Crosswind stability and critical safe speed of medium-duty box trucks on expressways under strong-wind conditions

1 Introduction

Road traffic safety remains a fundamental objective in highway planning, operation, and intelligent transportation management. Recent studies have increasingly emphasized that crash risk should not be understood only from aggregated crash counts or general roadway indicators, but should be refined into mechanism-level safety problems that can support targeted interventions. For example, Wu et al. (2026) developed a multimodal framework that moved from crash reports to mechanism-oriented road safety diagnostics by explicitly linking crash narratives, diagrams, and street-view information to underlying contributing factors. Similarly, Zhang et al. (2025) showed that safety-oriented engineering decisions can be improved when crash risk is considered together with operational objectives in a multi-objective route-searching framework. These studies indicate an important methodological trend in road safety research: the analysis should progress from broad safety concerns to specific, operationally controllable risk mechanisms. In high-wind regions, one such mechanism is the crosswind-induced instability of high-sided freight vehicles operating on expressways.

With the rapid development of integrated transportation systems, expressways have increasingly extended into regions with complex natural environments, where the impact of strong winds on vehicle safety has become progressively more pronounced (Li et al., 2025). In open terrains and typical high-wind regions, crosswinds can generate significant aerodynamic forces, which may induce lateral displacement, yaw instability, and even rollover of high-sided vehicles (Shen et al., 2024), thereby constituting a critical factor affecting traffic safety. Previous studies have demonstrated that meteorological conditions significantly influence both the occurrence and severity of accidents involving heavy vehicles, among which crosswinds play a particularly prominent role (Uddin and Huynh, 2020). Under strong crosswind conditions, heavy trucks are more prone to instability, with rollover exhibiting a higher probability compared with other instability modes (Hu et al., 2021). Moreover, accident statistics indicate that although heavy trucks account for a relatively small proportion of total crashes, they are disproportionately represented in fatal accidents (Schindler et al., 2020). Due to their large lateral windward area and relatively high centre of gravity, medium-duty box trucks are especially sensitive to crosswind disturbances. Therefore, the general road safety problem can be further refined into a specific engineering question: under strong crosswinds, what combinations of wind direction, wind intensity, loading condition, and vehicle speed cause medium-duty box trucks on expressways to approach rollover-critical states, and how can these conditions be translated into practical speed-control thresholds?

Extensive research has been conducted worldwide on vehicle stability under crosswind conditions. Baker (1993) classified crosswind-induced accidents into sideslip, rollover, and excessive yaw, providing a foundational framework for subsequent studies. Building on this framework, numerous studies have investigated the coupled effects of aerodynamics and vehicle dynamics using multibody dynamic models, numerical simulations, and wind tunnel experiments, systematically examining the influence of wind speed, vehicle speed, and structural parameters on vehicle stability (Du and Wang, 2022; Gu and He, 2014). Gu and He (2014) employed numerical simulations combined with an Adams/Car model to analyze the effects of crosswinds on the driving stability of high-speed container trucks and proposed safe driving speeds under Beaufort level 6 to 7 wind conditions. Wang and Hu (2013) utilized TruckSim to simulate crosswind-induced rollover of box trucks, analyzing the influence of wind speed, vehicle speed, and center-of-gravity height on lateral stability, thereby providing intuitive guidance for safe operation under crosswind conditions. Dorigatti et al. (2012) conducted similar investigations under turbulent wind conditions and expanded aerodynamic coefficient databases for high-sided road vehicles on long-span bridges. Stoyanoff et al. (2016) proposed an extended rolling model to evaluate vehicle stability under strong gust conditions for different vehicle types.

For medium- and heavy-duty trucks, existing studies have employed tools such as Adams/Car and TruckSim, along with wind tunnel experiments, to quantify the effects of crosswind speed, vehicle speed, and center-of-gravity height on lateral and roll responses. In some cases, safe driving speeds have been recommended for specific scenarios. Wind tunnel experiments and wind engineering studies conducted by Cheli et al. (2011), Xu and Guo (2003), Guo and Xu (2006), and Baker (1988) further indicate that high-sided vehicles are more susceptible to instability under turbulent or gusty wind conditions, and that loading conditions may significantly alter the critical wind speed for rollover (Baker and Soper, 2019; Brandt et al., 2022). Meanwhile, recent studies incorporating realistic gust profiles and high-fidelity vehicle models in one-way coupled simulations have revealed that the most adverse crosswind yaw angle may not necessarily correspond to the conventional 90 ° pure crosswind condition; instead, higher rollover risk may occur near approximately 120 ° (Alrejjal and Ksaibati, 2023). These studies provide an important basis for understanding crosswind-induced vehicle instability. However, most of them focus on bridge environments, articulated or heavy vehicles, container trucks, turbulent-wind databases, or isolated critical scenarios. Their findings cannot be directly converted into a loading-sensitive speed-control framework for medium-duty box trucks operating on straight expressway sections in high-wind regions.

Despite these advances, several limitations remain in the existing literature. First, many crosswind safety studies are conducted in bridge or special road environments, whereas long, straight expressway segments in open high-wind regions represent another typical exposure scenario that has received less focused attention. Second, existing analyses often emphasize a single loading state or vehicle configuration, leaving insufficient understanding of how unladen, half-laden, and fully laden box trucks differ in mass distribution, tyre load transfer, and rollover resistance. Third, although wind yaw angle directly affects the decomposition of aerodynamic side force and roll moment, the most adverse wind direction for medium-duty box trucks across a wide yaw-angle range remains insufficiently clarified. Fourth, previous studies often stop at qualitative stability interpretation or scenario-specific safe speeds, while engineering practice requires a multi-factor safety boundary that links wind level, wind direction, loading condition, vehicle speed, and rollover-risk indicators. Therefore, the novelty of the present study is grounded in these unresolved gaps rather than in the use of TruckSim alone.

Xinjiang, as one of the most representative high-wind regions in China, is characterized by high wind speeds, long durations, and complex spatial distributions. Taking the “Hundred-mile Wind Zone” along the Hami-Tuyugou section of the G30 Expressway as an example, extreme wind speeds can reach 30–37 m/s, with frequent occurrences of strong crosswinds, posing severe challenges to highway operational safety. In such environments, box trucks traveling at high speeds on long, straight road sections are more susceptible to rapid load transfer and increased rollover risk under crosswind effects. This practical context further sharpens the research problem addressed in this study: the stability of medium-duty box trucks on expressways in high-wind regions cannot be reliably managed by general wind warnings or uniform speed limits alone; instead, it requires a vehicle-type-specific and loading-state-sensitive evaluation of crosswind rollover thresholds.

