Evaluation of Ride Comfort and Driving Safety for Moving Vehicles on Slender Coastal Bridges

In recent decades, an increasing number of slender bridges has been built worldwide with long spans in many coastal areas to cross straits or seas, linking cities or islands. These long-span bridges are usually highly flexible with low structural damping, which in turn makes them more vulnerable to vibrations due to the ambient environmental excitations. Nevertheless, serving as the backbone of the transportation system in coastal areas, these long-span bridges usually could carry a high volume of traffic on a daily basis. During extreme weather-related events, such as hurricanes, associated with strong winds, flooding, or storm surges, the safety and reliability of the bridges as well as the moving vehicles in evacuations on the bridges are of great concern [1,2]. The complex dynamic coupling effects among the flexible supporting long-span bridge, running vehicles, and wind and/or wave were found to have significant effects not only on the bridge performance, but also on the vehicle ride comfort and driving safety [3,4].

The vehicle ride comfort is directly associated with the vehicle vibrations that are transmitted to occupational drivers in various directions as a result of their contact with the seat, back, and footrest. Exposure to excessive whole-body vibration from the vehicles may lead to short-term discomfort and long-term physical damage such as the musculoskeletal pain and back pain, especially for the drivers with long-distance driving or with higher responsibilities, e.g., for public transportation and large cargo trucks [5,6]. Therefore, as one of the important performance of vehicle, the vehicle ride comfort plays a critical role in determining the driving satisfaction as well as the driving safety and long-term health of the drivers. In addition, when vehicles are traveling through long-span bridges, the vehicle accident risks are found to increase considerably due to the complex dynamic interactions among the wind, vehicles, and the flexible supporting bridge structures [7]. There have been several reports about the accidents for various types of moving vehicles on bridge due to the strong lateral winds. For example, on August 11, 2004, seven high-sided road vehicles on the Humen suspension bridge in China were blown over due to the strong wind gust right before a strong typhoon landing [8]. Similar accidents were also reported for a semi-truck on the Mackinac Bridge in U.S. on July 18, 2013, at the wind speed over 65 miles per hour during a severe storm [9]. To ensure the safety of the drivers and passengers, it is essential to perform the vehicle ride comfort and safety evaluation.

Active research has been carried out regarding the vehicle ride comfort and driving safety assessment based on vehicle-bridge-wind coupled system in the last decade [3,4,10,11]. The current existing studies on ride comfort in terms of evaluating human exposure to whole-body vibration and repeated shock are primarily based on several existing standards [12–14]. Xu and Guo [15] investigated the ride comfort of heavy vehicles on a long-span cable-stayed bridge under crosswind based on the root-mean-square (RMS) value with respect to one-third octave-band frequency recommended by ISO 2631/1 [13]. This specification was later updated to a newer version [14] in which the frequency-weighted RMS values are evaluated for ride comfort based on multi-axis whole-body vibrations. Later on, this newer version was adopted by Yin et al. [16] to evaluate the ride comfort of a single truck on a high pier, multispan continuous bridge based on lateral vehicle responses. Most recently, Zhou and Chen [4] evaluated the ride comfort based on vehicle multi-axis vibration responses by incorporating the stochastic traffic flow and wind effects. In addition to the ride comfort, the vehicle driving safety issue may also arise for highway vehicles under hazardous driving environments, e.g., strong crosswind and/or slippery road surface. Guo and Xu [7] conducted vehicle safety analyses of high-sided road vehicles based on coupled vehicle-bridge-wind interactions in which the vehicle accidents including the overturning, excessive sideslip, and exaggerated rotation were investigated. Chen and Chen [17] conducted accident risk assessment under comprehensive hazardous driving conditions by combining a local single-vehicle accident model with established accident criteria. Later on, Chen and Chen [18] further improved the efficiency of the proposed accident risk assessment framework by using the response surface method. Recently, Zhou and Chen [3] incorporated the stochastic traffic flow simulation into the vehicle-bridge-wind system for more advanced traffic safety assessment. Chen et al. [11] further investigated the influence of wind barrier on the vehicle driving safety based on wind tunnel experiments and numerical simulations.

