Investigation of multi-lane factor models for bridge traffic load effects using multiple lane traffic data

Structures 24 (2020) 444–455

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Investigation of multi-lane factor models for bridge traffic load effects using multiple lane traffic data

T

⁎

Junyong Zhoua, , Zhixing Chena, Jiang Yia, Haiying Mab a b

College of Civil Engineering, Guangzhou University, Guangzhou 510006, Guangdong, China College of Civil Engineering, Tongji University, Shanghai 200092, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Bridge Traffic load Multi-lane factor Lane load disparity WIM data

Multi-lane factor (MLF) is a crucial component of the traffic load model in design specifications for bridges. Due to the significant changes in highway freight transportation over the past two decades, it is urgent to propose precise MLFs for bridge assessment under onerous traffic loading. This study reviews the underpinning approaches of existing MLF models and uses numerical examples and site-specific weigh-in-motion data to investigate the performance of these approaches. First of all, three MLF underpinning approaches described in the literature, i.e., the multi-presence truck weights approach, the multi-presence truck load effects approach, and the coincident lane load effects approach, are illustrated and discussed. Then, numerical examples of multi-lane bridge traffic load effects without lane load disparity are used to compare the MLFs calibrated by the three approaches. Finally, realistic weigh-in-motion data that have a strong lane load disparity are employed to calibrate MLFs by these approaches, and the MLFs are compared with those in bridge design codes. Results show that the three approaches have large deviations on MLFs calculation based on the same traffic data and studied bridges. In general, the coincident lane load effects approach is flexible for MLF calibration of traffic data with and without lane load disparity. However, its calculation procedure is relatively complicated, especially when the number of traffic lanes is large. The multi-presence truck load effects approach gives comparable results of traffic data without lane load disparity but overestimates MLFs when there is lane disparity of traffic loads over multiple lanes. The multi-presence truck weights approach needs to be improved for traffic data with and without lane load disparity otherwise generates large deviations. The selection and improvement of the three approaches are further discussed. This work highlights the importance of MLF approaches to consider the lane disparity of traffic loads over multiple lanes in practical engineering.

1. Introduction In the design of multi-lane bridges, the traffic load model is placed to the most unfavorable position of each traffic lane to calculate the most significant load effect (LE). However, the fact is that the probability of adverse traffic loads over multiple lanes act simultaneously on the most unfavorable positions of the bridge is very low. To account for this, a multi-lane factor (MLF) is often introduced to consider the probability reduction, which forms a crucial component of the traffic load model in many design specifications for bridges [1–2]. The general equation of the MLF could be formulated as follows:

MLF =

SN N ·S1

(1)

where, SN and S1 are the characteristic total traffic load (effect) and

⁎

reference-lane traffic load (effect) respectively under a defined load return period. N is the number of traffic lanes. In this way, the standard traffic load model is determined based on the LE of a single lane (or the reference lane), and the MLF is adopted on this basis for the calculation of total LE for bridges carrying multiple traffic lanes. Due to the clear physical meaning and convenient implementation, the concept of MLF has been widely accepted for engineers and researchers [3–4]. Nowadays, many multi-lane bridges have been constructed to cater for the rapid development of road transportation. Along with the incorporation of the intelligent transportation system and bridge infrastructures [5–6], it is of great importance to understand the mechanism of multi-lane bridge traffic loading, i.e., MLF. The accurate estimation of MLFs would provide deep knowledge on how the performance of bridge components are influenced by traffic loads on different lanes, and offer support for the intelligent management of

Corresponding author. E-mail address: [email protected] (J. Zhou).

https://doi.org/10.1016/j.istruc.2020.01.031 Received 11 August 2019; Received in revised form 23 January 2020; Accepted 23 January 2020 2352-0124/ © 2020 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

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2. Review of MLF approaches

random traffic loads on multi-lane bridges for structural safety. In design codes for bridges worldwide, the values of MLF are quite different, depending on the different underpinning approaches used. For the Chinese and Canada bridge design codes [7–8], the MLFs (also known as multi-presence reduction factors) are derived based on the “multi-presence truck weights approach” initially proposed by Bakht and Jaeger [9]. While, the MLFs (also known as multiple presence factors) in AASHTO [10] were established utilizing the “multi-presence truck LEs approach”, which were mainly sourced from the framework proposed by Nowak [11–12]. For BS5400 [13], Eurocode [14], and ASCE guideline [15], however, the MLF was determined by numerous simulations of traffic LEs on multi-lane bridges, which incorporated many traffic behavior assumptions. Along with the advances of sensor and computing technologies, it is convenient now to measure site-specific traffic loads for bridges. Wherein, weigh-in-motion (WIM) technology could collect vehicle load sequences without traffic interruption. The long-term and high-precision WIM data provides new solutions to understand the realistic distribution of on-bridge traffic loads and resultant traffic LEs [16–18]. Site-specific WIM data have motivated the study on multi-lane bridge traffic loading. However, it can be of a challenge to propose a code-level MLF model. First of all, the influence factors are of a wide range (such as bridge span length, structural effect type, features of traffic data). Moreover, the MLF model should be simple and easy for practical engineering. Hence, most of the continued studies in this field use measured traffic data to update the calculation of MLFs based on the existing approaches. Based on the multi-presence truck weights approach proposed by Bakht and Jaeger [9], Yang et al. [19] utilized measured traffic data to analyze the MLF for composite girder bridges with varying span lengths. Yin et al. [20] calculated MLFs of bridges of various traffic lanes based on the approach but used realistic multi-peak Gaussian distribution of truck weights from measured WIM data. Following the multi-presence truck LEs approach proposed by Nowak [11–12]; Gindy and Nassif [21] investigated the occurrence probability of multiple truck presence (including following, side by side, and staggered) based on 25 WIM sites spanning 10 years; Fu et al. [2] used 68 million trucks of 436 months gathered from 43 sites to calculate the MLF for both strength and fatigue limit states; Gil and Kang [22] and Hwang et al. [23] collected WIM data from Korea to calculate the multiple truck presence probabilities and calibrate MLFs for short and medium span bridges. Van der Spuy et al. [18] directly used long-term multi-lane WIM data to determine MLFs based on the extreme extrapolation of total-lane and reference-lane LEs. More recently, Zhou et al. [1] proposed a novel multi-coefficient MLF model from extreme events of coincident lane LEs using multivariate extreme value theory, sitespecific WIM data were used to calibrate and validate the proposed model and approach (termed as “coincident lane LEs approach” hereon). This paper aims to investigate the underpinning approaches of existing MLF models using multi-lane traffic data. Three approaches are discussed, i.e., the multi-presence truck weights approach, the multipresence truck LEs approach, and the coincident lane LEs approach. The comparative results are obtained to improve the understanding of these approaches, and to reveal the mechanism of multi-lane traffic loading on bridges. The organization of this paper is as follows. In Section 2, these three approaches of MLFs are reviewed in terms of the assumptions and numerical procedures. In Section 3, numerical examples of multi-lane traffic LEs are used to compare the MLFs calibrated by the three approaches, where traffic loads over multiple lanes are identical. In Section 4, realistic two-lane, three-lane, and four-lane unidirectional WIM data are employed to calibrate the bridge MLFs, where traffic loads over multiple lanes are significantly different. Based on the MLFs of the numerical study and WIM data calibration, a comprehensive comparative study is performed to illustrate the features of these approaches and reveal the mechanism of multi-lane traffic loading on bridges.

