Limited Response of Curve Safety Level to Friction Factor and Superelevation Variation under Repeated Traffic Loads

Abstract

Although road horizontal curves are high-risk sections for accidents, current road safety assessments often neglect the dynamic evolution of superelevation and the friction factor. The connotation for road safety level was clarified by examining the significance of road factors in traffic safety through the systemic characteristics of roads. Among these characteristics, curve safety level is determined by the ratio of the supply and demand of the lateral friction factor. On the basis of international standards and specifications, this study clarified the design supply and demand of friction factors for curve by considering the distribution of tangential and lateral friction factors. Expanding on the steady-state bicycle model while accounting for road geometric parameters and vehicle operation characteristics, the lateral friction factor demanded for vehicles was quantified. Meanwhile, the characteristics of the friction factor supplied and the superelevation variation were analyzed by using the road service life as a variable, along with their influence on the actual supply of the friction factor and the curve safety level. The results of the analysis indicate a rapid decrease in curve safety level during the first two years of road utilization, followed by a slower declining trend, with a significant 27% reduction in curve safety level by the end of the second year. Furthermore, the decline in the curve safety level is mainly attributed to variations in the road surface friction factor, whereas the influence of superelevation variation on the curve safety level is restricted. In the absence of maintenance interventions, the curve safety level will decrease by over 30% after three years of operation. Controlling operational speed is one of the effective measures for ensuring traffic safety. Meanwhile, the impact of the friction factor and the superelevation variation on the curve safety level accumulates over time, thus causing drivers to have difficulty perceiving these alterations. Therefore, dynamic safety evaluations that account for the fluctuation in the friction factor and superelevation induced by repetitive vehicle loading must be undertaken.

1. Introduction

Road safety is a prominent concern among transportation experts globally. The World Health Organization has documented a persistent surge in fatalities stemming from road traffic accidents, culminating in an annual toll of 1.35 million lives lost due to vehicular collisions on roads [1]. Notably, over 25% of fatal accidents are associated with curved road sections [2]. Previous studies have shown that accident rates increase as the radius of curves diminishes. For instance, the accident rate on curves with a radius less than 200 m is at least two times higher than that on curves with a 400 m radius [3]. Consequently, investigating the safety of small-radius curve sections holds critical significance in enhancing the overall safety standards of roadways.

Numerous empirical and theoretical investigations have been undertaken to enhance the traffic safety of curve sections. Diverse accident theories have been developed to explicate the causes of accidents, such as stochastic theory [4], propensity theory [5], causation theory [6], systemic theory [7], and behavioral theory [8]. While these theories lay the theoretical groundwork for traffic safety evaluations, their independent application proves inadequate for the precise inspection of road engineering projects. This deficiency is notably characterized by the lack of explicit evaluation standards and the absence of assessment metrics within these theories. In response to this issue, countries have extensively explored methods for road traffic safety evaluation, thus yielding positive outcomes. Among them, the criteria for road traffic safety evaluation presented by Lamm [9] are considered highly exemplary, thereby serving as the foundation for such assessments in various countries. According to Lamm [9], evaluating road traffic safety should encompass inspections across three critical domains: design consistency, operating speed consistency, and driving dynamic consistency. In addition, several studies have proposed an array of assessment indicators and criteria, including speed profile [10] and driver expectancy [11].

The above criteria evaluate the relationship between geometric design indicators and traffic safety to ensure the safety of curve sections. Further analysis of these measures has exclusively focused on the design state. As such, it has disregarded the dynamic variations in road safety levels influenced by environmental factors and vehicular loading. The repetitive impact of traffic loads induces dynamic variation in the road surface friction factor and superelevation. For instance, Zador et al. [12] observed a gradual reduction in the road surface friction factor under the action of vehicular loads, noting that curves with larger curvature experience a faster decline in the road surface friction factor than straight segments. Mayora [13] found that raising the average road surface friction factor from below 0.5 to above 0.6 resulted in a 68% reduction in collision accidents. A decrease in the friction factor will adversely affect road traffic safety. Additionally, Montella et al. [14] found that the superelevation in curves often falls below the designated design levels during the service life of highways. Cafiso et al. [15] also observed instances of reduced superelevation in curve sections with small radii while adjusting collision functions on the basis of road surface conditions and geometric design indicators. The decrease in superelevation heightens the requisite friction factor for safe driving, which has a detrimental effect on driving safety. However, limited attention has been given to the dynamic variation in the friction factor and superelevation. The existing research cannot comprehensively address the extent of the impact of these dynamic changes in road components on the safety of road systems. As such, further study must explore how the variations in the road surface friction factor and superelevation caused by repeated vehicle loads affect the curve safety level.

This study addresses the inadequacy of existing road safety evaluations in considering the dynamic evolution of the road surface friction factor and superelevation. Firstly, the connotation of road safety level was elucidated from the systemic characteristics of roads, with the curve safety level being determined by the ratio of the supply and demand of the lateral friction factor. Subsequently, a method for determining the design for the curve safety level was proposed on the basis of international standards and specifications by considering the allocation of lateral and tangential friction factors. Taking the number of vehicle load actions as a variable, a method for determining the actual curve safety level was formulated. Finally, the limited response of the curve safety level to the friction factor and superelevation variation was analyzed by employing the service life of roads to represent the repeated action of vehicle loads.