The above research on the vehicle ride comfort and driving safety assessment are all based on the vehicle-bridge-wind system. Nevertheless, for slender coastal bridges, in addition to the wind actions, the wave actions could also contribute to the bridge dynamic responses significantly [19], which may in turn have influences on the vehicle ride comfort and driving safety. However, the investigations of wave effects on the vehicle ride comfort and driving safety in the context of vehicle-bridge-wind-wave (VBWW) dynamic interactions are very limited in the literature. In the present study, the vehicle ride comfort and driving safety evaluation are performed for slender coastal bridges considering the dynamic interactions between the bridges and the surrounding environmental loads, such as wind and wave loads. First, an analytical numerical framework of the VBWW system is introduced, which can be used to obtain the dynamic responses of each individual vehicle under various wind and wave conditions. Based on the vehicle dynamic responses, the ride comfort is evaluated based on the frequency-weighted RMS values of the vehicle whole-body vibration responses through the frequency weighting and averaging techniques. Meanwhile, the vehicle driving safety analysis is also evaluated based on two predefined safety criteria, i.e., the roll safety criteria (RSC) and the sideslip safety criteria (SSC). To demonstrate the methodology, a coastal slender bridge is modeled to form the coupled VBWW system. Parametric studies are carried out to investigate the influence of multiple variables on the vehicle ride comfort and driving safety, e.g., the wind and wave load combinations, the vehicle moving speeds, traffic lane locations, and vehicle types.

The coupled VBWW dynamic system consists of bridge and vehicles as well as the external wind and wave loads that are depending upon the vibrational states of bridges or vehicles. Because the complicated dynamic coupling effects among the bridge, vehicles, wind, and wave cannot be modeled appropriately using the existing finite element (FE) software, an analytical framework of the coupled VBWW dynamic system is established using the commercial FE software ANSYS [20] and the computer programming language matlab [21]. The general flowchart of the entire simulation process consists of three steps, as summarized in Fig. 1. First, the numerical models for both the bridge and vehicles are developed to extract the initial coefficient matrices. Second, the stochastic wind and wave fields are simulated which are used to calculate the wind and wave forces on the bridge/vehicle. Third, the governing equations of the coupled VBWW system are developed to facilitate the dynamic analysis. Subsequently, the dynamic responses of the running vehicles subject to various wind and wave loading conditions can be predicted, which are used for further ride comfort evaluation and safety analysis. The details of these procedures are elaborated in this section.

Wind and wave could be correlated as the wind is one of the major driving forces for the surface wave generation. During the past few decades, many wind-wave models, ranging from simple formulas to more sophisticated numerical models (e.g., SWAN, WAM, and SLOSH), have been proposed for various applications. Nevertheless, the sophisticated numerical models are developed and orientated for applications from ocean to coastal scales in general in which the spatiotemporal resolutions of the output wind and wave fields are not appreciable for the dynamic analysis of coastal infrastructures [22]. As an alternative, some spectrum-based methods, such as spectral representation method (SRM), are still used widely for structural engineering field, which utilizes only a few parameters as input (e.g., wind speed and fetch) for generating the wind and wave fields with small spatiotemporal scale [23]. In the present study, the correlated wind and wave fields are generated by using the SRM, which is presented in detail in this section.

where U₁₀ is the mean wind speed at elevation of 10 m above the SWL; the surface roughness z₀ is estimated by the Charnock expression [25] as z₀ = (α₀/g)(KU₁₀/ln(10/z₀))², where K = 0.4 is the Von Karman constant and α₀ is an empirical constant ranging from 0.01 to 0.02. The mean wind speed is used to obtain the wind and wave spectra, which will be further used to simulate the wind and wave fields based on the SRM [26].

where u* is the shear velocity of the flow given by u* = KU(z)/ln(z/z₀).

where η(y, t) is the surface elevation along the wave propagation direction (i.e., y) on a still water plane as a function of y and time t, as shown in Fig. 2; $ai=2Sη(ωi)Δω$ is the wave amplitude of each individual wave component; N_f is the number of frequencies; S_η is the wave spectrum; ω_i= [iΔω + (i − 1)Δω]/2, where Δω = (ω_u − ω₀)/N_f is the frequency resolution, and ω_u and ω₀ are the upper and lower cutoff frequencies, respectively; $ω̃i$ is a random number between ω_i-1 and ω_i; ε_i is N_f sequences of random phases distributed uniformly between 0 and 2π; k_i=2π/λ_i is the wave number determined from the dispersion relationship $ωi2=kigtanh(kih)$⁠; λ_i is the wavelength of the ith wave; g denotes the acceleration of gravity; u_x(y, z, t) and $u˙x(y,z,t)$ are the components of water particle velocity and acceleration in the wave propagation direction; (h + z) is the vertical distance of any point along the pile height from the sea bed; and y is the horizontal distance of the point in the wave from a reference point.

where α_s = 0.076(U₁₀/Fg)^0.22 is the coefficient; γ is the peak enhancement factor; F is the fetch distance; σ = 0.07 when ω ≤ ω_p, else σ = 0.09 when ω > ω_p; ω_p=22(g²/U₁₀F)^1/3 is the peak frequency; and $φ(k0h)=(tanh2k0h)/(1+2k0h/sinh2k0h)$ is a factor considering the shallow water effect on the spectrum and k₀ is the wave number.

