Development of a Multiple Truck Presence Model

In bridge design and evaluation , t he characteristics of t ruck traffic is part icularly important . Truck weights and axle configurations are directly used in calculations of maximum load effects including posit ive bending moments and shear forces in simply-supported bridges as well as negative bending moments of cont inuous spans. The loading event that is likely to govern a bridge design is t he simultaneous occurrence of two or more heavily loaded t ruckscalled Mult iple Presence. Consequently, the statistics of different loading patterns as well as the weight correlation among coincident t rucks are equally important to live load analysis. However , studies of Mult iple Presence events are either convolut ion models or models based on visual observation of the traffic, but very little research has been undertaken to analyze Weigh-in-Motion data. In this study, Weigh-in-Motion data collected on highways is analyzed in order to determine the occurrence of Mult iple Presence events incorporating two t rucks and t heir weight correlation . Based on t he analysis, a prediction model is developed to estimate the frequency of t he Mult iple Presence events using known site parameters. Also, probabilit ies for full weight correlations are found for t he Mult iple Presence loading cases. A special remark is given to the influence of the Gap distances between the t rucks involved into a Mult iple Presence event . It is found that t he Gaps do not have a threshold distance for t he influence of the lighter t ruck that could be ut ilized to reduce t he Mult iple Presence occurrence probabilit ies.

loading patterns as well as t he weight correlation among coincident t rucks are equally important to live load analysis. However , studies of Mult iple Presence events are eit her convolut ion models or models based on visual observation of the traffic, but very little research has been undertaken to analyze Weigh-in-Motion data.
In t his study, Weigh-in-Motion d ata collected on highways is analyzed in order to determine t he occurrence of Mult iple Presence events incorporating two t rucks and t heir weight correlation . Based on t he analysis, a prediction model is developed to estimate t he frequency of t he Mult iple Presence events using known site parameters. Also, probabilit ies for full weight correlations are found for t he Mult iple Presence loading cases. A special remark is given to t he influence of t he Gap distances between t he t rucks involved into a Mult iple Presence event . It is found t hat t he Gaps do not have a t hreshold distance for t he influence of t he lighter t ruck t hat could be ut ilized to reduce t he Mult iple Presence occurrence probabilit ies.  Guzda et al. (2007) 3.4 MP Occurrence Probabilties after Gindy and Nassif (2007) 4.  Ghosn and Moses, 1986) 25 3.4 ormalized Moments vs. Span Lengths (after Nowak and Hong, 1991) . . 27 3.5 ormalized Moments vs. Span Lengths for various degrees of correlation (after Nowak and Hong, 1991) . . . . . . . . . . . . . . . . . . 29 3.6 Daily Maxima by Event Type (after Caprani et al. , 2007) . . 36 3.7 Gap Distances for Following events (after Guzda et al. , 2007) 37 3.8 MP Loading Cases (after Gindy and Nassif, 2007) . . . . . . 39 3.9 The Probability of Occurrence plotted against the Truck Volume for a 120ft Span (after Gindy and Nassif, 2007)  The loading event that is likely to govern a bridge design, however, is the simultaneous occurrence of two or more heavily loaded trucks-called Multiple Truck Presence (MP).
Consequently, the statistics of different loading patterns as well as the correlation among coincident trucks are equally important to live load analysis.
Other studies of MP events have been based on either simulation of traffic or limited visual observations, but very little research has been undertaken to analyze Weigh-in-Motion data.
Weigh-in-Motion data is traffic data which is recorded by sensors that are directly installed on the roadway. The data is collected when the vehicles pass the devices without 1 requiring them to stop. The collected data contains various information about the vehicles and data is widely available.
The aim of the this study is to analyze Weigh-in-Motion data collected on highways to determine the occurrence of Multiple Presence events and their weight correlation.
Based on the analysis, a prediction model will be developed to estimate the frequency of the Multiple Presence events using known site parameters.

Methodology
In the whole area of Civil Engineering and many other areas of expertise, models are based on probability methods with a certain level of conservativeness. Both sides, the Load side and the Resistance side of a model follow certain statistical distributions. The means of those distributions are mostly far apart, but the upper tail of the Load and the lower tail of the Resistance might overlap for extreme cases, as graphically shown in Figure 1.1 for normally distributed variables. In such an unusual case the Load is greater than the Resistance and that would inevitably lead into system failure.
For the resistance side, extensive material tests have been performed and t he results are implicated into the building codes. For any material the given values in the codes are, in fact, a certain quantile that provides a balanced proportion between sufficient safety and economical factors.
For the load side, the accurate determination is far more complex. The variance of the loads is likely to be very large and cannot be found by simple tests. Current bridge load models are based on a range of assumptions that can be considered overly conservative.  (Nowak and Hong, 1991, i.a.) , for example, bases all his Multiple Presence Occurrence Probabilities and Full Weight Correlation Probabilities on assumptions after analyzing 10,000 heavy trucks from an Ontario Truck Survey (Agarwal and Wolkowicz, 1976).
The main objective of this study is to identify the probability of occurrence for Multiple Presence events on highway bridges and further the probability of trucks being fully correlated in weight, as this combination-Multiple Presence with fully correlated trucks-is most likely to induce the largest bridge responses.
This work is divided into two main parts. The first part is of rather theoretical nature, as an introduction is given of the statistical methods, that have been used in the literature and throughout the development of the model. Also, an extensive literature review is done on the various existing life load models for highway bridges. This review introduces the different approaches to determine the controlling load and reveals that many models 3 use Multiple Presence probabilities and weight correlation probabilities that are either based on simulation, engineering assumptions or visual evaluation of recorded traffic.
Additionally, an introduction to Weigh-in-Motion (WIM) systems and data collection is given.
The second part deals with the analysis of Weigh-in-Motion data. This data contains detailed information about the vehicles that pass over the sensors, including a very accurate time-stamp and weight information. First, the data needs to be filtered as only trucks are relevant for inducing maximal bridge responses. Also, many errors in the entries have to be removed before analysis is performed.
In a next step, the weight distribution of the NJ database is evaluated and an algorithm is implemented to determine/simulate the occurrence probabilities of Multiple Presence events for several span lengths from the filtered WIM data using the t imestamp of the trucks. Based on the outcomes, a prediction model for Multiple Presence occurrence probabilities is developed by utilizing the various site parameters available from the WIM data using regression analysis methods.
A special remark is given to the influence of the gap lengths for maximum bridge responses. Gap lengths denote the distance from the rear bumper of the first truck to the front bumper f a following truck.
Analysis is performed to identify the probabilities of a full weight correlation between the two trucks involved in a Multiple Presence event, as an event with two fully correlated trucks is likely to represent the most significant loading case. 4 The model is discussed in t he conclusions, and recommendations for furt her studies are made. Appendix A gives an example of t he practical application of t he model developed for a usual lOOft highway bridge.

Statistical Methods
The various live-load models that will be presented in Chapter 3 are all more or less based on statistical methods, and in order to analyze the Weigh-in-Motion data, many statistical methods have to be utilized.
This Chapter describes some of the statistical methods, that have been used in the literature and that have been applied to the data. The general procedure for regression analysis is explained including the description of variable types and generalization for more complex models. It is also explained how the parameters can be tested for significance and how the whole model is validated. A special remark is given to the Coefficient of Determination as an indicator for the model accuracy.
Furthermore, it is described how to use the models found for actual prediction and how to detect lacks of fit using Residual Analysis. The last part deals with methods of how to carry out actual regression analysis and parameter testing describing the two main methods that have been utilized in this thesis. 6

General Linear Regression Models
The idea of regression analysis is basically testing whether one variable is dependent on or can be expressed by using other variables. Hence, the variable that is to be predicted is called dependent or response variable, whereas the other variables are called independent or predictor variables.
Expressing the above in terms of a formula, we get To solve the problem, (k + 1) equations 1 have to be solved, which is very complex.
Therefore, regression analysis is often carried out using a computer. To numerically solve the equation, matrix algebra is applied, using the following matrices: The simple reason for this is, that only one {3-coefficient has to be calculated for the variable. Using different encoding is possible but would lead to two coefficients -one for each level. If one level is set to zero, the coefficient for this level can be omitted. For more than two levels, an incremental encoding is recommended, with one coefficient for every level.

