A practical multi-lane factor model of bridges based on multi-truck presence considering lane load disparities

Abstract

Many bridge design specifications consider multi-lane factors (MLFs) a critical component of the traffic load model. Measured multi-lane traffic data generally exhibit significant lane disparities in traffic loads over multiple lanes. However, these disparities are not considered in current specifications. To address this drawback, a multi-coefficient MLF model was developed based on an improved probabilistic statistical approach that considers the presence of multiple trucks. The proposed MLF model and approach were calibrated and demonstrated through an example site. The model sensitivity analysis demonstrated the significant influence of lane disparity of truck traffic volume and truck weight distribution on the MLF. Using the proposed approach, the experimental site study yielded MLFs comparable with those directly calculated using traffic load effects. The exclusion of overloaded trucks caused the proposed approach, existing design specifications, and conventional approach of ignoring lane load disparity to generate comparable MLFs, while the MLFs based on the proposed approach were the most comprehensive. The inclusion of overloaded trucks caused the conventional approach and design specifications to overestimate the MLFs significantly. Finally, the benefits of the research results to bridge practitioners were discussed.

Graphical abstract

Keywords

bridges / multi-lane factor / traffic load / lane load disparity / multi-truck presence / weigh-in-motion data

1 Introduction

1.1 Background

In designing and evaluating bridges, traffic loads are applied to the most adverse positions of the structure for the maximum load effect (LE). However, the probability of such traffic loads co-occurring over multiple lanes is low. Therefore, the multi-lane factor (MLF) or multi-presence factor represents the reduced probability of extreme events co-occurring in adjacent lanes. The MLF has a clear physical significance and is convenient for engineering applications, making it an essential part of the traffic load model in many bridge design codes [ 1, 2]. Generally, a standard lane load model is established from the characteristic load (or LE) of a single lane, with MLFs added as adjustment factors to account for the characteristic loads (or LEs) of all the lanes. Recently, the measurement of vehicle weights in operation has enabled a more accurate evaluation of bridge traffic LEs [ 3– 5]. It has also generated data for developing a refined MLF model that reveals the mechanism of multi-lane traffic loadings on bridges [ 1, 2, 6– 9].

1.2 Literature review of multi-lane factor

Currently, there are four primary approaches used to determine the MLFs of bridges [ 1, 8].

The first approach (Approach 1) involves the application of probabilistic statistics of multi-truck presence and the corresponding loads, as initially proposed by Jaeger and Bakht [ 10, 11] and employed under the Canadian and Chinese design codes [ 12]. Under this approach, the MLF is defined as the ratio of the average multi-truck-presence weight to the maximum observed truck weight at a specific probability of occurrence calculated based on independent repeated trial events of multi-truck presence. Although this approach is practical and straightforward, it is based on many statistical assumptions: the truck weights follow a normal distribution, the maximum observed truck weight is 3.5 standard deviations above the mean, the lane loads are identically and independently distributed (IID), and the emergence of trucks obeys the Poisson distribution with its parameters being related to the truck traffic volume, truck speed, truck length, etc. Moreover, the approach assumes that bridges with a span length of up to 125 m are governed by free-flowing traffic.

The second approach (Approach 2), which was developed primarily by Nowak [ 13] and is applied by the American Association of State Highway and Transportation Officials (AASHTO), involves modeling the LEs arising from multi-truck presence. This approach produces the most adverse loading cases for multi-lane bridges, including the side-by-side or staggered passage of trucks in multiple lanes, from which the MLF can be determined based on the respective probabilities of the passages. However, the probability of occurrence of such events strongly depends on the traffic characteristics, bridge configuration, and the assumed correlations between following, staggered, and adjacent vehicles [ 2, 13, 14].

The third approach (Approach 3) is the simulation and evaluation of traffic LEs on multi-lane bridges with different span lengths, numbers of lanes, and LE types, among other parameters. The approach is adopted under the ASCE guidelines and Eurocodes [ 15, 16]. However, it is based on many assumptions about traffic behavior, such as heavy vehicles to cars ratio is 25:75 under the Eurocodes [ 16].

The fourth approach (Approach 4) involves the multivariate modeling of coincident lane LEs, as Zhou et al. [ 1, 9] proposed. It employs a multi-coefficient MLF model to reveal the mechanism of extreme coincident lane LEs. To this end, it first determines the MLF coefficients by applying multivariate and univariate extreme-value-extrapolation algorithms. In this approach, the MLF is transformed from a single coefficient to multiple coefficients, clarifying the contribution of traffic load on each lane to the resultant total bridge LE. However, in bridge engineering, multivariate extreme-value-extrapolation algorithms are complicated and require sufficient sampling of coincident lane LEs. Moreover, these LEs must be significant nonzero values, or the governing event should be single lane loading. Thus, this approach requires long-term traffic load simulation, which may be infeasible in practical application.

These four approaches quantify the MLF model from different mechanical perspectives based on either traffic loads or traffic LEs. Nevertheless, the objective of these approaches is the same, namely to determine the MLF model used to design or evaluate the LEs of bridges. The four approaches are briefly compared in Table 1. For engineering applications, the MLF model and its calculation method should be sufficiently accurate and straightforward to use, and complex computational procedures should be avoided whenever possible. The probability statistics of the multi-truck presence method (Approach 1) is based on a clear physical concept and relatively simple. However, the following assumptions on which the approach is based are inconclusive. For the normal distribution assumption of the gross weight of trucks, the statistics of measured traffic loads on sites reveal that the truck weight has a multi-peak distribution [ 1, 3, 6, 17]. In the multi-peak distribution pattern of all truck weights, the gross weights of these heaviest trucks (i.e., the last peak) are often taken to be normally distributed [ 1, 3, 13, 17, 18]. For instance, Nowak [ 13] manually sketched the best fitting straight lines for the upper tail data of bridge LEs on normal probability papers. OBrien et al. [ 17] used a best-fit normal distribution to model the upper tail of the probability density histogram of truck weights. Anitori et al. [ 18] found that the upper 5% of the data of bridge LEs could be well fit by a normal distribution function. Zhou et al. [ 1, 3] used the Gaussian mixture model to fit the multi-peak distribution of truck weights with satisfactory accuracy of capturing the tail tendency of the data. However, parameters in such a normal distribution are often obtained by tail fitting rather than by using the normal distribution to directly fit all the truck weights as in previous studies [ 10– 12]. Therefore, an accurate mathematical model describing the multi-peak features of the gross weights of all trucks is required. For the assumption related to the identical independent distribution of lane loads, the measured data indicates that traffic loads in different traffic lanes vary significantly, which is based on the traffic rule of lane separation for various vehicle types [ 6– 9, 19]. For the assumption that the maximum observed truck weight is 3.5 standard deviations above the mean value, many measurements revealed that the relationship is significantly different from site to site [ 8, 13, 14]. Therefore, it is necessary to improve this approach as a practical means of establishing MLFs.

