# Feasibility of a Quasi-Static Approach in Assessing Side-Wind Hazards for Running Vehicles
## 3 Feasibility of the Quasi-Static Approach Under a Uniform Wind
### Comparison of Tire Forces Under Various Wind Speeds  
Vertical reaction and lateral friction forces were evaluated for two vehicle types under a driving speed of 50 km/h on a flat straight road (\(R~{}=~{}\infty\), \(\alpha~{}=~{}0\%\)). Since an initial vibration occurred in the dynamic approach, stabilized results were obtained by providing a sufficient simulation time of 30 s.  
Uniform wind speeds were gradually increased at 10 m/s until one of the vertical tire forces reached zero. Figures 6 and 7 show the variations in the tire forces for the van and trailer, respectively. The quasi-static approach provided tire forces that compared to those of the dynamic approach. The vertical reaction forces showed a certain difference between the quasi-static and dynamic approaches, while the lateral friction forces were quite similar in both approaches. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Type** & \(a\)** (m)** & \(b\)** (m)** & \(c\)** (m)** & \(h_{c}\)** (m)** & \(h_{T}\)** (m)** & \(A\)** (m\({}^{2}\))** & \(m_{s}\) (ton)** & \(m_{f}\)**(ton)** & \(m_{r}\)**(ton)** \\ \hline Van & 1.35 & 1.75 & 1.63 & 0.71 & - & 6.8 & 2.10 & 0.15 & 0.53 \\ Trailer & 5.22 & 4.78 & 1.86 & 1.94 & 1.10 & 7.2 & 6.00 & - & 0.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions and parameters of vehicle models. Figure 5: Aerodynamic coefficients according to wind direction for: (**a**) van and (**b**) tractor-trailer. Figure 6: Tire forces of windward front axle of the van in uniform wind for: (**a**) vertical reaction force, (**b**) lateral friction force. The differences in vertical reaction forces for the two vehicle types were further investigated. One interesting point was a tendency for differences according to the wind speeds in the two vehicle types. In the case of the van, the gap between the two approaches remained constant to the wind speed whereas the gap for the trailer increased with increases in the wind speed. This differing tendency implies that the main cause of the gap was from different sources for the two vehicle types. In the case of the van, this gap was induced by the difference in suspension modeling between the two approaches. The dynamic approach with TruckSim analysis reflects the type of suspension (independent or solid), the differences in stiffness values for the front and rear axles, and the changeable location of the suspension for the solid type. Since the present study utilized the default van model provided by TruckSim, the stiffness and location of the suspensions were different in the front and rear axles. However, the quasi-static approach adopts an assumption of equal suspension stiffness and location for both the front and rear axles, which resulted in compatibility, as shown in Equation (8). Accordingly, both approaches showed a difference in the vertical reaction forces for the van, as shown in Figure 6a. In order to support this inference, we modified the van model for the dynamic approach so that the same stiffness values and location of suspension could be applied to both axles. Figure 8 shows similar vertical reaction forces in both approaches, which confirms that the difference in vertical reaction forces was induced from the suspension modeling between the two approaches, particularly for the van. On the other hand, the gap for the trailer, as shown in Figure 7a, was due mainly to the additional rolling moment induced by the lateral displacement of the trailer body in the dynamic approach. According to the TruckSim analysis, the roll angle at the roll center and the lateral displacement of  
Figure 8: Windward vertical reaction force of the van with equal stiffness of suspension in front and rear axles. Figure 7: Tire forces of the trailer in uniform wind: (a) vertical reaction force of windward wheel, (b) lateral friction force. the trailer body were evaluated at approximately 3 degrees and 0.05 m, respectively, when the wind speed reached 19 m/s.