Modeling and Classification of Random Traffic Patterns for Fatigue Analysis of Highway Bridges

Abstract

With the increasing severity of traffic congestion, the impact of random traffic patterns has emerged as an indispensable factor in the fatigue design and assessment of highway bridges. In this study, an analytical approach has been proposed for modeling the effects of random traffic patterns on fatigue damage. A fatigue damage ratio, referred to as RPEF, is introduced to establish the correlation between damage and traffic characteristics. Two quantitative parameters representing two characteristics of traffic loads, namely the average loading occurrence number (scale parameter) and the vehicle headway COVs (shape parameter), have been found to be excellent indices for clustering the different dimensional randomness of RPEFs. Based on a comprehensive case study, the realization of using equivalent RPEFs to evaluate bridge fatigue damage under mixed traffic conditions was explored. The results indicate that the actual fatigue damage of bridges can be evaluated through the superposition of different traffic pattern effects, provided that the pattern classification number fits the fluctuations in traffic flow. It is necessary to ensure the rationality of traffic pattern classification for structures with spans greater than 50 m, as an overly simplistic traffic pattern classification may lead to an underestimation of fatigue damage.

1. Introduction

Traffic loads constitute the primary consideration in the fatigue analysis of highway bridges. For decades, while considerable attention has been devoted to the detailed investigation of vehicle loads, relatively limited focus has been placed on the classification of random traffic patterns and their impact on structural fatigue damage. The current fatigue design standards primarily rely on observed free-flow conditions prevalent on most expressways, employing standardized vehicle models and traffic volumes for damage evaluation [1,2,3]. However, as congestion issues become increasingly severe, the spatiotemporal distribution of traffic loads on specific bridges is experiencing potential alterations. Conducting systemic research on the effects of random traffic patterns is of great importance for improving the accuracy of fatigue analysis.

During the service life of bridges, traffic loads, similar to other environmental factors, exhibit significant randomness and uncertainty [4,5]. With the development of IoT sensing technology [6,7,8], traffic information is being comprehensively grasped. In terms of classifying random traffic patterns, traffic statistical models with 2 or 3 states are commonly used, such as the velocity-based statistical model incorporating states of free flow, following flow, and congested flow [9,10], or the density-based statistical model incorporating states of low to high densities [11]. This simplified classification has, to some extent, facilitated traffic simulation. However, due to the vagueness in state statistics, it may not accurately capture the intrinsic randomness of real-world traffic conditions. Yang et al. [12] constructed four different traffic patterns and used them for the fatigue reliability assessment of a long-span suspension bridge. Their research showed that the structural fatigue reliability would decrease with the random traffic flow simulation without differentiation of traffic states might lead to dangerous analysis results.

Historically, there exist two perspectives regarding the impact of random traffic patterns on bridge fatigue damage. The first perspective, originating from the Eurocode developers, argues that, in comparison to the free flow state, the congested traffic state, due to its reduced traffic efficiency, may not necessarily increase the fatigue damage rate per unit time for structures [13]. The second perspective holds that congested traffic intensifies the load superposition effect, thereby increasing the fatigue damage caused by each vehicle on average [14]. Maljaars [15] investigated the bridge fatigue damage under stop-and-go traffic congestion patterns, finding that these patterns can reduce the temporal fatigue damage rate but have an inapparent impact on the average damage per vehicle. Baptista [16] discovered that, within a span of 40 m, there is no difference in the fatigue damage caused by different traffic conditions. However, as the span increases, the fatigue damage under congestion conditions can be up to several times higher than that under free-flow conditions, and a mixed traffic condition becomes more unfavorable than pure congestion conditions in long-span bridges for a significant reduction in the number of rain-flow cycle statistics. Wysokowski [17] conducted an analysis of bridges with spans exceeding 50 m and found that, at a speed of 30 km/h, the fatigue durability of the analyzed bridges can be up to 23% lower than that calculated according to the standard procedures outlined of Eurocode3. The existing research primarily focuses on the influence of congestion patterns on fatigue damage. However, due to the variability in traffic conditions, some findings have not yet reached consistent views, and no theoretical model has been developed to explain more generalized effects. For engineering practice, the available guidance and support are somewhat limited.

This study aims to propose an analytical method for modeling the effects of random traffic patterns on bridge fatigue damage and conduct a feasibility study on fatigue damage estimation under mixed traffic conditions. To this end, a damage ratio reflecting the traffic randomness is defined, and theoretical analysis and numerical simulation studies are carried out around the establishment of its modeling method. Then, an estimation method for the equivalent damage under mixed traffic conditions is proposed based on traffic state decomposition, and its rationality is demonstrated through the analysis of a measured traffic case. The technical framework proposed in this paper offers a multidimensional perspective for addressing the random traffic pattern effects on fatigue damage and provides a theoretical basis to improve the fatigue analysis precision for highway bridges.

2. Concept and Framework

Previous studies on traffic pattern classification for structural fatigue analysis have primarily relied on experience. Apart from comparing simulations with real traffic, there appeared to be no better means for rational verification. To address this issue, it is crucial to gain a deeper understanding of the operational characteristics of traffic states and their impact on structural fatigue damage, and to develop a model that can effectively quantify and correlate the randomness of both traffic and structural damage. The construction begins with the definition of the damage ratio k_r, as shown in Figure 1. k_r is used to quantify the traffic randomness effects on structural damage, referred to as the random process effect factor (RPEF). The definition is as follows:

$$k_r={\displaystyle \frac{{\bar{D}}_{\mathrm{RLM}}}{{\bar{D}}_{\mathrm{SLM}}}}$$