1.3 Motivation and objective

The multi-coefficient MLF model proposed by Zhou et al. [ 1, 9] clarifies the differences between lane loads and their contribution to the bridge LEs, thereby providing a deep understanding of multi-lane traffic loading on bridges. However, the calculation of MLF coefficients requires traffic simulation and univariate and multivariate extreme value extrapolation, which is a prohibitively complicated task in bridge engineering practice. This study aims to establish a practical multi-coefficient MLF model based on an improved approach involving the application of multi-truck presence probability statistics (Approach 1). Although the proposed MLF model has the same form as in Refs. [ 1, 9], the underpinning approach for MLF coefficient calculation is fundamentally different and relatively simple for engineering applications.

The improved approach is general and applicable to all truck-weight-distribution patterns and observed truck weight features. The remainder of this paper is organized as follows. In Section 2, the multi-coefficient MLF model based on the improved approach with the probability statistics of multi-truck presence is mathematically derived. In Section 3, a sensitivity analysis of the model is conducted. Section 4 describes the results of an experiment performed to demonstrate site-specific traffic data to the development of a practical MLF model for the assessment and maintenance of bridges. The results of the proposed method are then compared to those of previous approaches. Section 5 discusses the significances of the results and how bridge practitioners can benefit from this study. Finally, Section 6 concludes the paper by presenting major findings.

It should be noted that the dynamic effect of traffic load on bridges is generally not considered in the study of MLFs as it is considered a separate problem [ 1, 2, 6– 16]. As the ultimate goal of MLF is to calculate the bridge LE, it is necessary to verify that the proposed MLF model based on the statistics of traffic loads can also be used to characterize bridge LEs. In this study, the assumption of linear correlation between truck weight and maximal truck LE as in Approach 1 is still adopted. In reality, truck LEs are influenced by truck weights as well as configurations. In general, heavy trucks that control the maximal LEs of bridges may have similar configurations, and thus a linear correlation between truck weight and maximal truck LE may be acceptable, as found by Kwon et al. [ 20]. However, there can be cases in which heavy vehicles are long with lighter axle loads or short with heavier axle loads. In this approach, they are treated as being the same based on the simplifying assumption.

Nevertheless, the rationality of the assumption is carefully validated in Section 4. Moreover, this study establishes MLFs based on the probabilistic statistics of multi-truck presence events (Approach 1). Such multi-truck presence events in free-flowing traffic govern the maximal traffic LEs of short- and medium-span bridges with lengths of up to 60 m [ 13, 18, 20, 21], and therefore, the proposed approach applies to such structures.

2 Model and methodology

2.1 Model form

For a given bridge with N traffic lanes, a multi-truck presence event under the model developed in this study is defined as the occurrence of overlapping between trucks in other traffic lanes with a truck in the reference lane (e.g., Fig. 1(a), in which lane 1 is the reference lane). The truck’s gross weight in lane i ( i = 1,…, N) in a multi-truck presence event is denoted as W _i, as shown in Fig. 1(b). The MLF is applied to represent the reduction coefficient under the standard single-lane load model to a multi-lane bridge. Accordingly, for a specified load return period, it is defined as the ratio of the average truck weight in a multi-truck presence event to the maximum truck weight on the reference lane:

(1) M L F = \frac{W_{\mathrm{N}}^{\mathrm{c}}}{N\cdot W_{\mathrm{s}}^{\mathrm{c}}} = \frac{W_1^{\mathrm{e}}+W_2^{\mathrm{e}}+\cdot \cdot \cdot +W_n^{\mathrm{e}}}{N\cdot W_{\mathrm{s}}^{\mathrm{c}}} ,

where the subscript ‘s’ represents the standard (or reference) lane, i.e., the traffic lane loaded with the maximum truck weight; N is the number of traffic lanes; W _N is the total weight of trucks in the multi-presence event within the overlap range (one truck length). The superscript c is used to represent various characteristic values, e.g., W_{\mathrm{s}}^{\mathrm{c}} is the characteristic weight of trucks on the reference lane during the return period of traffic loading; W_{\mathrm{N}}^{\mathrm{c}} is the characteristic total weight of trucks over multiple lanes during a multi-presence event during the return period. The superscript ‘e’ refers to extreme values, e.g., W_i^{\mathrm{e}} is the extreme truck weight on lane i during the multi-presence event corresponding to W_{\mathrm{N}}^{\mathrm{c}}. Clearly, W_i^{\mathrm{e}} is the proportion of the characteristic total-lane vehicle weight W_{\mathrm{N}}^{\mathrm{c}} (i.e., W_{\mathrm{N}}^{\mathrm{c}} = {\displaystyle {\sum }_{i=1}^N{W}_i^{\mathrm{e}}}), which is not larger than the characteristic reference-lane vehicle weight W_{\mathrm{s}}^{\mathrm{c}}.

Under the bridge design codes used in Canada, China, and the United States, the lane load model is the same for all traffic lanes; therefore, lane load disparity is not explicitly reflected in these codes. For other bridge design codes such as Eurocode, BS5400, and the ASCE-recommended model for long-span bridges, different values of the lane load model are applied to different traffic lanes. However, the full lane load model is still applied in the most adverse traffic lane depending on the calculated bridge component. Therefore, these models also fail to reflect the lane load disparity because this factor (non-uniform distribution of traffic loads across lanes) is objective and independent of the location of the calculated bridge component. Under the proposed MLF model, r_i = W_i^{\mathrm{c}}/W_{\mathrm{s}}^{\mathrm{c}} is applied as a lane-correction coefficient to reflect the unevenness of traffic loads across multiple lanes. This coefficient represents a reduction of the characteristic loads of each lane to that of the reference lane. A value of r close to 1 indicates that the characteristic truck weights in different traffic lanes are similar, implying identical traffic loads over different traffic lanes.