In the present study, a prototype coastal slender cable-stayed bridge (see Figs. 4 and 7) is modeled in 3D using finite element method. Bridge deck, tower, pier, and pile foundation are modeled with three-dimensional spatial beam elements based on Timoshenko beam theory. Cables are modeled as link elements with the modified modulus of elasticity accounting for the cable sag effect. The initial stain is also applied on each cable element according to the designed pre-tension force. All the six bearings between the bridge deck and the pier/tower are modeled by swing rigid links and horizontal rigid links to allow free longitudinal motion of the bridge deck. The piers and pile foundations are fixed at the bottom. The soil surrounding the pile foundations is simplified as spring elements. Rayleigh damping is adopted to construct the bridge damping matrix as a function of the stiffness and mass matrices, and two structural damping ratios associated with two specific vibration modes of the bridge.

In studying the interaction between the vehicles and structures, the vehicle model is usually simplified but maintaining all the relevant essential information [2,24,31]. Following their work, in the present study, the vehicles are modeled as a couple of rigid bodies, suspension systems, and tires that are connected by a series of springs and dampers. The vehicle bodies and the tires are modeled as the rigid bodies, whereas the elasticity and dissipation capacities of both the suspension system and the tires are idealized as springs and viscous dampers, respectively. For example, a typical two-axle four-wheel road vehicle is modeled with five rigid bodies and 16 sets of springs and dampers, as shown in Fig. 3 [24,31]. As shown in Fig. 3, 13 degrees-of-freedom (DOFs) are assigned for the road vehicle, among which 5DOFs are assigned for the vehicle body including two translation DOFs (i.e., vertical Z_v and lateral Y_v) and three rotational DOFs (i.e., rolling φ_v, yawing ϕ_v, and pitching θ_v), and the remaining 8DOFs are assigned for the four tires (i.e., vertical Z_si and lateral Y_si (i = 1∼4)). In addition, the bridge deck and the tires of the vehicles are assumed to be point-contact.

The stochastic wind and wave fields provide the time histories of the wind speed and wave velocity/acceleration, which will be used to simulate the dynamic interactions among the vehicles, bridge, wind, and wave. The modeling of wind forces on the bridge and vehicles, and the modeling of the wave forces on the bridge pile-group foundation are elaborated in this section.

where the subscripts st, se, and b refer to the static, self-exited, and buffeting wind force components, respectively; and L, D, and M refer to the lift, drag, and torsional moment, respectively. Different from the bridge deck, the wind effects on the bridge tower mainly consist of the static and buffeting wind forces, as the bridge towers are usually much stiffer and the self-excited wind forces can be ignored. The formulations of the lift force, drag force, and torsional moment of each wind force component in Eq. (11) are not presented for the sake of brevity. Interested readers are referred to the literature [24,31] for more details.

where F_S, F_L, F_D, M_P, M_Y, and M_R are the side force, lift force, drag force, pitching moment, yawing moment, and rolling moment acting on the vehicle, respectively; C_S(ψ), C_L(ψ), C_D(ψ), C_P(ψ), C_Y(ψ), and C_R(ψ) are the corresponding wind aerodynamic coefficients; A₀ is the frontal area of the vehicle; h_v is the distance from the vehicle gravity center to the road surface; U_r is the relative wind velocity to the vehicle; and ψ is the yaw angle defined as the angle between the mean wind direction and the vehicle driving direction.

where C_wD and C_wM are, respectively, the drag and inertia coefficients which are taken as C_wD = 1.2 and C_wM = 1.5 [33]; ρ_w is the water density; D₀ and A are the diameter and section area of the pile, respectively; u_w and $u˙w$ are the water particle velocity and acceleration obtained using Eqs. (8) and (9), respectively; and $ub$ and $u˙b$ are the velocity and acceleration of the pile, respectively.

where K_z and K_g are, respectively, shelter and interference coefficients defined as the force ratio, F_Group/F_Single, with F_Group being the wave force on a slender pile within a pile-group and F_Single being the wave force on a single isolated pile determined using Eq. (13); S_G/D₀ is relative spacing in which D₀ is the pile diameter and S_G is the gap between the surfaces of two neighboring piles in a pile-group. It is worth mentioning that the effects of the currents are not considered in the present study as the currents are found to be weak near the prototype bridge foundation.

where K_l and C_l are the coefficients of the vehicle's lower spring and damper; Z_a is the vehicle-axle-suspension displacement; Z_b is the displacement of the bridge at road-tire contact points; r(x) is the road surface profile; and $r˙x$ = (dr(x)/dx)(dx/dt)=(dr(x)/dx)V(t) with V(t) being the vehicle velocity.