(1)

where ${\bar{D}}_{\mathrm{RLM}}$ and ${\bar{D}}_{\mathrm{SLM}}$ represent the average fatigue damage per vehicle of the fatigue load spectrum under the random loading mode (RLM) and the sequential loading mode (SLM), respectively. Their formulas are as follows:

$${\bar{D}}_{\mathrm{RLM}}={\displaystyle \frac{1}{T_D\cdot ADTT}}\sum {\displaystyle \frac{n_i\Delta {S_i}^m}{C}}$$

(2)

$${\bar{D}}_{\mathrm{SLM}}=\sum w_{V,j}{\displaystyle \frac{\Delta {S_{\mathrm{eq},j}}^m}{C}}$$

(3)

where T_D is the duration of fatigue analysis (in days), ADTT is the average daily heavy vehicle volume in a single lane, ΔS_i is the i-th-level stress range generated by random traffic flow, n_i is the number of cycles corresponding to the i-th-level stress range, ΔS_eq,j is the equivalent stress range generated by the j-th vehicle type of fatigue load spectrum, w_V,j is the weight coefficient corresponding to the j-th vehicle type of fatigue load spectrum, and m and C are constants from S–N curves.

The fatigue analysis of both loading modes is performed using the linear cumulative damage rule in combination with the rain-flow stress cycle counting method. In addition to defining the RPEFs, the framework illustrated in Figure 1 also outlines the modeling and characterization of RPEFs, as detailed in Section 3 and Section 4. The methodology involves utilizing the dimensionless characteristics of traffic load effects, as reflected in the damage ratio, to formulate a quantitative expression by introducing suitable shape parameters derived from the headway probability distribution. However, real-world traffic conditions are highly diverse. Although theoretical relationships exist among flow rate, velocity, and density, differences in vehicle composition and probabilistic models introduce a degree of uncertainty into the characterization process. These challenges are addressed through extensive numerical simulations. Section 5 presents the application of RPEFs in fatigue damage assessment under mixed traffic conditions. It also discusses the rationale underlying different random traffic pattern classification methods, including those based on time scale, velocity, and flow rate, as well as strategies for optimizing the classification work.

In terms of a top-down fatigue analysis, the calibrated RPEFs can be employed to adjust the simplified results derived from fatigue load models, similar to the damage correction factors associated with different design parameters recommended in current design specifications [1,2,3]. This framework primarily focuses on the longitudinal loading superimposition effects of traffic loads on bridges. The transverse loading effects, which account for the influence of structural lateral system and lane configuration, can be adjusted through multi-lane effect factors [2,3,18,19,20]. Another aspect beyond the scope of this research is the vehicle-induced impact effect, which can be practically estimated using dynamic amplification factors [1,2,3,21,22].

3. Modeling of RPEFs

This section involves modeling RPEFs to facilitate the clustering analysis of random traffic patterns. It primarily addresses the complexity associated with the sequence of load occurrences, which can be effectively managed through the application of scale-free stochastic process similarity principles and mathematical probability transformation approaches.

3.1. Similarity Principle of Filtered Stochastic Processes and Its Inference

The stress order in the stress history is widely acknowledged to have a significant impact on the counting results of rain-flow cycles. In the case of a filtered stochastic process, such as the bridge stress history induced by vehicle loads, the stress order primarily relies on the stochastic nature of vehicle arrivals, which is typically characterized by specific vehicle spacing or the time headway probability model. The establishment of a vehicle arrival model involves simplifying vehicles into point loads without considering load distribution, which allows for the analysis of probabilistic characteristics of structural stress. This approach facilitates the application of relevant theories on filtered stochastic processes [23,24]. Building upon this concept, the randomness of vehicle arrival can be quantified to reflect the stress order, allowing for the observation of its correlation with RPEFs. As shown in Figure 2, traffic load flow is typically idealized as a constant-speed model. Based on the probability statistics of headway, vehicle loads are simplified into concentrated loads positioned at the front of the vehicles, forming a point random loading mode (PRLM).

Generally, for the same structural system, their internal force influence lines at the same cross-section location exhibit shape similarity. Suppose that the influence line function for a specific characteristic location is given as h(x). If it originates from bridges of different lengths, it can be standardized into h(x) = A_Lθ(x/L), where A_L is the amplitude of the influence line related to the bridge length L. The domain of the shape function θ(x) is from 0 to 1, and its function value is 0 outside this range. Based on this, the stress response z(t) caused by any randomly arranged load flow can be expressed as

$$z(t)=A_L\cdot \sum _{i=1}^{\infty }{y}_i\cdot \theta \left({\displaystyle \frac{x_i\left(t\right)}{L}}\right)$$

(4)

where y_i is the ith point load and x_i(t) is the the ith load position at time t, starting from the entrance of the bridge at x = 0.

As demonstrated in Equation (4), while keeping other parameters fixed, the impact on z(t) is determined by the ratio between the temporal–spatial position of the vehicle x_i(t) and the bridge length L. Considering the point load flow as an ideal random process and assuming the statistical independence of its load and spacing variables, there theoretically exists an infinite number of similar stress sequences, each generated by the same scale-ratio random variable χ_i = x_i/L. Actually, without considering cut-off limits, these similar random point load processes exhibit identical RPEFs (ratio between the average single-point load fatigue damage ${\bar{D}}_{\mathrm{PSLM}}$ under random loading modes and the average single-point load fatigue damage ${\bar{D}}_{\mathrm{PSLM}}$ under sequential loading modes), which is determined by the rain-flow cycle counting rules and the fatigue damage accumulation criterion. To distinguish from k_r for vehicle loads, the RPEFs in point load mode are denoted as k_r,p.