2.1. Multi-presence truck weights approach Multi-presence truck weights approach was initially proposed by Bakht and Jaeger [9]. There were three essential assumptions for this approach: The weights of fully loaded trucks follow a normal distribution, which was validated through site observations in the early 1980 s [24–25]. Trucks appear independently, and the weights of trucks on each traffic lane follow the same normal distribution. The maximum observed load in the population of fully loaded trucks is 3.5 standard deviations above the mean value, i.e., Wmax = μ + 3.5σ. Following these assumptions, the MLF is the ratio of the average maximum truck weight on the multi-lane roadway to the maximum truck weight on the reference lane

MLF =

μ + r·σ Wa max 1 + r · Cv = = Wmax μ + 3.5σ 1 + 3.5Cv

(2)

where, Wamax is the average maximum truck weight over the multiple traffic lanes. μ and Cv are the mean value and the coefficient of variation of the normal distribution of truck weights. r represents the standard deviations above the mean of the normal distribution for Wamax. The probability of the simultaneous presence of the average max-

(

Q·g (r )

)

n

imum truck weight over multiple lanes is pn = (p1 )n = 86400 Δt . where, n is the number of traffic lanes; Q is the average daily truck volume of a single lane; g(r) is the probability that truck weight is no less than Wamax; Δt is the time duration that a truck passes the critical bridge section. Moreover, the number of tests for that event is m = 365·T ·(86400 Δt ) , where T is the design lifetime of the bridge structure. Therefore, the occurrence of the event follows a binomial distribution given by B(m, pn), in which pn is a very small value, and m is a very large value. According to the principle that Poisson distribution is the limit distribution of the binomial distribution, the probability distribution of the event is further formulated by

P (X = k ) =

i

m−i

(mk) ⎛⎝ mm·p ⎞⎠ ⎛⎝1 − mm·p ⎞⎠ n

n

=

e−(m·pn ) (m ·pn ) k . k!

(3)

It is required that the occurrence probability of the event in bridge lifetime should be no more than a guaranteed rate, e.g., 5% guaranteed rate in 100 years design period specified in MCT [8]. Therefore, it is concluded that the expectation value of the Poisson distribution should be no more than 0.05, i.e., m ·pn ⩽ 0.05. Hence, g(r) could be calculated by

g (r ) ⩽

1 86400 ⎛ ⎞ Q ⎝ Δt ⎠

n−1 n

1

0.05 n ⎞ ·⎛ ⎝ 365·T ⎠

(4)

With the known value of g(r), r is determined, and the MLF is further derived based on Eq. (2). Within Eqs. (2) and (3), Q, △t, T, and Cv are critical coefficients that influence the MLF. △t is determined by the speed and length of trucks, which are diverse for different tuck types, and generally given by average statistical values. T, the bridge lifetime (unit: year), varies in different national design specifications. The remaining parameters of Q and Cv are derived by statistics of traffic data on specific sites, which are highly random and pose a significant impact on the calibrated MLF. The assumptions in the multi-presence truck weights approach are questionable considering real traffic situations. Many recent measured traffic data indicate the distribution of truck weight does not follow a simple unimodal distribution [1,20]. In particular, the weights of trucks in each lane may be not identically distributed, since trucks are generally required to travel in their target traffic lanes based on their types and weights [26–27]. Moreover, the presence of trucks in each lane may be not independent as similar trucks may tend to form into 445

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constraint condition ∑j = 1 αj, i = 1. When the bridge deck is supported by a single component, i.e., j = 1 andαj, i = 1, Eq. (6) is further simplified as:

platoons or groups because drivers of heavy loaded trucks are ape to travel together [1,28]. Moreover, the approach uses the probability reduction of simultaneous presences of average maximum truck weight to determine MLF. This assumption is generally true for short span bridges as a single truck and side-by-side trucks govern the loading effects of single-lane and multi-lane bridges, respectively. However, for medium and long span bridges, the governing traffic loading scenario is much more complicated with more trucks involved. As a result, the probability reduction of truck weights may not precisely represent the reduction of traffic LEs that directly control the safety of bridge components. Nevertheless, the approach is simple and easy to understand, so it is still utilized in some design codes [7,8] and recent research work [19,20].

MLF =

S1e (q)

The framework of the approach was proposed by Nowak [11,12], where the MLF was defined as following

SNc (q) N ·S1c (q)

(5)

where, q is the quantile (similar to the guaranteed rate in Section 2.1) to determine the characteristic value. S1c (q) is the characteristic LE of the reference lane. SNc (q) represents the characteristic LE of the total (N) lanes, which is determined by the probability of occurrence of side-byside trucks and their loading effects on bridges. In the approach, moments and shears of simple- and continuousspan bridges with a wide range of span lengths are utilized to analyze traffic LEs. Three types of correlations between truck weights in the same lane and adjacent lanes are considered: ρ = 1.0 (full correlation), ρ = 0.5 (partial correlation), ρ = 0 (no correlation), where ρ is the Pearson correlation coefficient. The occurrence probabilities of different truck weight correlations in a single lane and adjacent lanes are determined through site observations. Lane LE is regarded to be caused either by a single truck or two (or more) trucks following behind each other at a varied distance from 5 m to 30 m. The characteristic value of reference-lane LEs is determined by the quantile through fitting a normal distribution to the tail of these LEs. Characteristic total-lane LE is calculated by the simultaneous presence of side-by-side trucks over multiple lanes according to the three types of correlations. The multi-presence truck LEs approach determines the MLF based on truck LE rather than truck weight, which is more rational and has been utilized as the underpinning approach for MLFs in AASHTO [10]. Furthermore, many subsequent works attempt to refine the occurrence probabilities of truck weight correlations from measured traffic data [16,21–23]. The approach is applicable to short and medium span bridges since multiple trucks presence governs the traffic LEs of these bridges. However, it also does not readily apply to long span bridges, since congestion governs the traffic LEs while the number of simultaneous presences of trucks is difficult to quantify.