2. Methodology

Currently, speed and driving dynamic indicators are commonly used to evaluate road safety internationally. The fundamental idea is to gauge the safety of roads on the basis of design speed and operating speed differentials, or their corresponding driving dynamic indicators. This examination can help determine the rationality of the design indicators used during the design phase. However, as the service life of roads extends, repeated vehicle loads cause variations in pavement friction factor and superelevation, thus leading to a reduction in the supply of the lateral friction factor. Furthermore, as operational speeds of vehicles increase, the demand for the lateral friction factor to ensure safe travel also increases. This trend further results in reduced safety levels, thus potentially reaching a critical state for stable vehicle passage. Considering the limitations of the existing indicators for safety evaluation that capture the dynamic alterations in road conditions during operation, this study uses the ratio of the supply and demand of the lateral friction factor to characterize the curve safety level. It aims to thoroughly consider the impact of dynamic changes in the road surface friction and superelevation on curve safety.

2.1. Curve Safety Level

The criteria for safety levels vary across distinct industries. Within the medical domain, the safety range is defined as the span between the maximum effective dose and the minimum toxic dose of a drug, thus signifying the drug’s safety level. Typically, a greater span implies a safer pharmaceutical drug. In the construction industry, structural reliability theory deals with the probability of a structure to sustain loads and endure environmental factors over the designated period and specified conditions. The designated period corresponds to the standard design lifespan of the structure, while the specified conditions encompass the structure’s typical design, construction, utilization, and environmental circumstances. However, the fundamental nature of road safety within the transportation field is not entirely clear. Drawing inspiration from medical and construction industries and considering the unique characteristics of road traffic safety, this study introduces and refines the definition of road safety levels. Road safety level refers to the ability of roads, under customary construction and operational conditions, to endure diverse interactions involving individuals, vehicles, road infrastructure, and the environment. Additionally, it encompasses the ability of roads to maintain the necessary overall stability and traffic service capability in the event of accidental occurrences and their consequences. Figure 1 illustrates an example of the relationship between the designed and the actual states of road safety levels.

Individuals, vehicles, road infrastructure, and environments constitute the elements of the road system. Each entity fulfills distinct roles and exerts varying influences on curve safety. Centrifugal force is generated as vehicles travel on curved sections. Some of this force is offset by the superelevation of the road, while the rest is balanced by the lateral friction factor between the tire and the road surface. Road geometric design theory employs the point mass model (Equation (1)) to limit the demand for the lateral friction factor by setting the curve radius and road superelevation, thus ensuring the safe driving of vehicles.

$$f_y^{PM}=\frac{v^2}{gR}-i_h$$

(1)

where $f_y^{PM}$ is the demand of the lateral friction factor for point mass model; $v$ is the vehicle speed (m/s); $g$ is the gravitational acceleration (m/s²); $R$ is the curve radius (m); $i_h$ is the superelevation (%). Road geometric design theory indicates that the main road factors affecting driving stability are the tire–road friction factor and the sideway force coefficient, namely, the supply and demand of the lateral friction factor. The safety criterion Ⅲ proposed by Lamm is a quantitative measure for the designed curve safety evaluation. It is calculated using the safety margins, which is defined as the difference between the maximum supply of the lateral friction at design speed ($f_R$) and the demand of lateral friction by the vehicle to negotiate a horizontal curve at $V_{85}$ ($f_{RA}$). Lamm’s safety criterion Ⅲ also prescribes limiting values for different design levels: $f_R-f_{RA}\ge 0.01$ for good design; $-0.04\le f_R-f_{RA}<0.01$ for fair design; $f_R-f_{RA}<-0.04$ for poor design. The models of parameters related to the safety criterion Ⅲ are as follows [16]:

$$f_R=0.38-3.82\times {10}^{-3}\times V_d+0.98\times V_d^2$$

(2)

$$f_{RA}=\frac{V_{85}^2}{127R}-i_h$$

(3)

$$V_{85}={10}^6/(8270+8.01\times \frac{63700}{R})$$

(4)

where $V_d$ is the design speed (km/h); $V_{85}$ is the expected 85th-percentile speed for design.

Lamm’s safety criterion Ⅲ is based on the curve design state, which the supply and demand of friction is a fixed value under the determined curve radius and superelevation. However, the performance of road inevitably undergoes variation during road operation. Any change in the components of the road system (geometric elements, pavement characteristics, vehicles, drivers, and environment) will have an impact on the curve safety. The interactions among individuals, vehicles, road infrastructure, and the environment within the road system can be quantified through the lateral friction factor. The ability of the road to endure various interactions refers to the supply of the lateral friction factor, which is related to the repeated action of vehicle loads. Meanwhile, the capability of the road to maintain overall stability refers to the demand of the lateral friction factor by vehicles, which is associated with speed, road superelevation, and curve radius. Curve safety level is the relationship between the supply and demand of the lateral friction factor for safe vehicle travel. The mathematical expression for the curve safety level can be articulated as follows:

$$f_p=\frac{f_s}{f_d}$$

(5)

where $f_p$ is the curve safety level; $f_s$ is the supply of the lateral friction factor; $f_d$ is the demand of the lateral friction factor. $f_p\ge 1$ indicates that the road can provide sufficient lateral friction factor for vehicle stability, where a larger value of $f_p$ corresponds to a safer road. Conversely, $f_p<1$ implies that the road is unable to provide adequate lateral friction for vehicles, thus raising the potential risk of skidding or rollovers.