In recognizing that the governing equations given in Eq. (16) contain a large number of DOFs and motion-dependent load vectors, i.e., $Fvwi$⁠, F_bwave, $Fbvi$⁠, and $Fvbi$⁠, the separation iterative method is applied to solve the two equations separately at each time-step, and then to achieve the solution through the equilibrium iterations based on the coupling relationship of the two subsystems [31,38]. The Newmark-β method is adopted to solve the differential equations, in which an integration time-step of 0.01 s is selected to provide accurate dynamic responses, according to the preliminary sensitivity analyses.

where λ₁ and λ₂ are the parameters related to the driver behavior, and λ₁ = λ₂ = 0.3 are adopted in the present study as suggested by Chen and Cai [31]; Δ_y and $Δ˙y$ are the relative lateral displacement and velocity between the vehicle center and the bridge; δ is the driver's steering angle, which is usually adopted to simulate the driver's adjustment of driving when there are constant lateral impacts from gusting winds; δ is incorporated into the vehicle governing equations, i.e., Eq. (16a) during the simulation.

Due to the differences of individual's tolerance of accelerations that could be affected by individual's age, health, or psychological conditions, setting the ride comfort criterion could be complicated despite many efforts that have been made in recent years. In the present study, the whole-body vibration measures recommended by ISO 2631-1 [14], which has been widely used as the criterion for health, comfort, perception, and motion sickness in practice, are adopted for vehicle ride comfort evaluation.

where a_ij (i = v, l, p, r; j = s, b, f) denotes the acceleration: the first subscript indicates the response direction such that “v,” “l,” “p,” and “r” represent vertical, lateral, pitching, and rolling direction; and the second subscript describes the axis location such that “s,” “b,” and “f” represent seat, backrest, and floor, respectively; $Z¨v$⁠, $Y¨v$⁠, $θ¨v$⁠, and $φ¨v$ are the vertical, lateral, pitching, and rolling accelerations at the vehicle centroid, respectively; d_s, y_s, and h_s are, respectively, the longitudinal, transverse, and vertical distances between the vehicle centroid and the seat.

As suggested by the standard, the accelerations should be frequency-weighted in order to model the vibrating frequencies of the human body more realistically [14]. First, the time histories of the original accelerations at all participating axes are transformed into frequency domain using the discrete Fourier transform (DFT) and then multiplied by the corresponding frequency weighting factor. After frequency weighting, those frequency-weighted responses in the frequency domain are converted back into the corresponding frequency-weighted responses in the time domain by applying the inverse DFT. Figure 6 shows the frequency weighting factor W_i (i = k, d, e, c) as recommended by the standard in which the subscript “k” represents the vertical vibration at the seat and both vertical and lateral vibrations at the floor, the subscript “d” represents lateral vibration at the seat and floor, the subscript “e” represents the rotational vibrations at the seat, and the subscript “c” represents the vertical vibration at the backrest. Those weighting factors serve as filters to the original responses such that the effects of low and high frequency contents of the original responses are reduced. The frequency range of the frequency-weighted responses is set as 0.5–80 Hz for the ride comfort analysis.

where RMS_ij (i = v, l, p, r; j = s, b, f) denotes RMS values of the frequency-weighted acceleration response, and the subscripts “i” and “j” have the same definitions as those of the acceleration responses; and M_i (i = k, d, e, c) are the multiplying factors for each frequency-weighted response to compensate the varying vibrating effects due to different locations and directions, as shown in Table 1. The subscripts of the multiplying factors have the same definitions as those of the subscripts for the frequency weighting factor.

To facilitate the ride comfort evaluation, the comfort criterion is also provided in which six different comfort levels are defined based on various ranges of OVTV values, as shown in Table 2.

where F_vli and F_vri denote the vertical contact forces at the left and right wheels of the ith axle, respectively, and k is the number of the axles (k ≥ 2). Equation (21) indicates that the rolling accident will occur as the value of RSC is less than the threshold value of 1.2.

where F_vl and F_vr are the vertical contact forces at the left and right wheels of an axle, respectively; F_hl and F_hr are the lateral contact forces at the left and right wheels of an axle, respectively.