3. Numerical study 3.1. Data description In this section, numerical data are used to calibrate MLFs based on the above three approaches. Numerical data are generated assuming that the truck loads and truck volumes on different traffic lanes are identical without any disparity. This complies with the assumption of identical lane truck loads used in the multi-presence truck weights approach and the multi-presence truck LEs approach. Fig. 1 describes the truck model that represents all the vehicle types carried by the studied bridges in their lifetime. Simply-supported bridges with span length ranged from 10 to 50 m are studied and the girder bending moment in the middle span is investigated. The truck configuration is a 55 t (ton) standard truck load model based on the Chinese design code [8]. For the numerical studies, the gross weight of the truck is assumed to follow a normal distribution with a mean value of 44 t. The axlespacings and axle weights of the truck are also shown in Fig. 1. It is known from the three approaches that the average daily truck volume (ADTV) and the coefficient of variation (Cv) of truck weights are two essential parameters affecting the results of MLFs. Therefore, three truck volume conditions with 1000, 1500, and 2000 trucks per lane per day, and three truck weight conditions with the coefficient of variation being 0.1, 0.2, and 0.3, are considered. The truck speed is assumed to follow a normal distribution with the mean value and the standard value of 50 km/h and 10 km/h, respectively.

2.3. Coincident lane LEs approach The coincident lane LEs approach was proposed by Zhou et al. [1]. Supposing there are several components supporting the multi-lane bridge, the MLF of a certain component is calculated by

MLFj =

S jc, N (q) N ·S1c (q)

=

e N S je, i (q) 1 S j,1 (q) · c · ∑ αj, i e N S1 (q) i = 1 S j,1 (q)

(7)

S1c (q)

where η = is defined as a multi-lane combination coefficient, which indicates the reduced probability of occurrence of coincident extreme events; γi = Sie (q) S1e (q) is defined as a lane correction coefficient, which describes the relationship of the LE between the considered lane and the reference lane. In the approach, the critical is to calculate the multi-lane combination coefficient and the lane correction coefficient. Since S1e (q) is independent of S1c (q) , conventional univariate extrapolation algorithms can be used to calculate the combination coefficient. Whereas, Sie (q) and S1e (q) are correlated since they jointly contribute to SNc (q) , and should be determined from the extremes of coincident event lane LEs, which is the scope of bivariate extreme value theory. The multi-presence truck weights approach and multi-presence truck LEs approach use a single coefficient to encompass all the information about the extreme value of multiple lane traffic loading effects. However, the MLF model based on the coincident lane LEs approach uses a lane correction coefficient and a combination coefficient to show how the proportion of traffic loads in each lane contributes to the resulting total LE for any given component, which was stated to provide better insight into the underlying mechanism of multi-lane traffic loading. It is acknowledged that the coincident lane LEs approach considers the disparity of lane traffic loads, and is applicable to any bridge span.

2.2. Multi-presence truck LEs approach

MLF =

1 ·η ·(γ1 + γ2 + ....+γN ) N

3.2. Calibration procedures (6)

To ensure that the calibrated MLFs using different approaches have the same guaranteed rate, a design lifetime of 100 years and a traffic load with a return period of 1950 years based on the Chinese bridge design code are investigated. Moreover, the initial calculated MLFs are normalized (MLF1 = 1.0) for an equivalent comparison. In the multi-presence truck weights approach, MLF is determined only by the feature of truck weight, which is not related to the

where, the superscripts e and c represent the extreme value and characteristic value in bridge lifetime, respectively; the subscript j is the label of the concerned component in a multi-component supported bridge; S jc, N is characteristic total LE for component j; S je, i is the LE of lane i for component j in the same extreme event forming S jc, N ; αj, i is the proportion of the ith lane LE that component j shares, which follows the 446

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Fig. 1. The layout of axle weight and axle spacing of the truck.

(2) Extract the daily maxima of reference-lane LEs and the traffic lane which produces the most adverse LEs is set to be the reference lane. Fit a generalized extreme value distribution to the daily maxima, and obtain the characteristic reference-lane LE, S1c (q) , considering a return period of 1950 years. (3) Extract the daily maxima of the coincident lane LEs. Fit a bivariate extreme value copula to the correlations between LEs of the reference lane and other lanes in the coincident events, and reproduce the extreme coincident lane LEs in bridge lifetime to calculate Sic (q) (i = 1, 2, …, N). (4) Calculate the multi-lane combination coefficient, η = S1e (q) S1c (q) , and the lane correction coefficient, γi = Sie (q) S1e (q) . Compute the MLFs for various bridge span lengths based on Eq. (6).

structural information such as bridge type, span length, or influence line type. Detailed procedures are as follows: (1) Calculate g(r) based on Eq. (4) with T = 100, n = {1, 2, 3, 4}, Q = {1000, 1500, 2000}, and △t = 12.8 m/(50 km/h/3.6) = 0.92 s. (2) Compute the initial MLFs based on Eq. (2) with r, μ = 44 t, and Cv = {0.1, 0.2, 0.3} and, the observed maximum truck weight for a single lane is Wmax = μ×(1 + 3.5Cv). (3) Place the truck (shown in Fig. 1) weighted MLF1 × Wmax to the most adverse position of the concerned bridge influence line to derive the characteristic reference-lane LE. Where, MLF1 and r1 are the calculated initial MLF and deviation for a single lane. (4) Normalize the MLFs based on their corresponding single-lane MLF1.

It is noted the calibrated MLFs are related to bridge span length and influence line type in the multi-presence truck LEs approach and coincident lane LEs approach. However, for the multi-presence truck weights approach, the MLFs are only related to truck weight information. This procedure is not used for calibration of traffic load model in any bridge standards. However, it is only utilized for the comparison with the other two approaches in the study.