Accordingly, this study focuses on a two-axle medium-duty box truck and develops a vehicle dynamics model under crosswind conditions using TruckSim. The study advances previous work in four main aspects. First, it refines the broad issue of road safety in high-wind regions into a specific mechanism-level problem: crosswind-induced load transfer and rollover risk of medium-duty box trucks on straight expressways. Second, it systematically evaluates the coupled effects of wind intensity, wind yaw angle, vehicle speed, and loading condition, rather than examining only isolated factors or a single loading state. Third, it combines the minimum tyre vertical load and the maximum Load Transfer Ratio to identify both incipient wheel lift-off and continuous rollover-risk evolution, with separate consideration of front-axle, rear-axle, and whole-vehicle load transfer. Fourth, it translates simulation results into critical safe speeds and differentiated engineering control thresholds for unladen, half-laden, and fully laden vehicles. Through these contributions, the study extends existing crosswind-stability research from scenario-specific dynamic response analysis toward practical, loading-sensitive speed management for medium-duty box trucks operating on expressways in high-wind regions.

2 Methodology

2.1 Definition of wind direction and wind intensity

Wind conditions were classified based on levels 6–12 of the Beaufort scale, as specified in the Chinese standard GB/T 28591-2012 (State Administration for Market Regulation and Standardization Administration of the People's Republic of China, 2012), which correspond to wind speeds capable of significantly affecting vehicle stability. In this study, the term wind yaw angle (θ) is used consistently to describe the relative orientation between the incoming wind and the vehicle. It is defined as the angle measured from the vehicle longitudinal axis to the incoming wind direction in the top-view plane; 0 ° corresponds to a headwind, 90 ° to a pure lateral crosswind, and 180 ° corresponds to a tailwind. The investigated wind yaw angles range from 30 ° to 150 °. The corresponding Beaufort wind-speed ranges are reported in both m/s and equivalent km/h in Table 1 to avoid confusion with vehicle speed units.

TABLE 1. Beaufort wind scale: wind speed ranges and corresponding descriptions.

To avoid terminology ambiguity, the expression wind yaw angle is used throughout the manuscript. Five characteristic wind yaw angles - 30 °, 60 °, 90 °, 120 °, and 150 ° - are selected to cover oblique headwind, near-crosswind, pure crosswind, and rear-quartering wind conditions, as illustrated in Figure 1. This angular range not only includes the near-90° crosswind condition commonly regarded as critical in engineering practice, but also accounts for recent findings suggesting that the most adverse wind yaw angle may deviate from 90 ° (Alrejjal and Ksaibati, 2023).

2.2 Input–response mechanism of crosswind-induced instability

Previous studies have shown that the instability process of vehicles under crosswind conditions can be generally described as an “input–response–failure” chain, consisting of wind load input, vehicle dynamic response, and stability loss (Baker, 1986; Chen and Cai, 2004; Han W. et al., 2022).

At the input stage, strong crosswinds act on the lateral surface of the vehicle with a given wind speed and yaw angle, generating aerodynamic side forces and rolling moments. During the response stage, the vehicle reacts dynamically to these external disturbances through mechanisms such as tire lateral slip and suspension roll motion. The response capability is governed by vehicle geometric parameters (e.g., center-of-gravity height), operating conditions (e.g., vehicle speed and loading condition), and road characteristics. When the lateral aerodynamic force exceeds the tire–road friction limit, the vehicle experiences sideslip. When the rolling moment induced by crosswind exceeds the restoring moment provided by vehicle weight, the windward-side tires progressively unload and may eventually lose contact with the ground, indicating that the vehicle has reached a rollover-critical state. Under crosswind conditions, the instability thresholds vary significantly across vehicle types. Passenger cars, characterized by a lower center of gravity and smaller lateral windward area, typically exhibit lateral deviation or loss of directional control prior to rollover. In contrast, high-sided vehicles such as box trucks, due to their large lateral exposed area and elevated center of gravity, are more prone to rollover, which generally constitutes the dominant mode of instability (Guo M. H. et al., 2025; Huang et al., 2018).

2.3 Rollover risk evaluation indicators and critical state identification

To accurately identify rollover-critical states and quantitatively assess rollover risk, the minimum tyre vertical load (F_{z, min}) and the load transfer ratio (LTR) are adopted as evaluation indicators in this study. The minimum tyre vertical load is used to determine the onset of rollover, while the LTR is employed to characterize the continuous evolution of rollover risk.

2.3.1 Minimum tyre vertical load (F_{z, min})

The vertical tire loads were extracted from TruckSim. The normal driving condition and rollover state are illustrated in Figures 2 and 3, respectively. The studied vehicle, a two-axle box truck, is equipped with a single-tyre front axle and a dual-tyre rear axle. The vertical loads are denoted as F z , L 1 t (front-left) and F z , R 1 t (front-right). The aggregated vertical loads for the rear-left (F z , L 2 t) and rear-right (F z , R 2 t) dual-wheel assemblies are calculated as shown in Equations 1 and 2: where: F z , L 2 i t, F z , L 2 o t are the vertical loads on the inner and outer tires of the rear-left wheel assembly, respectively. F z , R 2 i t, F z , R 2 o t are the vertical loads on the inner and outer tires of the rear-right wheel assembly, respectively.

F z , L 2 t = F z , L 2 i t + F z , L 2 o t (1)

F z , R 2 t = F z , R 2 i t + F z , R 2 o t (2)

TABLE 2. Classification of rollover risk zones based on LTR_max and F_z,min.

LTR_max, denotes the maximum absolute value among the front-axle, rear-axle, and whole-vehicle LTR, values.