Based on the equivalence of k_r,p between similar filtered processes under point load modes, an analysis of the probabilistic characteristics among them is conducted. Here, the influence line shape function θ(x) is taken as a given parameter, with the primary emphasis being on the randomness of the ratio χ_i = x_i/L. To establish a connection with the probability characteristics of traffic flow, it is replaced with the ratio of vehicle headway to bridge scale. Specifically, we define the random variable for spatial headway d as H_D and the random variable for temporal headway τ as H_T. Additionally, define a dimensionless variable Ξ = H_D/L= H_T/T, where T is the travel time of a single vehicle on a structure with length L, i.e., T = L/v. Here, both H_D and H_T are assumed to be time-invariant. Therefore, the transformation relationship between the probability density functions f_HD, f_HT, and f_Ξ of H_D, H_T, and Ξ is as follows:

$$f_{HD}(d)=f_{\Xi }\left({\displaystyle \frac{d}{L}}\right){\displaystyle \frac{1}{L}}$$

(5)

$$f_{HT}(\tau )=f_{\Xi }\left({\displaystyle \frac{\tau }{T}}\right){\displaystyle \frac{1}{T}}$$

(6)

It should be noted that the probabilistic transformations like Equations (5) and (6) do not alter the shape parameters of the original probability models, such as the parameter α of the Gamma distribution and the parameter σ of the lognormal distribution [25]. Additionally, some generalized shape parameters, like the coefficient of variation (COV) and standard central moments of order 3 or higher, also remain invariant under the transformations. The remaining parameters, apart from the shape parameters of a probability model, are commonly referred to as scale parameters and can vary in the aforementioned transformations. Therefore, the dimensionless variable Ξ can be regarded as a set of random variables with identical shape parameters when provided with a point spacing/temporal probability model. This facilitates pattern classification, indicating that, through the anchoring of the shape parameters, only the scale parameters change within the Ξ probability model. To quantify the scaling effect, weighted-average operations are conducted on Equations (5) and (6) to derive the following outcomes:

$$\rho L=\lambda T=n_{\Xi }$$

(7)

where $\rho ={{\bar{H}}_D}^{-1}$, $\lambda ={{\bar{H}}_T}^{-1}$, and $n_{\Xi }={\bar{\Xi }}^{-1}$ are the reciprocals of the mean values of their respective variables. Specifically, ρ is called the traffic density and λ is called the traffic flow rate or vehicle average occurrence rate.

Essentially, ρL and λT denote the average occurrence of vehicle loads acting on the structure, while n_Ξ represents the average occurrence of vehicle loads after scaling the spatial–temporal dimension. Equation (5) possesses significant theoretical extensibility, as it bridges the concepts of density and flow rate in traffic flow theory [26]. Additionally, it is supported by renewal theorems, manifesting as an important harmonious property of random processes. The definitions of load density ρ and flow rate λ can vary depending on different traffic types. In this paper, they are defined as two types: one corresponds to the entirety of traffic, marked by density ρ_W and flow rate λ_W, while the other corresponds to the heavy vehicle traffic excluding cars, marked by density ρ_HV and flow rate λ_HV. Each type possesses its own vehicle headway statistical properties.

3.2. Functional Form of RPEFs and Its Limits

Since the RPEFs of ideal filtered stochastic processes are primarily determined by two shape functions abstracted above, i.e., the influence line shape function θ and the non-dimensional load spacing interval probability density function f_Ξ, they can be employed to characterize the RPEFs. To parameterize features beyond shape, f_Ξ can be regarded as a generalized probability model Ω_Ξ,PMS with a given shape parameter, releasing a scale parameter n_Ξ. If the influence line shape θ is treated as a known condition, k_r,p under point load modes can be expressed in the following form:

$$k_{\mathrm{r},\mathrm{p}}=f_{\theta }\left(n_{\Xi }|f_{\Xi }\in {\Omega }_{\Xi ,\mathrm{PMS}}\right)$$

(8)

Extending Equation (8) to the filtered stochastic processes generated by vehicle loads can be expressed as follows. Taking into account the differences between real vehicle loads and point loads, “≈” is used to describe the non-strict functional relationship between the k_r coefficient and f_θ:

$$k_r\approx f_{\theta }\left(\rho L|f_{HD}\in {\Omega }_{HD,\mathrm{PMS}}\right)$$

(9)

$$k_r\approx f_{\theta }\left(\lambda T|f_{HT}\in {\Omega }_{HT,\mathrm{PMS}}\right)$$

(10)

When the vehicle occurrence is selected as a characterizing parameter, it becomes feasible to derive a limit for the RPEFs. Specifically, in cases where the total length of the bridge is relatively short or the traffic density is low, there tends to be minimal vehicle occurrence. As a result, under such circumstances, the damage induced by RLM approaches that of SLM:

$$\underset{\rho L\to 0}{lim}{k}_r\approx \underset{n_{\Xi }\to 0}{lim}{k}_{\mathrm{r},\mathrm{p}}=1$$

(11)

4. Numerical Analysis of RPEFs

This section undertakes numerical study to evaluate the validity of the proposed modeling method for RPEFs. The influence of the probability distribution of vehicle headway will be thoroughly investigated under the given load and resistance conditions.

4.1. Objectives and Solutions

As shown in Figure 3, the regularity of RPEFs is divided into three progressive research objectives. Objective 1 aims to explore the shape effect of random processes. In this part of the simulation, the impact of vehicle type is completely ignored, and the correlation between RPEFs and the shape parameters of the headway probability distribution is theoretically demonstrated through dimensionless point loading simulations. Objective 2 aims to cluster the RPEFs under macroscopically stable traffic flow by utilizing the characterization conclusions derived from Objective 1. In this part of the simulation, a constant traffic speed is employed, taking into account the vehicle type and safe driving spacing. Objective 3 aims to investigate the impact of microscopic traffic behaviors on the clustering of RPEFs. In this part, the cellular automaton method is employed for simulation, allowing vehicles to change speed and lanes. From an application perspective, the clustering analysis in Objective 2 is a focal point of interest, as macroscopic traffic flow modeling remains the main approach for fatigue analysis. Objectives 1 and 3 provide theoretical and practical perspectives, respectively, serving as a foundation for unraveling the inherent scale-invariance of RPEFs and validating the applicability of the proposed modeling method.

