«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»
4.3.3 Time History Simulation of Road Surface Roughness The road surface roughness is usually assumed to be a zero-mean stationary Gaussian random process and can be expressed on the inverse Fourier transformation on a power spectral
density function (Huang and Wang 1992) as:
N r ( x) = ∑ 2S (φk ) ∆φ cos ( 2πφk x + θ k ) (4.63) k =1
where φ0 is the discontinuity frequency of 1/2π (cycle/m) and Ar is the roughness coefficient (m3 /cycle) related to the road condition.
4.3.4 Quasi-static Wind Effect on the Vehicle Wind action on a running vehicle includes static and dynamic load effects. The quasi static wind forces on vehicles are adopted since a transient type of force model is not available (Baker 1999):
where Fwx, Fwy, Fwz, Mwφ, Mwθ and Mwz are the drag force, side force, lift force, rolling moment, pitching moment and yawing moment acting on the vehicle, respectively. CD, CS, CL, CR, CP and CY are the coefficients of drag force, side force, lift force, rolling moment, pitching moment and yawing moment for the vehicle, respectively. “A” is the frontal area of the vehicle; hv is the distance from the gravity center of the vehicle to the road surface; Ur is the relative wind speed to
the vehicle, which is defined as (Fig. 4.3):
where V is the driving speed of vehicle; U and u(x, t) are the mean wind speed and turbulent wind speed component on the vehicle, respectively; β is the attack angle of the wind to the vehicle, which is the angle between the wind direction and the vehicle moving direction (Fig.
4.4); ψ is usually between 0 to π.
Fig. 4.4 Velocity diagram on vehicles 4.3.5 Numerical solutions To simulate the process of vehicles running on the bridge, the position of any vehicle changes with time. Correspondingly, the coefficients of the coupled equations are also time dependent. Therefore, the matrices in Eq. (4.55) should be updated at each time step after the new position of each vehicle is identified. In the present paper, both the Wilson-θ method and the second-order Rouge-Kutta approach are tried and Rouge-Kutta method is finally chosen for the virtues of convenience and accuracy. A computer program based on Matlab is developed to solve the differential equations.
4.4 Numerical Example 4.4.1 Prototype bridge and vehicles The Yichang Suspension Bridge located in a typhoon zone of south China has a main span of 960 m and two side spans of 245 m each. The height of the bridge deck above 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 et al. 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 et al. 1998). Except explicitly specified, the wind attack angle for the bridge is zero.
72 As shown in Section 4.2, the introduced generalized vehicle model is applicable in simulating various vehicles, from two-axle cars, trucks, buses to five-axle tractor-trailer. To consider the wind effect on one particular vehicle, wind tunnel test should usually be conducted to obtain the wind force coefficients. Unfortunately, available wind tunnel data are limited to very few kinds of vehicles (Baker, 1991a,b). In the present study, one four-wheel vehicle is used in the numerical example. A sketch of the vehicle model is shown in Fig. 4.6 and the main parameters are listed in Table 4.2. The reason to choose this vehicle is mainly because this fourwheel vehicle has quite similar dimensions to that used in Baker’s work (Coleman and Baker 1990), where wind force coefficients from wind tunnel test are available. This vehicle has seven degrees-of-freedom, three for the rigid body of vehicle (vertical displacement Zv1, pitching about y axis θv1 and rolling about x axis φv1) and the other four for the vertical displacements of the four wheels (ZaL1, ZaR1, ZaL2, ZaR2) (Fig.4.1).
Wind force coefficients can be expressed as follows (Baker 1987):
where α1 to α 6 equal to 5.2, 0.94, -0.5, 2.0, -2.0 and 7.3, respectively (Coleman and Baker 1990).
To properly simulate the traffic on the bridge, traffic flow information is required to define the actual vehicle type and distribution. At any different moment, vehicles on the bridge may have quite different numbers and distribution patterns and locate in different lanes randomly.
Since each vehicle may have at least several degrees of freedoms, it is still technically difficult to conduct a coupled analysis on all vehicles together based on simulated real traffic flow using current computer techniques. It is especially true when the bridge is long and the traffic is busy.
The common practice in analyzing the interaction between vehicles and bridges is to choose only one vehicle or a series of identical vehicles in one line (Yang and Yau 1997; Guo and Xu 2001).
In the present study, only one line of up to three vehicles is assumed to be evenly distributed along the side lane with an interval of 5 m and the lateral distance to the torsion center of the bridge dLq1= dLq2= 5m (Fig. 4.3). The position of vehicles and the interval setting are decided based on some considerations to preliminarily simulate the crowded transportation reality during a hurricane evacuation process. The effect of the number of vehicles will also be studied in the following numerical analysis.
The turbulent wind velocities are simulated (see Eq. (4.61)) along the bridge span with the simulation interval ∆=25 m, corresponding to the length of four finite elements along the main span. The total number of frequency intervals Nf equals 1024 and the upper cutoff frequency equals to 2π. Figure 4.7 gives the time histories of wind velocity in the middle point of main span when the average wind speed on the deck U= 40 m/s. The roughness displacement is also derived after the adoption of roughness factor of 20×10-6 m3/cycle for good road condition as shown in Fig. 4.8 (Huang and Wang 1992). Second-order Rouge-Kutta approach is adopted to solve the differential equations with an integration time step of 0.015 second and a total steps of 3000.
With zero initial conditions, the time history of response for the bridge only and the 75 vehicles on the road are predicted under the wind action. It is assumed that after 50 seconds, the vehicles enter the main span of the bridge, the integration starts with the initial conditions of the moment after 50 seconds. Only the dynamic responses of the bridge and vehicles during the period of the vehicles running on the bridge are demonstrated hereafter. The assumption of 50 seconds before the vehicles enter the bridge is to make sure the bridge has enough time to be excited by the wind with stable response.