Based on this, the minimum vertical load of the dual-wheel assemblies on either side is defined in Equation 3:

F z , min t = min F z , L 1 t , F z , R 1 t , F z , L 2 t , F z , R 2 t (3)

This metric represents the minimum load-bearing capacity on either side of the vehicle when subjected to lateral aerodynamic and inertial forces. A reduction of the vertical load on any tire or wheel assembly to zero indicates that the corresponding side is completely unloaded. At this point, the vehicle is supported by only one side, signifying the attainment of the critical rollover threshold. Consequently, the condition F_{z, min}(t)≤0 is established as the physical criterion for identifying both tire lift-off and the incipient rollover state.

2.3.2 Load transfer ratio (LTR)

To continuously characterize the process of wheel load transfer and the evolution of rollover risk under crosswind conditions, this study adopts the Load Transfer Ratio (LTR) as the primary evaluation metric. LTR is defined as the ratio of the difference in vertical loads between the vehicle’s left and right sides to its total vertical load, thereby quantifying the asymmetry of the vertical load distribution.

It is pertinent to note that the TruckSim software outputs the vertical load for each individual tire. Given that the vehicle’s rear axle is equipped with a dual-tire configuration and that rollover is fundamentally a macroscopic instability phenomenon governed by the imbalance of overall support forces between the two sides, a methodological choice was made. Specifically, the vertical loads of the inner and outer tires on the same side are aggregated into a total wheel-assembly load for each side. The LTR is then calculated based on these aggregated loads. This approach more accurately reflects the vehicle’s overall load-bearing state under lateral forces and mitigates potential inaccuracies arising from localized fluctuations in single-tire loads.

Furthermore, considering that the front and rear axles of the subject dual-axle van truck exhibit distinct responses to crosswind, the LTR is subsequently defined separately for the front axle, the rear axle, and the vehicle as a whole, as shown in Equations 4 –6. where: F z , L 1 t and F z , R 1 t are the vertical loads on the left and right wheels of the front axle, respectively. F z , L 2 t and F z , R 2 t are the vertical loads on the left and right wheel assemblies of the rear axle, respectively.

L T R f t = F z , L 1 t ‐ F z , R 1 t F z , L 1 t + F z , R 1 t (4)

L T R r t = F z , L 2 t ‐ F z , R 2 t F z , L 2 t + F z , R 2 t (5)

L T R v t = F z , L 1 t + F z , L 2 t ‐ F z , R 1 t + F z , R 2 t F z , L 1 t + F z , L 2 t + F z , R 1 t + F z , R 2 t (6)

The front-axle LTR (LTR f t) is utilized to characterize the load variation on the steering axle, while the rear-axle LTR (LTR r t) characterizes that of the drive axle. The whole-vehicle LTR (LTR v t) reflects the overall trend of the vehicle’s load transfer.

Established research indicates that during the evolution of a rollover event, a local axle often reaches its stability limit prior to the vehicle’s overall response. For heavy vehicles, such as articulated trucks, wheel lift-off typically initiates at the rear axles and subsequently propagates to the front axles (Hung et al., 2022; Hung and Khanh, 2024). Therefore, to effectively prevent the underestimation of risks associated with this localized, incipient instability, the maximum of the individual axle LTRs is adopted as the comprehensive evaluation metric, as formulated in Equation 7.

L T R max t = max L T R f t , L T R r t , L T R v t (7)

2.4 Engineering speed limit determination based on rollover risk zoning

2.4.1 Basis for the selection of risk thresholds

To facilitate standardized assessment across different operating conditions, this study adopts the Load Transfer Ratio (LTR) as a normalized indicator of lateral load transfer. Physically, |LTR| = 0 represents a symmetric left-right vertical load distribution, whereas |LTR| approaching 1 indicates that the vertical load on one side tends toward zero and the vehicle approaches wheel lift-off or rollover. Therefore, LTR is suitable for both continuous risk evaluation and engineering threshold definition.

The thresholds used in this study follow commonly adopted engineering interpretations in rollover and heavy-vehicle stability studies. The range |LTR| < 0.6 is regarded as a conservative safe zone because a sufficient tyre-load reserve remains on both sides of the vehicle. The range 0.6 ≤ |LTR| < 0.8 is defined as a warning zone, indicating that lateral load transfer has become significant and that the stability margin is being rapidly consumed. The range 0.8 ≤ |LTR| < 1 is defined as a high-risk zone because one-side tyre unloading is already severe and the vehicle is close to the physical rollover boundary. Finally, F_z,min ≤ 0 or |LTR| approaching 1 is used to identify the critical rollover state. These thresholds are consistent with previous LTR-based rollover risk classifications and provide an engineering basis for graded warning and control (Wang et al., 2022; Chen et al., 2023; Guo Q. et al., 2025; Chen and Chen, 2010).

It should be emphasized that the thresholds of 0.6 and 0.8 are not intended to represent additional physical rollover boundaries. Instead, they are used as conservative engineering thresholds for graded risk warning before complete wheel lift-off occurs. The value of LTRmax = 0.6 was selected as the boundary between the safe and warning zones because, beyond this level, the left-right vertical load imbalance becomes significant and the remaining stability margin begins to decrease rapidly. The value of LTRmax = 0.8 was selected as the boundary of the high-risk zone because the vertical load on the windward-side wheels is already substantially reduced and the vehicle is close to the incipient rollover condition. Therefore, these two thresholds provide a practical basis for distinguishing normal operation, warning, and high-risk states, and for deriving conservative critical safe speeds under different wind and loading conditions.

2.4.2 Definition of critical rollover speed

For any given operational scenario, defined by a specific combination of loading condition (L), wind intensity (G), and wind yaw angle (θ), the critical safe speed V_cr is defined as the highest vehicle speed that satisfies both the physical rollover criterion and the conservative engineering safety criterion within the simulation window, as shown in Equation 8.

V c r L , G , θ = max ⁡ { V L T R max V , L , G , θ < 0.6 (8)

Specifically, the physical criterion requires that the minimum vertical tyre force remains positive throughout the evaluation period (F_z,min > 0), indicating that wheel lift-off does not occur. In addition, for engineering speed-limit determination, LTR_max < 0.6 is adopted as the conservative safety criterion. This choice introduces a safety reserve before the vehicle enters the warning zone and prevents the recommended operating speed from being located too close to the rollover boundary. When LTR_max ≥ 0.6 at a given speed, that speed is treated as unsuitable for normal operation under the corresponding wind and loading condition, even if physical wheel lift-off has not yet occurred.