4.2. Simulation Parameters

(1)

Fatigue load spectrum

The fatigue load spectrum (

Table 1. Fatigue load spectrum parameters.

Heavy Vehicle Type		Percentage (%)	Average Axle Weight (tf) and Axle Spacing (m)
V21		8.75	1.89 (3.0) 2.3
V22		45.11	3.40 (4.7) 7.74
V23		30.30	5.05 (6.6) 9.84
V31		2.79	4.36 (2.1) 4.48 (5.8) 1.37
V32		3.09	7.68 (5.0)13.66 (1.5) 13.74
V41		4.59	6.85 (2.0) 7.44 (4.8) 13.97 (1.4) 15.25
V42		1.82	4.1 (3.8) 12.03 (6.8) 11.96 (1.4) 11.87
V51		1.74	5.55 (3.7) 12.53 (6.6) 11.47 (1.4) 10.75 (1.4) 11.23
V61		1.03	4.24 (1.9) 4.4 (2.7) 11.29 (6.6) 9.61 (1.4) 9.71 (1.4) 10.49
V62		0.78	5.59 (3.5) 8.42 (1.5) 7.98 (7.0) 9.59 (1.4) 9.24 (1.4) 10.3

Note: The hollow circle symbolizes the headway side wheels, while solid circles symbolize the rear side wheels.

) used in this article is derived from traffic statistics of 176,470 vehicles surveyed by the WIM system on a suburban highway in Hangzhou, China [27]. This is a typical load spectrum affected by regional traffic congestion. It contains 10 types of heavy vehicles, including 3 types of 2-axle vehicles (V21, V22, V23), 2 types of 3-axle vehicles (V31, V32), 2 types of 4-axle vehicles (V41, V42), 1 type of 5-axle vehicle (V51), and 2 types of 6-axle vehicles (V61, V62). The vehicles included in this spectrum are limited to those with a gross vehicle weight (GVW) exceeding 3 tons. Table 1 provides their relative proportions, average axle loads, and average wheelbase (in parentheses). The proportion here is the relative proportion between heavy vehicles, not including cars.

Table 2. Parameters of the mixed probability distribution for GVWs.

Heavy Vehicle Type	Distribution Type	First Distribution			Second Distribution			Third Distribution
		p1	μ1	σ1	p2	μ2	σ2	p3	μ3	σ3
V21	Lognormal	0.35	1.14	0.04	0.46	1.32	0.12	0.19	1.87	0.42
V22	Lognormal	0.20	1.26	0.12	0.76	2.37	0.52	0.04	3.43	0.14
V23	Lognormal	0.15	2.18	0.41	0.83	2.73	0.17	0.02	3.36	0.23
V31	Normal	0.54	12.56	3.16	0.43	25.52	7.16	0.03	47.15	8.44
V32	Normal	0.45	16.8	4.86	0.27	35.11	7.23	0.27	65.16	10.11
V41	Normal	0.16	14.29	1.47	0.58	35.97	12.86	0.26	77.78	6.72
V42	Normal	0.12	13.68	1.19	0.63	44.9	7.24	0.26	39.92	13.46
V51	Normal	0.12	17.32	1.96	0.83	54.81	11.11	0.05	74.25	11.02
V61	Normal	0.19	17.71	1.63	0.8	56.85	12.98	0.01	100.61	3.41
V62	Normal	0.26	20.22	2.68	0.72	61.27	14.87	0.01	117.69	5.87

shows the probability distribution parameters of GVWs, which are fitted by three peak normal distributions or logarithmic normal distributions $f_{\mathrm{GVW}}\left(x\right)=\sum p_i{f}_i\left(x\right)$. For vehicles weighing less than 3 tons, they are assumed to be cars with a length of 5 m, and only their spacings, not their weights, are considered in the simulation.

(2)

Influence lines

Two typical influence line shapes are used for fatigue analysis: the mid-span bending moment influence line M1 of simply supported structures (Figure 4a), and the support negative bending moment influence line M2 of two-span continuous structures (Figure 4b). They represent the two most common forms of bridge unfavorable section response under moving loads, and many influence lines within the bridge critical lengths can be seen as approximations of their shapes. According to the methodology outlined, L refers to the total length of the influence line, which is set from 10 to 200 m.

(3)

S–N curve type

As a condition for critical damage, the S–N curve type is closely correlated with the calibration results of RPEFs. The linear S–N curve type N·S^m = C with slope m = 3 is chosen for this study, which is widely used in some design codes [1,28]. Considering the continuous reduction in fatigue limit during service, the damage caused by stress amplitudes below the constant amplitude fatigue limit (CAFL) is counted for this type of S–N curve [29]. This significantly facilitates the calculation work for this study, as the damage ratio can be rapidly determined without the need to specify the particular fatigue strength parameter C. The Eurocode bilinear S–N curve [30] can also be used for RPEF calibration, but it requires a comparatively complex procedure to determine the equivalent fatigue detail strength, as detailed in previous study [31].

(4)

Vehicle headway distributions

The Gamma distribution and the lognormal distribution are chosen to characterize the probability models of whole traffic vehicle headways due to their widespread usage. Equations (12) and (13) are their shifted probability density functions. The Gamma distribution is often employed to characterize headway statistics in low traffic density, whereas the lognormal distribution is commonly applied to medium and high traffic density. In situations of low traffic density, the distribution of vehicle headway tends to follow a negative exponential distribution, which is a specific instance of the Gamma distribution (with α = 1). The shift parameter x₀ is a minimum headway parameter that some studies consider the vehicle length and safe driving distance, but, in most cases, it is fitted to 0. Given the whole traffic headway parameters and the heavy vehicle mixing rate p_HV, the temporal and spatial distribution of heavy vehicle loads is obtained by Monte Carlo simulation. The p_HV is set from 0.1 to 0.5 to ensure its adaptability to various traffic conditions. However, there might be a certain degree of simulation instability in some headway cases when p_HV exceeds 0.4.