The multiple relationships between $f_s$ and $f_d$ in characterizing the curve safety level offer an intuitive representation of the dynamic variations in road safety during usage in contrast to the safety criterion Ⅲ proposed by Lamm [9]. The variations in the lateral friction factor and superelevation induced by repeated vehicle loading will be reflected in the interaction between the road and vehicles during road operation. A reduction in the supplied lateral friction factor implies a diminished capacity for interaction between vehicles and the road. Additionally, an increase in the demanded lateral friction factor during instances of overspeed driving leads to a decrease in the road’s overall capability for stability. However, neither a decrease in road capacity nor a decrease in its capability for stability necessarily mean that a traffic accident will occur. The probability of an accident only significantly increases when the actual curve safety level falls below 1. The designed curve safety level is fixed with its design specifications, which generally exceed 1 by a sufficient margin to ensure that the road can endure various interactions. The safety level of the curved road undergoes dynamic changes during operation. As such, the designed curve safety level and the actual curve safety level must initially be elucidated to conduct an analysis on the curve safety level.

2.2. Designed Curve Safety Level

The designed curve safety level depends on the relationship between the speed and the curve radius. It is coupled with the consideration of superelevation to determine the lateral friction factor, which is in accordance with the principles of road geometric design theory. The lateral friction factor prevents vehicles from sliding when traveling on a curve segment due to centrifugal force. Considering driver comfort, numerous countries have derived lateral friction factor design values for various design speeds on the basis of the point-mass model by assessing the demand of lateral friction factor. Hence, the ratio of the supply and demand of the lateral friction factor in the road design can serve as the designed curve safety level, which can be expressed as follows:

$$f_{pd}=\frac{f_{sd}}{f_{dd}}$$

(6)

where $f_{pd}$ is the designed curve safety level; $f_{sd}$ is the designed lateral friction factor supply; $f_{dd}$ is the designed lateral friction factor demand.

The designed lateral friction factor supply ($f_{sd}$) is not specified in the standards of various countries, which poses a challenge in determining the designed curve safety level. Nevertheless, international standards specify the maximum longitudinal friction factor supply for different design speeds by considering the necessary visibility for vehicle travel, thus providing insights for determining $f_{sd}$. As vehicles traverse curved road sections, the force interaction between tires and the road surface primarily involves tangential and lateral forces. The steering and braking actions of vehicles induce variations in tangential and lateral forces, which are reflected in the tangential and lateral friction coefficients of the road surface. Utilizing friction in one direction will decrease the reserve of friction in the other direction, where the relationship between tangential and lateral friction forces is known as the friction ellipse [16]. The lateral friction coefficient utilization ratio was proposed by Germany to characterize the relationship between lateral and tangential friction factors, as shown in Equation (7) [17]. The design of the lateral friction factor supply can be formulated on the basis of the theory of friction ellipses.

$$f_s=n\times 0.925f_x$$

(7)

where $f_x$ is the supply of the tangential friction factor; $n$ is the utilization ratio. $n$ is 70%, which means that 71% of friction will be available in the tangential direction.

The $f_{dd}$ can be considered from the perspective of setting the minimum radius of the curve. An essential condition for the stable movement of vehicles along curved segments is that the lateral force coefficient cannot exceed the lateral friction coefficient between the road surface and the tire. Countries take into account a balance between the range of tire–road friction coefficients and the comfort of drivers and passengers when determining the lateral force coefficient. The lateral force coefficients specified in the design standards effectively avert the danger of vehicle skidding. Consequently, such coefficients from various countries have been adopted as the design for the lateral friction factor demand.

Taking the design parameters specified in the Chinese design specification as an example, the curve safety levels at various design speeds are calculated according to Equations (6) and (7), as shown in

Table 1. Designed curve safety level at various design speeds.

Design Speed (km/h)	Longitudinal Friction Factor	Lateral Friction Factor	Designed Curve Safety
60	0.33	0.15	1.42
80	0.31	0.13	1.54
100	0.30	0.12	1.62
120	0.29	0.10	1.88

. Notably, the designed curve safety level in Table 1 is based on the minimum radius corresponding to the design speed and the friction factor of the road surface in wet conditions. In practice, the values employed in the design are typically larger than the minimum radius, where the friction coefficient of the road surface in dry conditions is also greater than that in wet conditions. Therefore, the designed safety level corresponding to each speed in Table 1 should be considered the minimum value of the designed curve safety level.

2.3. Actual Curve Safety Level

The actual curve safety level is defined as the ratio of the actual supply of the lateral friction factor on the road surface to the actual demand of the lateral friction factor for the safe driving of vehicles during operation. The expression is as follows:

$$f_{pa}^n=\frac{f_{sa}^n}{f_{da}^n}$$

(8)

where $f_{pa}^n$ is the actual curve safety level in the n-th year of operation; $f_{sa}^n$ is the actual supply of lateral friction factor in the n-th year of operation; $f_{da}^n$ is the actual demand of lateral friction factor for safe driving in the n-th year of operation.