2.4.3 Principle for determining engineering speed limits

To ensure operational safety in practice, the recommended engineering speed limit V_limit is determined conservatively and must not exceed V_cr. In other words, the selected speed limit is the highest discrete tested speed at which both F_z,min > 0 and LTR_max < 0.6 are satisfied. This treatment distinguishes the physical rollover threshold from the operational safety threshold and provides a margin suitable for real-world traffic control.

An important exception is defined for extreme-risk conditions. If LTR_max is already 0.6 or greater at the minimum tested vehicle speed of 40 km/h, the vehicle has entered at least the warning zone even under the lowest simulated speed. In this case, no acceptable normal operating speed is recommended for that scenario, and vehicle passage should be restricted or prohibited depending on the wind level and loading condition.

3 Development of the vehicle model based on TruckSim simulation

3.1 Vehicle model and simulation setup

In this study, a dynamics model of a dual-axle, medium-sized box truck was constructed using TruckSim. The fundamental vehicle parameters were derived from publicly available model data (as shown in Table 3). To reflect the variations in dynamic characteristics under different operational loading states, three distinct loading conditions were established: unladen, half-laden, and fully laden.

TABLE 3. Main vehicle parameters.

As the primary objective of this study is to investigate the rollover response of the vehicle under crosswind conditions, the simulations were conducted under an open-loop, straight-line driving scenario. The steering input was set to zero, and the vehicle was operated at a constant speed, thereby excluding any active driver intervention. This configuration eliminates the influence of driver control inputs and allows for a focused analysis of the effects of crosswind loading, vehicle speed, and loading condition on vehicle stability.

In the TruckSim model, the crosswind condition was defined directly in the corresponding task window by specifying the wind speed and wind direction for each simulation scenario. After these wind parameters were prescribed, TruckSim generated the aerodynamic forces and moments acting on the vehicle body through its built-in aerodynamic wind-load framework. Therefore, the wind load was not introduced as a manually defined external force or moment time history. No additional CFD simulation or wind tunnel test was conducted to obtain vehicle-specific aerodynamic coefficients; instead, the aerodynamic treatment followed the TruckSim wind-load setting framework and simplified engineering assumptions.

3.2 Calculation of load and center of gravity position

The rollover risk of a vehicle under the influence of crosswinds is primarily governed by the equilibrium between the aerodynamic forces and the vehicle’s anti-overturning capability. The position of the vehicle’s center of mass is a critical parameter in determining this anti-rollover capacity. Aerodynamic side forces, along with yaw and roll moments, act directly upon the vehicle body, which alters the vehicle’s state of forces, modifies its state of motion, and ultimately affects its stability (Wang et al., 2025; Zhang et al., 2021; National Highway Traffic Safety Administration (NHTSA), 2015).

Therefore, this paper employs the method of static moments to calculate the position of the vehicle’s overall center of mass for various loading conditions. The total mass and the center of mass of the vehicle are determined by the superposition of the unladen vehicle’s mass and the cargo’s mass. The formula for calculating the resultant center of gravity is presented in Equations 9–11. where: m_v is the mass of the unladen vehicle; m_c is the mass of the cargo; x_v and z_v are the longitudinal position and height of the center of gravity of the unladen vehicle, respectively; x_c and z_c are the longitudinal position and height of the center of gravity of the cargo, respectively.

x = m v x v + m c x c m v + m c (9)

y = 0 (10)

z = m v z v + m c z c m v + m c (11)

To simplify the analysis, it is assumed that the cargo is uniformly distributed within the cargo box and is stacked evenly in the vertical direction. Based on the vehicle’s structural dimensions and the arrangement of the cargo box (see Figure 4), the position of the cargo’s center of mass and the combined center of mass for the entire vehicle were calculated for different loading states (see Tables 4–6).

TABLE 4. Calculation parameters and cargo CG Coordinate (unit: mm).

TABLE 5. Summary of unladen/half-laden/fully laden masses.

TABLE 6. Vehicle mass and CG location under different payload conditions.

Furthermore, to facilitate practical engineering applications, the loading status is classified into three categories: a cargo mass of less than 1/4 of the rated load capacity is defined as the unladen state; a mass between 1/4 and 3/4 is defined as the half-laden state; and a mass greater than 3/4 is defined as the fully laden state. The corresponding total vehicle mass and center of mass parameters for each of these conditions are presented in Table 6.

3.3 Road model

To isolate the independent influence of aerodynamic crosswind effects and eliminate the interference of road geometry on lateral stability, this study constructed an idealized road model in TruckSim. The model consists of a 1-km-long, straight, and level expressway section. The roadway was assumed to be flat in the transverse direction, with zero superelevation, zero horizontal curvature, zero longitudinal grade, and no surface irregularities. The pavement surface was defined as dry asphalt with an adhesion coefficient of μ = 0.8–0.9 (Guo M. H. et al., 2025), representing typical dry pavement conditions. These assumptions were adopted to establish a controlled baseline in which the effects of wind speed, wind yaw angle, vehicle speed, and loading condition on rollover stability could be compared consistently.

The key assumptions and parameter settings of the road model are summarized in Table 7. In the present model, superelevation was not considered. Superelevation, or banking on curves, may influence crosswind stability because it introduces a lateral component of gravitational force and changes the distribution of tyre vertical loads between the two sides of the vehicle. Depending on the curve direction, banking angle, vehicle trajectory, and wind direction, superelevation may either partly counteract or amplify crosswind-induced lateral load transfer. In particular, when a crosswind-induced rolling moment acts in the same destabilizing direction as the load-transfer tendency associated with road banking and curvature, wheel unloading and rollover risk may increase. Therefore, the results of this study should be interpreted as baseline estimates for flat and straight expressway segments, while the coupled effects of superelevation, curvature, and crosswind should be examined in future work.

TABLE 7. Key assumptions and parameter settings of the idealized road model.

Building upon this road model, the vehicle operational state was set to open-loop, constant-speed, straight-line driving. The steering input was set to zero, and no acceleration or deceleration manoeuvre was imposed. This configuration eliminates the effects of driver’s closed-loop corrections and transient speed-control behaviours, thereby providing a comparable simulation environment for revealing the coupling dynamics between crosswind loading and vehicle rollover response.