Gamma distribution:

$$f(x)={\displaystyle \frac{{\beta }^{\alpha }}{\Gamma \left(\alpha \right)}}(x-x_0{)}^{\alpha -1}{e}^{-\beta (x-x_0)}$$

(12)

Lognormal distribution:

$$f(x)={\displaystyle \frac{1}{(x-x_0)\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{[\mathit{ln}(x-x_0)-\mu {]}^2}{2{\sigma }^2}}}$$

(13)

where α and β are the characteristic parameters of the Gamma distribution; σ and μ are the characteristic parameters of the lognormal distribution; x₀ is the minimum fitting parameter; and $\Gamma \left(\alpha \right)$ is the gamma function with an expression of $\Gamma \left(\alpha \right)={\int }_0^{+\infty }{x}^{\alpha -1}{e}^{-x}dx$.

(5)

Vehicle minimum driving spacing

For Objectives 2 and 3, the minimum driving spacing between different types of vehicles is calculated using Equations (14) and (15) [32], ensuring the prevention of spatial overlap between vehicles. However, the utilization of an ideal probability model (Equations (12) and (13)) inevitably generates random numbers that fail to meet the safe driving requirements. The proposed strategy is to prioritize allocating the random numbers that satisfy the distance conditions to heavy vehicles, while distributing the remaining numbers to cars, which contribute less to the stress response. Finally, the random numbers that do not meet the distance requirements are replaced with the minimum driving distance. However, the generation of a vehicle convoy sample is not possible if the safe distance between heavy vehicles cannot be guaranteed. For microscopic traffic simulations employing lane changing strategies, there is a probability of 0.8 for small cars to change lanes when the driving distance is not met [33].

For heavy vehicles,

$$x_{\mathrm{HV},min}=1.5v+7.5$$

(14)

For cars,

$$x_{\mathrm{CAR},min}=1.4v+6$$

(15)

where the units for safe driving spacing x_{HV, min} and x_{CAR, min} are m, while the unit for vehicle speed v is m/s.

Figure 5 shows the simulation flowchart for macroscopic and microscopic traffic. The vehicle spacing judging condition for microscopic traffic is continuously updated in real time, serving as a fundamental basis for determining speed and facilitating lane changes. Additionally, microscopic traffic simulation needs to pay attention to the issue of traffic overflow. This situation occurs when vehicles that have already entered the bridge maintain a slow following, causing the waiting vehicles to be unable to meet the minimum spacing required by the initial headway. In this situation, the subsequent simulation is interrupted to maintain the stability of headway probability.

4.3. Shape Effects of Arrival Probability Model

Based on the headway probability model and the condition of the shift parameter x₀, four sets of dimensionless spacing probability distribution are designed: GD1 (Gamma distribution, x₀ = 0), GD2 (Gamma distribution, $x_0=0.1\overline{X}$), LD1 (lognormal distribution, x₀ = 0), and LD2 (lognormal distribution, $x_0=0.1\overline{X}$). Among them, the shift parameter $x_0=0.1\overline{X}$ for GD2 and LD2 is a designed dimensionless variable based on $\overline{X}$, which is the ratio of the vehicle average spacing to the bridge length. Here, the case where x₀ is less than 1 (i.e., $\overline{X}\le 10$) is investigated. Then, extensive numerical analysis has been conducted to reveal that the correlation between RPEFs and the shape parameters (α for Gamma distribution or σ for lognormal distribution) of the headway probability distribution. Figure 6 shows the point load RPEFs calibrated for the M1 under a 30% mixing rate, which were obtained from at least 100,000 random spacing samples. It is observable that the RPEFs tend to decrease as the shape parameter α of the Gamma distribution increases, whereas the relationship between RPEFs and the shape parameter σ of the lognormal distribution is inverse. The RPEFs curves of the Gamma distribution exhibit significant variations when comparing α = 1 and α = 2, indicating a drastic impact on RPEFs due to changes in traffic conditions from low to medium density. Moreover, as the shape parameter changes to a certain extent, the RPEFs of both probability models tend to converge, indicating that there will be no significant alteration in the fatigue damage effect when traffic conditions approach an extremely traffic dense state. Furthermore, upon comparing GD1 (solid line in Figure 6a) with GD2 (dashed line in Figure 6a) and LD1 (solid line in Figure 6b) with LD2 (dashed line in Figure 6b), it becomes apparent that a common shift condition has minimal influence on RPEFs. Thus, the assignment of x₀ = 0 is a reasonable and relatively safe parameter choice.

As mentioned in the modeling section, apart from the original shape parameters of the headway probability model that characterize RPEFs, there are also additional generalized shape parameters independent of the model, such as the coefficient of variation (COV), kurtosis (γ₃), and skewness (γ₄). However, the generalized shape parameters of headways in established probability models can be directly derived from the specific shape parameters, eliminating the necessity for additional analysis within the whole traffic model. The main emphasis shifts towards the generalized shape parameters of heavy vehicle headways derived through statistical analysis. Figure 7 shows the correlation between the three types of generalized shape parameters and RPEFs under different average loading occurrence n_Ξ. The figure illustrates a significantly stronger correlation between the COV and RPEFs compared to kurtosis γ₃ and skewness γ₄, while maintaining consistency across different probability models. This finding is intriguing as it suggests that the COV may exhibit insensitivity to headway probability models in RPEF characterization, indicating its potential for further exploitation.