One cable-stayed bridge that consists of five spans with a span arrangement of 60 + 176 + 700 + 176 + 60 m is used as the simulation example in the present study, as shown in Fig. 7(a). The main span and two inner approach spans of the bridge are made of steel, and the two outer approach spans are made of prestressed concrete. The streamlined box girder is 40 m wide and 3.5 m high, which supports six traffic lanes as shown in Fig. 7(b). The bridge girder is supported by 176 stayed cables along the bridge span and six bearings implemented at the towers and the piers. The bridge deck is 45 m above the SWL. The two H-shaped concrete bridge towers are 186 m tall, supported by the pile foundation that consists of 30 piles with a radius of 3.0 m and a length of 32 m in the water. A concrete mass pile cap is adopted to connect all the piles, which is 73.05 m in the transverse direction and 24.5 m in the longitudinal direction. The elevation of the bottom of the pile cap is 8 m above the SWL. The layout of the tower along with the pile-group foundation is shown in Fig. 4. The wind field is simulated for both the bridge deck and the tower such that a total number of 83 and 11 uniformly distributed wind points are assigned along the bridge deck axis and the height of each bridge tower, respectively. Meanwhile, the wave fields at the two pile foundations with 700 m apart are simulated independently due to their weak correlations.

For the simulation of the correlated wind and wave fields, the time interval is selected as 0.01 s in accordance with the integration time-step used to solve the governing equations. The upper cutoff frequency and the frequency interval are set as 2 Hz and 0.002 Hz, respectively, to ensure that the simulated wind and wave fields can accurately represent their characteristics [26]. As an illustration, Fig. 8 shows the time histories of the correlated wind and wave fields at selected points with the mean wind velocity U = 10 m/s and U = 20 m/s (here and hereafter, the mean wind refers to the mean wind at the deck level, unless otherwise noted). As shown in Fig. 8, the higher mean wind speed can generate relatively larger wind and wave time histories compared with the lower mean wind speed. The mean wind is assumed to act on the bridge deck, and the tower and the wave are assumed to act on the pile foundation. Both of them are in the lateral direction, which are believed to represent a worse scenario than the other wind or wave directions. However, the effects of wind-wave loading directionality will not be discussed in the present study.

Two types of vehicles are adopted in the analysis, i.e., the sedan car and light truck, with the corresponding parameters shown in Table 3. For the traffic simulation, the number of vehicles in each traffic lane is set as 30 with an equal distance of 30 m. The 15th vehicle is selected as the representative vehicle and the corresponding dynamic responses as the vehicle travels on the bridge are used for the following vehicle ride comfort and safety evaluation. All the analyses are based on the VBWW system, i.e., the correlated wind and wave fields are applied on the vehicle-bridge system, unless otherwise noted.

Note that the original vehicle accelerations corresponding to each axis of consideration need to be frequency-weighted with corresponding frequency weighting curves before calculating the OVTV and performing the ride comfort evaluation. The frequency weighting and averaging procedures are illustrated in detail using a case study in which a vehicle fleet consisting of 30 light trucks moves at a speed of 15 m/s (54 km/h) on the third traffic lane with a good road condition. The necessary parameters for the correlated wind and wave fields simulation are given as U = 10 m/s, γ = 3.3, h = 32 m, and F = 200 km, respectively (here and hereafter, same γ, h, and F are used for the wave field simulation, unless otherwise noted). For brevity, only the vertical acceleration at the seat position for the 15th vehicle, i.e., a_vs, is presented. Figure 9 shows the power spectrum density (PSD) of the original vertical acceleration response and the frequency-weighted PSD after applying frequency weighting curve W_k. Through the comparison between Figs. 9(a) and 9(b), the amplitudes of the original PSD at the frequency range below 2 Hz are significantly reduced, while the amplitudes of the original PSD at the frequency range from 3 Hz to 6 Hz are not affected too much. This is due to that the value of the filter W_k is very small for smaller frequency, i.e., W_k < 0.25 for f < 2 Hz, and then W_k quickly reaches to a large value with higher frequency, i.e., W_k remains around 1.0 at the frequency range 3–6 Hz, as shown in Fig. 6. The updated time history for the acceleration, therefore, can be obtained based on the inverse DFT using the frequency-weighted PSD. Figure 10 compares the vertical acceleration before and after the frequency weighting. The comparison results indicate that the original acceleration is significantly reduced after frequency weighting, i.e., the corresponding RMS is reduced from 0.257 to 0.077, with a reduction of 70.2%.

Following the same procedure, the RMS for the original and frequency-weighted accelerations at all participated axes can be obtained which are shown in Table 4 for comparison. As shown in Table 4, the RMS of the accelerations are all reduced at various degrees after applying frequency weighting, except for the vertical acceleration at the backrest, which has only a reduction of 1.0%. All the other accelerations have a reduction rate ranging from 33.3% to 73.9%. The reason for only a 1.0% reduction for a_vb is that the frequency weighting curve W_c remains constant at around 1.0 in the frequency range of 1–7 Hz, which covers the dominant frequency of the original PSD, as shown in Fig. 6. Finally, the frequency-weighted RMS for all axes are modified by applying the multiplying factors to obtain the OVTV in order to evaluate the ride comfort level. The OVTV of the representative light truck in the case study is obtained as 0.165 m/s², and the corresponding comfort level is classified as “not uncomfortable” based on comfort criteria in Table 2.