The supply of the lateral friction factor plays a vital role in ensuring vehicle stability. The repetitive impact of vehicle loading on the aggregate polishing of road surface leads to a decrease in the supplied lateral friction coefficient. Consequently, reductions in the capacity for vehicle–road interactions adversely affects the curve safety level. Moreover, the curve superelevation partially offsets some centrifugal forces, thus reducing the demand of the lateral friction factor to some extent. A reduction in superelevation diminishes the capacity of the road to counteract centrifugal forces, thereby increasing the demanded friction factor. The uneven settlement of roads leads to variations in superelevation, which further affects the curve safety level under the long-term effects of soil consolidation and vehicle loads during road operation. The variation characteristics of pavement friction factor and superelevation under repeated vehicle loading are discussed in this section.

2.3.1. Actual Supply of Lateral Friction Factor under Repeated Vehicle Loads

Do et al. [18] initially proposed a predictive model for friction coefficients on the basis of analysis of the evolution of friction coefficients of asphalt pavements under the influence of accumulated traffic volume. Subsequently, they further improved the predictive model for friction coefficients by incorporating polishing speed and tire–road contact pressure, which can be utilized to analyze the impact of friction coefficients on trucks and passenger cars [19]. However, the models above were both based on data obtained in the laboratory. They disregarded the relationship between laboratory data and cumulative traffic volume, thus making them less directly applicable in real-world scenarios [20,21]. Conducting on-site assessments to measure pavement friction coefficients across multiple operational years provides an accurate depiction of the relationship between cumulative traffic volume and pavement friction coefficients. However, such measurement requires high data accuracy and volume. Hofko et al. [22] utilized an indoor testing apparatus to replicate the polishing effect of cumulative traffic volumes ($CTV$) and average annual daily truck traffic (${AADT}_{HGV}$) on the pavement friction coefficients. A prediction model of the SMA asphalt pavement under dry conditions was proposed to depict the relationship, as described in Equations (6) and (7). This study focuses on the impact of a decreased friction factor due to abrasion caused by cumulative traffic volumes on the curve safety level. The friction factor model by Hofko provides an effective means to circumvent the need for extensive measurement efforts. As the primary parameter for evaluating the safety level of circular curves, pavement friction coefficients can be obtained through actual measurements or alternative models without affecting the reliability of the method used in this study to evaluate road safety on the basis of safety level.

$$f_{60}=-0.039\ln \left(PP\right)+0.7357$$

(9)

$$PP=5336.6·\frac{CTV·{AADT}_{HGV}}{{10}^6·{10}^4}-5099.5$$

(10)

where $f_{60}$ is the pavement friction coefficient at a speed of 60 km/h; $PP$ is the number of polishing times; $CTV$ is the cumulative traffic volumes; ${AADT}_{HGV}$ is the average annual daily truck traffic.

The American Society for Testing and Materials developed a general model correlating the pavement friction coefficient supply with driving speed through regression analysis on statistical data [23]. Considering the high efficiency of passenger car tire–pavement friction coefficient utilization, Olson et al. recommended utilization values for different vehicle types [24]. The supply of the friction factor for various driving speeds of a passenger can be expressed as follows:

$$f_V=1.2f_{60}{e}^{-0.00642(V-60)}$$

(11)

where $V$ is the vehicle speed (km/h); $f_V$ is the pavement friction coefficient at a speed of $V$ km/h.

On the basis of Equations (7) and (9)–(11), coupled with specific traffic parameters for the road section, the supply of the friction factor can be obtained for roads over different operational years and at varying speeds for passenger cars.

2.3.2. Variation in Superelevation under Repeated Vehicle Loads

Variation in superelevation is the consequence of lateral uneven settlement in roads, which is affected by various factors, including soil consolidation and vehicular loads. Numerous studies have analyzed road settlements and revealed the impact of consolidation settlement caused by subgrade filling and cumulative deformation due to traffic loading on road settlement. The finite element method, supported by the mechanical theory, offers advantages such as high prediction accuracy without the need for prolonged observational data, which is widely used in the research of subgrade consolidation settlement prediction [25,26,27]. The empirical method possesses a rapid and intuitive advantage, thus making it more suitable for predicting settlement under the dynamic influence of traffic loads [28]. Among those presented by existing studies, the empirical equation proposed by Xu et al. [29] effectively considers the combined effects of traffic loads and soil consolidation on highway settlements. However, given that settlement resulting from traffic loads diminishes with increasing depth, the empirical equation accounting for soil consolidation follows the same pattern. A solitary empirical equation proves insufficient for predicting settlements in embankment roads. Throughout the studies on road settlement prediction, limited attention has been given to the impact of vehicle loads on the variation in pavement cross-slopes from the traffic safety perspective. Our previous study developed a combined empirical equation and numerical method approach to predict the cross-slope variation in embankment highways. It incorporates a finite element model to account for soil consolidation and calibrates dynamic stress parameters in the empirical formula for settlement due to traffic loads [30]. The settlement prediction model is as follows:

$$S=S_1+S_2=\stackrel{n}{\sum _{i=1}}\left[a_i{\left(\frac{q_{d_i}^{{\chi }_i}{q}_{s_i}^{1-{\chi }_i}}{S_{\sigma i}\left(q_{{ult}_i}-c_i{q}_{d_i}^{{\chi }_i}{q}_{s_i}^{1-{\chi }_i}\right)}\right)}^{m_i}{\left(\frac{p_i}{p_a}\right)}^{n_i}{e}^{-{\left(\frac{k_i}{N}\right)}^{b_i}}\right]h_i+S_2$$