3.4 Design of simulation scenarios

To systematically investigate the influence of various factors on the crosswind stability of a box truck, this study establishes a multi-factor simulation matrix. The experimental design includes four variables: vehicle speed, loading condition, Beaufort wind level, and wind yaw angle. Vehicle speed is expressed in km/h, while Beaufort wind speed is reported in m/s and, where used in fitted equations or operational tables, also converted to equivalent wind speed in km/h using W_eq = 3.6 W. The factor levels are listed in Table 8.

TABLE 8. Factor-level table.

4 Simulation data output and analysis

4.1 Influence of wind yaw angle and identification of the most adverse angle

To identify the most adverse wind yaw angle for a box truck under crosswind conditions, simulations were first conducted for the unladen vehicle under different Beaufort wind levels, vehicle speeds, and wind yaw angles. The LTR was used as the primary evaluation metric to compare vehicle stability across the tested angular conditions.

Figure 5 focuses on the unladen vehicle under Beaufort level 11 wind. It compares the time-history/peak LTR response at wind yaw angles of 30 °, 60 °, 90 °, 120 °, and 150 ° for four vehicle speeds of 40, 60, 80, and 100 km/h. Under the same wind level and vehicle speed, the LTR varies nonlinearly with wind yaw angle.

Figure 6 further compares LTR_max under Beaufort levels 9–12, different vehicle speeds, and the same five wind yaw angles for the unladen condition. The purpose of this figure is to show whether the ranking of adverse wind yaw angles remains consistent as wind intensity and vehicle speed increase. The results indicate that vehicle speed and wind intensity jointly amplify LTR_max, while 120° remains the most unfavorable wind yaw angle in the investigated range.

As the wind yaw angle increases from 30 ° to 90 °, LTR gradually increases but remains lower than that at 120 °. At 120 °, LTR reaches its maximum, indicating the most severe wheel-load transfer and the highest rollover tendency. When the wind yaw angle further increases to 150 °, LTR decreases but remains higher than the values observed at 30 ° and 60 °.

This phenomenon demonstrates that the most adverse wind yaw angle does not strictly correspond to the conventional 90 ° pure crosswind. Instead, the most unfavorable condition occurs under rear-quartering inflow around 120 °. Under this condition, the lateral aerodynamic component and the longitudinal wind component jointly increase the effective roll tendency and wheel-load transfer, leading to a higher rollover risk.

More specifically, the variation of rollover risk with wind yaw angle can be explained by the combined change in the side-force component, longitudinal wind component, and effective rolling moment. When the wind yaw angle increases from 30 ° to 90 °, the lateral component of the incoming wind increases, resulting in a larger aerodynamic side force and stronger lateral load transfer. However, the maximum rollover risk does not occur at the pure 90 ° crosswind condition. At the 120 ° rear-quartering wind condition, the vehicle is still subjected to a large side-force component, while the longitudinal wind component modifies the relative airflow around the box body and contributes to a larger effective rolling moment about the leeward-side wheels. Because the medium-duty box truck has a large side-projected area and a relatively high centre of gravity, this effective rolling moment leads to more severe wheel unloading on the windward side and a higher LTR response. When the wind yaw angle further increases to 150 °, the side-force component and effective rolling tendency decrease, and thus the rollover risk becomes lower than that at 120 °.

Table 9 summarizes the risk zones of the unladen vehicle under coupled wind yaw angle and vehicle speed conditions. The risk zones are classified according to the LTR_max and F_z,min criteria in Table 2. The results confirm that the 120° condition is the most critical angular condition and that increasing vehicle speed lowers the wind level required to reach the high-risk or critical rollover zones.

TABLE 9. Crosswind stability risk matrix of the unladen box truck under coupled wind yaw angle and vehicle speed conditions.

TABLE 10. Scenarios for the analysis of loading factors.

In summary, within the simulated range, 120 ° is identified as the most adverse wind yaw angle for the box truck. This result is consistent with previous studies indicating that the most unfavorable aerodynamic condition for high-sided vehicles may deviate from the nominal 90 ° crosswind and shift toward a rear-quartering direction.

4.2 Influence of vehicle speed

After identifying 120 ° as the most adverse wind yaw angle, the influence of vehicle speed was analyzed under the unladen condition. Figure 7 presents LTR variations for vehicle speeds from 40 to 100 km/h and Beaufort levels 6–12, with the wind yaw angle fixed at 120 °.

For a given Beaufort wind level, LTR increases with vehicle speed, indicating that vehicle speed is a key factor governing rollover risk. The increase becomes more pronounced at higher wind levels because the relative wind velocity and the resulting aerodynamic roll moment increase nonlinearly.

At Beaufort level 6, the unladen vehicle remains in the safe zone over the tested speed range. As wind level increases, the same vehicle speed corresponds to a higher risk zone. For example, when LTR_max exceeds 0.6, the vehicle enters the warning zone; when LTR_max exceeds 0.8, the vehicle enters the high-risk zone. These risk-zone definitions correspond to the thresholds listed in Table 2.

The results also show that a speed that is acceptable at lower wind levels may become unsafe under stronger wind. Therefore, the critical safe speed must be determined by considering wind intensity and vehicle speed simultaneously, rather than by applying a fixed speed threshold independent of wind conditions.

The fundamental mechanism underlying this non-linear growth is that the aerodynamic loads experienced by the vehicle—particularly the side force and rollover moment—are proportional to the square of the relative velocity between the vehicle and the air. Consequently, an increase in vehicle speed disproportionately and sharply amplifies the rollover moment, leading to a drastic transfer of wheel load to the leeward side.

In summary, Figure 7 quantifies the speed effect under the most adverse wind yaw angle and unladen condition. Increasing vehicle speed accelerates the deterioration of crosswind stability, particularly when the wind level is Beaufort level 8 or above.

To visualize the coupled effect of wind intensity and vehicle speed, Figure 8 presents a heatmap of LTR_max for the unladen vehicle at a fixed wind yaw angle of 120°, while Figure 9 converts the same operating conditions into risk zones based on the thresholds in Table 2.

Figure 8 shows that the high-LTR_max region expands from high vehicle speeds toward lower vehicle speeds as the Beaufort wind level increases. This indicates that strong wind can substantially reduce the speed range in which the vehicle remains within the safe zone.