Model Quality and Parameter Testing
After the general methodology of regression analysis has been shown in the previous sections, we now want to determine whether our solutions are accurate and how to test for the significance of single parameters as well as for the whole model.
The value of r always ranges from -1to1 , independent of the units of the variables. Values close to -1 or 1 denote a very strong linear relationship between the two variables, as all data points fall on the least square line. Values around 0 imply very little to no relation.
Negative values of the Coefficient of Correlation express a decreasing value of x with an increasing value of y. In the literature, this is often referred to as Negative Correlation.
The Coefficient of Correlation can only be utilized for linear relationships between two 11 variables. For more complex regression models or mult iple variables, t he Coefficient of Determination (Section 2.2.2) is used instead.

Coefficient of D etermination
To determine t he overall accuracy of a regression model, t he {multiple) coeffi cient of determination, R 2 , gives a measure of t he fit . Assuming t hat t he parameters Xi are not significant for t he prediction of y, a good prediction for y would be just t he mean y.
Hence, t he sum of squares of t he deviations ( S Syy) for every Yi would be almost equal to t he sum of squares error (S SE) (see Section 2.1).
The mult iple coefficient of determination is t hen defined The R 2 values expresses t he accuracy of fit , whereas 1 denotes a perfect 2 fi t and 0 implies a total lack of any fit . For simple linear regression models wit h only two variables, t he Coefficient of Determination is equal to t he square of t he Coefficient of Correlation (Section 2.2.1) R 2 = r2 .
The disadvantage of t his coefficient is t hat the data has to have significantly more data points t han parameters, otherwise t he R 2 -values is forced to 1 (perfect fit ), only if t he number of data points equals t he number of parameters.
2 That would imply, t hat t he regression line found would pass t hrough every data point.
For t his reason, another measure for t he accuracy of fit of a regression model is int roduced, which is based on t he R 2 value: t he adjusted R~ -value, which is defined R2 = 1n -1 (1 -R2) a n-(k+l) (2.14) where n denotes t he number of data points and k the number of parameters used in t he model. It should be noted t hat, because of t he first t erm, t he R~-value can not b e forced to a perfect fit, as it is penalized by t he number of variables that are used in t he model. Therefore, it provides a better measure, especially for smaller datasets.

P arameter Test (t-Test)
To test if a parameter has a significant contribut ion on t he regression model, a so called t-Test is utilized. Wit h t his test it is possible to test for the significance of t he parameters.
To set up the test, it is hypot hesized that a parameter Xi has no contribution of any significance to t he regression model. It should be noted that a prediction t hat is outside the range of t he regression model is dangerous, as the outcomes might be erroneous and do not represent t he actual data.
For example, a term x 2 might give an accurate curvature for the range of t he data.
However-for a greater range, t his term represents a parab ola as shown in Figure 2.5.

Stepwise Regression
To find an appropriate model for t he regression analysis, t he t- where SSER is the sum of squares error of the reduced and SS Ee is the sum of squares error of the complete model and MS Ee is the mean squares error of the complete model.
(k-g) denotes the number of added (3 in the complete model, (k+ 1) is the total number of (J's in the complete model and n denotes the total sample size. The F-value calculated is tested against an Fa-value, which is retrieved from the F-table or software with 111 = kg degrees of freedom for the numerator and 112 = n -( k + 1) degrees of freedom for the denominator.
This Fa-value represents the significance level that we want to test for. If F > Fa, the Null Hypothesis is rejected and at least one of the added parameters in the full model has significant impact and therefore makes the model valid.

21
Chapter 3 Review of the Literature As bridges are costly structures with high maintenance costs, but also high significance for public life, designers try to optimize bridges in terms of cost-effectiveness and safety.
For that reason, the assumptions of the live load are crucial to be most accurate, but still provide a high level of safety. The problem about bridge loading is that precise traffic modeling is a very complex task. The significant increase of traffic during the last decades and changes in the traffic pattern and composition make this task even more difficult.
Hence, a lot of research has been done on the determination of bridge loading. evertheless, studies only dilatory deal with the most relevant loading case explicitly, the

Multiple Truck Presence (MP) .
Multiple Presence means the simultaneous presence of more than one truck on a bridge at the same time. For single lane bridges, the only MP loading case occurs, when one truck is followed by one or more other trucks with a gap distance smaller than the span length-so that all the trucks participate in the induction of structural response. For more than one lane, additional loading cases exist for multiple trucks being in adjacent lanes. One of the most influential loading cases is the Side-by-Side event, where two 22 ks ar e located next to each other on a bridge. If the trucks are no exactly next to true each other, the trucks can be considered Staggered (Gindy and Nassif, 2007).

23
Another crucial point is the extrapolation of the maximum response per unit time analyzed to the life-span maximum response of the bridge. Several different methods will be introduced as they are an essential part of the models.

3.1.l Ghosn and Moses
In their study (Ghosn and Moses, 1986) , a load model for short to medium span bridges is proposed. For this purpose, a convolut ion model is found that randomly generates traffic data based on traffic data sets to predict the number of loading cases utilizing a truck slot model. Each lane of a short to medium span bridge is separated into two 38ft slots, where a truck can be placed and a gap of 30ft which is observed to be the minimum gap between the trucks (Moses and Garson, 1973) .   Ghosn and Moses, 1986) 25 trucks per day (Moses and Ghosn, 1983). For higher volumes, the model is extrapolated.
For low volume sites, the number of maximum loading events reduces.
With an average of 2000 trucks per day, the maximum loading case for the life-span (in this case 50 years) of the bridge can be found by extrapolating the number of loading events to approximately N = 35 x 10 6 . Applying this value to the distribution of the maximum loading moment, they get where Fx(x) denotes the cumulative distribution function for one event. With the assumptions and the preliminary work described above, a formula is developed to calculate the median value of the maximum bending moment in 50 years. The formula reads where H is the variable of interest for MP events. It is a random variable that describes overloading and incorporates MP loading of vehicles. H depends on truck volume and span length.
Based on these results, Ghosn and Moses develop a method to adjust the AASHTO Bridge Design Code with new safety factors and design loads.

N owak
A statistical method for the assumption of Bridge Live-Load was developed by Andrzej S. Nowak, who widely published on the topic of load modeling. Many publications by For multiple presence loading, Nowak and Hong consider Following and Side-by-Side multiple truck occurrences. For both cases, assumptions (based on observations) of the frequency of occurrence and the weight correlation are made for the trucks involved. It is stated that every 10th truck is followed by another truck with a headway distance of 50ft or less. These trucks are uncorrelated with regard to weight and are assumed to be the 7 .5 year maximum truck and an average truck. Every 50th truck is involved in a following event and partially correlated to the other, where one is the maximum 1.5 year truck and the other on the maximum daily truck. Every lOOth truck is followed by another one with a full wight correlation, where both are the maximum nine-month trucks. Table 3.1 gives a brief summary of the MP occurrence frequencies used by Nowak.
Plotting the normalized moment against the span for a single truck and the correlation assumptions made for 15ft and 30ft headway distance, it is found that for shorter spans to about 120ft, a single truck triggers the largest reaction of the bridge, whereas -depending on the headway -for longer spans, two fully correlated trucks controlled (Nowak and Hong, 1991) , as shown in Figure 3.5.
For Side-by-Side occurrences of trucks, two trucks travel in adjacent lanes with a reduced weight compared to one single truck. From observations it is concluded, that every 50th to lOOth truck is involved in a Side-by-Side event ( owak and Hong, 1991). Is is also noted that every 50th truck has no correlation (with regard to weight) to the truck in the adjacent lane, therefore one truck is assumed to have the maximum  Calculating the corresponding moments for the load pattern, it is found out that the loading case of two fully correlated trucks controls. Nowak and Hong state that the maximal 1.5-month truck used for a fully correlated Side-by-Side occurrence induces a moment of about 853 compared to the maximal 75-year truck, which is used for the Single Truck load modeling.