4.4. Cluster Analysis of RPEFs Under Macroscopic Traffic Flow

The analysis objective of this section shifts to actual vehicle loads. The generalized shape parameter COV of heavy vehicle headway continues to be employed for RPEF clustering analysis based on actual traffic flow parameters. As shown in

Table 3. Parameters of headway probability distributions.

Probability Model	Source	No.	Headway Type	α or σ	β or μ	x0	λW (veh/h)	ρW (veh/km)	v (km/h)	COV(HW)	COV(HHV)
											pHV = 10%	pHV = 20%	pHV = 30%	pHV = 40%
Gamma distribution	[9]	G1	HT	1.85	0.093	0	180.97	3.31	54.7	0.73	0.984	0.942	0.925	0.889
		G2	HT	1.38	0.114	0	297.39	5.12	58.1	0.85	0.981	0.965	0.929	0.905
	[34]	G3	HT	0.9	0.04	0	160.00	2.04	78.4	1.05	0.998	0.980	0.993	0.971
		G4	HT	12.9	7.23	0	2017.67	66.48	30.4	0.28	0.912	--	--	--
	[35]	G5	HT	1.548	0.263	0.207	590.80	13.13	45.0	0.78	0.955	0.894	0.887	0.831
	[36]	G6	HT	1.269	1.225	0.69	2085.85	26.28	79.4	0.53	0.907	0.805	0.664	--
Lognormal distribution	[9]	L1	HT	0.878	1.56	0	514.53	11.22	45.9	1.08	0.999	0.977	0.976	0.959
		L2	HT	0.58	0.83	0	1326.75	29.46	45.0	0.63	0.929	0.876	0.792	0.683
	[37]	L3	HT	0.55	0.882	0	1281.05	24.13	53.1	0.59	0.945	0.832	0.789	0.720
		L4	HT	0.533	0.821	0	1374.22	38.91	35.3	0.57	0.929	0.825	0.750	0.639
	[38]	L5	HD	0.497	1.811	4.5	657.64	87.70	7.5	0.32	0.931	0.852	0.785	0.713
		L6	HD	0.426	3.027	4.5	1922.99	36.90	52.1	0.37	0.907	0.808	0.690	--

, statistical parameters of measured vehicle spacing/time headway based on two probability models are extensively gathered, including six groups of Gamma distribution parameters and six groups of lognormal distribution parameters. Additionally, the information on flow rate λ_W, density ρ_W, headway COV(H_W), and average speed v of traffic as a whole is provided. Four heavy vehicle mixing rates p_HV are taken into account for analysis, which are 10%, 20%, 30%, and 40%. Simulation is carried out based on the combination of all the above parameters. Table 3 also presents the simulated heavy vehicle headway COV(H_HV) under these combined conditions. It is noticeable that COV(H_HV) decreases as the mixing rate and traffic density increase. Overall, COV(H_HV) are significantly higher than the whole traffic headway COV(H_W), indicating a higher freedom in the spatial distribution of heavy vehicles. For each RLM simulation, it comprises at least 100,000 vehicle samples.

The RPEFs of two typical influence lines are illustrated in Figure 8, presented in three parameter systems. The first system corresponds to the influence line length L (Figure 8a,b), with the legend indicating the mixing rate p_HV. It can be observed that the dispersion in this coordinate is quite significant, making it impossible to clearly distinguish the data of the mixing rate. The second system, represented by the average occurrence of heavy vehicle ρ_HVL (Figure 8c,d), with the legend indicating that the heavy vehicle mixing rate p_HV can clearly distinguish the data of mixing rate. However, due to the inclusion of different headway cases within the same mixing rate, leading to a slight data overlap in the cases of medium to low density and high mixing rates. In the third system, which is also represented by the average occurrence ρ_HVL (Figure 8e,f), with the legend indicating the COV of heavy vehicle headway, all RPEFs are categorized into five groups with different intervals of COV(H_HV). The clustering effect of third system is more refined than the second one, which verifies the rationality of the proposed characterization Equations (8)–(10) directly, allowing for a more precise distinction of RPEFs under the same mixing rate. However, it is worth mentioning that the first system, which suffers from a significant loss of accuracy in RPEF characterization, is most commonly used in existing specifications to set fatigue damage correction coefficients. This also implies that, if the coefficients are calibrated solely based on free-flow traffic conditions, it may lead to an underestimation of fatigue damage in congested situations. The proposed modeling method, in contrast, offers a unique combination of precision and flexibility in fatigue design, making it a potential approach for standardization.

From Figure 8e,f, it can be observed that the maximum damage efficiency occurs when the COV(H_HV) is between 0.75 and 0.8. This indicates that the presence of a correlation between traffic density and headway variation is evident, in contrast to the flexibility of assigning parameters in an ideal point load model. Although dense traffic results in a larger ρ_HVL, its headway variation is constrained (smaller COV(H_HV)), which may lead to a reduced number of peak-to-valley cycles in the stress history, thereby impacting damage efficiency. Conversely, in a free-flowing traffic state (larger COV(H_HV)), the situation is contrary. Therefore, either extreme smooth or extreme congestion are not representative for actual traffic situations. The fatigue analysis work must comprehensively consider the influence of different traffic conditions and the interrelationships among them.

4.5. The Impact of Microscopic Traffic Behavior on RPEFs

Given that constant speed traffic conditions are ideal, it is necessary to further investigate the variations of RPEFs under microscopic traffic behaviors to clarify the applicability of the proposed modeling method. Recognizing the heavy vehicle number has great impact on RPEF results, the investigation focuses on two types of microscopic traffic behaviors. The first type involves only speed variation without lane changing behavior, wherein all vehicles travel on the designated influencing lane. When encountering a slow-moving vehicle ahead, vehicles decelerate to keep a safe following state with minimum driving spacing (Figure 9a). The second type involves both speed variation and lane changing behavior. The speed adjustment strategy aligns with the first type, while the lane changing strategy applies only to cars. Once the distance between a car and the preceding vehicle reaches a safe following distance, the car has a certain probability to switch lanes (Figure 9b). To acquire an unfavorable scenario, the analysis omitted instances where vehicle acceleration causes a widening of the following distance.