Figure 9 provides the engineering interpretation of Figure 8 by mapping continuous LTR_max values into discrete control zones. Compared with the heatmap, the matrix more directly indicates the vehicle-speed ranges that require warning, speed restriction, or prohibition under each Beaufort wind level.

From a mechanistic standpoint, an increase in vehicle speed significantly raises the relative velocity between the vehicle and the airflow. This, in turn, causes the lateral aerodynamic force and the resulting roll moment to grow in a non-linear fashion. As the roll moment increases, the disparity in load distribution between the left and right wheels is exacerbated, intensifying the wheel load transfer effect and ultimately causing the LTR to rise rapidly, approaching the critical rollover state.

In summary, vehicle speed is one of the critical factors influencing the crosswind stability of a box truck. Under the same wind level conditions, the higher the vehicle speed, the more susceptible the vehicle is to entering a high-risk or even a critical rollover state. Concurrently, under high-wind-level conditions, relying solely on a reduction in vehicle speed is insufficient to completely eliminate the risk of rollover. This implies that vehicle speed control must be considered synergistically with the prevailing wind speed conditions.

4.3 Influence of loading status on crosswind rollover stability

The risk of vehicle rollover is not uniform under different loading conditions. To isolate the loading effect, simulations were conducted at the most adverse wind yaw angle of 120 ° for three loading conditions: unladen, half-laden, and fully laden. The specific scenarios for the loading factor analysis, including the loading status, wind level, and vehicle speed, are summarized in Table 10. Figure 10 compares LTR_max among the three loading conditions under Beaufort levels 9–12 and vehicle speeds of 40, 60, 80, and 100 km/h, while Figures 11, 12 present fitted response surfaces for the half-laden and fully laden cases.

Figure 10 shows that, under identical wind level and vehicle speed, LTR_max decreases as the loading condition changes from unladen to half-laden and fully laden. Thus, this figure is used primarily to compare the loading effect, rather than to repeat the wind-speed and vehicle-speed trends already shown in Figures 7–9.

In this paper, G denotes the Beaufort wind level, W denotes wind speed in m/s when referring to the Beaufort scale, and W_eq denotes the equivalent wind speed in km/h. The conversion is W_eq = 3.6 W. In the fitted response-surface equations, W_eq is used so that the wind-speed variable and vehicle speed V are both expressed in km/h. This notation is adopted to avoid confusion between wind speed and vehicle speed units. The fitted response–surface equation for LTR_max is given in Equation 12:where V represents the vehicle speed in km/h, W_eq denotes the equivalent crosswind speed in km/h converted from the Beaufort-scale wind speed in m/s, and LTR_max is the maximum absolute peak value among the front-axle, rear-axle, and whole-vehicle LTR values.

L T R max = 0.4284 − 0.001596 V − 0.007279 W e q + 3.901 × 10 − 5 V W e q + 2.672 × 10 − 5 V 2 + 5.915 × 10 − 5 W e q 2 (12)

The fitting results demonstrate that this model possesses a high degree of accuracy, with an R² value of approximately 0.997 and a Root Mean Square Error (RMSE) of approximately 0.0094. It is therefore capable of effectively describing the variation trends of LTR_max within the scope of this research.

It is important to note that this equation represents an empirical response surface fitted from discrete simulation data points. Its primary purpose is to describe the trends in LTR_max as a function of vehicle speed and wind speed and to provide estimations within the investigated range. It does not represent a rigorous analytical model of the vehicle’s crosswind rollover mechanics. Based on this fitted equation, the following figure was plotted.

Similarly, the LTR_max for the fully laden condition was fitted, and the resulting curve is given in Equation 13:

L T R max = 0.4815 − 0.001239 V − 0.008410 W e q + 1.866 × 10 − 5 V W e q + 3.357 × 10 − 5 V 2 + 6.466 × 10 − 5 W e q 2 (13)

The fitting results indicate that this model also possesses a high degree of accuracy (R² ≈ 0.9989, RMSE ≈ 0.0050). As can be observed from the figure for the fully laden condition, the critical wind speed and the corresponding vehicle speed are clearly discernible within the simulated scenarios.

An analysis of the response surface reveals that the maximum Lateral Load Transfer Ratio (LTR_max) exhibits a significant non-linear growth trend with increases in both vehicle speed and wind speed, and a distinct coupling effect exists between these two variables. In the region of low wind speed and low vehicle speed, LTR_max remains at a low level, corresponding to the safe zone. As either vehicle speed or wind speed increases, the contour lines progressively shift towards the lower vehicle speed region, indicating that the vehicle can enter the warning or high-risk state at lower speeds.

From the perspective of risk distribution, the LTR_max = 0.6 and LTR_max = 0.8 contour lines shift comprehensively towards the lower vehicle speed region. This demonstrates that under strong crosswind conditions, merely reducing vehicle speed is insufficient to completely avoid rollover risk. The distribution of these contour lines shows marked differences across the various loading conditions: the high-risk area is largest and extends furthest into the low-speed region under the unladen condition, followed by the half-laden condition, while the risk area is noticeably contracted under the fully laden condition.

The rollover risk assessment results for the different loading conditions are compiled and presented in Tables 11–13.

TABLE 11. Rollover risk matrix under different wind levels and vehicle speeds (unladen condition).

TABLE 12. Rollover risk matrix under different wind levels and vehicle speeds (half-laden condition).

TABLE 13. Rollover Risk Matrix under Different Wind Levels and Vehicle Speeds (Fully laden Condition).

The most adverse wind levels for each respective condition are compiled in the table above. A clear trend emerges: as the vehicle’s load is reduced, the maximum wind level it can safely withstand is critically diminished. Specifically, when the vehicle is in an unladen state and traveling at a speed of 80 km/h, it becomes susceptible to rollover risk upon encountering a Level 9 wind. Consequently, the unladen state is identified as the most unfavorable condition for crosswind stability.

To complete the analysis at the upper boundary of the wind scale, the Lateral Transfer Ratios (LTR) corresponding to the maximum wind speed of Level 12 were also determined. These findings are summarized as follows.

