«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»
overturning accident, rotational (yawing) accident, and side slipping accident (Baker 1991b). All the accident criteria are decided based on some practical considerations. For lateral side slipping and rotational accident, the criteria can be set to avoid the vehicle sliding into other lanes, which causes potential accidents, such as impacting other vehicles, curbs, soft ground or tripping the vehicle into rollover (Gillespie, 1992). Theoretically, a overturning or rollover accident is to happen only when the C. G. of the vehicle is raised up to the rollover point and an experienced driver may be able to stabilize the vehicle when one wheel loses contact with the road surface (Gillespie, 1992). However, for ordinary drivers, setting more conservative criteria still makes sense.
Baker et al. once gave some guidelines for accident identifications (1986, 1991), which will be adopted here as follows: within some distance of the vehicle entering a sharp edged gust, the overturning accident is said to happen when one of the tire reaction forces Nj fell to zero, or side slipping accident is said to happen when the lateral response Y of the vehicle exceeds 0.5 m, or 112 the rotation accident is said to happen if the yawing displacement ψ v exceeds 0.2 radius. Such sharp-edged wind field exists when the vehicle just passes the bridge tower which used to block the wind action on the vehicle, or there is a strong gust acting on the vehicles and the bridge suddenly.
In the real world, the strong wind usually may not reign the driving for a very long time.
Besides, the strong wind itself will naturally remind the driver to concentrate more on driving.
So it is maybe acceptable for ordinary drivers to continually adjusting steering angle for a not too long period. On the other hand, the low wind speed is relatively common in ordinary days and it is not very realistic for drivers to adjust the steering angle in a very high frequency for a long time. Since there is no sufficient available information, it is suggested to set a steering angle adjusting frequency limitation of 1 Hz when wind speed is lower than 15 m/s and 2 Hz when wind speed is 15 m/s above. This additional criterion eliminates some results which are actually impractical in the real world.
5.4 Numerical Example 5.4.1 Prototype Bridge and Vehicles The Yichang Suspension Bridge located in the south of China has a main span of 960 m and two side spans of 245 m each. The height of the bridge deck above the water is 50 m. The sketch of the bridge is shown in Fig. 4.5 and its main parameters are shown in Table 2.1 (Lin and Chen 1998). Based on a preliminary analysis with modal coupling assessment technique (Chen et al., 2004) and the observation of wind tunnel results, the four important modes considered in the present study are shown in Table 4.1 to consider the three-direction response of the bridge (Lin and Chen, 1998). In this example, the attack angle of the wind to the vehicle β=90o, i.e., the wind is acting perpendicular to the driving direction.
The vehicle model shown in Fig. 4.6 has seven degree-of-freedom, three for the rigid body of the vehicle (vertical displacement Zvr, pitching about the y axis θvr and rolling about the x axis φvr) and the other four for the vertical displacements of the four wheels (ZaL1, ZaR1, ZaL2, ZaR2).
Wind force coefficients used by Coleman and Baker (1990) are adopted here:
For 0 ≤ χ π / 2
where a1-a6 equal to 5.2, 0.93, 0.5, 2.0, -2.0 and 7.3 for such vehicle , respectively.