(12)

where $S_1$ is the settlement induced by traffic load (m); $i$ is the soil layer; $q_d$ is the traffic-load-induced dynamic deviatoric stress (kPa); $q_s$ is the initial static deviatoric stress (kPa); $q_{ult}$ is the undrained shear strength of soft subsoil; $S_{\sigma }$ is the stress sensitivity; $\chi$ is the weighting of the effect of dynamic and static deviatoric stresses (kPa); $p$ is the confining pressure (kPa); $p_a$ is the standard atmospheric pressure (kPa); $N$ is the number of repeated traffic load applications; $h$ is the soil layer thickness (m); $a$, $b$, $c$, $m$, $n$, and $k$ are constants, which were acquired from reference [30]; $S_2$ is the settlement due to consolidation (m), which was obtained by a finite element model.

The previous research results on the variation in cross-slope indicate that as the service life increases, the cross-slope for the outer lanes increases, while that of the inner lanes decreases. Following the previous approach, this study further analyzes the influence of vehicle loads on the dynamic changes in superelevation within curved segments. The method framework is depicted in Figure 2.

2.3.3. Actual Demand of Lateral Friction Factor for Vehicles

The actual demand of the lateral friction factor for vehicles is based on various factors, such as vehicle model, speed, curve radius, and superelevation. Variations among different vehicle models become apparent in the considerations and complexities of models. The vehicle model must first be determined to ascertain the demand of lateral friction factor for vehicles. The point mass model employed in road geometric design theory considers that the demand of the lateral friction factor is related to the curve radius, operating speed, and road superelevation. However, the point mass model does not account for force distribution across vehicle axles. When the average friction demand per axle is less than the available tire–road friction supply, it may lead to a situation where the friction demand on one axle significantly exceeds the supply while another axle experiences a surplus of friction supply. The steady-state bicycle model considers the influence of curve radius, vehicle speed, vehicle type, gradient, superelevation, and acceleration to determine the demand of the lateral friction factor for each vehicle axle. It provides an accurate representation of the actual demand of the lateral friction factor during vehicle operation. Such demand for the front and rear axles of a vehicle can be expressed with the steady-state bicycle model as follows [31]:

$$f_{yf}^{BM}=\frac{b\frac{v^2}{R}-g{i}_{h}b}{gb-\left(a_x+gi\right)h}$$

(13)

$$f_{yr}^{BM}=\frac{a\frac{v^2}{R}-g{i}_{h}a}{ga+\left(a_x+gi\right)h}$$

(14)

where $f_{yf}^{BM}$ and $f_{yr}^{BM}$ are the friction demand in tangential direction on front and rear axles, respectively; $a_x$ is the acceleration rate (m/s²); $i$ is the grade (%); $a$, $b$, and $h$ are the vehicle length parameters (m), which were adopted as 1.05 m, 1.61 m, and 0.65 m, respectively.

Equations (13) and (14) delineate the demand of the lateral friction factor for vehicles navigating curved road sections. They account for variables such as curve radius, vehicle speed, vehicle type, longitudinal slope, superelevation, and acceleration. Upon specification of these parameters, the computation yields the requirements for lateral friction coefficients for any given scenario. To validate the applicability of the steady-state bicycle model, the case of a 6% designed superelevation was considered. The minimum radius and maximum longitudinal slope corresponding to different design speeds outlined in Chinese standards were selected. As such, this configuration considered the most unfavorable conditions. The acceleration rate of −0.85 m/s² was selected because it is the widely used in the literature for slight deceleration in curves [32]. The demand of the lateral friction factor for different vehicle models was obtained and is presented in

Table 2. Demand of lateral friction factor for different vehicle models.

Design Speed (km/h)	Curve Radius (m)	Superelevation (%)	Acceleration Rate (m/s2)	Gradient (%)	Point Mass Model	Steady-State Bicycle Model
						Front Axle	Rear Axle
60	135	6	−0.85	−5	0.150	0.158	0.164
80	270	6	−0.85	−5	0.127	0.134	0.138
100	440	6	−0.85	−5	0.119	0.125	0.130
120	710	6	0.85	−5	0.100	0.106	0.109

, where the point mass model calculation results represent the design specifications outlined in the standard.

The point mass model underestimates the demand of lateral friction factor for vehicles because of the lack of consideration of the effects of longitudinal slope and braking. Additionally, research has demonstrated that when decelerating on a downhill curved road, weight transfer causes the demand of the lateral friction factor for the vehicle’s front axle to be lower than that of the rear axle [32]. From the perspective of traffic safety, the rear axle of the vehicle is selected as the analysis object in this study.