Figure 11 shows the OVTVs for both of the light truck and sedan car driven on lane 1 at a speed of 25 m/s (90 km/h) with and without considering the wave loads. Figure 11 shows that, for the same type of vehicle under the same wind speed, the difference of the OVTVs for the cases with and without considering the wave loads is all within 7%, indicating that the effects of wave on the vehicle ride comfort are much less than those from the wind. It is also shown in Fig. 11 that the OVTVs increase with the wind speed for both types of vehicles. The OVTVs for the two types of vehicles first increase with a steady rate when the wind speed is lower than 20 m/s and then increase faster as the wind speed continues to increase. In addition, the light truck has a higher OVTV than that of the sedan car under the same wind speed, indicating that the light truck is more prone to the potential ride comfort issue. A threshold value of 0.315 based on the ISO ride comfort criteria is also shown in Fig. 11. The value below the threshold line indicates that the driver will not feel uncomfortable based on the defined criteria. Accordingly, the critical wind speed at which the driver will began to feel uncomfortable for the light truck and the sedan car is 12.5 m/s (45 km/h) and 22 m/s (79.2 km/h), respectively.

In order to show the influence of vehicle speed on the ride comfort, Fig. 12 shows the OVTVs for the light truck with various vehicle speed driven on lane 1. Again, it should be noted that all the case studies in the present study are based on the VBWW system, unless otherwise noted. It is shown in Fig. 12 that the OVTVs increase as the vehicle speed increases under the same wind speed. Figure 12 also shows that the critical wind speeds for the three vehicle moving speeds are 22.5 m/s for V = 15 m/s (54 km/h), 20 m/s for V = 20 m/s (72 km/h), and 12.5 m/s for V = 25 m/s (90 km/h), indicating the higher vehicle speed and the higher value of OVTV. The same trend can also be observed for the sedan car. For the sake of brevity, the results of the sedan car are not presented here.

It is noted that the eccentricity of the traffic lane has effects on the bridge dynamic characteristics, which may in turn affect the vehicle dynamic characteristics. Figure 13 compares the OVTVs for the light truck driven at a speed of 25 m/s (90 km/h) on three different traffic lanes. As shown in Fig. 13, lane 3 is the most unfavorable lane for the light truck. For example, the driver may feel a little uncomfortable on lane 3 at a wind speed of 10 m/s, while the driver will not feel uncomfortable under the same wind speed when the driver drives on lane 1 or lane 2. Again, the same trend is also found for the sedan car.

It should be noted that the previous results are all based on a representative vehicle travelling on the bridge with the simultaneous presence of multiple other vehicles, which is the most common scenario on the bridge. As mentioned before, the coupling vehicle-bridge effects have large influences on the vehicle dynamics. Consequently, the ride comfort condition for a vehicle in the presence of multiple vehicles may be different from the single vehicle scenario. Figure 14 compares the OVTVs for both the light truck and the sedan car with and without considering the presence of other vehicles. The case considering the traffic flow uses the same traffic fleet as described earlier and the moving speed is taken as 25 m/s (90 km/h). As shown in Fig. 14, the OVTVs for both the types of vehicles in a single vehicle scenario are smaller than those when considering the traffic flow. For instance, under the wind speed of 15 m/s, the reduction ratio of OVTVs from traffic flow scenario to single vehicle scenario is 25.6% and 17.6% for the light truck and the sedan car, respectively. Accordingly, the critical wind speed increases from 12.5 m/s to 20 m/s for the light truck and from 22 m/s to 23.8 m/s for the sedan car. The results also indicate that the light truck is more likely to be affected by the presence of multiple other vehicles.

Similar to the ride comfort evaluation, a parametric study is also performed to study various factors on the vehicle driving safety. Figure 15 compares the safety indicators, i.e., RSC and SSC, of the vehicle driven on lane 1 at a speed of 25 m/s (90 km/h) with and without the wave loads for both the types of vehicles. It is shown in Fig. 15 that the difference of the safety indicators in the cases with and without considering the wave loads is less than 5% under all the wind speeds for both the light truck and the sedan car. This suggests that the wave load has minor effects on the vehicle driving safety compared with the wind load. Based upon the simulation results from this prototype bridge, the evaluation results of both the ride comfort and the driving safety considering the wind and wave loads suggest that the wind loads have more significant effect than the wave loads. Two reasons may explain why wave loads have negligible influences on the vehicle dynamics. First, the wave surface elevation could not reach the pile cap in general, since the still water level is 6.7 m below the bottom surface of the pile cap, as shown in Fig. 4. In addition, considering that the water depth at the bridge foundation is 32 m, the wave-induced forces on these piles may not be sufficiently large enough to cause appreciably bridge dynamic responses. The other reason is that a large rigid concrete mass pile cap is adopted to connect all the 30 piles as a whole, which makes the pile-group very stiff. Consequently, the wave-induced dynamic effects on the bridge deck of this prototype bridge are not significant, which may in turn have negligible effects on the running vehicles.