Another coefficient, η, is introduced to characterize the relationship between extreme truck weights in the multi-truck presence event and characteristic truck weights, which is defined by

(2) \eta = \sum _{i=1}^N{W}_i^{\mathrm{e}}/\sum _{i=1}^N{W}_i^{\mathrm{c}} .

It should be noted that the characteristic total-lane load W_{\mathrm{N}}^{\mathrm{c}} is unaffected by this mathematical transformation. In other words, the simplification does not have a significant effect on the MLF calculation. However, the use of the same coefficient η for all traffic lanes has a clear physical significance in that the lane loads of the multi-truck presence event are equivalently reduced from their corresponding characteristic lane loads. In this study, η was defined as the multi-lane combination coefficient, which represents the reduction of the characteristic total-lane load under a multi-truck presence event to the sum of the characteristic lane loads. When η tends to 1, the probability of occurrence of a multi-truck presence event in which heavy trucks are simultaneously loaded at adverse positions on a multi-lane bridge is high.

Using the lane-correction coefficient r and multi-lane combination coefficient η, the MLF model can be further formulated as follows:

(3) M L F = \frac{1}{N} \cdot \eta \cdot \left(r_1+r_2+\cdot \cdot \cdot +r_{\mathrm{n}}\right) .

Equation (3) denotes a multi-coefficient MLF model with the same form as that in Refs. [ 1, 9]. However, the underlying principles used in this study for determining the coefficients are fundamentally different from those used in the previous studies and are based on the probability of multi-truck presence.

2.2 Improved methodology

This section introduces the probabilistic statistics of multi-truck presence as an improved methodology that avoids the statistical assumptions considered in the initial Jaeger–Bakht method. It is generally assumed that the presence of a vehicle is independent of the preceding vehicle [ 10– 12]. Given this assumption, it should also hold true that the simultaneous entry of vehicles in multiple lanes within the overlapping range is an independent event, enabling the application of the mathematical theory of independent repeated trials. If the average daily traffic volume of each lane is given by Q _i ( i = 1,…, n) and the cumulative probability distribution function of truck weights in each lane is F _i(·), then, for a given lane i, the probability of the presence of heavy trucks heavier than \eta \cdot W_i^{\mathrm{c}} in a multi-presence event (i.e., that the truck is present in the overlapping range in Fig. 1(a)) is

(4) p_i = \frac{Q_i\cdot \left(1-F_i\left(\eta \cdot W_i^{\mathrm{c}}\right)\right)}{86400/\mathrm{\Delta }t} ,

where ∆ t is the average time required for a truck to pass through a specific cross-section of the bridge, which is measured in s/veh (where “veh” stands for vehicle). In this case, ∆ t = L/ v, where L and v are the average truck length (m/veh) and average truck speed (m/s), respectively. The numerical factor ‘86400’ is the number of seconds per day (24 h). Here, Q_i \cdot \left(1-F_i\left(\eta \cdot W_i^{\mathrm{c}}\right)\right) is the number of heavy trucks (weight greater than \eta \cdot W_i^{\mathrm{c}}) in a day. 86400/∆ t implies the number of occurrences in a day that a truck is present in the overlapping range.

For multi-lane bridges, the assumption of event independence can be applied to express the probability of occurrence of on-bridge multi-presence trucks with gross weights all greater than \eta \cdot W_i^{\mathrm{c}} as follows:

(5) p = \prod _{i=1}^n p_i = {\left(\frac{\mathrm{\Delta }t}{86400}\right)}^n \prod _{i=1}^n \left[Q_i\cdot \left(1-F_i\left(\eta \cdot W_i^{\mathrm{c}}\right)\right)\right] .

In bridge design, a reference period T (such as 100 or 75 yr) is commonly defined. Thus, the number of occurrences of multi-truck presence over such a reference period is given by m = K \cdot T \cdot 86400/\mathrm{\Delta }t, where K indicates the number of valid days over a year. Based on independent repeated trials, the probability of occurrence of multi-truck presence within a design reference period would follow a binomial distribution B ( m, p). Given that m is a large value and p is close to zero, this binomial distribution can be approximated by the Poisson distribution given by P( X = l) = λ ^l × e ^−λ / l!, ( l = 0,1,…) with intensity λ = m × p.

In designing or evaluating bridges subjected to traffic loads, the characteristic traffic LEs are defined as occurring at a quantile over a period. The Chinese bridge design code requires a quantile of 95% of the maximal value within a design reference period of 100 yr. Therefore, the corresponding return period is calculated as R = 1/(1−0.95 ^1/100) = 1950, i.e., R = 1950 yr, corresponding to a 95% quantile assurance over the design period T = 100 yr. Therefore, the probability of non-occurrence of the Poisson distribution should be 95% or greater, i.e., P( X = 0) = e ^−λ ≥ 0.95, and the intensity should be λ = m × p ≤ −ln (0.95) = 0.0513. Thus, the multi-truck presence event can be expressed by the following quantile assurance equation:

(6) \prod _{i=1}^n \left[Q_i\cdot \left(1-F_i\left(\eta \cdot W_i^{\mathrm{c}}\right)\right)\right] = \frac{\lambda }{K\cdot T} {\left(\frac{86400}{\mathrm{\Delta }t}\right)}^{n-1} .

Given that F _i(·) is a monotonically increasing function, where the left-hand side of Eq. (6) is uniquely correlated with the multi-lane combination coefficient η. Therefore, for a given reference period, η can be directly determined using Eq. (6). Besides, the lane-correction factor r _i is calculated based on the characteristic lane load, which can be determined using the cumulative distribution function (CDF) of the lane truck weights given by

(7) W_i^{\mathrm{c}} = F_i^{-1} \left(1-\frac{1}{K\cdot R\cdot Q_i}\right) .