Moses
In the NHCRP Report 454 (Moses, 2001) , Moses reviews the Live-Load Model developed by Nowak in a former HCRP Report (Nowak, 1999) and calibrates the live load factors for Highway Bridges based on the truck database by Nowak (Section 3.1.2). Also a model for the occurrence of Side-by-Side Multiple Truck Presence is developed.
First, Moses introduces the proposed 3S2-design truck (AASHTO, 1994) and compares it to the Ontario Truck Database (Agarwal and Wolkowicz, 1976) , that owak utilized for his work (Section 3.1.2). Even though the truck data is not fully normally distributed and the trucks are not 3S2 vehicles, these assumptions can be made and Moses calculates the AASHTO 382 design truck weights for multiple span lengths and find a good fit with the average weight W = 68 kips and a standard deviation of ow = 18 kips, which is 't ·mi'lar to T owak's outcomes using the HS20 design truck times the average moment qui e s1 response.
For Multiple Truck Presence events, namely the Side-by-side events, he recapitulates the assumptions as introduced in Nowak and Hong (1991) (see Section 3.1.2), stating that no references were given in the paper , no studies have been made on truck weight correlation and that the Ontario data represents a very high average daily truck traffic (ADTT) , that makes Multiple Presence events more likely.
Despite these objections, he incorporates the assumptions and finds the governing loading case to be Side-by-Side event with two fully correlated trucks. After Nowak and Hong (1991), every 15th truck is involved in a Side-by-Side events and every 30th Side-by-Side events incorporates two fully correlated truck, implying the probability of P = 1/450. Hence, the maximum weight in this case is represented by the weight of the maximum 2-month weight for both trucks.
The number of MP events can be calculated where ADDT/5 represents the upper 203 of the trucks and P 5 ; 5 the probability of MP events (1/15 after Nowak and Hong (1991)). The variate t of the probability (l/NMP) is found to be 4.09 in a normal distribution.
With those values, the maximum weight for a 75-year Side-by-Side event for a 3S2 truck is calculated This is in a reasonable range, as t he t ot al load after Nowak (Nowak, 1999) is 286 kips.

Compared t he AASHTO Bridge Design (AASHTO , 1998) -where no reduction factor
is applied for multiple presence loading -the t otal load is 384 kips.
As stated above, owak's assumption for t he frequency of mult iple t ruck presence events are based on engineering assumptions. Stating t hat every 15th heavy truck (upper 203 of truck population) is involved in a Side-by-Side event would imply, t hat every t hird truck of the whole population is involved. This seems by far too high , as Moses (Moses and Ghosn, 1983) finds t hat only 13 to 23 of the trucks occur at the same time. For this reason, he int roduces a simple occurrence model which is dependent on t he average daily truck traffic (ADTT). This model omits factors like traffic speed, road grade, platooning etc., but gives a numerical approach of calculating t he number of multiple occurrences.
Assuming that t rucks can move freely in the right two lanes of the traffic stream, t hat the average speed is 60 mph and that the length of an average truck is 60ft , t he average spacing is Avg. Spacing (ft) = 88ft /sec x 108, 000/ ADTT (3.5) and the average number of slots between two t rucks is Ps/s = 158 400 = 1.578 x 10 x ADTT ' (3 .7) Table 3.2 shows t he probabilities for Side-by-Side events for various ADTT-values. These values are very low compared to t he 0.33 assumed by Nowak (Nowak and Hong, 1991).
Moses states t hat platooning of t ruck (average 5 t rucks per platoon) could increase t he values by a factor of five. Moses conservatively proposes the probabilities as shown in the third column of

Crespo-Mingui116n and Casas
Proposed Crespo-Minguill6n and Cases present a model which is completely based on a simulation method of t raffic t hey developed (Crespo-Minguillon and Casas, 1997) . Their work can be subdivided into t he actual simulation of t he t raffic and t he ext rapolation of t he extreme values to determine t he maximum bridge responses .
The aut hors give an overview over t he current t raffic simulation methods and generally classify into three categories: • Theoretical Models • Simulation of static traffic (mostly based on observed data) • Simulation of real traffic fl ow ' 33 where the latter are assumed to be t he most complete models-even t hough many models are not general models but are developed for a certain purpose such as t he prediction of maximum load effects or t he dynamic analysis of bridge structures.
The authors develop a simulation model of real traffic which is suitable for a broad variety of applications, which is globally valid and adaptable to site-specific circumst ances, but simple enough not to over-capacitate modern computers. For t hat purpose, they find t he correlations for t he most important parameters, like vehicles in one lane, vehicles' type, etc. For the site-specific adapt ion, the average daily traffic (ADT) and t he percentage of trucks are ut ilized , as t hey can be easily found in WIM datasets. As the overlapping of vehicles in a model is a commonly known source of error in t raffic simulation , t he algorithm developed avoids t hose situations. The aut hors summarize all t he requirements stated above into two basic t asks: • Estimation of t he load effect s due t o real traffic simulation and • Extrapolation of t he maximum values found per reference t ime unit (one week) to the life-span of t he structure (100 years).
The model developed for t he simulation of traffic accounts the mean daily t raffic flow (ADT) and calibrates it for daily and furthermore for hourly variation. For t he now hourly traffic, a binomial decision is made whether the highway is congested or not. This decision is influenced by t he previous (weekday and hour) and the intensity of traffic for each is derived from theoretical density curves. Further t he types of vehicles are chosen randomly by a Markov-chain process which is merely influenced by the site parameter given (percentage of t rucks, determined from WIM dat a). For t he last step, headways, velocities, geometries and weights are assigned. Additionally t he authors implemented an algorithm that allows overtaking and passing of vehicles, making the model even more realistic.

.1.5 Caprani, O 'Brien and McLachlan
Caprani, O'Brien and McLachlan (Caprani et al. , 2007) develop a statistical method to calculate accurate load extrapolations and to determine governing loading cases.
Evaluating the conventional method of linearly extrapolating the cumulative distribution functions (CDF) of all bridge responses (as, for example, utilized by Nowak and Hong (1991)) , the authors identify that the assumption of independent and identically distributed (iid) loading events is violated, as, for example, loading events with multiple trucks (Multiple Presence events) are more complex than loading events with only one truck. The MP events incorporate distributions for the number of trucks, the geometric properties and the location of the trucks on the bridge. Therefore, the conventional methods might be inaccurate for extreme value extrapolations.
Conventionally (as also described in Ghosn and Moses (1986)), the bridge span is partioned into j slots on the bridge, with a maximum number of trucks nt. The probability of load event i being the maximum loading event for a time period, S is less than some values. For nd loading events per day, the probability equates to where Fj(s) denotes the cumulative distribution function of s and fj is the probability of occurrence for event involving j trucks. Even though the number of loading events per day varies, the utilization of the average gives a sufficient accuracy. 1 ·ng their method for the evaluation of W1M data, they find that MP events in-App y1 1 . g 4 trucks are not governing for maxima of shorter periods of time, but become vo vm more significant over the period of time, as shown in Figure 3.6-this is a very important finding that should be considered in the future calibration of Bridge Design Codes. Their study is .based on a total of 2.5 hours of videotaped traffic on a highway near a bridge in Delaware. All gap distances are not exactly measured but estimated during the analysis of the videotapes. This might be source of inaccuracies.
Stating that MP events with gap (rear bumper to front bumper) distances of more than two truck lengths do not have an impact on the maximum load, they found that 6.43 of nt ed trucks are involved in a following event. Figure 3. 7 shows the gap distances the cou of the Following events from their study. Defining Side-by-Side as two trucks in adjacent lanes with a headway (front bumper to front bumper) separation of maximal two truck length, it was found that 7.63 of all trucks were involved in a Side-by-Side event. For smaller headways they found the Side-by-Side events to involve 6.03 of all trucks when the maximum headway was 1.5 truck lengths and 4.43 with a headway of one truck length, respectively. All the MP occurrence probabilities are summarized in Table 3.3. Even though it is stated that this Poisson model might not be accurate on multi-lane highway traffic, they find their outcomes to be in the same range as the analytical solution.

.1.7 G indy and Nassif
In their paper, Gindy and Nassif (Gindy and assif, 2007) develop a method to determine Multiple Truck Presence statistics from Weigh-in-Motion data. Their database contains of 48 directional WIM sites, geographically dispersed all over Iew Jersey. This is a previous version of the database used in this study and described in greater detail in Section 4.3. The average daily truck traffic (ADTT) is derived from the WIM data and the sites are categorized into Light (ADTT<l ,000 truck per day), Average (1,000< ADDT<2,500) and Heavy (ADTT>2,500).
For the multiple truck presence, Gindy and Nassif consider the four most common loading cases for bridges: Single, Following, Side-by-Side and Staggered shown in Figure 3.8 and also address, that parameters like truck volume, span length, area-and road type etc.
have influence on the occurrence of multiple truck presence events. From their data analysis, a mean weight Wµ = 45 kips and a 95th percentile weight Wg 5 = 79 kips are determined, which is about the same range as other studies (i.e. Nowak and Hong, 1991;Moses, 2001) .
To determine the probability for the Multiple Presence events, the truck traffic is simu-  truck and therefore provides a high accuracy for the determination of entrance a nd exit times when simulating the traffic over a bridge. This accuracy is about 11.4 in for a truck with a speed of 65 mph. Rounded to the next full second, the accuracy would decrease to 95ft at the same speed and would not be suitable for the analysis.
Events with more than two simultaneous trucks on the bridge are discarded as they a re not part of the study.   Gindy and Nassif (2007) For the weight correlation between two trucks involved in a Multiple Presence loading case, it was found out t hat average volume sites have a higher tendency of scattering than sites with lower or higher volumes.