According to the traffic parameters provided in Table 3, microscopic traffic simulations were conducted with a controlled vehicle speed variation coefficient of 0.1~0.3 and a heavy vehicle mixing rate of 0.3. To ensure consistency, both strategies were simulated with the identical arrival vehicle parameters, and the average heavy vehicle flow rate λ_HV and travel time T of dynamic traffic were estimated based on the statistics from their bridge sections (beginnings, midspans, and ends). Figure 10a illustrates the relationship between COV(H_HV) and λ_HVT of Strategy-1 and Strategy-2, with a traffic passage distance of L = 200 m. It can be seen that, as the traffic flow rate increases, the interaction between vehicles becomes more pronounced, and the effect of lane changing behavior of Strategy-2 on the statistical parameters of heavy vehicle traffic becomes more significant. However, given the negative correlation between COV(H_HV) and λ_HVT, the fatigue analysis under medium- and high-traffic-density conditions (Figure 10b) suggests that there may not be significant differences in damage between the two types of microscopic traffic behavior. Compared to macroscopic traffic, microscopic traffic can increase the dispersion of RPEFs to a certain extent. However, this discreteness can generally be explained by the macroscopic traffic RPEFs within the corresponding COV(H_HV) range. This suggests that the varying dimensions of traffic randomness exhibit a degree of consistency in their impact on bridge fatigue damage.

5. Fatigue Damage Assessment Under Mixed Traffic Conditions

The actual traffic conditions often experience significant fluctuations throughout the day, influenced by rush hours. In the following discussion, we elaborate on the utilization of the proposed method for estimating structural fatigue damage under mixed traffic conditions, aiming to establish the comprehensive technical framework of this study.

5.1. Equivalent RPEFs

For actual traffic, despite its vast variety of changes, it can always be classified into several patterns. Based on the proportion of each traffic pattern, damage accumulation principles can be employed to predict structural fatigue damage D_d under mixed traffic conditions. Therefore, the expression of D_d is as follows:

$$D_d=\sum _{i=1}^N{n}_i{k}_{r,i}{\bar{D}}_{\mathrm{SLM}}$$

(16)

where N is the number of traffic patterns; n_i is the number of heavy vehicles at the ith traffic state; k_r,i is the RPEFs at the ith traffic state, which are expressed as

$$k_{\mathrm{r},i}=f_{\theta }\left({\rho }_{\mathrm{HV},i}L\left|\mathrm{COV}\right(H_{\mathrm{HV},i})\right)$$

(17)

The above equation incorporates the modeling method proposed, which utilizes heavy vehicle traffic density ρ_HV and the COV of heavy vehicle headway as calculation parameters for their relationship with RPEFs, which is more direct. The reason for choosing density ρ_HV instead of flow rate λ_HV for calculating the average loading occurrence effects is primarily due to the monotonic relationship between density and traffic speed, whereas flow rate does not exhibit such a correlation. For convenient application, an equivalent RPEF is introduced to represent the weighted impacts of different traffic states, transforming Equation (16) into the following form:

$$D_d=N_d{k}_{\mathrm{r},\mathrm{eq}}{\bar{D}}_{\mathrm{SLM}}$$

(18)

where k_r,eq is the equivalent RPEF, which is expressed as

$$k_{\mathrm{r},\mathrm{eq}}=\sum w_i{k}_{\mathrm{r},i}$$

(19)

where $w_i=n_i/\sum n_k$ represents the heavy vehicle volume weight of the i-th traffic state.

Given the absence of a probability model for heavy vehicle headway, a method for estimating equivalent RPEFs is proposed through a correlation analysis of overall traffic statistics. As shown in Figure 11, the solution of equivalent RPEFs is divided into three main paths: scale parameters, shape parameters, and weight parameters. The flowchart includes the following steps: a. Statistics collection, including average flow rate, speed, loading time, and mixing rate for each traffic state; b. Calculation of scale parameters, including the whole traffic density ρ_W, heavy vehicle traffic density ρ_HV, the average loading occurrence ρ_HVL for each state; c. Calculation of shape parameters, including the whole traffic headway COV(H_W) and the heavy vehicle headway COV(H_HV) for each state (these parameters exhibit a significant correlation with density and mixing rate); d. Calculation of heavy vehicle volume weight coefficient w for each state; e. Calculation of equivalent RPEFs under mixed traffic conditions, including RPEFs of each state and their weighted-average computation, i.e., k_r,eq = Σw_ik_r,i.

The achievement of this process relies on the establishment of three key functional relationships, namely Fun1~Fun3, as detailed in Appendix A. Respectively, Fun1 is the functional relationship between the coefficient of variation of headway distance COV(H_W) for the whole traffic flow and the traffic flow density ρ_W (Figure 12a), Fun2 is the functional relationship between the COV(H_HV) for heavy vehicles and the entirety of traffic COV(H_W) as well as the heavy vehicle mixing ratio p_HV (Figure 12b), and Fun3 is the k_r calibration function for different influence lines (Figure 8e,f). The establishment of these functional relationships enables a nearly direct estimation of structural fatigue damage under mixed traffic conditions, bypassing the need for extensive simulation processes.