In the multi-presence truck LEs approach, the MLFs are calculated based on the bridge LE, and the calibration procedures are summarized as follows: (1) Load a single truck or two (or more) trucks on the most adverse locations of the bridge influence lines to extract the reference-lane LEs, where the occurrence probabilities of following trucks for ρ = 0, ρ = 0.5, and ρ = 1.0 are 1/50, 1/150, and 1/500, respectively according to Nowak [12]. The axle-to-axle gap between successive trucks varies from 5 m to 30 m to obtain the most adverse LE. (2) Plot the reference-lane LEs in a normal probability way, and fit a linear line to the tail of the LEs. Obtain the characteristic value with the ordinateS1c = Φ-1(1/(365 × T × Q)). (3) Load additional side-by-side trucks in the adjacent lanes, where the occurrence probabilities of side-by-side trucks are assumed to be 1/ 15, 1/150, and 1/450 respectively for ρ = 0, ρ = 0.5, and ρ = 1.0 according to Nowak [12]. Calculate the characteristic total LEs according to the three kinds of truck weight correlations, and the most adverse one is SNc . (4) Calculate MLFs for various bridge span lengths based on Eq. (5).

3.3. Results The equivalent uniformly distributed loads (EUDLs) based on the characteristic reference-lane LEs are first calculated for various bridge span lengths, as shown in Fig. 2. Then, the normalized MLFs are calculated and compared, as shown in Fig. 3. Finally, the calculated total LEs of the three approaches through multiplying unified EUDLs and MLFs in Fig. 3, are compared to the numerical solutions by extreme extrapolation of simulated total LEs, and the results are shown in Table 1. Fig. 2 shows the EUDLs of S1c significantly decreases with the increases of bridge span length for the three approaches. On the other hand, ADTV has a limited influence on the EUDLs, with relative errors less than 10%. This is because the number of trucks per day produces little deviations on the quantile value for determining the characteristic single-lane LE. Cv, which represents the features of site-specific traffic loads, has a great impact on the EUDLs, and EUDLs of S1c increases significantly with Cv. As comparing three approaches, the EUDLs of the reference lane calibrated by the multi-presence truck LEs approach and coincident lane LEs approach are approximate to each other. It indicates that the use of different extreme value extrapolation algorithms on sufficient underlying data (5000-day LEs) yields a similar prediction (see Fig. 4). However, the EUDLs calibrated by the multi-presence truck

For the coincident lane LEs approach, the MLFs are derived based on the multivariate extreme value relationships of lane LEs, as follows: (1) Simulate multi-lane traffic loads on bridges with different span lengths using in-house self-developed software “MSCA” (multi-axle single-cell cellular automaton) [27,29,30]. The daily truck volume is evenly distributed throughout a day in each lane, and the laneselection probability is the same to different traffic lanes. Totally 5000-day traffic loads are modeled. 447

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Fig. 2. EUDLs of the characteristic reference-lane LEs using (a) multi-presence truck weights approach, (b) multi-presence truck LEs approach, and (c) coincident lane LEs approach.

occurrence probability of multiple presences of adverse truck loads over multiple lanes. Fig. 4 gives an example of extrapolation of characteristic total LE, in which the total LEs are determined from the simulation on the four-lane bridge with a span length of 40 m and traffic load condition of Cv = 0.3. Two extreme extrapolation algorithms, Gaussian tail fitting recommended in multi-presence truck LEs approach and GEV fitting on block maxima recommended in coincident lane LEs approach, are compared. The characteristic value is determined by a quantile of 1950 years return period. It is found the two approaches yield a similar result, indicating the prediction results are insensitive to the extrapolation algorithm when the underlying data is sufficient (5000-day LE). Similar findings are also found for other cases. For simplification, the GEV fitting approach is utilized to obtain the characteristic totallane LEs in the subsequent sections. To further illustrate how these three approaches perform, the total LEs by multiplying unified EUDLs of S1c and MLFs are compared to the numerical solutions (extrapolated characteristic total LEs based on multi-lane traffic load simulation). On the numerical solutions, the total LEs are extracted from the traffic load modeling using MSCA [27,29,30]. Table 1 gives the comparison results of total-lane LEs between calculation by the three approaches and numerical solutions. Numerical solutions could be regarded as the exact solution to these

weights approach are all larger than those by the other two approaches, especially when Cv is high. That is because the results are directly related to truck weight instead of truck LE. Fig. 3 gives the calibrated MLFs using these three approaches. Results show that MLF slightly increases with the increase of ADTV. That is because the increase in the number of trucks slightly raises the occurrence probability of multiple truck presences, and thus results in higher MLFs. The variability of truck loads, represented by Cv, again has more significant influences on MLFs. With the increase of Cv, the MLFs significantly decrease. This is because the strong variability of truck loads reduces the occurrence probability of multiple presences of heavy trucks. Moreover, the MLFs calibrated by different underpinning approaches are significantly different. MLFs produced by the coincident lane LEs approach are the largest, while the multi-presence truck weights approach is the smallest. This is related to the assumptions of how the occurrence probability of simultaneously adverse lane LEs is considered. The MLFs calibrated by the multi-presence truck weights approach are smaller than these from the multi-presence truck LEs approach. That is because the truck loads are assumed independent in each traffic lanes in the multi-presence truck weights approach, whereas trucks load in adjacent lanes are considered to have correlations in the multi-presence truck LEs approach. It indicates that a strong correlation of truck load in adjacent lanes would increase the 448

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Fig. 3. Calibrated MLFs using (a) multi-presence truck weights approach, (b) multi-presence truck LEs approach, and (c) coincident lane LEs approach.

