lifetime load effect (static load effect plus that due to dynamic interaction), the results of dynamic simulations based on 10-years of static results are used in a multivariate extreme value analysis. This form of analysis allows for the inherent correlation between the total and static load effect that results from loading events. A distribution of dynamic amplification factor and estimates for a site dynamic allowance factor are made using parametric bootstrapping techniques. It is shown that the influence of dynamic interaction decreases with increasing static load effect.
### Outline of Research

*This is an extract from Chapter 1 of the dissertation, and is more informative as to the specifics of the research than the abstract.*

**Traffic modelling and simulation**

The research presented herein is heavily reliant on the software tools developed as part of this work. The efficacy and power of the software has implications for the manner in which bridge load research is carried out: for example, larger sample sizes generally result in more accurate load effect prediction. Accordingly, in this work, the object-orientated approach to programming is used. An explanation of this method, and the programs based upon it, is given in Chapter 4. As a result of these developments, it is now possible to simulate 5 years of traffic for a typical heavily trafficked European trunk motorway on a typical high-specification desktop personal computer. This traffic may be used to assess load effects from any form of influence line or slices of an influence surface. The statistical analysis outlined later may then be applied to the complex of results gathered.
**Headway modelling**

Intuitively, the gap between successive trucks has important implications for the quantity of load that may be applied to a bridge: this work describes, in quantitative terms, the implications for various lengths and load effects. It is found that existing headway (gap plus the lead truck length) models do not focus on the small headways that are critical for bridge loading events. A new method of modelling headway for this critical range is presented: it exhibits less variability in load effect estimation; conforms to the physical requirements of traffic; and preserves measured headway distributions. This method is described in Chapter 5, along with comparisons to existing methods.
**Composite distribution statistics**

The load effect output from the process of measurement, modelling, and traffic simulation, requires a statistical analysis to permit estimations of future load effect values. Extreme value analysis assumes that the data to be analysed is independent (or, at most, has minor dependence) and identically distributed. When daily maxima (for example) are considered as the basis for further statistical analysis, it is shown here that although this data is independent, it is not identically distributed. Physically, this is manifest as the difference in load effect between 2- and 3-truck crossing events, for example. Intuitively, such events are not identically distributed, and as such, should not be mixed as a single distribution in an extreme value statistical analysis. A method termed composite distribution statistics is presented which accounts for the different distributions of load effect caused by different event types. Exact equations are derived, as well as asymptotic expressions which facilitate the application of the method. The method is checked against results derived from the exact distribution, and compares favourably. Also, the method is applied to the output from the simulation process and compared with the traditional approach. It is shown that the composite distribution statistics method can give significantly different results.
**Prediction of extreme load effects**

The *raison d’être*of the bridge loading model, and subsequent statistical analysis, is the prediction of extreme, or maximum lifetime, load effects. Basic prediction techniques are outlined in Chapter 3, but more advanced methods are required to reflect the complexity of the underlying process and its model, such as the method of composite distribution statistics developed as part of this work. Such extrapolation methods, are subject to substantial variability: different samples give different estimates of lifetime load effect. To allow for this variability, the method of predictive likelihood is used in this work. This is a relatively new area of Frequentist statistics and is not yet adopted in many practical fields of research. Predictive likelihood yields many benefits for the bridge loading problem. Most importantly, the variability of the predicted extreme can be calculated. Further, sources of uncertainty, such as the random variation of the data and of the parameter fits to the data, can be taken into account. Therefore the result of a predictive likelihood analysis gives a measure of the uncertainty inherent in the bridge loading problem, and enables this uncertainty to be taken into account.

Colin

Just found your website. I also graduated from Bolton St ’99. Am in Australia working as a Bridge Engineer since 2001. What is your e-mail address and I’ll send you an e-mail outside this website?

Regards

Ken