3. Results

3.1. Supply of Friction Factor and Superelevation Variation Prediction

A typical section of the Rongwu Expressway in Shandong Province, China, was selected as a case study to evaluate the impact of friction factor and superelevation variation on curve safety level. This section is a two-lane, soft-soil-based embankment with a large proportion of heavy trucks, which can more fully reflect the needs of the pavement friction factor and superelevation variation evaluation. The observed traffic volume from the monitoring station on the section in 2020 was considered as the base year for traffic volume. The forecast results of the traffic volume, with an 8% of design traffic growth rate, are presented in

Table 3. Cumulative traffic volume and average annual daily truck traffic of Rongwu Expressway.

Traffic Volume Type	Road Operation Time (Years)
	1	2	3	4	5	6	7	8	9	10
$CTV$ (107 veh)	5.5	11.6	17.9	24.8	32.2	40.4	49.1	58.6	68.8	79.8
${AADT}_{HGV}$ (veh)	2617	2826	3052	3297	3560	3845	4153	4485	4844	5231

. Assuming that the pavement does not undergo maintenance treatment, the supply of the lateral friction factor for vehicles at different speeds during the operation was obtained on the basis of Equations (7) and (9)–(11), with the results shown in Figure 3. Taking the operation to the end of first year as an example, it can be obtained the $f_{60}$ is 0.43 based on Equations (9) and (10) with the traffic volume in Table 3 as the parameters. Subsequently, the $f_V$ at different speed V km/h can be calculated according to Equation (11), which serve as the $f_x$ in Equation (7). Finally, the $f_s$ at different speed can be obtained from Equation (7). At the end of first year, the supply of lateral friction factors for vehicle speeds of 60 km/h, 80 km/h, 100 km/h and 120 km/h were 0.33, 0.29, 0.26 and 0.23, respectively.

Figure 3 shows the variation in the supply of the lateral friction factor on the road surface as the operational lifespan increases for different operating speeds. The trend exhibited in the figure indicates a consistent reduction in the lateral friction factor with the increment in operational years, irrespective of the operating speed; this trend aligns closely with the findings of Lamm [33]. In terms of the variability in the lateral friction factor at a specific operating speed, in the absence of road maintenance, the decline in the supply of the lateral friction factor is more rapid during the initial years of road operation, followed by a slower decline. Regarding the variability in the lateral friction factor at a particular operating speed, without road maintenance, the supply of the lateral friction factor decreases faster during the initial years of road operation, followed by a slower decline. For instance, when considering an operating speed of 80 km/h, after a decade of road operation, the maximum supply value of the lateral friction coefficient decreases by over 40%, which is consistent with the result of Kane [19]. Hence, the progressive reduction in the supply of friction factor on the road surface caused by repetitive vehicular loading poses a significant impact on the safety of circular curves, thus demanding careful consideration.

Vehicle load is one of the main factors leading to variations in superelevation, with the extent of variations being associated with the magnitude and positioning of vehicle load [30]. Trucks are assumed to be occupying the outer lane, while passenger cars have unrestricted movement to account for the influence of varying vehicle load positions on superelevation variations. Employing the superelevation prediction framework as depicted in Figure 2, year-to-year changes in road superelevation were calculated through Equation (12). Parameters were calibrated by applying the finite element model. The results are presented in

Table 4. Parameters related to long-term settlement prediction.

Carriageway	Design Superelevation	Road Operation Time (Years)
		0	1	2	3	4	5	6	7	8	9	10
Inside lane (%)	6.00	6.51	6.53	6.54	6.55	6.55	6.56	6.56	6.56	6.56	6.56	6.51
Outside lane (%)	6.00	5.85	5.86	5.87	5.87	5.88	5.88	5.88	5.88	5.88	5.88	5.85

.

Vehicle loads will result in an increase in the inside lane superelevation and a decrease in that of the outside lane after the road is put into operation. However, the increase in the inside lane superelevation will not exceed 10%, and the decrease in the outside lane superelevation does not exceed 3%. The variation in superelevation mainly occurs in the early stages of road operation, with the superelevation basically stabilizing after four years of operation. This outcome implies a minimal impact on driving safety because of the superelevation variation caused by vehicle load. However, in cases where the maximum design superelevation is implemented in a curve, the superelevation of the inside lane may exceed standard regulations within one year of road operation. Particular attention is required for the safety of slow-moving vehicles, particularly in regions susceptible to freezing and snow accumulation.

3.2. Impact of Friction Factor and Superelevation on Curve Safety Level

Lateral friction and superelevation are the primary parameters used to determine curve design specifications, with their variations directly affecting the curve safety level. Considering a curve radius of 270 m, a superelevation of 6%, a longitudinal slope of 5%, a vehicle speed of 80 km/h, and an acceleration of −0.85 m/s² as an example, the impact of the road surface friction and superelevation variation on the curve safety level can be determined on the basis of the steady-state bicycle mode. The results are depicted in Figure 4.

A significant difference can be observed in the impact of pavement friction variation and superelevation variation on the curve safety level. In the absence of road maintenance, variation in the friction factor on the road surface has an adverse impact on the curve safety level, thus resulting in a continuous decline in the curve safety level. The variation in road surface friction leads to a reduction of over 14% in the curve safety level at the end of first year in operation, while the reduction approaches 40% after five years operation. The impact of superelevation variation on the curve safety level is related to the carriageway lanes, which is beneficial to the inner lane but detrimental to the outer lane. However, the superelevation variation effect on the curve safety level remains limited, not surpassing 5% for either lane. In summary, the variation in pavement friction has a much more significant impact on the curve safety level than the variation in superelevation, which provides effective evidence for the focus on the friction coefficient in road operation and management.