If the underlying data of lane truck weights is sufficient for the determination of the high quantile in Eq. (7), the characteristic lane load can be directly interpolated. In general, however, the traffic load measurement period is relatively short. Therefore, univariate extreme value statistical extrapolation is required for the prediction of the quantile. An examination of the literature reveals that many tail extrapolation algorithms have been applied in the study of bridge traffic loads (or LEs). For example, Nowak [ 13] plotted the bridge LEs on a graph of normal probability and extrapolated the characteristic value by sketching the best fitting curve for the upper tail data, i.e., tail fitting using the normal distribution. In this study, the least-squares method was applied to the tail data of a normal probability graph to obtain a linear best fit, and the resulting linear function was used to extrapolate the characteristic values of the given reference periods, as would be shown later in Section 4.2.

In summary, if the lane traffic volume Q _i, CDF of the lane truck weights F _i(.), and design reference period T are known, the multi-coefficient MLFs for bridges can be calculated using Eqs. (6) and (7). Moreover, when n = 1, MLF = 1, which is in accordance with the definition of the MLF in this paper. When there is no difference in loadings between lanes, the Jaeger–Bakht method can be expressed as Eq. (6), which indicates that the proposed approach is a more general method. Notably, the conventional approach indicated in the study is the MLF calculation but without considering the lane load disparity, i.e., the parameters in Eqs. (6) and (7) are the same for each lane.

3 Model sensitivity analysis

Mathematically, the multi-coefficient MLF equation results are dependent on several parameters, including Δ t, R, Q, and F(·). Research has shown that the MLF decreases significantly with truck volume and that an increment in the variation of truck weight also leads to a decrease in the MLF [ 10– 12]. To complement these findings, the influence of the classical parameters Δ t, R, Q, and F(·) on MLF are analyzed in this section, and the truck volume and weight distribution differences across multiple lanes not considered in the conventional approach are carefully investigated.

In the analysis, the following values were adopted for these parameters when they are not involved in the sensitivity studies. The daily truck volume of the reference lane Q ₁ was assumed to be 5000 veh/d, and the truck weights within an individual lane were assumed to follow a normal distribution with a COV of 0.3 [ 8, 12], which was assumed to be constant for different traffic lanes. A return period of 1950 yr, as defined in the Chinese bridge design code (MOT 2015), was set ( R = 1950 yr), and each year was assumed to contain 365 valid days ( K = 365). The average time for truck passage through a bridge cross-section was defined as 0.75 s (Δ t = 0.75), which considered a mean truck speed of 48 km/h and a mean truck length of 10 m. The correlation between truck length and weight was not established due to its complexity. However, the effect of varying Δ t caused by the randomness of truck length is investigated in Section 3.1. The values of R, K, Δ t, and COV represent the adverse case of traffic loading on bridges. Note that we assumed the truck weight follows a normal distribution. This represents the upper mode near the legal limit of the entire (multi-modal) truck weight distribution [ 13, 17, 18], which simplifies the numerical analysis. Moreover, the qualitative relationships between the investigated variables and the resulting MLFs are not influenced by simplification.

3.1 Influence of classical parameters

The effects of the classical parameters (i.e., Δ t, R, Q, and F(·)) on the MLFs are shown in Fig. 2. The truck weights in multiple traffic lanes were assumed to be identical and to follow a normal distribution with a mean value and COV of 40 t and 0.3, respectively. The figure demonstrates the following findings about the nature of the MLF: 1) it increases with truck volume and stabilizes at Q ≥ 3000 veh/lane/d and N ≤ 4; 2) it decreases significantly as the variance in truck weight increases; 3) it is not sensitive to changes in the load return period for R ≥ 100 yr and N ≤ 6 (this also means that it is insensitive to the number of valid days per year ( K)); 4) it increases with the average truck passage time and stabilizes at Δ t ≥ 0.5 s and N ≤ 4; 5) in the descending order of sensitivity, the impact of the parameters on the MLF is COV > Q > Δ t > R. Furthermore, the maximal MLFs for two-, three-, and four-lane configurations are 0.75, 0.62, and 0.53, respectively, which can be observed consistently from the four sub-plots. These values differ from those in previous studies [ 11, 12] because these values in the study are based on applying different reference-lane characteristic truck weights for MLF normalization. In the previous studies, the reference-lane characteristic truck weight was set empirically as 3.5 standard deviations above the observed mean value; whereas, in this study, it was calculated using Eq. (7). Nevertheless, the influence rule obtained for the classical parameters is consistent across all the studies.

3.2 Influence of truck-volume lane disparity

To consider differences in truck volume across multiple lanes, we introduced the following normalization factor:

(8) {\xi }_{\mathrm{n}}^i = \frac{Q_i}{Q_{max}} ,

where Q _i is the truck volume in lane i and Q _max = max ( Q _i) for 1 ≤ i ≤ n.

Figure 3 illustrates the effect of lane disparity in terms of truck volume on the MLF. As indicated in the Figure, the MLF depends significantly on the differences in the truck volume between lanes, particularly for ξ ≤ 0.2, with the influence increasing as the number of traffic lanes increases. As the normalization factor decreases, the MLF decreases at an increasing rate; at ξ ≥ 0.4, the MLF reduction does not exceed 8%. When there is no difference in truck volume between lanes (i.e., ξ = 1.0), the maximum MLFs for two-, three-, and four-lane configurations were 0.75, 0.62, and 0.53, respectively, which is consistent with the findings in Fig. 2. The minimum MLFs at two, three, and four lanes, respectively, can be achieved by setting the value of ξ close to 0. Because the MLF is extremely sensitive to ξ as the latter tends to 0, the simulation step for ξ in this study was set to 0.01 to reduce the computation time. Nevertheless, the minimum MLFs shown in Fig. 3 may vary with different simulation steps (such as 0.001). Overall, these findings indicate that when there is a significant variance in truck volume over multiple lanes, the normalization factor should be taken into account for determining the MLF to avoid significantly overestimating the coincident lane LEs.