Summary and Discussion
Many different methods have been introduced which present effective ways of calculating bridge live-loads. The methods represent a good cross-section of the different types of models available. All of the methods explicitly or implicitly show that the multiple presence of two or more trucks will trigger the maximum structural responses of bridges.
Even though a variety of different models exist, many of them incorporate assumptions, which have not been validated but considered to represent the reality with a good accuracy. Nowak .. ~L . c, 1···- Figure 3.9: The Probability of Occurrence plotted against the Truck Volume for a 120ft Span (after Gindy and Nassif, 2007) Methods which simulate t he t raffic or the truck t raffic respectively, are mostly based on Monte Carlo Simulations. The parameters simulated are fit ted distributions of measured or observed data. In many cases, · w:rM data is ut ilized for that purpose. The model have the advantage t hat they accurately reflect t he t raffic and therefore can be used not only for the load estimation but also, for example, for the design of signals or ot her traffic-related issues. Simulation models show increased accuracy for higher complexit ies.
In that context, Crespo-Minguill6n and Casas present a very complex method, t hat considers many parameters like e.g. daily and hourly t raffic or congestion and free-fl.ow cases when simulating traffic fl.ow.
A big disadvantage of this is, that due to t he need for high complexity for accurate results, the simulation requires many parameters which are not necessarily available. Also, . 1 ti· on algorithms generate a very high processing load even on high-end computers, s1rnu a so that large-scale simulations can take weeks or longer and t herefore can become costly.
MP events are implicit ly accounted for when the bridge responses are simulated.
Hence, an approach to directly determine the probabilit ies of MP events, without t he inaccuracy of major assumptions, that bypasses the utilization of complex traffic flow rnodeling would provide a fast , accurate and cost-efficient way of predicting site-specific expected live-loads.
Guzda, Bhattacharya and Mertz present a method that is based on visual count ing of MP events from a videotape of a traffic camera. This method is limited to only 2.5 hours of traffic in one site, as t he data analysis is very complex task with many possible sources of errors and additionally extremely t ime-consuming. Utilizing this method for longer periods of time and/or multiple sites is not practicable.
The study of Gindy and assif also targets at the determination of MP probabilities.
In their study, the t ime-stamp of WIM datasets is used to identify MP events. This method proves to be effective for larger amounts of data from multiple sites and over longer periods of t ime. Their methods and algorithms will be utilized for the analysis of WIM data further in t his study.
All of the above studies that identify MP occurrence probabilities deliver results in the range of around 8-10% for Following events and around 1-4% for the Side-by-Side events (compare Tables 3.1, 3.2, 3.3 and 3.4) , depending on the exact definition of t he various MP loading cases and t he underlying data.

Weigh-in-Motion Systems
To develop a Multiple Truck Presence Model, Weigh-in-Motion (WIM) data is used to assess the traffic volume and the associated trucks weights. WIM data is widely available nearly throughout all U.S. states and has been collected for more than two decades.
The main advantage of · w:rM systems for our purpose is, that they provide unbiased truck data. Heavy vehicles often avoided or bypassed the traditional weigh stations located at the highways. However, the WIM sensors are often times not even recognized by the drivers. Another advantage accompanying this is that the trucks do not have to stop, as the data is collected while the vehicles are in motion. This is beneficial as the processing rote increases (a sensor measures within a fraction of a second, while the clearance at weigh stations is a matter of minutes) and the vehicles can avoid unnecessary stops.
WIM stations also require less personnel and therefore have lower maintenance costs than traditional weighing stations.
A slight disadvantage of WIM stations is, that they are less accurate than the traditional scales. After initial calibration, the wheel load is accurate to about ± 13 , according to the National Bureau of Standards (ASTM, 2002). During the lifecycle of a system the inaccuracy will increase to about ±2%, before maintainance and re-calibrat ion is required (McCall and Vodrazka, 1997;ASTM, 2002).
Another disadvantage is the reduced informat ion that is collected at the WIM stations.
Information about engine (fuel) type, year and model of t he vehicle or origin and destination are not available. As this information is neither required nor relevant for this study, they can be neglected.
The following Sections describe t he various WIM systems t hat are most commonly used and the site parameters necessary for inst allation. Also, an int roduction to t he 'NIM database will be given t hat has been used for t his study.

WIM Sensors
Even though many different methods of measuring traffic flow and hence various types of sensors exist, t he most commonly ut ilized sensors t hroughout t he U.S. are

Piezoelectric sensors
Piezoelectric sensors detect t he vehicle loads by a change in voltage t hat is induced when the sensor is pressurized by t he wheels. From t he change in voltage t he dynamic and static loads can be calculated using t he calibration and speed data.   (1997) A typical WIM station setup normally includes one or two Piezoelectric sensors for t he loads as well as one or two inductive loops for system initialization and speed detection.

Load Cell sensors
A single Load Cell sensor normally contains two scales next to each other, measuring axle weights simultaneously by summing up the values of the two scales. A normal system setup consists of a Load Cell sensor, an off-scale sensor to determine whether vehicles do not pass the Load Cell correctly and at least one inductive loop for speed measurement.
A normal layout mostly also has an axle sensor and two inductive loops for the same reasons stated in Section 4. When developing a traffic prediction model, these factors may have a significant impact on a model found.
Vehicle Information is data, that is collected by the sensors. The following list is an excerpt of the most relevant data that is collected by t he sensors • Timestamp (with an accuracy to 1/lOOth of a second), • Speed, • Travel Lane, • Vehicle Class,
The data collected by the sensors is mostly logged into a continuous binary file.
The data can be downloaded from the stations using a data modem and then be converted to a ASCII-standard comma separated file.

WIM Database
The For this analysis, only two consecutive months of truck data were utilized, as the processing of the data is very time-consuming. Hence, seasonal trends are not eliminated from the data. The database for the two months includes a total of about 1.4 million trucks and includes WIM stations with a broad variety of parameters as shown in Table 4.1 and is therefore a good basis for the development of a prediction model. Some sites are at the same geographical location but are separated into directional sites. This was done to be able to identify directional trends in the weight distributions.

Data Filtering
The logging systems collect data of all the vehicles that pass the sensors without in-depth consistency checks. Even though the systems work with a high accuracy, some of the collected data is subj ected to errors due to (for example) a sudden change of speed while passing the sensors, a change of lanes within the measuring section or slow traffic due a congested road.
Furthermore, for this study, only truck data was analyzed so that a set of filters had to be applied to the raw data, as described below.
Error Code = 0 The WIM systems are capable of diagnosing the sensors. If an error occurs, an error code is written into the collected dataset. An error code 0 means that no errors occurred.
Speed <10 mph For very slow traffic, the WIM sensors tend to be inaccurate, so very slow traffic is filtered out. 50 Figure 4.3: WIM Stations in New Jersey (after Gindy and Nassif, 2007) Speed > 120 mph It is very unlikely for a truck to travel faster than 120 miles per hour and so it is assumed that this data is subjected to errors.

Class > 13
Class 13 is the highest class of the FHWA classification system. So if a truck is classified higher than Class 13 it is likely to be an exceptional heavy vehicle that needs a special permit and is not relevant for common traffic or the dataset is subjected to errors.
GVW <15 kips If the Gross Vehicle Weight (GVW) is smaller than 15 kips the vehicle should not be considered a truck but a heavy panel truck and is not relevant for this study.
AXW <2 kips If the weight of one axle (AXW) is less than 2 kips, the vehicle can be considered a car or a panel truck and again is not relevant. AXW >70 kips An axle weight of more than 70 kips is most likely an exceptional heavy vehicle with a special permit or the dataset is subjected to an error.

Sum of AXS >LE
If the sum of all axle inter-spaces (AXS) is greater than the total length of the vehicle (LE), the dataset is considered erroneous.
The filtered data was then separated by direction as the original data was recorded in one file with a parameter for the direction. This filtering is interesting as directional differences in the traffic pattern occur (i.e. commuter traffic etc.).

Filter Results
Applying the filters to the raw WIM data from the NJ DOT, around 753 of all datasets were filtered out meeting one of the criteria stated above. About three percent of the data was filtered out because of errors detected by the WIM-system and an error-flag set to other than zero. The speed filters and the vehicle filter only filtered out a marginal share of far less than one percent each. Around 553 of all the WIM data was filtered out by the Gross Vehicle Weight (GVW) filter. This is due to light truck traffic that is usually lighter than 15 kips. For the axle-weight filters, less than one percent of the data was filtered out. With the filter of the axle-weight being higher than 70 kips, no entries were detected.