A quantitative analysis of the data presented in Tables 11–13 reveals the distinct differences in the critical wind level among the various loading conditions. The results show that the unladen condition exhibits the poorest stability. At a vehicle speed of 80 km/h, Beaufort Level 9 wind already places the vehicle in the warning zone, whereas Beaufort Level 10 and above lead to rollover. Under the half-laden condition, rollover does not occur until Beaufort Level 11 is reached. Under the fully laden condition, rollover occurs only at the upper limit of Beaufort Level 12 wind speed (approximately 133 km/h). These results indicate that the critical wind level that the vehicle can withstand increases significantly with increasing load.

From a mechanical perspective, the rollover resistance of the vehicle is primarily governed by the competition between the aerodynamic roll moment and the gravitational restoring moment, while also being affected by variations in the vertical tyre load distribution. The gravitational restoring moment can be expressed, as shown in Equation 14:where m denotes the vehicle mass, g the gravitational acceleration, and z c g the height of the center of gravity. As the load increases, the total vehicle mass rises, thereby enhancing the gravitational restoring moment and improving resistance to crosswind-induced roll motion. Under crosswind excitation, the lateral aerodynamic force and the associated roll moment cause a redistribution of the vertical tyre loads between the left and right sides, thereby intensifying wheel load transfer. The load transfer ratio (LTR) can be used as a normalized indicator of this moment imbalance and thus reflects the relative magnitude of the aerodynamic roll moment and the gravitational restoring moment. As both wind speed and vehicle speed increase, the stability differences among loading conditions become increasingly pronounced. Under Beaufort Level 9 wind, all three loading states retain a certain safety margin; however, when the wind level reaches 10 or above, the unladen condition is the first to enter the rollover region, followed by the half-laden condition, whereas the fully laden condition still maintains a relatively high stability margin.

M g r a v = m g z c g (14)

The fundamental reason for this trend is that, in the unladen state, the vehicle mass and the baseline vertical tyre loads are comparatively low. Consequently, under the same crosswind aerodynamic excitation, the vehicle is more susceptible to larger relative wheel load transfer, which causes earlier one-side wheel unloading and a more rapid approach to the rollover threshold. By contrast, as the load increases, both the gravitational restoring capacity and the overall load-carrying support of the vehicle are enhanced, leading to reduced wheel load transfer and improved vehicle stability.

4.4 Operational characteristics, critical safe speed, and engineering control of box trucks under crosswind conditions

4.4.1 Operational characteristics under different wind levels

Table 14 summarizes the qualitative operational response of box trucks subjected to Beaufort wind levels 6–12 under unladen, half-laden, and fully laden conditions. The wind speed is expressed by Beaufort level; for the two representative level-12 cases, the equivalent km/h values are also given together with their m/s values to maintain consistency with the fitted equations.

TABLE 14. Qualitative operational characteristics of box trucks under Beaufort wind levels 6–12 for different loading conditions.

“Warning” denotes a transitional state approaching rollover instability. Beaufort level 12 was examined at two representative equivalent wind speeds: W_eq = 120 km/h (33.3 m/s) and W_eq = 133 km/h (36.9 m/s), corresponding approximately to the lower and upper bounds of the level-12, range.

4.4.2 Critical safe speed and risk classification

To quantify the safety boundary, Table 15 reports the critical safe speed, V_cr, together with the associated risk level for each wind level and loading condition. Here, V_cr is defined as the highest operating speed that satisfies the prescribed crosswind safety criterion within the evaluation window, thereby providing a conservative control threshold for traffic management. As expected, V_cr declined markedly with increasing wind level and, for a given wind level, followed the order unladen < half-laden < fully laden. For the unladen vehicle, V_cr decreased from 100 km/h at Beaufort level 6 to 80, 60, 40, and 20 km/h at levels 7–10, and no acceptable operating speed was identified at levels 11–12. For the half-laden vehicle, V_cr remained at 100 km/h up to level 7, decreased to 80 km/h at levels 8–9, then fell to 40 and 20 km/h at levels 10 and 11, with no acceptable speed at level 12. The fully laden vehicle exhibited the highest tolerance, with V_cr maintained at 100 km/h through level 9, decreasing to 60, 40, and 20 km/h at levels 10–12, respectively. Overall, the risk classification shifted from controllable or moderately risky conditions at low wind levels to high or extremely high risk, and finally to non-recommended operation, as wind intensity increased.

TABLE 15. Critical safe speed and corresponding risk classification of box trucks under different wind levels and loading conditions.

“—“ indicates that no acceptable operating speed is recommended under the specified condition. Note. The value of 20 km/h was not directly simulated in the TruckSim matrix. It was adopted as a conservative low-speed control recommendation based on field experience from expressway reconstruction and expansion construction management in high-wind regions. Accordingly, it should be interpreted as an engineering traffic-control speed rather than a simulated critical rollover speed.

4.4.3 Safety control and engineering implications

For operational control, an additional safety margin was introduced in accordance with the conservative principle recommended by the International Maritime Organization (IMO) (Han Y. et al., 2022; Ferreira et al., 2024), such that the allowable operating wind level was taken as one Beaufort level lower than the critical design wind level, as shown in Equation 15.

B a l l o w = B c r i t − 1 (15)

On this basis, the speed limits listed in Table 16 were formulated as conservative engineering recommendations and were selected so as not to exceed the corresponding critical safe speeds in Table 15. To avoid unit ambiguity, Table 16 reports the Beaufort wind-speed range in m/s together with the corresponding equivalent range in km/h in the same wind-speed column. The recommended control strategy is strongly loading-dependent. Under Beaufort level 6, normal operation may be maintained with enhanced crosswind monitoring. At level 7, differentiated speed control should be initiated, particularly for unladen vehicles. At levels 8–9, progressively stricter speed limits are required, especially on bridges and other wind-exposed sections. At level 10, unladen vehicles should be prohibited, whereas half-laden and fully laden vehicles should be restricted to 40 and 50 km/h, respectively. At level 11, only fully laden vehicles should be permitted, and only under temporary traffic control at 40 km/h. At level 12, road closure should be adopted as the default measure for all box trucks; if emergency passage of fully laden vehicles is unavoidable, the operating speed should not exceed 40 km/h under strict real-time monitoring and sectional control.

TABLE 16. Conservative recommended speed limits and engineering control measures for box trucks under crosswind conditions.

The recommended speeds are conservative operational limits and do not exceed the corresponding critical safe speeds listed in Table 15. Equivalent wind speed in km/h is converted from the Beaufort-scale wind speed in m/s using W_eq = 3.6 W.