5.2. Optimization of Traffic Pattern Classification

When employing the superposition effect of multiple steady-state traffic flows to approximate actual non-steady-state traffic flow, the issue of optimal approximation arises. It is important to recognize that, due to the competitive relationship between different patterns in the statistical analysis of fatigue damage events, a classification containing numerous unfavorable states will inevitably diminish the damage effects of other states. Consequently, the optimal approximation should converge towards achieving the maximum damage effect. In other words, there is an upper limit to the equivalent RPEFs, which also reflects the steady effect within the sample space modeled from actual traffic conditions. To facilitate the statistical analysis, an equivalent density index, ρ_HV,eq, was proposed to identify the potentially most unfavorable pattern classification:

$${\rho }_{\mathrm{HV},\mathrm{eq}}=\sum _i^N{w}_i\cdot {\rho }_{\mathrm{HV},i}\cdot {\displaystyle \frac{\mathrm{COV}\left(H_{\mathrm{HV},i}\right)}{\mathrm{COV}\left(H_{\mathrm{HV},\mathrm{Free} \mathrm{Flow}}\right)}}\to \max$$

(20)

where COV(H_{HV,Free Flow}) denotes the coefficient of variation of heavy vehicle headway under free-flow conditions, which is theoretically assumed to be 1.

The equivalent density ρ_HV,eq essentially scales the density of different traffic states uniformly to the density at the free-flow damage rate. Certainly, while optimizing the equivalent RPEFs to their maximum potential, it is also desirable to minimize N for pattern classification, thereby reducing the statistical workload as much as possible. This is discussed further in the subsequent case study.

5.3. Case Study

A China national highway traffic dataset is used in the case study, which was collected using millimeter wave and video recognition systems with a minimum statistical duration of 5 min. The daily traffic volume in the main lane of this highway is approximately 16,000 vehicles, with a heavy vehicle mixing rate of 40%. To achieve a discretized analysis, three traffic pattern classification strategies are adopted (Figure 13). These strategies are flow-rate-based (FB), velocity-based (VB), and time-scale-based (TB). The FB strategy categorizes traffic conditions by analyzing the variations in traffic flow rates across different time intervals within a day (Figure 13a), roughly dividing the day into seven segments with time scales ranging from 2 to 6 h. VB strategy is based on vehicle speed (Figure 13b), considering speeds above 50 km/h as smooth state and speeds below 50 km/h as congested state. Figure 13c shows temporal variation in the mixing rate of heavy vehicles with a 5 min interval. Based on the given traffic data, the equivalent density ρ_HV,eq of different pattern classification methods was calculated (Figure 13d). The TB strategy demonstrates the outcomes across different time intervals, including 1 h, 2 h, 3 h, 4 h, 6 h, 8 h, 12 h, and 24 h. It can be seen that, as the number of TB segments increases (i.e., the time scale decreases), the value of ρ_HV,eq increases. The FB and VB strategies also follow this trend. Once the number of segments N exceeds 7, the value of ρ_HV,eq tends to stabilize.

The results of k_r,eq for different traffic pattern classification strategies are shown in Figure 14. For comparison, a series of 100 microscopic traffic simulation experiments (MS) were conducted, with Strategy-1 and Strategy-2 each comprising 50%. It can be seen that there is a significant difference in k_r,eq values. The values estimated by the TB strategies will decrease as the time scale increases, with its most unfavorable situation is TB-24h, corresponding to a complete daily average traffic state. FB and VB strategies demonstrate similar trends, i.e., when the number of divisions is few or their durations are large, the corresponding k_r,eq tends to be relatively small. The changes presented by k_r,eq are basically corresponding to the equivalent density ρ_HV,eq. The k_r,eq evaluated by the strategies of TB-1h, TB-2h, TB-3h, and FB are very close and demonstrate an upper-limit level. If TB-1h is taken as the maximum statistical reference, the ρ_HV,eq values of these strategies differ by less than 10%. Compared to MS, these results are relatively conservative for the adoption of a 95% survival rate in the construction of Fun3. The instability of a small number of results (M2, L = 50~100 m) may be related to an insufficient number of calibration samples for specific structural responses. Overall, the results of the case study validate the hypothesis that the maximum effect of traffic pattern classification reflects the actual traffic effect. Additionally, the study elucidates the reasons why previous research has indicated that traffic pattern classification should not be overly simplistic. Especially for structures with spans greater than 50 m, it is necessary to ensure the rationality of traffic pattern classification. The FB strategy could serve as an effective solution for traffic statistics, warranting further validation and optimization.

6. Conclusions

This study has proposed an analytical approach for modeling the effect of random traffic patterns on fatigue damage for highway bridges, aiming to improve the accuracy of traffic pattern clustering and fatigue assessment under mixed traffic conditions. The methodology can be summarized as introducing damage ratio RPEFs to characterize the impact of traffic randomness. This is accomplished by applying the similarity principles of scale-free stochastic processes for modeling purposes. Two key parameters quantifying RPEFs have been proposed: the shape parameter of the headway probability model and the average occurrence of loading events. These encompass the essential aspects of traffic randomness on bridge fatigue damage, namely, load intensity and variation. The numerical results demonstrate that both the effects of macro-traffic states and micro-traffic behaviors conform to the same principles. The case study of mixed traffic shows that the actual fatigue damage of bridges can be evaluated through the superposition of different traffic pattern effects, provided that the pattern classification number fits the fluctuations in traffic flow. Overly rough division will lead to an underestimation of actual damage. Certainly, there remains a necessity for further optimization and validation efforts within this domain.

It is reasonable to posit that the proposed method can serve as theoretical guidance for refining damage correction factors of specification fatigue load models. This is especially pertinent concerning the influence of traffic volume, which should encompass the impact of variations in traffic states rather than being overly simplified to an accumulation of load cycles. The practicality of the method enables standard developers to apply the calibrated RPEFs in formulating damage correction factors, thereby eliminating the need for extensive simulations and facilitating the identification of more generalized traffic conditions. From a standardization perspective, numerous significant initiatives remain to be undertaken to further advance this field. These efforts encompass the investigation of the impact of the spatial distribution of heavy vehicles under different traffic states, taking into account more influence line types, multi-lane effects, and the development of the fatigue reliability evaluation method within the proposed framework.