52
The consistency check for t he t ruck lengths filtered out about 15% of the data. For many k th e overall length was not equal to the sum of t he axle spacings. This might be true s, due to a change in speed while passing the measuring section or to lane changes, where the trucks did not pass the sensors properly. The remaining about 25% of the data can be considered clean truck data, that will be used for further analysis. Table 4.2 summarizes t he filter results of the New J ersey dat a. The weight distribution of t ruck traffic is important for estimating t he weight correlation among multiple trucks. A first visual impression of t he weight distribution is given in Even though, the weight data of t rucks is not normally distributed , many models fit the weight data to a normal distribution with the parameters µ (mean) and <J (standard  This Chapter defines various loading patterns and describes an algorithm to determine those events from the New Jersey WIM data. The outcomes are presented and discussed. This analysis considers MP events with only two trucks, as this is the most likely to be the governing loading case for short to medium span bridges (40-200ft).

Multiple Presence Loading Cases
To identify loading patterns involving multiple trucks, these loading patterns have to be clearly defined. In this study, the detection of five basic loading patterns is implemented.
These are namely Single Truck: A Single Truck event occurs, when only one truck at a time passes the bridge. Jo trucks in the vicinity are close enough to be on the bridge at the same time.
Following: Two trucks pass the bridge at the same time. Both of the trucks travel in the same lane with a Gap 1 distance, which is less than t he bridge span.

Side-by-Side:
In a Side-by-Side event, two trucks pass the bridge at the same time but travel in different lanes with an overlap of at least half of the body length of the leading truck. These events are most likely to generate the maximum expected bridge responses.

Staggered:
A Staggered event is basically the same as the Side-by-Side event with the exception that the overlap is less than half of the body length of the leading truck.
Other: Events that involve at least three or even more trucks are not considered in this study. For further studies t hese events are nevertheless detected and classified as "Other".
An overview of all the loading cases considered can be found in Figure 5.1.

Multiple Presence Detection Algorithm
With the loading cases clearly defined in Section 5. 1, the algorithm from Gindy and Nassif (2007)  utilized. The timestamp can be used to calculate the entrance and exit times for a simulated bridge as accurate as 11.4 in for a speed of 65 mph when it is exact to 1/lOOth of a second. A timestamp only being precise to the full second would produce an error of more than 90ft , which exceeds the smallest chosen span length by a factor of more than two.
Assuming that the trucks maintain constant speed and stay in the travel lane in which they entered the bridge, the entrance and exit times of each truck can be calculated for various simulated span lengths. If one truck is still on the simulated bridge while another truck enters, a MP event is detected.

As the \;v!M data also contain information about the lane of travel, it can be determined whether an MP event is a Following event (trucks are in the same lane) or if it is a
Side-by-Side or Staggered event. The latter two cases are detected by calculating and comparing the overlap distance between the two trucks to the length of the leading truck. If the overlap is greater than or equal t~ half the truck length, the loading case is considered a Side-by-Side event. For a gap distance smaller than half the truck length, the event is considered a Staggered event (cp. Section 5.1).
Trucks that have been involved in one MP event are discarded by the algorithm and can not be involved in further MP events to avoid a numerical exaggeration. MP events where three or more trucks were detected to be on the simulated bridge at the same time were considered as "Other". These other events were not analyzed in this study.
The algorithm was run for every truck and every WJM location in the database with span lengths from 40ft to 200ft with steps of 20ft, so that sufficient data is available for every location and for multiple span lengths. Figure 5.2 shows a flowchart of the algorithm implemented.

Results of the Analysis
The algorithm was run with all datasets available from the NJ DOT and therefore a representative number of Multiple Presence events was detected. As some parameters of the WIM station were known-namely the average daily truck traffic ( ADTT) , the number of lanes, the area in which the WIM station is located (rural or urban) and the importance of the road type (major or minor highway)-the next step was to identify their association with the occurrence of Multiple Truck presence events. As the outcomes  The most obvious parameter is t he truck volume t hat passes a bridge. As shown in Fig-5 3 the occurrence of Single t rucks decreases wit h increasing t ruck volume, whereas ure · , the Multiple Presence events increase. This is logical as t he space on a highway is limited and with increasing t ruck volume, t rucks are likely to be closer together and t he Gap distances decrease, respectively. were added to visualize similarit ies or differences between the parameters. Excep t for the Following events, t he t rendlines seem to have t he same parameters and t herefore no significant influence of t he area type is evident. A det ailed analysis is done in the Model Development (Section 5.4) .
The same was done for t he importance parameter (major or minor highway). In t his case, the trends vary significantly except for t he Staggered events. This denotes a significant influence of t he highway type. Again, a more detailed analysis is performed in Section 5.4. It should be also noticed, t hat for higher t ruck volumes of more t han -1,700, no data is available for minor highways. This is obvious as t he classification of highway importance highly depends on t he traffic volume. Therefore, t he trend line for minor highway may not be accurate for volumes greater than -1,700, as regression extrapolation by be erroneous (see Section 2.3).
The influence of t he span length is shown in Figure 5.6. For Single events t he rate of occurrence decreases wit h longer spans, as expected, because t he longer t he span, t he P robable is t he occurrence of two or more t rucks, as t he t rucks remain on t he more bridge for longer time when assuming constant speed. For t he Following and Side-by-Side events, an increase of t he MP probability is detect ed for longer spans for the same reason. oticeable is t he slight decrease of t he occurrence probabilities for longer spans for Side-by-Side events. This phenomenon is due to t he fact t hat for longer spans, t he arrangement of t he t rucks on t he bridge is more variable t han for shorter spans. Hence, the probability of overlapping by more than half t he truck length becomes less likely, and more complex loading patterns occur as t he span lengths increase.
Even though t he number of lanes is another parameter t hat has to be considered in t he development of a prediction model, no other study was found where analysis of Multiple Presence events was performed for more t han two lanes (cp. Chapter 3). As only one site in the database had four lanes and six sites had t hree lanes, a graphical t rend is not noticeable, as seen in Figure 5. 7. Nevert heless, t he number of lanes will be included and tested for significance in Section 5.4. The algorit hm introduced by Gindy and Nassif (2007) (Section 3.1.7), which utilizes t he time-stamps to ident ify MP events was implemented and t he New J ersey WIM database was evaluated for Multiple Truck P resence events. The results show t hat t he probability of Multiple Presence events is far below t he assumpt ions made be Nowak and Hong (1991) (Section 3.1.2) and vary according to t he loading cases defined in Section 5. 1.
The Side-by-Side event , which is commonly referred to as t he governing loading event, is least probable according to t he outcomes. This is due to t he definit ion t hat t he overlap of the two trucks in adj acent lanes has to be more t han half of t he t ruck length of t he leading truck to be considered as Side-by-Side. The probabilities found maximize at around two percent . These values are also less t han t he values observed by Guzda et al. ( 2007 ) (Section 3.1.6). In t he latter case, this might be due to t he different definition of Side-by-Side events in their study (Headway distance of maximal two t ruck lengths) .
Tables 5.1 , 5.2 and 5.3 give an overview over the average MP occurrence probabilities found for each MP loading case for multiple spans and multiple ADTT ranges 2 .
Also, some par ameters and their influence were ident ified. As expected, truck volume, the highway importance and the span-length seem t o have t he most influences. The location of the WIM site seems less important .
2 Truck Volume Ranges are Light (ADTT< lOOO) , Average (1000 < ADTT< 3000) and High (ADTT > 3000)  Distributions can either be obtained from the literature or-even more accuratelydetermined from WIM data. The latter gives more accurate results for the site-specific requirements.
In the previous Section 5.3 , the probability of occurrence for Multiple Truck Presence loading cases was derived from WIM data. The underlying WIM database from the NJDOT has a great variety of sites, each with different parameters (see Section 4.3).
Therefore, this WIM data will be used to develop a site-specific prediction model for Multiple Presence events based on regression methods in the following Sections.