At Beaufort level 12, ≤40 km/h applies only to emergency passage of fully laden vehicles under strict real-time monitoring and sectional control; general road closure remains the preferred measure.

The results further indicate that the common assumption that an unladen vehicle is inherently more stable because of its lower center of gravity does not hold for high-sided vehicles under strong crosswind loading. Instead, the unladen condition represents the most critical operating scenario, which is consistent with previous studies on heavy-vehicle crosswind stability (Cioffi et al., 2023). Accordingly, traffic management in high-wind regions should adopt a loading-based priority framework, a wind-level-triggered control strategy, and stricter operational restrictions for unladen vehicles. These findings provide a quantitative basis for dynamic speed-limit setting, loading-sensitive warning systems, and differentiated traffic management in wind-prone expressway sections.

5 Discussion and limitations

This study quantified the crosswind stability of a two-axle medium-duty box truck under coupled wind yaw angle, wind intensity, vehicle speed, and loading conditions, and further established a risk-based framework for critical safe speed evaluation. The results show that rollover propensity is governed by the balance between the crosswind-induced aerodynamic rolling moment and the gravitational restoring moment, while the resulting load redistribution is effectively captured by LTR_max and F_{z, min}. A notable finding is that the most adverse wind direction did not occur at the conventional 90° pure crosswind, but at approximately 120° rear-oblique inflow. This suggests that, for high-sided box trucks, the combined action of lateral aerodynamic force and longitudinal wind component may produce a larger effective rolling tendency than a purely lateral inflow. In addition, the unladen condition was consistently identified as the most critical operating state, despite its lower centre of gravity compared with the laden cases. This indicates that the increase in vehicle mass and baseline vertical tyre loads under laden conditions provides a stronger restoring effect than the destabilizing influence associated with the rise in centre-of-gravity height. Therefore, for box trucks operating in high-wind regions, rollover risk should not be assessed solely on the basis of centre-of-gravity height, but rather from the combined effects of mass distribution, tyre load transfer, and aerodynamic loading.

From an engineering perspective, the proposed F_{z, min}-LTR_max framework provides a practical basis for differential speed control and traffic management under crosswind conditions. In particular, the results support the use of load-state-dependent control strategies, in which unladen vehicles are subject to stricter speed restrictions or operational prohibitions than half-laden and fully laden vehicles under the same wind environment. This is particularly relevant for expressways in high-wind regions, where dynamic warning systems and differentiated access control may be required for high-sided freight vehicles.

6 Conclusion

A TruckSim-based crosswind dynamic model of a two-axle medium-duty box truck was established to quantify the coupled effects of wind yaw angle, wind intensity, vehicle speed, and loading condition on rollover stability. Using the minimum tyre vertical load (F_{z, min}) and the maximum load transfer ratio (LTR_max) as evaluation criteria, the following conclusions are drawn.

• The most critical crosswind yaw angle is 120 °, rather than the conventional 90 ° pure crosswind.

Within the investigated range of 30 °–150 °, the hazardous yaw-angle interval was identified as 90 °–120 °, with 120 ° producing the most severe rollover tendency. For the unladen vehicle, the 120 ° condition led to warning, high-risk, and critical rollover states at 40, 60, and 80–100 km/h, respectively, whereas 30 ° and 60 ° remained safe over the full tested speed range. This indicates that rear-quartering winds can be more destabilizing than pure lateral winds for high-sided box trucks.

• 2. Rollover risk increases nonlinearly with wind intensity and vehicle speed, and the critical safe speed decreases rapidly as wind level rises.

At the most unfavorable yaw angle of 120 °, the directly simulated safe-speed boundary of the unladen truck decreased from 100 km/h at Beaufort scale 6 to 80, 60, and 40 km/h at Beaufort scales 7, 8, and 9, respectively. Under Beaufort scale 10, normal operation at 40 km/h approached an extremely high-risk condition; therefore, a supplementary conservative control speed of 20 km/h was recommended based on field traffic-management experience rather than direct simulation. For the half-laden truck, the directly simulated safe-speed boundary was 100 km/h at Beaufort scales 6–7, 80 km/h at Beaufort scales 8–9, and 40 km/h at Beaufort scale 10, while a supplementary 20 km/h control speed was recommended under Beaufort scale 11. For the fully laden truck, the directly simulated safe-speed boundary remained 100 km/h at Beaufort scales 6–9, decreased to 60 and 40 km/h at Beaufort scales 10 and 11, respectively, and a supplementary 20 km/h control speed was recommended under Beaufort scale 12. These results confirm a strong coupled amplification effect of wind speed and vehicle speed on rollover propensity.

• 3. Loading condition is a dominant factor in crosswind stability, and the rollover risk consistently follows the order unladen > half-laden > fully laden.

At 80 km/h, the unladen vehicle entered the high-risk zone at Beaufort scale 9 and reached the critical rollover state at Beaufort scale 10 and above; the half-laden vehicle did not roll over until the upper limit of Beaufort scale 12; and the fully laden vehicle rolled over only at the upper limit of Beaufort scale 12 combined with 100 km/h. Although loading increases the centre-of-gravity height, the associated increase in vehicle mass and baseline vertical tyre loads provides a larger restoring moment and reduces the relative load transfer, thereby improving rollover resistance.

• 4. The combined Fz, min-LTRmax criterion provides a practical basis for differential speed management of box trucks in high-wind regions.

Based on this criterion, all loading states can operate at 100 km/h under Beaufort scale 6; under Beaufort scale 7, the unladen truck should be limited to 80 km/h; under Beaufort scales 8-9, the recommended speed limits are 60/40 km/h for the unladen truck, 60/50 km/h for the half-laden truck, and 80/60 km/h for the fully laden truck; under Beaufort scale 10, unladen trucks should be prohibited, while half-laden and fully laden trucks should be restricted to 40 and 50 km/h, respectively; under Beaufort scale 11, only fully laden trucks are recommended to operate, at speeds not exceeding 40 km/h; and under Beaufort scale 12, road closure should be adopted in principle.

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Footnotes

Edited by: Yajie Zou, Tongji University, China

Reviewed by: Nengchao Lyu, Wuhan University of Technology, China

Amin Moeinaddini, Amirkabir University of Technology, Iran

Ting Fu, Tongji University, China