# Feasibility of a Quasi-Static Approach in Assessing Side-Wind Hazards for Running Vehicles
## 4 Feasibility of the Quasi-Static Approach Under Turbulent Winds
### Comparison of Tire Forces Under Various Mean Wind Speeds  
Variations in tire forces were further compared under various mean wind speeds. The data from 1 min fluctuations in horizontal wind speeds were artificially generated for various mean wind speeds by utilizing a wave-superposition approach to the Von-Karman spectrum [14]. Values for a turbulence intensity and length scale of 13.1% and 100 m, respectively, were assumed to approximate the approaching wind conditions for a sea-crossing bridge deck at the level of 50 m [15]. The mean wind speed was increased by an increment of 1 m/s. By applying the same random number for different wind speeds, we generated a similar pattern of time histories for turbulent wind speeds, as shown in Figure 15.  
Figure 14: Windward vertical reaction force of the enhanced quasi-static trailer model for (**a**) whole period and (**b**) first 10 s.  
Figure 15: Generated wind speed data for the mean wind speeds of 11 m/s-16 m/s. Minimum vertical reaction forces and maximum friction forces were obtained during 1 min patterns of turbulent winds for each mean wind speed. Moreover, based on the variations in tire forces, the critical wind speed was determined for both side-slip and overturning accidents. The limits for the states of the two accident types are defined as follows [2]:  
* Side-slip: when one of the axles reaches the friction limit
* Overturning: when one of the vertical reaction forces of the tires becomes zero  
The tire friction forces for the van were calculated, and the critical wind speed was determined regarding side-slip accidents, whereas the trailer was examined for an overturning accident based on the vertical reaction forces. The stability of each vehicle was evaluated under conditions whereby the vehicle was running on the same curved road described in Section 4.1. The coefficient of friction, \(\mu\), between the tire and the road surface was assumed to be 0.8 for a dry road condition [16]. The vehicle speed was set at 40 km/h, and the oncoming wind had an incident angle of 90 degrees to the vehicle direction as it entered the curved road (\(R~{}=~{}700\) m, \(\alpha~{}=~{}2\%\)). The variations in the tire friction forces and critical wind speeds of the van were evaluated for the possibility of a side-slip accident. The front axle was examined for the possibility of a side-slip accident because the front axle reaches the friction limit state earlier than the rear axle does. The definition of the normalized friction force, \(\overline{F_{f}}\), was introduced in Equation (29) as the friction force divided by the vertical tire forces. According to [17], the friction force increased in a linear fashion until the occurrence of tire slip, at which point it was saturated and road grip was lost when the normalized friction force reached friction coefficient, \(\mu\). Therefore, the friction limit state can be defined by Equation (29). \[\overline{F_{f}}~{}=~{}\frac{\sqrt{F_{yf}^{2}+\left(F_{x1}+F_{x3}\right)^{2}} }{F_{z1}+F_{z3}}>\mu \tag{29}\]  
Figure 16 shows the normalized friction force of the van for the turbulent wind at different mean wind speeds. The normalized friction force was determined as the maximum value from among the 1 min applications of turbulent winds. The normalized friction force obtained from the quasi-static approach showed good agreement with that from the dynamic approach for all wind speeds. The critical wind speed was determined using Equation (29) for the occurrence of a side-slip accident for the van, which was determined to be approximately 21.0 m/s using both approaches. In order to estimate the critical wind speed for trailers regarding an overturning accident, the vertical reaction forces of the windward tire were obtained for various mean wind speeds. Figure 17 compares the vertical reaction forces estimated using the quasi-static, the enhanced quasi-static, and the dynamic approaches.