Data Encoding
The parameters used in the regression models are described above. For better clarity, these parameters are abbreviated in the following Sections as shown in Table 5.4.
Span-length, truck volume and number of lanes are quantitative parameters, while Areaand Roadtype are qualitative parameters and hence have to be encoded into levels (appropriately 0 or 1) for numerical analysis for t he reasons stated in Section 2 .1.1. To develop an appropriate prediction model for each of the three loading cases, regression analysis is utilized. To check for the significance of each parameter, the stepwise regression methods are used as described in Section 2.4.l. After each step, a t-test (Section 2.2.3) is performed to test for the significance of the parameter added. The Null hypothesis, that the (Ji of the added parameter is equal to zero is tested against the alternative hypothesis f3i # 0. The percent values for "Significance" given in Tables 5.5, 5.6 and 5.7 denote the probability that rejecting the Null Hypothesis was right , according to the t-test.
The significance threshold for added parameters is set to 95%, meaning t hat all added meters with a probability of the right decision to reject Ho of less than 953 will para be declared as "not significant" and omitted in the next step . This threshold value is commonly used and provides a sufficient level of accuracy.
After a model has been found , the overall model is assessed with an F-test (see Section 2.2.4) and t he model quality is ident ified using the adjusted R~-value (see Section 2.2.2). A residual analysis reveals unequal variances and hence t he need for a stabilizing transform ation .

Following
The first stepwise regression was performed for the Following events, as shown in Table   5.5. In the first step, t he Span length is significant within a level of more than 993.
Hence, this parameter is also included in the next st ep. In step four, the added parameter Areatype failed to reach t he predefined 953 significance and is considered insignificant .
This means that the contribution of this variable is negligible for the model accuracy.
Therefore it is omit ted for further steps.  To validate the overall model legitimation, an F-Test was performed. The Null Hypothesis that all f3i = 0 is tested against t he Alternative Hypothesis, that at least one f3i # 0.
The F-value of the model found in interation 5 equates to 353.6 which is much higher than the value found from the F-Distribution, which is 2.39. Therefore the ull Hypothesis is rejected and the overall model validity is confirmed. Replacing the f3is with the values from the regression model (rounded to three decimal places) , the equation The coefficients are rounded to four decimal places and the equation returns the percentage of trucks involved in a Side-by-Side event with a residual standard error of 0.185.

Staggered
Also, for the Staggered event, a linear regression model was found by adding variables step by step as shown in Table 5.7. This analysis shows, t hat for this loading case the 74 Truck Volume is the most important predictor again. The influence of the Span Length tends to be slightly smaller than for the Following case but considerably higher than for the Side-by-Side events. This behavior was expected after the discussions above and in Section 5.3. Again, the Areatype does not have a great contribution to the overall accuracy of the model and therefore can be omitted.
0.8014 4 The convergence plots in Figure 5.10 are a good indicator for the importance of the Volume as a model parameter. For this loading case it can also be seen, that the Number of Lanes have a greater contribution compared to the other loading cases. However, this influence can be questioned as only one site with four lanes and ten sites with three lanes exist in the database, while all other sites have two lanes.
Validating the model found in iteration 5, significant evidence is found that at least one of the ,Bi-coefficients is unequal to zero, as the computed F-value of 521.5 exceeds the value from the F-distribution (2.39) by far. The model equation after substituting the ,Bis with the values found (rounded to four decimal places) from the model becomes

Regression with Logarithmic Terms
The occurrence probabilities of Multiple Presence events are not assumed to increase linearly over increasing Traffic Volume or Span Length, but are expected to converge against a maximum value. Hence, additional Logarithmic terms for these parameters were introduced into the regression model. The logarithmic terms are based on the natural logarithm based on e. The terms are added to the best model found in Section 5.4.2 and tested for legitimation using the F-Test introduced in Section 2.2.4 with the methods for the nested models, introduced in Section 2.4.2.
In the latter test, the Null-Hypothesis, that the added logarithmic terms do not contribute to the model accuracy, is expressed by assuming that the ,Bi-coefficients for the added logarithmic terms are zero. This hypothesis is then tested against the Alternative Hypothesis, that at least one of the added logarithmic terms is legitimate and contributes to the model. That means, in mathematical terms that at least one ,Bi-coefficient is un-equal to zero. The hypotheses are tested by comparing the calculated F-Values against the Fa-values from the F-Distribution with a significance level of a= 0.05.

Following
Adding logarithmic terms to the Following event data shows a slight increase in the R2-value. The equation for t he regression model for the Following events becomes a (5.6) The R~-value increases from 0.8331 found in the linear model to 0.8493 when the logarithmic terms are used. The Null Hypothesis is rejected, as the found F-value of 27.3 is much greater than the a = 0.05 significance level of the F-Distribution, which is 3.0.
This means that at least one of the two added logarithmic terms contributes to the accuracy of the model.

Side-by-Side
For the Side-by-Side events the same analysis is performed, adding logarithmic terms for span-length and volume. Again, a slight increase in the model accuracy is found, as the R~-value increases from 0. 7752 to 0. 7907 when the logarithmic terms are added.
Performing the hypothesis testing, that the reduced model has the same accuracy as the complete model leads to a calulated F-value of 19.13 which is by far larger than the a= 0.05 significance level of the F-Distribution, which is 3.0. Therefore, the Null-Hypothesis is rejected and at least one of the two added logarithmic terms has significant contribution to the model.

Staggered
Also for the Staggered loading case, logarithmic terms were added and the overall model accuracy increased slightly from a R~-value of 0.8082 from the linear model to a R~-value of 0.8171 when adding logarithmic terms. The hypothesis testing is again performed on the Staggered data and the calculated F-value from the model was found to be 12.8. This is again greater than the a = 0.05 significance level from the F-Distribution, which is 3.0. Again, the run Hypothesis is rejected in support of the Alternative Hypothesis that at least one of the added logarithmic terms has a /3i-coeffi.cient unequal to zero.

Residual Analysis
One of the major assumptions of the validity for regression models is the so-called homoscedasticity of the residuals. Homoscedasticity implies that the residuals E all have the same variance a 2 . Unequal residual variances are called heteroscedastic.
As the WIM data is traffic data, it might follow a Poisson-Distribution. For the regression model above this becomes obvious if the residual variance increases for higher Volumes, as exemplary shown for the Following events in Figure 5.11 a). Hence, a heteroscedasticity is evident and the Following data follows a Poisson-Distribution. However, this phenomenon could not be observed for the Side-by-Side and Staggered loading patterns. To regain homoscedasticity, the dependent variable has to be transformed to stabilize the variance. A common transformation for Poisson data is a square-root transformation to achieve a approximately constant variance (Mendenhall and Sincich, 1996, p. 394) .
So, the transformed dependent variable becomes Utilizing the model for prediction, the values y* found have to be transformed back into the original units by using the reverse of the transformation, which is the square of y* for the Poisson data, so y = (y*)2 = .;;y2 (5 .9) The R~-values for the best models found in the Sections 5.4.2-5.4.2 are compared to the results for the transformed data in Table 5.8. As stated above, only the Following event data seemed to profit from the transformation to stabilize the variance. The variances of the Side-by-Side and the Staggered events seemed already constant and therefore a transformation had no reasonable advantage for the overall model accuracyactually, a slight decrease in the model accuracy is detected when the dependent variable was transformed.

Conclusions
The regression models found for the three different Multiple Truck Presence loading cases are developed, tested for legitimacy and provide a good base for predicting the MP occurrences when certain parameters are known.

80
The /3i coefficients found and the known parameters can now be inserted into the equation as described in Section 2.3 to compute the occurrences of the Multiple Presence events within the intervals also described that Section.
Adding logarithmic terms to the regression models increases the accuracy of all three models slightly but about 1-2%. As a traffic model for practical application should not be overly complex, the logarithmic terms should be omitted, as the extra work to incorporate these terms is disproportionate to the gains in accuracy which are negligible.
An extrapolation for Average Daily Truck Traffic Volumes higher than 5,000 should be executed with extreme caution and skepticism with regard to the results, as this is common pitfall in regression analysis: as the underlying regression model is a linear model, the MP probabilities could rise over 100% with a certain extemely high volume that is far out of range.
A logarithmic model approaches a maximal value asymtotically, which might be below 100%-but again: as this value is far out of range of the orginal regression region, it is most likely to be erroneous.
Looking at the residuals in the regression model, values of up to ±2% occur. The regression model found will graphically result in a line which is representative for the data, with the actual datapoints scattered around that line, as it can be observed similarly in the plots in Section 5.3. Considering this effect, a rather complex equation will lead to reasonable results but is not very practical as the ,Bi-coefficients found have numerous decimal places.
A good approach to solve this problem is implicitly given by Moses (see Section 3.1.3 and Moses, 2001), as he categorizes the Side-by-Side event probabilities according to 81 ADTT intervals (cp . Table 3.2). This can be done for arbitrary intervals and levels of conservativeness with respect to the MP probabilties.
An idea of the outline of a table for Multiple Presence occurrence probabilities is shown in Table 5.9. To use the Table, the ADTT interval is chosen and gives a base probability of the loading case, as the Volume was identified to be the most important parameter.
The other parameters are found on the right and give additional probabilities, which are to be added to the base probability.
No actual values are given in the table as it is just a suggestion how a table for practical engineers could be designed . For this study and the sake of consistency, the best linear models found for each loading case will be used for the further work and the applied example in Appendix A. 82