In addition to the variables of the vehicle shown in Table 4.2, some additional parameters
5.4.2 Vehicle Interaction Analysis with the Bridge As discussed earlier, with the bridge-vehicle interaction model, the time history of vehicle responses can be predicted for accident analysis. Interaction analysis starts from the first vehicle enters the bridge until the first vehicle leaves the bridge. Figs. 5.2-5.3 show the accelerations of vehicles in the vertical, rolling and pitching directions of the first vehicle with the same driving speed of 22 m/s versus the time under wind speed U=30m/s and U= 5m/s, respectively. The small sketch of the bridge indicates the corresponding location of the first vehicle on the bridge at any time. As will be seen later, vehicle accelerations in these directions will be used in the accident assessment. In Fig. 5.2, relatively larger accelerations can be observed when the vehicle moves to the middle range of the main span with 30 m/s wind speed. Such tendency, however, is not noticeable when the wind speed is as low as 5 m/s (Fig. 5.3). It is maybe because the acceleration contribution from the bridge to the vehicle is relatively small when wind speed is low. Comparing Figs. 5.2 and 5.3 suggests that the vertical and pitching accelerations decrease less significantly than the rolling acceleration does when the wind speed changes from 30 m/s to 5 m/s. It is possibly because that the accelerations of vehicles are contributed more significantly by high frequency vibration than by low frequency vibration. So the road roughness, as the high frequency excitation, may contribute more significantly to the total accelerations than the wind as a wide-band excitation does. Since the varying road roughness is only assumed on the vehicle moving directions (roughness is assumed the same in the bridge width direction), road roughness contributed much less to the rolling accelerations than in other two directions.
Figs. 5.4-5.5 give the relative responses of the vehicle in the vertical, rolling and pitching directions. Larger relative vertical response can be observed when wind speed U is 30 m/s compared with that when U=5 m/s with the same driving speed. Slightly larger relative responses can be observed when vehicles move in the middle-span region of the bridge compared with vehicles moving elsewhere on the bridge when wind speed U is 30 m/s (Fig. 5.4). Relative responses, however, have no obvious increment in the middle range of the main span when wind speed U is 5 m/s (Fig. 5.5). Such phenomenon shows again that the interaction effect exists between vehicles and the bridge, especially when wind speed is high. To avoid losing interaction information between vehicles and the bridge in the accident assessment process, vehicle dynamic results from interaction analysis seem to be very important.
20 10 0
-30 0 10 20 30 40 50 60 10 0
0.4 0.2 0.0
-0.4 0 10 20 30 40 50 60 10 0
0.0024 0.0022 0.0020 0.0018
Figure 5.5 Vehicle relative displacements when wind speed U=5 m/s and V=22 m/s 5.
4.3 Accident-Related Response of the Vehicle While the vehicle-bridge interaction analysis gives vehicle responses in several directions including vertical along axis z, rolling around axis x and pitching around axis y, responses in other directions, such as lateral (along axis y) and yawing (around axis z), are to be identified 118 separately. These response are called “accident-related response” since they are critical to the assessment of accidents.
The accident assessment is only given to the first vehicle starting from the moment when the vehicle just enters the main span. It is to simulate the situation that the vehicle just passes by the bridge tower, which may temporarily block the wind loading on the vehicles. Once the truck enters the main span (namely passes one bridge tower), the truck will experience a sudden wind action. In Eq. (5.54), only four unknowns are to be solved: the truck body lateral displacement Y, the truck body yawing angle Ψ and their respective velocities. The initial conditions for the differential equations are assumed to be zero. In any time step, the new steering angle will be decided and vector C in Eq. (5.54) should be updated.
The second-order Rouge-Kutta approach is chosen to solve the differential equations with the time step of 0.01 second. Since the bridge is symmetric, 500 meter driving distance, a little bit longer than one-half of the main span, was considered. The proposed model is first validated with the comparison with the results by Baker’s (1986) without considering driver behaviors of vehicles on the road. To enable a direct comparison, bridge and steering angle related terms in the formulations are omitted and the side slipping angle model in Eqs. (5.19-20) are changed correspondingly. It is found that the same results can be obtained as Ref. (Baker, 1986) with the same vehicle data.
Fig. 5.6 shows the lateral, yawing response and corresponding steering angle under 15 m/s (33 mph) wind speed and vehicle driving speed when no driver behavior is considered, namely with zero driver steering angle. It is found that both lateral and yawing responses keep increasing with the driving distance and quickly exceed the accident criteria. It suggests that a vehicle will quickly be off-lane and “sway the tail” on bridges and roads if no any driver interference is applied on the driving vehicle under moderate wind. While some difference for the lateral side slipping can be observed when the vehicle runs on the bridge or on the road, little difference can be observed for the vehicle yawing response for this moderate wind speed.