• The multi-presence truck weights approach is not appropriate to

examples. The calculated total-lane LEs by the three approaches are the product of characteristic single-lane LEs and MLFs corresponding. To exclude the influence of calibrated EUDLs of S1c by different approaches, unified values by the multi-presence truck LEs approach and coincident lane LEs approach are used. In doing so, the deviations of characteristic total-lane LEs between calculation and numerical solution are only influenced by the calibrated MLFs. It is shown from Table 1 that the multi-presence truck weights approach may underestimate total-lane LEs for bridges with multiple traffic lanes. Such large deviations on the calculation of total-lane LEs are unavoidable since this approach involves many questionable assumptions and the MLFs are calibrated only based on encountering events of trucks. Meanwhile, the multi-presence truck LEs approach yields reasonable total-lane LEs in most conditions, but may significantly underestimate the result for four-lane bridges. This indicates that the probability of occurrence of different correlations of trucks recommended in Nowak [12] may not be suitable for the cases under study. The coincident lane LEs approach yields the most accurate totallane LEs as numerical solutions with absolute relative errors less than 7%, indicating that the coincident lane LEs approach is capable to reflect the multi-lane loading mechanism on bridges. In summary, the main findings from the numerical study are as follows:

•

•

determine the characteristic reference-lane LEs, but the other two approaches are suitable and produce similar results. It is because the multi-presence truck weights approach utilizes the extrapolation of truck weights instead of truck load effects. The three approaches yield comparable MLFs when the coefficient of variation of truck weights is small. However, significant differences are observed when there is a large coefficient of variation of truck weight. The MLFs by multi-presence truck weights approach are far lower. The MLFs by multi-presence truck LEs approach and coincident lane LEs approach are comparable, especially when the number of traffic lanes is no larger than 3. The coincident lane LEs approach yields the most similar results to the numerical solutions, with relative errors of RMSE no higher than 3%. The results by the multi-presence truck LEs approach are comparable but may underestimate. It worth mentioning that when the two-lane MLF is set as 1.0 for the reference, the calculated MLFs for three- and four-lane by the three approaches are comparable.

Based on these findings of the numerical examples where truck loads over multiple lanes are identical, the following suggestions could be referred. (1) The multi-presence truck LEs approach or the coincident lane LEs approach is applicable to determine the characteristic reference-lane LEs. (2) The coincident lane LEs approach is feasible to 449

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Table 1 Comparison of chactertistic total-lane LEs between calculation by the three approaches and numerical solutions. Number of traffic lanes

Single-lane

Bridge length (m)

10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50

Two-lane

Three-lane

Four-lane

RMSE of relative errors

Cv = 0.1

Cv = 0.2

Cv = 0.3

A1

A2

A3

NS

A1

A2

A3

NS

A1

A2

A3

NS

64(0) 41(0) 34(−3) 29(−5) 23(−2) 107(−4) 70(−7) 57(−9) 48(−6) 39(−7) 145(−6) 94(−5) 77(−7) 65(−7) 53(−6) 177(−3) 115(−8) 94(−6) 79(−4) 65(−2) 5.37%

64(0) 41(0) 34(−3) 29(−5) 23(−2) 118(6) 77(3) 63(0) 53(4) 43(2) 162(6) 106(7) 86(4) 73(4) 60(5) 190(4) 124(−1) 101(1) 85(3) 70(5) 3.96%

64(0) 41(0) 34(−3) 29(−5) 23(−2) 113(2) 74(−1) 60(−3) 51(0) 42(−2) 156(1) 101(3) 83(0) 70(0) 57(1) 188(3) 122(−2) 100(0) 84(2) 69(4) 2.29%

64 41 35 30 24 111 75 63 51 43 154 99 83 70 57 183 125 100 83 67 N/A

84(−1) 55(0) 46(−1) 38(−3) 33(−2) 130(−8) 84(−1 0) 71(−1 1) 59(−1 1) 52(−1 2) 164(−1 6) 107(−1 8) 90(−1 6) 75(−1 7) 66(−1 8) 188(−1 8) 123(−1 8) 103(−2 1) 86(−23) 75(−1 9) 14.1%

84(−1) 55(0) 46(−1) 38(−3) 33(−2) 150(7) 98(4) 82(4) 68(3) 60(2) 197(1) 128(−2) 107(1) 90(−1) 79(−2) 211(−8) 138(−8) 115(−1 1) 96(−1 3) 84(−9) 5.53%

84(−1) 55(0) 46(−1) 38(−3) 33(−2) 146(4) 95(1) 80(1) 66(0) 58(−1) 198(1) 129(−1) 108(1) 90(0) 79(−1) 238(3) 155(4) 129(0) 108(−3) 95(2) 1.93%

85 55 46 39 34 141 94 79 67 59 196 130 107 91 80 230 149 130 111 93 N/A

106(−1) 70(−1) 58(−2) 48(−2) 41(−3) 155(−1 4) 102(−1 1) 85(−1 3) 70(−1 3) 60(−1 4) 188(−1 9) 124(−1 9) 103(−2 1) 86(−1 9) 73(−2 6) 206(−2 4) 135(−2 4) 113(−2 4) 93(−2 6) 79(−2 7) 17.6%

106(−1) 70(−1) 58(−2) 48(−2) 41(−3) 186(3) 122(7) 102(4) 84(4) 72(3) 236(2) 155(2) 130(0) 107(2) 91(−7) 239(−1 2) 157(−1 2) 131(−1 1) 109(−1 4) 92(−1 5) 7.1%

106(−1) 70(−1) 58(−2) 48(−2) 41(−3) 180(−1) 118(3) 98(1) 82(1) 69(0) 237(2) 156(2) 130(0) 108(2) 91(−7) 284(5) 187(5) 156(5) 129(2) 110(1) 2.9%

107 70 59 49 42 181 114 98 81 69 232 153 130 105 98 272 178 148 127 109 N/A

Note: A1-multi-presence truck weights approach; A2-multi-presence truck LEs approach; A3-coincident lane LEs approach; NS-numerical solution. ‘(+)’ (or ‘(−)’) is the relative error that is larger (or smaller) than the numerical solution.

4.1. The description of the WIM data

calibrate MLFs under any conditions, but the calculation procedures are a little complicated. (3) The multi-presence truck LEs approach is suitable to calibrate MLFs with the number of traffic lanes no more than 3 when using the occurrence probabilities of following trucks by Nowak [12]. (4) The multi-presence truck weights approach is better to infer MLFs when the two-lane MLF (other than the reference-lane MLF) is set as 1.0 for the reference.

The WIM data were collected from three bridge sites. Site 1 is the Second Daxie Bridge located in Ningbo, China. It has a main span of 392 m, and carries four-lane bidirectional traffic. Site 2 is the Hangzhou Bay Bridge, a sea-cross project in China. It has a main span of 448 m carrying six-lane bidirectional traffic. Site 3 is the Jiashao Bridge, the longest multi-pylon cable-stayed bridge in the world with a total length of 2680 m carrying eight-lane bidirectional traffic. In three bridge sites, WIM data in a single traffic direction are used, and the overall statistics features of the traffic loads are shown in Table 2. It is shown the truck volume, truck weight, truck speed, and the number of heavy trucks is significantly diverted in various traffic lanes, especially when the number of traffic lane is more than 2. Besides, there are many overloaded trucks with gross weight larger than 55 t, which induce very significant LEs. Moreover, it is found the average autocorrelation coefficients of truck weights in the same lane and adjacent lanes from the three datasets are close to 10%, indicating the occurrence of trucks

4. WIM data calibration Unlike numerical data which assume identical truck loads and truck volumes on different traffic lanes without disparity, realistic multi-lane WIM data show strong lane load disparity, thus are meaningful to validate the performance of the three approaches on MLF calibration of realistic multi-lane traffic data. In this section, site-specific WIM data are used to calibrate the MLFs based on the three approaches and the statistical features of WIM data are used to check the assumptions in these approaches.