Influence of the Gap Distances
Another phenomenon that has not been subject to research in the literature reviewed is the influence of the gap or headway distances. For a Side-by-Side event, the maximum headway distance is defined to be one half of the length of the leading truck, so the influence is very obvious.
For the other types of MP events (Following and Staggered), the gap or headway distances are more variable and might play an important role for the maximum bridge response. The two trucks produce higher responses when being closer together or further apart depending on the loads of the trucks and the static system. In this Chapter, the influence of the gaps and headways for the most common static systems that are used for small to medium span highway bridges-namely a simple supported beam and a continuous beam with one support-is analyzed.
The gaps or headways are normalized to the bridge span length to identify whether a threshold gap to span exists for the MP events to become irrelevant for the maximum responses. The analysis is performed for both Moments and Shear forces, and the outcomes are discussed. 84

.1 Static Systems and Influence Lines
The most common static systems for short to medium span highway bridges are either simply supported beams or continuous beams with one support at around midspan. This support is oftentimes the column between the directional lanes when one highway crosses another. These static systems are very simple to calculate and to design for. The gap or headway distances are-within their definitions-variable over the span length. Therefore, the influence lines of the moments and the shear forces can be utilized for evaluation. Influence lines represent the variation of the effects or responses ( deflections, moments, shear forces) at a certain point when a load is moved over the structure.
Multiple loads can simply be superimposed in linear systems.
The placement of the load on the structure that induces the greatest responses can easily be determined from the influence lines.

Algorithm
When a truck runs over the bridge, the resulting influence line can be superimposed with the influence line of the other truck that is involved in a Multiple Presence, as shown in Figure 6.3. To determine whether a MP event produces greater responses than one of the trucks on its own, the influence of the heavier truck is utilized and the influence of the lighter truck is added with a multiplication factor so that the total Response (R) becomes Rtotat = Rmax + a · R,nin (6 .1) This has the advantage that it is now easy to determine the influence of the lighter truck, as it is expressed by the factor a. For a equal zero, the influence of the lighter truck is negligible and the maximum response is merely produced by the heavier truck (i.e. at midspan for the simply supported beam). For a greater than zero, both trucks induce the greatest response together. The concentrated loads shown in Figure 6.2 and 6.3 were used for illustration of the theoretical background. For the algorithm, the actual truck configurations of the database were normalized to the HS20 and HL93 loads and utilized for the calculations.

Following
The analysis described above was executed for every Following event in the database for the maximum moment and maximum shear for a simply supported beam and for the maximum negative moment over the column for a continuous beam with one support. It should be noticed that in Figure 6.4 that only a-values are plotted that vary from zero due to clarity reasons. This represents only about 5-73 of all a-values found-93-953 of the a-values calculated are, in fact, equal to zero.
For the maximum moments of a simply supported beam this means that the threshold of the second truck being influential is when the gap is less than 0.5 of the span length of the bridge. This is comprehensible when looking at the influence lines in Figure 6.3.
For maximum shear for the simply supported beam, the threshold is voided again, as influences of the lighter truck occur for gap to span ratios longer than 0.5. This is also logical as it follows the influence line. Figure 6.5 shows a plot of a vs. gap to span for 40ft and 200ft for the shear. Again a-values equal to zero were omitted in the plot to retain clarity.
The influence of the lighter truck for maximum negative moments over a support of a continuous beam is also not limited to a certain threshold. This again is explainable with the shape of the influence line. Maximum negative moments are generated when each truck is at midspan of a field. Being closer together, the maximum moment becomes smaller. Hence, a wave-shape observed as shown in Figure 6.6 with the greatest influence of the lighter truck at around 0.5 to 0.8 gap to span. It should be noted, that for this loading case, no a-values were omitted in the plot, because of being zero. Hence for this loading case, the lighter truck is always influential.

Staggered
For the Staggered events, the same analysis was performed as was used for the Following events in Section 6.3. A threshold found is that the smallest gap to span ratio is 0.2 . Gap distances less than 0.2 gap to span can be considered as Side-by-Side events. Figure 6.7 also shows, that the influence threshold of the Moments is shifted to the right by about 0.2 gap to span compared to the Following event. The same shift occurs for the Negative Moments, as the wave-shape peaks at about 0.2 gap to span to the right . The 0.2 gap to span shift would also theoretically apply to the Shear as seen in Figure 6.7 b). As a gap to span ratio of greater than 1.0 is not possible, no actual threshold can be found.

Conclusions
The influence of the gap distance between two trucks in a Following Multiple Presence event (with two trucks) did have an influence on the maximum bridge response for the moments of a simply supported beam. A threshold could be found that denoted the maximum gap distance between two trucks to be influential. A greater distance means that the heavier truck has a greater effect on the moment response when at midspanregardless of the other truck. For this loading case, the approach would have been very promising, as the analysis shows that for a 200ft span only around 30% of the Following loading cases have a gap distance smaller than 0.5 gap to span, while 70% of the gap distances were found to be between 0.5 and 1.0 gap to span. For smaller spans, these values obviously tend to be even less for <0.5 gap to span and larger for >0.5 gap to span, respectively. This apparent advantage is voided again, when looking at the shear forces of the following events for a simply supported span bridge. Here, the influence of the second (lighter) truck is evident up to a threshold at about 0.9 gap to span. Analysis shows that for a 200ft span, around 90% of the gap distances are smaller than 0.9 gap to span, so that one can say that the threshold found is of rather theoretical nature and should not be included into a life-load model.
For the continuous beam with one middle-support, no threshold value can be found , as the lighter truck always contributes to the the maximum bridge moment responses.
The wave-shape of the plot in Figure 6.6 is obviously connected to the influence line in

94
For furt her analysis in t his study, t he full probabilit ies for MP occurrences will be used .
The t heoretical t hreshold for simply supported beams at about 0.9 gap t o span is omitted on the conservative side.

Weight Correlation
As stated in Section 3.2, not only the probability of occurrence of Multiple Truck Presence events are critical, but also the weight correlation between the two trucks involved, for determination of the maximum lifetime response. Nowak (Section 3.1.2 and Nowak and Hong (1991); Nowak (1993)) uses correlations that are assumed from the biased Ontario truck data (Agarwal and Wolkowicz, 1976) and therefore appear to be very conservative, as remarked by Moses (Section 3.1.3 and Moses (2001). Other models, based on simulation, treat the MP events entirely through the structural responses, without correlating the truck weights.
For models that explicitly incorporate the Multiple Presence occurrence probabilities, the correlation between the truck weight is a crucial factor for precise maximum load prediction. Therefore, the weight correlation between the two trucks involved is determined throughout this Chapter. To determine whether the weight correlations depend on the event type, the analysis is performed separately for each of the three event types with the two truck weights involved into the MP event.

Methodology
To find the correlation between the truck weights, a rather simple algorithm is utilized.
The loads (GVW) of the two trucks involved in the MP event will be divided to find a ratio between the truck weights. For a full correlation between the truck weights, this ratio will be exactly one.
Testing for the ratio to be exactly one would omit correlated weights that vary only marginally from the ratio to be exactly one, and the result will not re:(iect the real situation. Therefore, a threshold interval around one was implemented to also consider insignificant differences. An orientation for the threshold was given by the values that were found by Nowak (Section 3.1.2 and Nowak and Hong (1991) ). After a few tests, the most reasonable threshold was found to be around one percent difference. These values provide results that are in the same range as those found by Nowak. Weight differences from up to 0.5 kips for 50 kips trucks seems acceptable, as a like event will definitely induce a great structural response, even though the weights are not fully correlated by theoretical means.
To determine the percentage of fully correlated trucks, the weights of each loading case were divided and the ratio was checked against the interval. If the ratio is within the interval, a counter for fully correlated events is increased. After all occurrences of the particular MP event were checked, the ratio between the fully correlated events and the total number of events is calculated.