Fig. 5.7 shows the accident-related responses when the driver’s steering angle is applied on the driving vehicle. It is found that the lateral side slipping response has been well suppressed around the zero side slipping location (as shown on the top subplot of Fig. 5.7). The yawing response as shown in the middle subplot of Fig. 5.7 is also limited to some value much lower than the accident criteria. The corresponding steering angle is shown in the bottom subplot of Fig. 5.7. For comparison purpose, the results of vehicles on the road with the same wind speed and driving speed are also plotted in the figure. Similar to the case without driver behavior (Fig.
5.6), it is found that lateral side slipping response of vehicles on the bridge is higher than that of vehicles on the road and yawing response only show slight difference between vehicles moving on the bridge and on the road. Correspondingly, more significant change of steering angle is required when vehicles run on the bridge than on the road. The steering angle should be constantly changed during the process of driving (adjusting frequency is about 0.5 Hz from the figure). The coefficients λ1 = 0.1 and λ2 = 0.2 used in this example represent just only one type of driving behavior described in Eq. (5.59).
-0.1 0 100 200 300 400 500
1.2 1.1 1.0 0.9 0 100 200 300 400 500
Figure 5.9 Leeward wheel reactions of the truck when U=V=15 m/s (λ1=0.
2, λ2=0.1) The reaction forces of each wheel are critical to identify the risk of overturning accidents.
Fig. 5.8 and Fig. 5.9 show reaction force ratios for the windward and leeward wheels, which are normalized by the static reaction force of the corresponding wheel when the vehicle remains still on the ground. Comparison of results in Figs. 12 and 13 suggests that the reaction force ratios for the windward wheels are smaller than those of the leeward wheels. In addition, the mean value of the reaction force ratio for the windward rear wheel seems to be the smallest among the four wheels (about 0.7), but is still much higher than 0. Similar to the cases of displacements (Figs.
5.6 and 5.7), the reaction force ratios of vehicles on the bridge also have larger variations than on the road. As shown in Figs. 5.7-9, it can be concluded that it is safe for vehicles to run on both 123 the bridge and the road with the given steering angle shown at the bottom of Fig. 5.7 and with the wind speed and driving speed all equal to 15 m/s (33 mph). The absolute value and changing speed of the steering angle are quite low (adjusting frequency is about 0.5 Hz), which means an ordinary driver can achieve the similar maneuver as shown in Figs. 5.7-9.
Vehicles may have different performance under higher wind speed. Fig. 5.10 shows the accident-related responses and the steering angle versus the travel distance when wind speed is 35 m/s (78 mph) and driving speed is 15 m/s (34 mph). As shown on the top of Fig. 5.10, the lateral side slipping displacement is much higher comparing with that under the wind speed of 15 m/s and maintain around some value other than zero. The middle subplot of Fig. 5.10 shows the yawing response of vehicles and it has exceeded the accident criterion. The bottom figure of the steering angle shows very dense curves with large value, which suggests faster and larger change of steering angle is required when wind speed is higher.
The reaction force ratios are shown in Fig. 5.11 for the four wheels. It is found that the bolded curve (windward rear wheel) has turned to negative in some cases, which suggests the possibilities of overturning accidents. It can also be found that the windward wheels all lose some reaction forces compared with still vehicle situation and the windward rear wheel is most likely to lose contact with the road surface. The higher wind speed requires more frequent change of steering angle (adjusting frequency is about 2- 4 Hz) to control the vehicle, which also justify that vehicles are much more difficult to control in strong wind and an ordinary driver may be very prone to have an accident due to more possible mistakes in steering the wheels.
Vehicle performances under low wind speed and high driving speed are also investigated.