3.3 Influence of truck-weight distribution lane disparity

To consider the differences in truck weight between lanes, we introduced a mean truck-weight normalization factor:

(9) {\psi }_{\mathrm{n}}^i = \frac{u_i}{u_{max}} ,

where u _i is the mean truck weight in lane i and u_max = max( u _i) (1 ≤ i ≤ n). The influence of the mean truck weight on the MLF is shown in Fig. 4. The MLF depends significantly on the relative truck weight. When there was no lane disparity in truck weight distribution (i.e., ψ = 1.0), the maximum MLFs for two-, three-, and four-lane configurations were 0.75, 0.62, and 0.53, respectively, replicating the results in Section 3.1. However, if ψ is set close to 0, the minimum MLF for the two-lane configuration is 0.37 and not 1/ N = 0.50, with similar discrepancies occurring in the three- and four-lane cases. These discrepancies occurred because combining the reference and other lanes for obtaining the characteristic total truck weight makes the probability of the simultaneous presence of trucks very low, leading to a very small value of η. Therefore, it is necessary to quantify the differences in truck weight over multiple lanes.

4 Example application

This section presents a collection of multi-lane traffic data to investigate the truck-load characteristics over multiple traffic lanes. The data were used to apply a site-specific multi-coefficient MLF model to the in situ assessment of bridges. In this assessment, the MLFs were obtained using the proposed approach and validated by those directly calculated using traffic LEs. After that, the effect of weight restriction on MLF calculations was investigated. Finally, the advantages of the proposed multi-coefficient MLF model for bridge assessment were demonstrated through sample bridges, where the MLFs defined in the design specifications and those determined using the conventional approach were compared.

4.1 Statistics of multi-lane traffic data

Traffic data were collected from an expressway in Guangdong province, China. The expressway is a typical link connecting several medium-sized cities and carries eight-lane bi-directional traffic. Unidirectional weigh-in-motion (WIM) data were obtained from May 2014 to January 2015. Following common practice [ 1], unreliable data corresponding to the interruption of the WIM system for several months were removed. Given that the loads of lightweight vehicles have negligible effects on short-to-medium span bridges, only trucks with gross weights greater than 3.5 t were considered. Ultimately, WIM data covering a total period of 150 d were used. Notably, we calibrated the WIM data to provide only the static vehicle-axle weights by excluding the dynamic effect and temperature influence. The 150-d recording period for WIM data are acceptable in the traffic LEanalysis from a site-specific perspective as justified in previous studies [ 1, 19, 22].

Table 2 lists the statistical results for the recorded multi-lane traffic data. Large differences in lane truck volumes were observed: out of the 850162 trucks that were analyzed, measured, 30366, 173783, 397781, and 248232 were observed traveling in lanes 1 to 4, respectively. In terms of truck volume, lane 3 was considered the reference lane because it supported most of the trucks. Thus, the normalized truck volumes were {\xi }_4^1 = 0.08, {\xi }_4^2 = 0.44, {\xi }_4^3 = 1.00, and {\xi }_4^4 = 0.62. A total of 551 overloaded trucks with gross weights greater than 55 t were recorded. Although no overloaded trucks were found in the inner-most lane (lane 1), there were a considerable number of extra-heavy trucks (weights greater than 100 t) in lanes 3 and 4. Overall, the truck volume and load statistics exhibited significant lane disparities, indicating that they should be considered to determine site-specific MLFs.

Figure 5 illustrates the histograms of truck weight in each traffic lane. In each lane, the measured traffic comprises a mixture of vehicle types; consequently, the histograms of truck loads are in accordance with neither a single-mode distribution nor a typical two-peak distribution representing an unloaded or loaded truck weight. Nevertheless, significant differences were found in terms of truck loads among traffic lanes, with a considerably higher probability of overloaded trucks in the outer lanes than in the inner ones, which had a significant influence on the MLF.

4.2 Calculation of multi-coefficient MLFs

The method proposed for calculating the multi-coefficient MLF requires the inputs of Δ t, R, Q, and F(·). The lane truck volumes are listed in Table 2. A truck weight return period of 2000 yr was employed because variations in this value did not significantly impact the results (Fig. 2(c)). The value of Δ t was difficult to determine owing to the high degree of randomness in the truck length and speed data. It is seen from Fig. 2(d) that the rate of increase of the MLF gradually decreases to zero as Δ t increases. To conservatively analyze bridge LEs, a suitably increased MLF had to be obtained to account for the variation in Δ t values; therefore, a 95% quantile value of 0.79 s was employed to simplify the complex analysis of random Δ t values.

The most critical determining parameter was the distribution function of lane truck load, F(·). To better describe this distribution and predict the high quantile of data, the lower part of the data histogram was used to create the lower part of the distribution, and a normal distribution function that fits the tail of the data was used to create the upper part. The data of lane truck weights are plotted on a normal probability paper, and the fitting functions based on the above approach are also illustrated, as shown in Fig. 6. It is found that the truck weight data does not follow a straight line in the plot, i.e., the lane truck weight does not follow the normal distribution, which validates that the assumption of the normal distribution of truck weights in the conventional Jaeger and Bakht approach is rebuttable. Another finding is that the upper tail of lane truck weight data could fit a straight line (i.e., a normal distribution) with sufficient accuracy. This straight line is obtained through the best fitting of the tail data based on the conditions of R² ≥ 0.99 and RMSE ≤ 0.01, where only a small number of heavy trucks (nearly 2% of total data) were selected. This result validates the rationality of our previous assumption that the gross weights of these heaviest trucks (i.e., the last peak) can be taken to be normally distributed. As this fitting approach accurately describes the data in the lower part, and the trend in the tail, the fitting line was adopted as the distribution function of the lane truck load F(·). From our statistics, using these tail data for extrapolation indicated that 90% of the traffic contained two-axle trucks on lane 1, 45% was six-axle trucks on lane 2, 84% was six-axle trucks on lane 3, and 86% was six-axle trucks on lane 4. These results indicate that six-axle trucks constituted the most heavily loaded vehicle type in the outer three traffic lanes. The characteristic truck weights on the respective traffic lanes under a return period of 2000 yr were found to be W_1^{\mathrm{c}} = 33.6 t, W_2^{\mathrm{c}} = 157.5 t, W_3^{\mathrm{c}} = 152.5 t, and W_4^{\mathrm{c}} = 137.0 t. Here, we emphasize that the extrapolated characteristic truck weights may be high but are realistic, as reasoned in our previous studies [ 1, 19]. However, we recommend using 200 t as the upper limit because there are no reports of regular trucks heavier than that.