Fig. 4. Extrapolation of characteristic total LE using (a) Gaussian tail fitting algorithm and (b) GEV fitting algorithm. 450

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Table 2 Overall statistical features of the WIM data. Items

Site 1

Site 2

Lane2 No. of valid days Average autocorrelation coefficient of GVW (%) Average daily truck volume per lane Average GVW (t) Average truck speed (km/h) No. over 55 t

90 9.91 509 15.63 47.51 505

Lane1

Lane3

2123 27.82 43.13 11,817

400 10.66 57 14.42 79.35 681

Site 3 Lane2

2128 23.30 68.74 47,182

Lane1

Lane4

Lane3

Lane2

Lane1

2664 27.61 57.26 61,072

292 12.29 44 5.27 100.77 33

952 13.23 94.24 1289

2616 32.28 75.85 57,798

2281 37.80 67.81 56,173

Note: Lane 1 is the outermost lane farthest to the central divider of the roadway. Table 3 Statistics of truck correlations from three sets of WIM data. Traffic data

Site 1 Site 2 Site 3

Following (lane 1)

Side-by-side (lanes 1 and 2)

Side-by-side (lanes 1, 2 and 3)

Side-by-side (lanes 1, 2, 3, and 4)

ρ=0

ρ = 0.5

ρ = 1.0

ρ=0

ρ = 0.5

ρ = 1.0

ρ=0

ρ = 0.5

ρ = 1.0

ρ=0

ρ = 0.5

ρ = 1.0

1/71 1/40 1/52

1/167 1/72 1/99

1/411 1/267 1/364

1/58 1/15 1/21

1/172 1/25 1/60

1/1130 1/94 1/233

/ 1/212 1/60

/ 1/3346 1/266

/ 1/6876 1/9746

/ / 1/1906

/ / 1/37711

/ / 1/867350

Table 4 Gaussian fitting parameters and calibrated MLFs using the multi-presence truck weights approach. WIM data

Site 1 Site 2 Site 3

Note:

a

Gaussian fitting

Truck information

μ (t)

Cv

R

86.9 95.1 112.2

0.28 0.26 0.19

0.91 0.82 0.85

2

a

MLF

Q (veh/d)

Length (m)

Speed (km/h)

1

2

3

4

2632 4849 5782

8.27 8.50 8.46

50 62 85

1.35 1.34 1.30

0.99 0.99 1.01

0.79 0.80 0.86

0.63 0.65 0.74

the truck information is the statistical mean value from the WIM data.

investigated. The realistic WIM data are loaded on these bridges to obtain the history of traffic LEs.

is not independent. Therefore, WIM data contradict assumptions of identical distribution of truck loads over different traffic lanes in the multi-presence truck weights approach and multi-presence truck LEs approach. In addition, the truck correlations based on the WIM data are recalibrated as shown in Table 3. In the statistics, if the following or sideby-side trucks have approximate gross weight with relative errors no larger than 10%, the trucks are considered to have a full correlation. If the gross weight difference is larger than 10% but smaller than 50%, the trucks are considered to have a partial correlation. To determine the occurrence of following or side-by-side truck events, the axle-to-axle headways between trucks are defined as up to 30 m and −10 m, respectively. It is found the full correlation of following trucks is much higher than that observed by Nowak (1/500). However, the probability of full correlation of side-by-side trucks in a two-lane roadway is much lower than that revealed by Nowak (1/450). Moreover, when the number of traffic lane is large (e.g., greater than 2), the probability of side-by-side events with trucks loaded on all the traffic lanes is quite low, especially for full correlation condition. Therefore, the MLFs are calibrated by the multi-presence truck LEs approach using the statistical probabilities in Table 3. It is noted in the analysis of the approach hereafter the lane load disparity is not included. For instance, the combination of lanes 1 to N-1 is only analyzed for MLFN-1 in the N-lane WIM data, but other combinations such as lanes 2 to N (or lanes 1, 3 to N) are not considered.

(1) Multi-presence truck weights approach The statistics of WIM data contradict the basic assumptions in the approach. Nevertheless, the approach is used to calibrate MLFs and the results are then compared with those from other approaches. Preliminary analysis shows the distribution of truck weight is multimodal. Since bridge traffic LEs are generally governed by heavy trucks, hourly maximal truck weights are selected. It is found the hourly maximal truck weights follow a normal distribution. Table 4 gives the fitting parameters using the normal distribution as well as calibrated MLFs. The results indicate the MLF for a single lane is approximate 1.3, and it becomes 1.0 for the two-lane condition. The results of MLFs between Site 1 and Site 2 are the same, which is because their fitting Cv, and the truck volume per day per lane are equivalent. However, the results of MLFs from Site 3 show a little difference with these from the other two sites, which is also due to the considerable differences in Cv, and truck volume. The results again validate the findings in the numerical study that MLFs are significantly influenced by Cv. (2) Multi-presence truck LEs approach In this approach, truck LEs of the reference lane are first obtained and plotted in a normal probability paper. Then, fit a linear line to the tail of the LEs, and obtain the characteristic reference-lane LE, which is similar as shown in Fig. 4(a). Then, the additional truck LEs are determined by the probabilities of side-by-side trucks with different correlations as given in Table 3. Finally, the MLFs are calibrated based on Eq. (5), and the results are shown in Table 5.

4.2. MLFs calibration WIM data are used to calibrate MLFs using three approaches, and again simply-supported bridges with span length ranged from 10 to 50 m are studied and the bending moment in the middle span is 451

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Table 5 MLFs calibration by the multi-presence truck LEs approach using WIM data. WIM data

Site 1 Site 2

Site 3

Statistical parameters

MLF2 EUDL ofS1c MLF2 MLF3 EUDL ofS1c MLF2 MLF3 MLF4 EUDL ofS1c

Table 6 MLFs calibration by the coincident lane LEs approach using WIM data.