The vehicle safety indicators for both the light truck and the sedan car varying with the vehicle speed and the wind speed as the vehicle fleet moving on the first traffic lane are shown in Fig. 16. It is shown in Figs. 16(a) and 16(b) that the driving safety indicators of the vehicles decrease with the increase of the vehicle speed. Figure 16(a) shows that the RSC of the light truck at a moving speed of 25 m/s (90 km/h) approaches the threshold value of 1.2 under the crosswind U = 25 m/s and quickly drops below the threshold line as the wind speed continues to increase. However, the RSC for the sedan car remains above the threshold value for all the simulation cases, indicating that the wind speed less than 35 m/s may not cause rolling accident for the sedan car. The relative low critical wind speed for the rolling accident of the light truck compared with the sedan car may be due to the substantial lateral wind loads applied on the light truck due to its large side area. Different from the RSC, sedan car is more prone to have sideslip accidents in comparison with the light truck under the same moving speed and the same wind and wave load conditions, as shown in Fig. 16(b). For instance, for a moving speed of 25 m/s (90 km/h), the critical wind speed for the sedan car and the light truck is obtained as around 25 m/s and 32 m/s, by interpolating the SSC curves at the threshold value of 1.

Figure 17 compares the safety indicators of the vehicle driven on three different traffic lanes at a moving speed of 25 m/s (90 km/h). Similar to the ride comfort analysis, Fig. 17 also shows that lane 3 is the most unfavorable lane for the vehicle driving safety. For instance, when the traffic lane shifts from lanes 1 to 3, the critical SSC wind speed for the light truck and sedan car drops from 31.5 m/s to 27.5 m/s and from 25 m/s to 20 m/s, respectively, as shown in Fig. 17(b). Therefore, in the windy conditions, the drivers are suggested to drive their vehicles in lane 1 other than lane 3. Similar findings are also found by Chen et al. [11] that the outer traffic lane can pose more threats to vehicle sideslip risks than inner lanes under strong crosswind. Similarly, the critical RSC wind speed for both the types of vehicles also drops when the traffic lane shifts from lane 1 to lane 3, as shown in Fig. 17(a). This suggests that the coupling effects between the bridge and the moving vehicles have significant influence on the vehicle driving safety.

As an important factor for the vehicle-bridge interaction, the road surface roughness can also affect the vehicle dynamic characteristics. The road surface roughness can be generated by the inverse Fourier transformation. In order to investigate the influence of various road condition on the vehicle driving safety, three different road conditions, i.e., very good, average, and very poor, are adopted [41]. Figure 18 shows the safety indicators of the vehicle driven at a speed of 15 m/s (54 km/h) on lane 3 with various road conditions. It is shown in Figs. 18(a) and 18(b) that the driving stability of vehicles decreases with worse road surface conditions. In particular, the very poor road condition has significant influence on the driving stability of the sedan car. For instance, under the mean wind of 20 m/s, the RSC of the sedan car reduces from 4.1 for very good road surface condition to 2.7 for very poor road surface condition by 34%, while the SSC of the sedan car drops from 2.0 for very good road condition to 1.2 for very poor road condition by 40%.

This paper presents a comprehensive study on the vehicle ride comfort and driving safety evaluation for two types of vehicles based on the coupled VBWW dynamic system. Based on the VBWW framework, the effects of the correlated wind and wave loads on the vehicle performance are evaluated for the first time. After the evaluation criteria are defined, the influence of various factors, e.g., wind and wave excitations, vehicle moving speed, and location of the traffic lane, on the vehicle ride comfort and driving safety are investigated. The evaluation results show that the wind loads, compared with the wave loads, affect the ride comfort and the driving safety for the road vehicles more significantly. In addition, both the vehicle ride comfort condition and the driving stability will reduce or decrease with the wind speed and/or the vehicle moving speed. Under the same vehicle moving speed and load conditions, the light truck is more prone to the ride comfort issue and rolling accidents, while the sedan car is more prone to have sideslip accidents. Finally, the traffic lane and the road conditions are also found to have significant effects on the vehicle ride comfort and driving safety, indicating that the coupling vehicle-bridge effects significantly affect the vehicle ride comfort and driving safety as well.