Fig. 4.8 Simulated vertical road roughness for the bridge 4.4.2 Vehicle Dynamic Response Among the three vehicles, the first vehicle is chosen to monitor the response in the study.
Since the absolute response of the vehicles on the bridge includes the contribution from the bridge motion, the relative response of the vehicles to the bridge gives more valuable information for the vehicle safety. The relative response of vehicles is obtained through deducting the corresponding bridge response from the absolute response of the vehicles. Under wind speed U = 40 m/s, vehicles with speeds of V = 10 and 20 m/s are studied separately. In Fig.
4.9, the relative vertical displacement of the rigid body for the 1st vehicle is displayed. There is an increase of displacement when vehicle driving speed changes from 10 m/s to 20 m/s. Vehicles with different driving speeds have different location on the bridge at the same time, and it is observed that the vertical response of vehicles with 20 m/s driving speeds are increased when they are in the mid-span area (about 24-30 seconds in Fig. 4.9). Same tendency can also be observed for vehicles with 10m/s driving speed in the corresponding mid-span area (about 45 seconds in Fig. 4.9). The phenomena suggest that vertical motion of the vehicles have strong coupling with symmetric modes of the bridge. It is also noted that the mean value of the relative time history corresponds to the static response of the vehicle excited by static wind force 76 component. Since the static wind forces vary with the driving speed that affects the resultant of wind velocity Ur as shown in Eqs. (4.67) and (4.68), the mean values of the curves have slight shift vertically, which contributes partially to the difference of the response between different driving speeds shown in Fig. 4.9. However, the dynamic amplification of the vertical response contributed by the roughness, dynamic component of wind action on the vehicles, and their interaction with the bridge can obviously be observed.
The relative rolling displacement of the rigid body of the 1st vehicle is also shown in Fig.
4.10. Compared with Fig. 4.9, Fig. 4.10 shows a slight increase of rolling response of the vehicles in the location around the quarter span (about 12 seconds for V=20m/s or 24 seconds for V=10 m/s). No obvious difference exists when vehicles pass the middle of the main span. The results suggest that the difference of rolling response of the vehicle under different driving speed is relatively insignificant comparing with differences of vertical response. The moderate dynamic coupling effects exist between vehicle rolling motion and the asymmetric modes of the bridge.
As an important variable to identify the risk of overturning accident, the absolute rolling angular acceleration is also predicted under different wind speeds shown in Fig. 4.11(a) where the absolute angular acceleration changes little for different driving speed. The spectral density amplitude, shown in Fig. 4.11 (b) is obtained from spectral analysis of the time history. It is found that rolling acceleration of the vehicle is mostly dominated by the bridge response, which is indicated by several peaks including one corresponding to the torsional modal frequency of the bridge (marked with a circle in the figure).
The vertical relative response of vehicles on the highway bridge and road are compared with the same road surface conditions, same driving speed (V= 20 m/s) and same wind speed (U= 40 m/s). Results in Fig. 4.12 show that the vehicle dynamic response on the road is steadier, while its response on bridge increases significantly when time is between 15-30 seconds, which corresponds to the time when the vehicles pass the middle of the main span. In other time period, little differences on the vehicle relative response can be observed between the vehicles on the road and on the bridge. Such phenomenon shows again that the vertical relative response of the vehicle interacts with the bridge motion, especially with the symmetric modes.
The relative vertical response of the vehicle on the bridge under different wind speeds is also studied when the driving speed still remains at 20 m/s. Results displayed in Fig. 4.13 clearly suggest that the dynamic response of the vehicles under higher wind speed are much larger than that under low wind speed especially in some critical locations on the bridge (e. g., mid-span).
Again, such result further justifies the concern about the safety of the vehicles under strong winds.
In real traffic situations, many vehicles could run on the bridge at the same time. The relative dynamic response of the rigid body of the first vehicle is compared in Fig. 4.14 when there is only one vehicle and totally three vehicles on the bridge. The results show very insignificant difference on the vehicle relative displacement when the number of vehicles changes. However, the absolute response of the vehicle does decrease with the increase of vehicle numbers as shown in Fig. 4.15.
4.4.3 Bridge Dynamic Response Figs. 4.16 and 4.17 show the time histories of vertical and torsion dynamic response in the middle point of the main span under wind speed U = 40 m/s where two driving speeds V =10m/s 77 and V= 20m/s are studied separately and the results are also compared with the bridge response without any vehicle when wind speed remains at U= 40 m/s. It can be found that the bridge dynamic response decreases slightly with vehicles compared with that without any vehicle. Such decrease is more significant when the vehicles approach the middle area of the main span (t= 20second). Such decrease for the torsion response of the bridge is probably due to the eccentric weight effect of the vehicles locating on the windward side of the bridge, which suppresses the bridge response comparing with the bridge without any vehicle. For vertical modes, strong coupling and the weight of vehicles may all contribute to the decrease of the vertical response.
As stated before, for a given time, vehicles with different driving speed are located at different places on the bridge. For example, since the main span of the bridge is 960 m, vehicles with 20 m/s driving speed will roughly pass the middle of the main span at the time of 24 seconds, while vehicles with 10 m/s will pass the quarter point of the span (will need 48 seconds to pass the mid-span). It is found from the figures that only when vehicles are close to the place where the response is monitored (mid-point of the main span, in this case), the bridge response differs with different vehicle driving speeds.
0.1 0.1 Fig. 4.14 Relative vertical displacements for the rigid body of the 1st vehicle on the bridge with different number of vehicles when U=40 m/s and V=20 m/s
Fig. 4.15 Absolute vertical displacements for the rigid body of the 1st vehicle on the bridge with different number of vehicles when U=40 m/s and V=20 m/s
2 1 0