Bridge length (m)

WIM data

10

20

30

40

50

0.78 26.4

0.77 19.1

0.77 15.4

0.77 12.0

0.77 9.9

0.83 0.64 45.8

0.85 0.69 32.0

0.86 0.70 25.0

0.87 0.72 16.9

0.86 0.69 12.8

0.84 0.79 0.59 48.4

0.85 0.81 0.61 35.0

0.87 0.83 0.64 26.9

0.86 0.82 0.63 19.9

0.88 0.85 0.65 16.6

Site 1

Site 2

Site 3

It is revealed from Table 5 that the calibrated MLFs and EUDLs of S1c varied from site to site. The deviation of EUDLs of S1c is mainly due to the differences of truck weights among these sites. A large magnitude of truck weight at site 3 results in a greater EUDL of S1c . However, the variations of MLFs among these sites are mainly caused by lane disparity. A strong lane disparity between lanes 1 and 2 at site 1 indicates a smaller occurrence probability of simultaneous adverse lane LEs, producing a smaller MLF2 as compared with those of the other two sites. Moreover, the MLFs among different bridge lengths are equal, which is mainly because the used influence lines are similar, and the MLFs are hence only correlated with the characteristics of traffic loads over multiple lanes.

Statistical parameters

r1 r2 η MLF2 EUDL ofS1c r1 r2 r3 η MLF3 EUDL ofS1c r1 r2 r3 r4 η MLF3 EUDL ofS1c

Bridge length (m) 10

20

30

40

50

1.00 0.88 0.62 0.58 27.7

1.00 0.87 0.63 0.59 20.2

1.00 0.86 0.65 0.60 16.1

1.00 0.86 0.64 0.60 11.6

1.00 0.85 0.65 0.60 9.7

1.00 0.93 0.41 0.54 0.42 45.9

1.00 0.92 0.42 0.55 0.43 31.6

1.00 0.94 0.43 0.55 0.43 24.8

1.00 0.94 0.42 0.54 0.42 17.6

1.00 0.93 0.41 0.56 0.44 13.6

1.00 0.95 0.32 0.05 0.60 0.35 49.7

1.00 0.96 0.31 0.05 0.59 0.34 36.4

1.00 0.94 0.32 0.05 0.58 0.34 25.8

1.00 0.95 0.30 0.05 0.61 0.35 20.4

1.00 0.96 0.33 0.05 0.60 0.35 17.5

4.3. Discussions Based on the WIM data analysis, the simulated total-lane LEs could be used to infer their characteristic values. It is meaningful to investigate how these three approaches perform on the MLFs calibration of the WIM data. Hence, the calibrated MLFs and EUDLs of S1c by these approaches are multiplied corresponding, and the results are then again compared with the inferred characteristic total-lane LEs. Table 7 gives the calibrated MLFs using the three approaches and those from two national design codes, D60 and AASHTO. It is noted only MLFn in the N-lane condition is compared. All these values are normalized to the single-lane MLFs for equivalent comparison. It is revealed that the calibrated MLFs using the multi-presence truck weights approach and multi-presence truck LEs approach are larger than those using the coincident lane LEs approach. These MLFs using the three approaches are far smaller than those specified in the two design codes for the 2-lane and 3-lane conditions of site 1 and site 2. However, the differences become smaller for the 4-lane at site 3. Fig. 5 gives the calculated characteristic sing-lane LEs using the three approaches, D60, and AASHTO, where the initial single-lane MLFs (such as 1.20 for D60 and AASHTO) are involved into these values. It is revealed that the calculated EUDLs of S1c using traffic load models from design codes are far smaller than these by the three approaches. This is because there are many overloaded trucks for the WIM data, which is not covered by national traffic load models. Moreover, the EUDLs of S1c using the multi-presence truck LEs and coincident lane LEs are equivalent, because they are inferred by the same single-lane LEs although using different extrapolation algorithms, as explained by Fig. 2. However, the values using the multi-truck weights approach is extremely high. This is because they are directly calculated by the inferred maximum truck weights. The comparisons of EUDLs of S1c using these approaches indicate the single-lane traffic LEs from realistic WIM data are very large, beyond the range of traffic load models specified in design codes. It is desirable to recalibrate the standard traffic load model based on site-specific traffic load information. The calculated total-lane LEs are further compared to evaluate the capability of these three approaches and national traffic load models to describe the actual characteristic total-lane LEs from numerical solutions at three sites. Again, to exclude the influence of calibrated EUDLs of S1c by different approaches, unified values by the multi-presence truck LEs approach and coincident lane LEs approach are used. It is required that the calculated total-lane LEs should be close to the extrapolated characteristic total-lane LEs (the numerical solution), or

(3) Coincident lane LEs approach In this approach, the multi-lane combination coefficient is calibrated using block maxima based generalized extreme value distribution, whereas, the lane correction coefficient is calculated using bivariate copula and marginal generalized extreme value distributions. The relevance of these lane LEs can be fitted by the eight different forms of bivariate extreme value distributions provided by Stephenson in R package [31], except for the lane LEs of lane 4 at site 3. This is because there are few trucks on this lane so that there is scarcely any coincident lane LEs involved for daily maximal total-lane LEs. To simplify the calculation in this case, the lane correction coefficient of lane 1 is set to be 0.05, which is the ratio of average daily maximal lane LEs between lanes 1 and 4. Consequently, the lane correction coefficients and multilane combination coefficients are calculated, and the calibrated MLFs and EUDLs of S1c are given in Table 6. It can be seen from Table 6 that the lane correction coefficients varied from lane to lane, particularly for site 3 which carries more traffic lanes, indicating there are significant differences between lane LEs. Moreover, the multi-lane combination coefficient is around 0.64 for site 1 which carries two lanes, and the value becomes around 0.55 for site 2 with three lanes. This is because more traffic lanes involved reduces the occurrence probability of coincident lane LEs. However, the value increases to 0.60 for site 3 carrying four lanes, which is higher than that of site 2. This is because the lane correction coefficient of lane 4 at Site 3 is set to be a constant value of 0.05 and LEs of lane 4 are removed from the coincident lane LEs analysis so that only LEs of the other three lanes are used. Overall, the MLFs for Sites 1 to 3 are around 0.59, 0.43, and 0.35 respectively. Other similar findings could be figured out as also validated from Table 5 that, the EUDLs of S1c decrease with the increase of bridge length, and MFLs show little variances for different bridge lengths. It is noted in the N-lane site, the MLFs of 1 to (N-1) lanes could also be calibrated. For instance, MLF3 for site 3 can be calibrated for different combinations of traffic lanes, such as lanes 1–2–3, lanes 1–2-4, lanes 1–3-4, and lanes 2–3-4, using their corresponding coincident lane LEs. However, these issues are not considered in this paper and could be the research interest for further study. 452