The MLF calculation results are listed in Table 3. These values were derived based on the truck load in the reference lane (lane 2). Based on the extrapolated characteristic lane truck loads, the lane-correction coefficients for lanes 1–4 were determined to be r₁ = 33.6/157.5 = 0.21, r₂ = 1.00, r₃ = 152.5/157.5 = 0.97, and r₄ = 137.0/157.5 = 0.87, respectively. From Eq. (6) and the lane-load-distribution function expressed in Fig. 6, a multi-lane combination coefficient for four-lane loading of 0.17 was derived. Therefore, the MLF for four-lane loading was derived as MLF = η × ( r₁ + r₂ + r₃ + r₄) / N = 0.13. However, as the simultaneous loading of four traffic lanes does not always govern the design of bridge components, the MLFs for different combinations of lane loading (i.e., single, two, and three lanes) were investigated. Notably, the definitions of the reference lane in the normalization of lane truck volume (lane 3) and lane truck weight (lane 2) may seem to conflict but pose no impact on the results (see Eqs. (6) and (7)). The final MLFs are based on the reference lane that produces the most adverse lane load, i.e., lane 2.

The MLFs calculated using the proposed and conventional approaches were compared with the results of numerical solutions. In the conventional approach, the truck load distribution function in lane 2 was used, and the truck volume was set as the average truck volume over all four traffic lanes, i.e., 1417 veh/lane/d. The numerical results were obtained based on the following procedures: 1) the multi-lane WIM data was loaded on to simply supported beam bridges with span lengths of 10, 20, 30, 40, 50, and 60 m, and the time–history LEs of the mid-span bending moment and side-support shear force on different traffic lanes were collected. From this time history; 2) the daily maximal LEs for different combinations of traffic lanes were extracted; 3) a generalized extreme value distribution was fitted to the daily maximal LEs based on the classical extreme value theory as applied in other studies [ 1, 4, 16], leading to the conclusion that the distribution tended to Weibull distribution (the shape parameter is less than 0), and hence, was bounded; 4) the characteristic LEs were extrapolated over a return period of 2000 yr; and finally 5) the characteristic LEs on lane 2 as the reference values for normalization were used to calculate the MLFs. Notably, in selecting the daily maximal LEs of a combination of more than two traffic lanes, the corresponding LE of each traffic lane should be a nonzero value, i.e., the selected daily maxima should be induced by trucks over multiple lanes. During the passage of the truck over the bridge, it is found that the relationship between truck weight and the maximal bridge LE was approximately linear, especially over longer bridge spans (Supplemental Files). The results validate the suitability of the proposed approach based on the assumption of multi-truck weights present. Figure 7 depicts an example of the numerical calculation of a single-lane MLF; the results for other MLFs can be obtained using the same approach. For more details on the MLFs calculated from the numerical results obtained for different combinations of traffic lanes and at various LEs and span lengths, the reader is referred to the Supplemental Files. The mean MLFs under given traffic lane combinations are the numerical solutions (Table 3) and can be regarded as the reference MLFs.

The main findings of the comparisons above are summarized as follows. 1) The proposed approach yielded the closest MLFs to those obtained through numerical analysis. Any small deviations between them were likely the result of using a relatively small data set (≤ 150) for extreme value extrapolation. 2) The four-lane MLFs calculated using the proposed and conventional approaches were nearly equal. 3) The single-lane MLFs calculated using the conventional approach significantly overestimated the loading effects on lanes 1, 3, and 4. 4) The MLFs calculated using the conventional approach underestimated several combinations while overestimating those used for non-full-lane load situations. Overall, the proposed MLF model is more comprehensive than the conventional approach in describing the nature of the multi-lane loading mechanism. It should be noted that the MLFs listed in Table 3 were calculated under the assumption that all four traffic lanes on the highway were available. If a one- or two-lane closure occurs, the MLFs must be re-calculated based on the truck load characteristics for the restricted three- or two-lane highway.

4.3 Effect of weight restriction

The passage of overloaded trucks on highway structures has a significant impact on the structural safety of bridges. The use of truck weight restrictions is a standard policy for ensuring infrastructural safety. Accordingly, we modeled a strict weight restriction and analyzed its effect on MLFs. For the same WIM data set applied in the previous analysis, all overloaded trucks with gross weights greater than 55 t (the weight limit defined under Chinese bridge design code [ 23, 24]) were reset to 55 t based on their identical axle weight proportions. The distribution function of the lane truckload was then re-evaluated. The results are shown in Fig. 8, in which the left-truncated vertical line is used to predict the extreme value of lane load. As seen in the figure, the characteristic lane truckloads for lanes two, three, and four are all 55 t because of the strict weight restriction, whereas the characteristic lane truckload for lane 1 remains at 31.7 t.

The MLFs obtained using the proposed, conventional, and numerical solution approaches for all non-overloaded trucks are listed in Table 4. As with the results listed in Table 3, the proposed approach produced MLFs that most closely converged on the numerical analysis results. Although the MLFs calculated using the conventional approach were close to those calculated using the proposed approach in loading combinations with lane 1 excluded, the inclusion of lane 1 in multi- or single-lane-loading cases caused the conventional approach to overestimate the LEs.