The above discussions separately address the impact of variations in pavement friction or superelevation on the curve safety level. In practical situations, these variations often occur simultaneously. In such cases, the combined effect on the safety level of circular curves is noteworthy. Employing the same parameters as before for illustrative purposes, the curve safety level of an outside lane under the combined impact of pavement friction and superelevation variation is calculated and depicted in Figure 5.

Figure 5 shows that the curve safety level decreases year by year under the combined effect of pavement friction and superelevation variation. During the initial two years of road deployment, the curve safety level experiences a swift decline, after which the rate of decrease decelerates. Compared with the commencement of usage, the safety level shows a 27% reduction by the second year, which further decreases to 38% after five years. Consequently, the curve safety level falls below the designed value. However, this outcome does not imply an inability of the road to ensure the safety of vehicles. Until the end of the 10th year of use, the curve safety level remains above 1.25, which can meet the traffic safety under the dynamic influence of the repeated load of vehicles during the operation period. This indicates that the specifications have relatively high safety standards, which are beneficial for ensuring road safety during operation. The assessment of safety based on the safety level proposed in this study is in line with the concept of substantial safety.

In addition, for the combined effect of pavement friction and superelevation variation on the curve safety level, a decrease of 27% is slightly greater than the effect of pavement friction variation, which is a decrease of 26.2%. Meanwhile, the effect is significantly higher than the effect of superelevation variation on the safety level of circular curves, which shows a decrease of only 1.1%.

3.3. Impact of Vehicle Speed on Curve Safety Level

The curve safety level depends on the supply and demand of the pavement friction factor. Previous analysis has focused on the impact of pavement friction and superelevation variation on the supply of pavement friction factor. The demand of the friction factor also affects the curve safety level, which is related to the travel speed of vehicles. In accordance with vehicle dynamics theory, the centrifugal force acting on the vehicle increases as the vehicle operating speed accelerates, thus leading to a greater demand for the lateral pavement friction factor. The range of vehicle speed, which changes under different design speeds, should be defined to assess the impact of the vehicle operating speed on the curve safety level. Lamm’s safety criterion Ι stipulate the limiting values between the operating speed and the design speed to evaluate the design consistency level: $\left|V_{85}-V_d\right| \le 10 \mathrm{k}\mathrm{m}/\mathrm{h}$ for good design; $10 \mathrm{k}\mathrm{m}/\mathrm{h}<\left|V_{85}-V_d\right| \le 20 \mathrm{k}\mathrm{m}/\mathrm{h}$ for fair design; $\left|V_{85}-V_d\right|>20 \mathrm{k}\mathrm{m}/\mathrm{h}$ for poor design [9]. The difference between the operating speed and the design speed should be within 20 km/h, as poor design roads are avoided in actual engineering. Moreover, speeds exceeding the design speed adversely impact the curve safety level. The operating speed is selected in three intervals: $\mathrm{V}=V_d$, $\mathrm{V}=V_d+10 \mathrm{k}\mathrm{m}/\mathrm{h}$, and $\mathrm{V}=V_d+20 \mathrm{k}\mathrm{m}/\mathrm{h}$, which can reflect the speed selection interval of the driver. Focusing on the rear axle of the vehicle for analysis, with the road parameters from Table 2, the curve safety level for different design speeds is obtained following Equation (8). The results are illustrated in Figure 6. The horizontal axis in the graph represents different design speeds, while the vertical axis indicates the safety level of circular curves. The horizontal dashed line signifies a safety level of 1, while the horizontal solid line represents the designed safety levels for different design speeds, as derived from Table 1. Different colored data points denote the curve safety levels at varying operating speeds, arranged from top to bottom, which represent the safety levels from the 1st to 10th year of road deployment.

The curve safety level responds differently to changes in operating speed across various design speeds. With higher design speeds, the designed safety level increases, thus indicating a greater capacity of the road to accommodate changes in vehicle operating speeds. The impact of design speed on the curve safety level is relatively minor when the operating speed does not exceed the design speed, where the curve safety level remains above 1.15 until the end of the 10th year of operation. When the operating speed surpasses the design speed, with an increase in the operational duration, the curve safety level has a noticeable decrease. Moreover, as the operating speed increases, the curve safety level decreases earlier to below 1. Taking the design speed of 120 km/h as an example, when the operating speed exceeds the design speed by 10 km/h, the curve safety level decreases to below 1 by the eighth year of operation. When the operating speed exceeds 20 km/h, the curve safety level decreases to below 1 by the second year of operation.

4. Discussion

In the current study, a curve safety level model was established to capture the essence of the dynamic relationship between the supply and demand of the lateral friction factor. The limited response of the curve safety level to friction factor and superelevation variation under repeated traffic loads was investigated.