Results
Analysis of correlation was performed for all MP events in t he database separated by the loading cases. To det ermine whether t he span length also has an influence on t he weight correlation, t he analysis was run again for two representative span lengths for each MP loading case.
For the Following MP event, t he percentage of fully correlated t rucks according to t he definition above was found to be 2.4955%, which is about a quarter of the findings of owak, who found t hat every tent h t ruck in a Following event is fully correlated. The difference might occur because t he Following event, according to Nowak, is defined by a maximum gap distance of 50ft (see Section 3.1.2) , while in this study the maximum gap distance for a Following event is t he bridge span (see Section 5.1). Trucks of equal weight might not be able to pass each other and remain behind each other in close distance.
This might explain why Nowak found the correlation to be as high as 10%. It should also be noted t hat owak only examined t he upper heavy 20% of the t ruck population and based his assumptions on this data and visual observat ion. In this analysis, the correlation is found to be vice versa: for smaller spans, the correlation was found to be less t han for longer spans. A 60ft span showed a full weight correlation in 2.043 of t he cases, while a 200ft span showed to have 2.537% of full weight correlation.
For the Side-by-Side events, a full weight correlation was found t o occur in 1.58% of t he Side-by-Side loading cases. This is about in the same range as in owak's work, who finds a full correlation in 3-5% of the Side-by-Side cases. likely to be in the process of passing the assumed heavier truck in the right lane. As the process of passing tends to happen faster when the passing truck is lighter, the chance of being detected as Staggered event is more likely than to be detected as Side-by-Side event. Therefore, a full correlation in weight might be less likely.
The difference in weight correlation only varies marginally with the span length. For a 60ft span , a full correlation probability was found to be 1.373 and for a 200ft span, this probability increased slightly to 1.403.
All the probabilit ies for a full weight correlation for the two trucks involved for each loading event are given in Table 7.1.

Conclusions and Recommendations
An extensive review of the literature revealed that basically three different approaches of live-load modeling for bridges exist, namely methods based on statistical methods (also referred to as "Convolution Models"), models based on simulation (i.a. Monte Carlo simulations) and methods that are based on sets of actual traffic data. The first two modeling methods normally involve deep knowledge of the statistical backgrounds (i.a. Moses and Ghosn, Section 3.1.1) or require a decent amount of numerical complexity (i.a. Crespo-Minguill6n and Casas, Section 3.1.4) to be applied for practical use. Methods based on traffic datasets (i.a. owak, Section 3.1.2) are found to be comparably easy to apply, but incorporate a lot of assumptions concerning the occurrence probabilities and the probabilities of a total weight correlation between the trucks involved in a Multiple Presence event. Other studies (i.a. Guzda et al., Section 3.1.6) develop methods to determine those probabilities from visual evaluation of videos of recorded traffic-which proves to be very time-consuming. Gindy and Nassif (Section 3.1.7) develop a method to determine the Multiple Presence probabilities from WIM data, which is widely available throughout the U.S.
Based on the findings of the latter study, a database was set up for ew Jersey WIM data. As only truck data is relevant for maximum live loads, the data was filtered to dispose of car data and faulty entries. The weight distribution was determined, and an algorithm based on the findings of Gindy and Nassif was implemented to detect the probabilities of Multiple Presence events in the NJ data. Based on the outcomes, a regression model was developed to estimate the MP occurrence probabilities based known parameters, such as traffic volume (ADTT), span length, number of lanes, area and road type (Section 5.4). Also, the probabilities of full weight correlation between the two trucks involved in a MP event were identified and found to vary slightly among the MP event types (Chapter 7).
A closer look was taken at the gap distances between the two trucks in a MP event to determine whether a certain gap to span ratio denotes a threshold where the heavier truck induces a greater response on its own than when considered in an MP event. This is done by comparison of the maximum values derived from influence lines of the MP events with both trucks and the heavier truck by itself. The outcomes showed the threshold for the moments of a simply supported beam to be at a gap to span ratio of 0.5. The advantage of the the eventually lesser probability of "true" Following events is voided when looking at the shear forces for t he same static system. For continuous beams with one support, no threshold could be detected. However, it was identified, that for this static system, the lighter truck is always influential. Similar results were found for the Staggered loading case (Chapter 6).
The overall findings in this study proved that the assumptions made by Nowak (Section 3.1.2) are overly conservative. The MP probabilities as well as the probabilities of full weight correlations were widely overestimated on the conservative side. Compared to the outcomes of the simple model developed by Moses (Section 3.1.3) for Side-by-Side events, the resulting MP probabilities found in this study are in about the same range but even tend to be slightly less. Compared to the findings of Guzda et al. (Section 3.1.6) the MP probabilities for Side-by-Side events found in this study are again in the same range but slightly less. This slight variation may be due to the different definitions of a Side-by-Side event.
Based on the findings in this study, a live load model was developed that only requires a few parameters to accurately predict the MP occurrence probabilities and full weight correlation probabilities. The regression model found can be replaced with a table for practical application, as proposed in Table 5.9. The probabilities of full weight correlation could be tabulated in a similar way for easy application.
The two components-MP and full weight correlation probabilities-are a crucial base to evaluate site-specific bridge loading and are of major importance for the accuracy of the overall results. Figure 8.2 gives an overview over the work flow for the evaluation of MP bridge loading. The components of this study are highlighted.
For further research, it is recommended to evaluate the loading cases that have been refered to as "Other" in the detection algorithm. These are loading cases with three or more trucks involved. Even though these loading cases are likely to have a small probability of occurrence, they might become relevant as maximum live-load for the life span of the bridge as identified by Caprani et al. (Section 3.1.5).
For the analysis, two consecutive months of data from New Jersey were used, so seasonal and regional trends are not eliminated from the data. To validate the results found , other WIM data can be used to check for those trends.
As for t he regression analysis, a closer look is to be taken whether the linear regression models can be replaced wit h Logistic Regression methods. Logistic Regression is a special model t hat is used to estimate or predict t he probability of occurrence of an event by fitting dat a to a logistic curve. A logistic curve or function is a sigmoid curve. That means, an S-curve is modeled wit h some set of P. It equates 1 P (t ) =l +e-t (8.1) First, t he growth is exponent ial and t han slows down and st ops at maturi ty as shown in These methods are extensively used in medical and social sciences to predict probabilities of events and t herefore seem suitable for t he estimation of MP occurrence probabilit ies.
The most apparent advantage of this method is that t he probability values cannot exceed a 1003 or become less t han 03 due to appropriate t ransformation of t he dependent variable P (Yi = 1).  Figure 8.2: Flowchart of a site-specific bridge load evaluation. This study covers the first three steps highlighted. The accuracy of these steps is one of the most crucial parts for an accurate bridge live-load evaluation.

Appendix A A Practical Example
In this Chapter, the outcomes of this study will be applied to evaluate the total probabilities for an exemplary bridge to show a practical application of the work. This example is not supposed to deliver accurate design values but is to exemplary show how the work from the previous Chapters can be applied to a practical problem.
In this example, a bridge is assumed that has that same weight distribution as the New Jersey data. The Average Daily Truck Volume (ADTT) is assumed to be 4,000 trucks per day. The highway is a major route, and the bridge is located in a rural area and has two lanes in each direction. The span length of the simply supported bridge will be lOOft.

A.1 Determination of Multiple Presence Probabilities
In a first step, the regression model found in Section 5.4 is utilized to determine the probability of the Multiple Presence events. With the parameters from above inserted in the Regression Model, following MP probabilities are found and shown in Table A.I. After Iowak (see Section 3.1.2) and Moses (see Section 3.1.3) , t he loading cases with t he largest bridge responses are t hose which incorporate two t rucks t hat are fully correlated in weight . Therefore, t he probability of full weight correlations was subject to evaluation in Chapter 7. The probabilities for full weight correlation for the three MP loading cases are determined from Table 7.1 in Section 7.2. To be slight ly conservative-and t hus on the safer side-the probabilities are assumed as shown in Table A  These assumptions are not supposed to be a recommendation for general use, but are merely used for this example to ret ain clarity t hroughout t his example.

A.3 Total Probabilities
The found probabilities from the last two Sections are now simply multiplied to determine the total probabilities of the Multiple Presence events with two fully correlated trucks. The total probabilities found denote the probability of a truck being involved in one of the three MP events and being fully correlated in weight to the second truck involved in the MP event. With these total probabilities, the number of MP events with two fully correlated trucks can be determined by multiplica tion with the total number of trucks expected for the lifetime of the bridge. With an Extreme Value Distribution fitted to the ECDF of the weight distribution, the GVW for the maximal trucks involved in such a MP event can be determined and the bridge can be designed according to the findings.
The extrapolation of the inverse ECDF of truck weight distributions and/or fitting of extreme value distributions to weight distributions is a wide field of research in bride design and covered by many of the literature introduced on Chapter 3. "Statistical thinking will one day be as necessary for efficient citizenship as the ability to read or write." -H.G. Wells