4.4 Application and comparison

The proposed multi-coefficient MLF model and its corresponding calculation method were applied to a specific site for verification. It should be noted that the MLFs were used in conjunction with the characteristic LE on the reference lane, which, when based on site-specific data, can differ from the characteristic LEs on the reference lane reported under different design codes. Moreover, the MLFs calculated using the proposed method can differ from those reported under the design specifications. Nevertheless, it is helpful to consider the resulting site-specific differences between MLFs. To compare the differences in MLFs alone, the influence of various standard lane models under various design codes and approaches should be excluded. In the following example, for simplicity, the bridge lane LEs induced by the standard lane load models based on design specifications and those induced by the proposed approach were assumed to be the same and set to one. This simplification did not influence the study’s outcome because the MLF model is a complementary approach to the standard lane load model applied to estimate multi-lane loading. Two types of bridges were analyzed: 1) bridges with integral girders and no significant differences in terms of influence lines over multiple lanes, and 2) bridges with multiple girders and differences in terms of influence lines over multiple lanes.

4.4.1 Case 1: Bridges with integral girders

Bridges with integral girders are not sensitive to the transverse loading of vehicles. When the lane LE is set to one for simplicity, the total LE of an integral girder bridge under multi-lane traffic loading can be expressed as N × MLF. The data listed in Table 5 compare the traffic LEs of a four-lane bridge with an integral girder under various design codes and the proposed approach. Two versions of the Chinese bridge design code, D60-2004 [ 23] and D60-2015 [ 24], the AASHTO standard [ 25], and the Canadian bridge design code (CSA) [ 26] were compared. The MLFs in these design codes are detailed in Ref. [ 1]. The MLFs calculated from the site-specific WIM data with and without overloaded trucks were also compared. As the MLFs calculated using the proposed approach were based on the four-lane traffic loading case, the total LE was the most adverse when the loading conditions of different numbers of traffic lanes were considered. For example, although many MLFs were calculated using the proposed approach to the same number of loaded traffic lanes, only the most adverse MLF was adopted.

It should be noted that the proposed MLFs were validated as approximates of the exact values achieved by the numerical solutions. Thus, the results of the proposed method successfully revealed the bridge LEs reflected in the in situ multi-lane traffic data. The primary findings from Table 5 can be summarized as follows. 1) When the overloaded trucks were excluded, both the proposed and conventional approaches yielded the same MLFs. 2) With the inclusion of overloaded trucks, the conventional approach produced smaller MLFs than the proposed approach, with a maximum difference of up to 24.3%. 3) With the exclusion of overloaded trucks, the two approaches and the design codes yielded comparable total LEs; however, the governing loading cases were different. 4) With the inclusion of overloaded trucks, the design codes significantly overestimated the MLFs, and the two-lane loading case was found to govern the total LE on the four-lane bridge. These results show that the multi-lane LE on an integral bridge can be appropriately evaluated by applying either the conventional approach or the MLFs defined under the bridge design codes when there are no overloaded trucks on the site. In the presence of many overloaded trucks, however, the conventional approach and design codes underestimate and overestimate the multi-lane LEs, respectively.

4.4.2 Case 2: Bridges with multiple girders

Bridges supported by multiple girders are highly sensitive to vehicle transverse loading. In this case, a simply supported multi-slab bridge with a length of 16 m was investigated, as shown in Fig. 9. The bridge supports four-lane traffic and has 12 slabs, which were pre-stressed and laterally connected via hinged post-cast concrete. The width of each traffic lane is 3.75 m. A finite element grillage model of the multi-slab bridge was established following the approach used in a previous study [ 3, 19]. In this model, the bending moment of each slab was obtained by applying a unit concentrated point load on each transverse finite element node of the bridge deck in the mid-span. The shared load of each slab was calculated based on the relationship between the concentrated load and the bending moment of a simply supported beam. From this, the transverse influence line of each slab can be obtained as the curve connecting the shared load value of each transverse node, as shown in Fig. 9. In this Figure, the horizontal axis represents the position of the unit point load, and the vertical axis the influence ordinate expressing the proportion of the unit point load shared by the slab. The following procedure can help determine the static traffic LEs of each slab in a multi-slab bridge: 1) the transverse influence line of each slab, as given in Fig. 9, is determined; 2) the trucks are loaded along the transverse influence line to achieve the load distribution factor for each slab; 3) the load distribution factor of each traffic lane is first multiplied by the lane correction coefficient, then the result of each lane is summed up, and finally, the results are multiplied by the multi-lane combination coefficient and standard-lane LE to obtain the total LE; 4) the final total LE is the largest of those under different lane combination loading scenarios. As mentioned earlier, the standard-lane LEs were assumed to be identical and set to 1 for simplicity. As a result, the total LEs were solely related to the MLF and the load distribution factor. The transverse loading of trucks on the bridge deck was modeled following the Chinese specification, i.e., a transverse wheel spacing and inter-vehicular gap of 1.8 and 1.3 m, respectively.

Figure 10 illustrates the calculated traffic LEs on each slab of the bridge using the proposed approaches and the design codes. As in the preceding modeling exercise, the proposed MLFs accurately revealed the bridge LEs by applying the multi-lane traffic data from the example site. The results led us to the following findings: 1) The proposed approach and the conventional approach yielded similar MLFs, except at the inner girders near the median divider of the highway; 2) The design codes produced comparable MLFs when overloaded trucks were excluded, although the governing loading cases differed; 3) With the inclusion of overloaded trucks, the design codes significantly overestimated the MLFs; 4) In cases involving both the inclusion and exclusion of overloaded trucks, the conventional approaches and design codes significantly overestimated the traffic LEs on the inner slabs; 5) The proposed approach revealed that the traffic LEs on the inner slabs were significantly lower than those on the outer ones, a result consistent with the reduced traffic loads in the inner lanes relative to the outer lanes; 6) The governing loading cases on the test bridge obtained by the proposed approach using site-specific data were the one- or two-lane loading cases. By contrast, the stipulated governing loading cases under the design codes were generally three- and four-lane loading cases.

The multi-coefficient MLF calculated using the proposed approach revealed the lane disparities in truck load and their contributions to the LEs on the various bridge components. In particular, the proposed approach is more versatile and accurate than the currently used single-coefficient model and enables a better understanding of multi-lane traffic loading on bridges. The results show that neither the conventional approach nor the current bridge design codes accurately represent the multi-lane LEs on a multi-slab bridge. In the absence of overloaded trucks, the conventional approach and design codes overestimate the LEs on the inner slabs next to the highway central divider because lane load disparity is not considered even though it significantly influences the LEs on those slabs. In the presence of many overloaded trucks, the conventional approach overestimates the LEs on the inner slabs, whereas the current design codes overestimate the LEs on all slabs. These findings highlight the need to establish a site-specific MLF model based on the features of multi-lane traffic data while carefully considering the influence of lane load disparity.