The curve safety level model considers the impact of repeated vehicle loading, road structure, road geometric parameters, vehicle models, and driving speed when determining the supply and demand of friction coefficients. Although the curve safety level model proposed in this study is inspired by Lamm’s driving dynamic criterion [9], fundamental differences emerge. Firstly, the supply of lateral friction in Lamm’s criterion is only related to the design speed. Meanwhile, this study considers the impact of repeated vehicle load actions during operation on the friction coefficient and superelevation, thus making it more suitable for dynamic safety analysis throughout the life cycle of the road. Secondly, the demand of lateral friction in Lamm’s criterion is based on $V_{85}$, which is a constant value for specific road sections. Meanwhile, this research considers various factors that affect the vehicle operating speed, such as vehicle characteristics and driver behavior. Thirdly, based on Lamm, the driving dynamic criterion believes that a decrease in the friction factor on road surfaces to a certain threshold will change the safety of the system. However, according to road geometric design theory, the stability of vehicle operation is related to speed, radius, and superelevation, the performance of which inevitably undergo variation during road operation. Only when the performance of the components in the system decreases to a certain threshold will traffic accidents occur. The curve safety level model proposed in this study aligns more closely with the concept of intrinsic safety, comprehensively reflecting the combined influence of human factors, vehicles, road design, and the environment on road safety.

Compared with superelevation variation, variation in pavement friction is the main factor that causes a decrease in the supply of lateral friction factor, resulting in a decrease in the curve safety level. This outcome shows a significant downward trend in the early stages and then a gradual slowdown. The aggregates within the asphalt mixture are affected by the effect of traffic polishing, yet the influence becomes inconspicuous after a certain number of traffic loads [34]. Meanwhile, the impact of operating speed on the safety level is reflected in the demand of the lateral friction factor. Higher operating speeds leads to an increase in the demand of the lateral friction, which is consistent with the results of literature [2,32]. The analysis results in this study obtain a comprehensive response of the curve safety level to the supply and demand of the pavement friction factor.

According to Figure 5, the curve safety level remains above 1.25 when the vehicle negotiating a curve at design speed, which the road can ensure the ability to endure various interactions. The designed curve safety level is 1.54, indicating that the friction factor considered in design are relatively conservative, since the friction factor influenced by various factors (driving behavior, vehicle load, environmental and so on) during the operation should be fully considered in the design state. It provides evidence for adjusting the limit values of parameters in the design of road curve sections. By comparing Figure 5 and Figure 6, the curve safety level is sensitive to changes in operating speed and pavement friction coefficient. To ensure traffic safety, the curve safety level can be enhanced by controlling the operating speed. However, the impact of pavement friction and superelevation variations persists as a long-term effect, that is, accumulating with the age of the road. Its impact on curve safety levels gradually accumulates, thus posing challenges for drivers to recognize these changes. The research findings offer compelling evidence for determining effective strategies concerning speed control and maintenance schedules for curved road sections.

Given that the curve safety level during operation needs to be analyzed in combination with specific road sections and traffic volume, a certain case in China was selected in this study, thus potentially limiting the universal applicability of the results. Nevertheless, the curve safety evaluation during operation can be investigated in subsequent studies on the basis of the curve safety level model of the current study. While the direct impact of superelevation variation on the curve safety level of remains limited, such variations can potentially result in inadequate lateral drainage on curve sections, thus diminishing the pavement friction and potentially adversely affecting the curve safety level. Further cases are required to investigate the impact of superelevation variation on curve safety levels within inadequately drained sections. Furthermore, the actual supply of the lateral friction of pavements can be obtained through measurements or models. This study draws on existing research models for prediction to prevent redundant research. The supply of friction coefficients model adopted in this paper is applicable to the analysis of friction coefficients on SMA asphalt pavement under dry conditions. Parameters of the model may differ from the environmental conditions, thus necessitating empirical verification of specific parameters.

5. Conclusions

On the basis of the dynamic relationship between the supply and demand of the lateral friction factor in the curved road section, the impact of pavement friction and superelevation variations caused by repeated vehicle loading on the curve safety level was given focus, thus unveiling the dynamic patterns of the curve safety level over the operational lifespan. The main conclusions of this research are as follows:

(1)

Under repeated vehicle loads, the supply of the pavement friction factor, which serves as the primary factor influencing the curve safety level, decreases rapidly in the early stages of road deployment, followed by a deceleration in a declining trend. Meanwhile, the superelevation increases on the inside lane, while that of the outside lane decreases. This variation and its impact on the curve safety level remain significantly limited.

(2)

The influence of pavement friction and superelevation variation on the curve safety level is reflected in the supply of the lateral friction factor. The curve safety level rapidly decreases within the first two years after road deployment, followed by a deceleration in a declining trend. Without any maintenance, the curve safety level will decrease by over 30% after four years of usage.

(3)

The impact of vehicle speed on the curve safety level is manifested in the demand for the lateral friction factor. As the vehicle speed increases, the demand for the lateral friction coefficient rises, thus reducing the curve safety level. The road’s overall capability to withstand over-speeding vehicles is compromised because of the pavement friction and superelevation variation caused by repeated vehicle loading. When drivers exceed the speed limit by 10 km/h, the curve safety level drops below 1 by the eighth year of operation. Exceeding the limit by 20 km/h results in the safety level dropping below 1 by the second year since deployment.

(4)

Strictly controlling operating speeds is one effective measure to enhancing the curve safety levels. However, the impact of pavement friction and superelevation variation on safety levels accumulates gradually. Under this scenario, drivers have difficulty perceiving these changes. As such, extensive attention should be given to pavement friction to ensure safe driving.