The proposed analytical numerical framework can be easily applied on other coastal bridges to predict the dynamic characteristics of its running vehicles, which may further provide guidance for vehicle ride comfort improvement or accident mitigation strategies under hazardous driving environments. In addition, further efforts are also needed to improve the vehicle driver behavior to integrate the drivers' planning capacities and decision making strategies for more comprehensive vehicle ride comfort and safety evaluations.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsors.

• U.S. National Science Foundation, Directorate for Engineering (Grant No. CMMI-1537121).

• National Natural Science Foundation of China (Grant No. 51708470).

a_i =

wave amplitude of ith wave

a_ij =

acceleration responses at the seat, backrest, and floor of the vehicle

A =

section area of the pile

A₀ =

frontal area of the vehicle

C =

coherence function

C_i(ψ) (i = S, L, D, P, Y, R) =

side/lift/drag/pitching/yawing/rolling wind force coefficient

C_l =

coefficient of the vehicle's lower damper

C_wD =

drag coefficient of Morison Equation

C_wM =

inertia coefficient of Morison Equation

C =

damping matrix

d =

displacement vector

d_s =

longitudinal distance between the vehicle centroid and the seat

D =

drag wind force on bridge

D₀ =

pile diameter

F =

fetch distance

F_i (i=S, L, D, P, Y, R) =

side/lift/drag/pitching/yawing/rolling wind force on the vehicle

F_bw =

wind force vector on the bridge deck and the tower

F_bwave =

wave force vector on the pile foundation

F_hl =

lateral contact force at the left wheels of an axle

F_hr =

lateral contact force at the right wheels of an axle

F_SR =

sideslip resistance

$F¯SR$ =

mean value of the sideslip resistance F_SR

F_vl =

vertical contact forces at the left wheels of an axle

F_vr =

vertical contact forces at the right wheels of an axle

$FvGi$ =

weight of the ith vehicle

$Fvwi$ =

wind force vector on the ith vehicle

$Fbvi$⁠, $Fvbi$ =

interaction force vector between the bridge and the ith vehicle

g =

acceleration of gravity

G_a =

gravity of the lightest axle

h =

water depth

h_s =

vertical distance between the vehicle centroid and the seat

h_v =

distance from vehicle gravity center to road surface

K =

von Karman constant

K =

stiffness matrix

k_i =

wave number

K_g =

interference coefficient

K_z =

shelter coefficient

K_l =

coefficient of the vehicle's lower spring

L =

lift wind force on bridge

M =

wind torsional moment on bridge

M =

mass matrix

M_i =

multiplying factors

N =

number of frequencies for wind spectrum

N_f =

number of frequencies for wave spectrum

r(x) =

road surface profile

RMS_ij =

root-mean-square value of acceleration response

S_G =

gap between the surfaces of two neighboring piles

S_η =

wave spectrum

S(ω) =

wind spectrum

u(t) =

lateral wind turbulence component

u_x =

water particle velocity

$u˙x$ =

water particle acceleration

u_* =

shear velocity

U₁₀ =

mean wind speed at elevation of 10 m above the still water level

U_r =

relative wind velocity to the vehicle

U(z) =

mean wind speed at elevation z above the still water level

v(t) =

longitudinal wind turbulence component

w(t) =

vertical wind turbulence component

W_i =

frequency weighting factor

y_s =

transverse distance between the vehicle centroid and the seat

$Y¨v$ =

lateral acceleration at the vehicle centroid

$Z¨v$ =

vertical acceleration at the vehicle centroid

z₀ =

surface roughness

Z_a =

vehicle-axle-suspension displacement

Z_b =

displacement of the bridge at road-tire contact points

α₀ =

empirical constant

α_s =

coefficient of TMA wave spectrum

γ =

peak enhancement factor

Δ =

distance between two adjacent wind points

Δω =

frequency interval

Δ_y =

relative lateral displacement between the vehicle center and bridge

$Δ˙y$ =

relative lateral velocity between the vehicle center and bridge

η =

wave surface elevation

$θ¨v$ =

pitching acceleration at the vehicle centroid

λ =

exponential decay coefficient

λ_i =

wavelength of the ith wave

λ₁, λ₂ =

parameters related to the driver behavior

μ_s =

sideslip friction coefficient of tire

σ =

coefficient of TMA wave spectrum

σ_SR =

mean square root of the sideslip resistance F_SR

ρ_w =

water density

φ =

shallow water coefficient

φ_ml =

random phase uniformly distributed between [0, 2π]

$φ¨v$ =

rolling acceleration at the vehicle centroid

ψ =

yaw angle

ω =

circular frequency

ω_p =

peak frequency

ω_u =

upper cutoff frequency

ω₀ =

lower cutoff frequency

$ω̃i$ =

random number at frequency interval [ω_i-1, ω_i]