5 Discussions

A multi-coefficient MLF model and an improved approach of modeling traffic loads on short- and medium-span bridges were developed in this study. As indicated in the case study, the MLF calibration was simple. It mainly involved establishing the distribution functions of lane truck loads and solving Eqs. (6) and (7) based on the statistics of site-specific traffic data. The proposed multi-coefficient MLF model clarifies the contribution of lane traffic load to the performance of bridge components, which provides a better understanding of the mechanics of multi-lane traffic loading on bridges.

Although the results are associated with the features of multi-lane traffic data obtained for a specific site, the outer traffic lane features reflecting a higher percentage of heavy trucks may be universal. If so, the calibrated MLFs for such sites should reflect the same features: a) outer traffic lanes with larger traffic LEs, and therefore much higher lane correction coefficients; b) lower multi-lane combination coefficients under the inclusion of inner traffic lanes that produce lower traffic LEs. Finally based on the results obtained for the experimental site, we shall discuss the benefits bridge practitioners can derive from the results of this study.

For the design of multi-lane bridges, the MLF models applied under current bridge design codes (such as AASHTO, Eurocode, BS5400, D60, CSA) can be modified to a multi-coefficient form, in which disparities in terms of lane load (lane correction coefficient) and the reduced probability of simultaneous multi-lane loading actions (multi-lane combination coefficient) are considered. The multi-coefficient MLF model is particularly applicable to bridges with large numbers of traffic lanes (e.g., ≥ 3) that carry overloaded trucks. For this design purpose, many case studies covering a wide range of traffic data and different numbers of traffic lanes should be conducted, and the resulting MLFs should be validated on various bridge spans and influence line types. Such efforts can be carried out following the procedures provided in Section 4. Then, MLF design tables and equations with different values of lane-correction coefficients and multi-lane combination coefficients related to the number of traffic lanes, lane truck volume, and lane truck weight statistical parameters can be established to be used by bridge designers. MLFs applied under the current bridge design codes and the conventional approach are still applicable in cases without overloaded trucks or traffic lane-load disparities over multiple lanes.

In assessing multi-lane bridges, the proposed multi-coefficient MLF model and approach can be directly applied based on the availability of measured multi-lane WIM data. The resulting multi-coefficient MLF model would be of particular interest in assessing multi-girder bridges or bridges carrying considerable volumes of heavy truck traffic. In cases without between-lane disparities in terms of truckloads, the conventional approach is still applicable. The WIM data used to establish the MLFs can be collected from the bridge or other sites with the same number of traffic lanes and similar features in terms of truckloads or by incorporating toll-by-weight and multi-lane camera data.

6 Conclusions

This study proposed a practical multi-coefficient MLF model of bridges based on an improved probabilistic statistical approach of modeling multi-truck presence. The proposed approach avoids statistical assumptions made by previous studies, many of which are refuted, and applies a more general method that considers lane disparity in truckload over multiple traffic lanes. The main findings are as follows.

1) A multi-coefficient MLF model, which applies a lane correction coefficient, r, to represent the lane-load difference and a combination coefficient, η, to denote the reduced probability of simultaneous lane loading, was derived. The model can be simply expressed as MLF = η × ( r ₁ + r ₂ + ··· + r _n)/ N, where N is the number of traffic lanes. This multi-coefficient MLF is more comprehensive than the current single-coefficient model when explaining the mechanism of multi-lane traffic loading on bridges.

2) An improved probabilistic statistical approach was proposed to model the multi-truck presence, in which lane disparity in terms of truckload over multiple lanes was considered. This approach produces a more general solution than the conventional one. The parametric evaluation results indicate that lane disparities in truck volume and weight distribution have a significant influence on the MLF.

3) The proposed model and the improved approach were applied to calculate the MLFs for an experimental site using four-lane traffic data. The proposed MLFs yielded results comparable to those directly calculated based on the traffic LEs of multi-lane WIM data, whereas the MLFs calculated using the conventional approach may induce large deviations.

4) Because the experimental site is subject to the passage of many overloaded trucks, a stringent weight restriction was imposed to evaluate its effects on the MLF. The weight restriction primarily caused the MLFs calculated by the proposed and conventional approaches and those based on current design codes to converge.

5) The MLFs calculated by the proposed method were applied to bridges with integral and multiple girders and compared with those based on several design codes and the conventional approach. The results revealed that the MLFs calculated by the proposed approach accurately reflected the significant difference between the traffic loads of the inner and outer girders on a multi-girder bridge. In contrast, the conventional approach and design codes produced significant deviations.

The proposed multi-coefficient MLF model and approach shed light on the potential pathways to optimizing bridge design and assessment for cases involving multi-lane traffic loads. This can lead to reductions in cost and effort in bridge design and maintenance. Furthermore, this study could also provide directions for revising multi-lane traffic load models in current bridge design specifications for considering lane load disparity. Besides, meaningful follow-up works could be performed, such as comparing MLFs obtained by these approaches using the same traffic data sets and establishing comprehensive MLF models based on a wide range of WIM data from different countries.

7 List of notations and nomenclatures

N: number of traffic lanes

W: truck gross weight

c: characteristic values

e: extreme value

r: lane-correction coefficient

η: multi-lane combination coefficient

Q: average daily lane traffic volume

F: cumulative distribution function

L: average truck length

v: average truck speed

T: bridge design period

K: number of valid days over a year

R: load return period

λ: Poisson intensity factor

i: lane label

u: mean lane truck weight

∆ t: average truck passage time through a cross-section

p: the occurrence probability of on-bridge multi-presence trucks

ξ: the normalization factor of lane truck volume

ψ: the normalization factor of lane truck weight

LE: load effect

MLF: multi-lane factor

WIM: weigh-in-motion

IID: identical independent distribution

CDF: cumulative distribution function

COV: coefficient of variance

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