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Table 7 Comparison of MLFs using different traffic load models for the WIM data. WIM data

Multi-presence truck weights approach

Multi-presence truck LEs approach

Coincident lane LEs approach

D60

AASHTO

Site 1/2-lane Site 2/3-lane Site 3/4-lane

0.73 0.60 0.57

0.78 0.64 0.59

0.58 0.42 0.35

0.83 0.71 0.56

0.83 0.65 0.54

Note: these values are normalized to the single-lane MLFs for equivalent comparison.

determine the characteristic reference-lane LEs. (2) The coincident lane LEs approach, which considers the lane disparity of truck loads, is more capable of calibrating rational MLFs. However, the calculations become complicated when the number of traffic lanes is larger than 3, where the coincident events that vehicles simultaneously load across multiple lanes are very rare. (3) MLFs in current design specifications cannot be directly applied to these sites and should be recalibrated. (4) Multipresence truck weights approach and Multi-presence truck LEs approach need to be improved to consider the lane disparity of traffic loads so that their calibrated MLFs could be used to these sites, which is a meaningful follow-up work.

otherwise, the calibrated MLFs are not accurate for the studied WIM data. Fig. 6 shows the comparison results for the three sites. It can be seen that only the coincident lane LEs yield similar results with numerical solutions, validating the feasibility of the coincident lane LEs approach to calibrate MLFs. However, the multi-presence truck weights approach and the multi-presence truck LEs approach produce significant higher MLFs. This is because the studied WIM data has strong lane load disparity, which conflicts with the assumption of the identical distribution of lane truck loads in these models. Such results highlight the importance to consider lane load disparity for MLF calibration. In summary, the main findings from WIM data calibration are as follows:

5. Conclusions

• The multi-presence truck weights approach is again verified not •

•

This paper reviews three main underpinning approaches of MLF models that were described in the literature, i.e., the multi-presence truck weights approach, the multi-presence truck LEs approach, and the coincident lane LEs approach. Numerical examples of multi-lane bridge traffic LEs without lane load disparity are utilized to calibrated MLFs by these approaches. Real-world 2-lane, 3-lane, and 4-lane WIM data that have strong lane load disparity are employed to calibrate MLFs by these approaches, and are then compared with national traffic load models. The results of MLFs are inter-compared to highlight the knowledge for understanding the application scope of these approaches and the mechanism of multi-lane traffic loading on bridges. Main findings are as follows:

appropriate to determine the characteristic reference-lane LEs, but the other two approaches are suitable and produce similar results. MLFs calibrated by multi-presence truck weights approach and multi-presence truck LEs approach are larger than those by coincident lane LEs approach. However, the coincident lane LEs approach yields the most comparable results of total-lane LEs with numerical solutions. It is because the strong lane disparity of truck loads of the multi-lane WIM data is well considered in the coincident lane LEs approach, but not included in the other two approaches. It highlights the importance of considering lane load disparity for MLF calibration. The MLFs in current design specifications such as D60 and AASHTO are higher than the results by coincident lane LEs approach. However, the calculated reference-lane LEs by specifications are lower. It is because there are many overloaded trucks for the WIM data, which is not covered by design specifications. It highlights the importance of recalibrating traffic load model based on site-specific traffic data.

(1) The multi-presence truck weights approach involves many assumptions that conflict with real-world observations, such as the normal distribution of truck loads, independent identical distribution of lane truck loads, and maximum observed truck weight being 3.5 standard deviations above the mean. The multi-presence truck LEs approach assumes truck loads are identical for different lanes, and relies on the occurrence probabilities of truck weight correlations in the same lane and adjacent lanes, which are strongly sitespecific. The coincident lane LEs approach proposed a multi-coefficient form of MLF which is capable of considering lane load

Based on these findings of the WIM data calibration, where truck loads over multiple lanes are significantly different, the following suggestions could be referred. (1) The multi-presence truck LEs approach or the coincident lane LEs approach is again applicable to

Fig. 5. Comparison of characteristic single-lane LEs for the three WIM data using different traffic load models. 453

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Fig. 6. Comparison of characteristic total-lane LEs for the three WIM data using different traffic load models.

Guangzhou Municipal Science and Technology Project (CN) [grant number 201904010188].

disparity, and is applicable to any bridge lengths. (2) Results of the numerical study indicate MLFs are significantly influenced by the characteristics of traffic loads but slightly affected by truck volume. The multi-presence truck LEs approach and the coincident lane LEs approach produce comparable results of characteristic reference-lane LEs and MLFs when there are sufficient underlying data. However, the multi-presence truck weights approach is not appropriate to determine the characteristic referencelane LEs and generates lower MLFs. The coincident lane LEs approach yields the most similar results to the numerical solutions; the multi-presence truck LEs approach gives comparable results; the multi-presence truck weights approach may produce large deviations. It worth mentioning that when the two-lane MLF is set as 1.0 for the reference, the calculated MLFs for three- and four-lane by the three approaches are comparable. (3) Results of application to WIM data shows the calculated MLFs by the multi-presence truck weights approach and the multi-presence truck LEs approach are larger than those by the coincident lane LEs approach. However, the coincident lane LEs approach yields the most comparable results with numerical solutions. It is because the strong lane disparity of truck loads of the multi-lane WIM data is well considered in the coincident lane LEs approach, but not included in the other two approaches. Moreover, the recommended MLFs in current design specifications show large deviations with the numerical solutions, and it is because there are many overloaded trucks for the WIM data, which is not covered by design specifications. These results highlight the importance of considering the lane disparity of traffic loads over multiple lanes and recalibrating traffic load model based on site-specific traffic data. The coincident lane LEs approach, which considers the lane disparity of truck loads, is more capable of calibrating rational MLFs. However, the other two approaches need improvement to consider the lane disparity of traffic loads so that their calibrated MLFs could be used to these sites, which is a meaningful follow-up work.

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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported by National Natural Science Foundation of China (CN) [grant number 51808148]; Natural Science Foundation of Guangdong Province, China (CN) [grant number 2019A1515010701]; 454

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