Dynamic Response of a Vehicle-Bridge Expansion Joint Coupled System

Automation Research Institute Co., Ltd. of China South Industries Group Corporation, Mianyang 621022, China School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China Technology and Equipment of Rail Transit Operation and Maintenance Key Laboratory of Sichuan Province, Mianyang 610031, China School of Mechanical Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450045, China


Introduction
Bridge Expansion joints are exposed to the environment of high humidity, large temperature difference, and strong vibration for a long time, which are potential weak spots. e performance of the modular bridge expansion joint (MBEJ) based on the static performance indices cannot meet the requirements of bridge heavy load and heavy traffic flow, and its life span is greatly reduced [1,2]. erefore, it is of great significance for the design and maintenance of expansion joints and bridges to study the dynamic characteristics of MBEJs under passing vehicles. e vibrations of MBEJs include vertical vibration and horizontal vibration. At present, there are few analyses on the dynamic performance of expansion joints. e impact effect of MBEJ vertical vibration is the main reason for the failure of center beam weld [3]. e horizontal vibration mainly affects the safety of the connection between the bridge deck and the MBEJ and the service life of the sliding bearing [4][5][6]. e research by Shun et al. [7] on the interaction between vertical vibration and horizontal vibration of MBEJs showed that the vibrations in two directions are independent of each other, so they can be studied separately. At present, there are few analyses on the vertical vibration performance of expansion joints. A study by Steenbergen [8] on the impact factor of an MBEJ showed that the impact factor of expansion joints can be as high as 1.7, which is much higher than those prescribed in the current design code. Ancich et al. [9] completed the experimental modal analysis and operational response shape studies on an MBEJ installed in the Anzac Bridge to indicate that the dynamic range factor can be reduced by introducing additional damping. Li [10] and Yan [11] analyzed the influence of vehicle velocity, joint gap width, and spring stiffness of the center beam bearings on the vertical dynamic characteristics of an MBEJ by using sine loads to simulate the action of vehicle.
e MBEJ is often regarded as an independent system to study its dynamic property, which can simplify the model to a certain extent. e interaction between the vehicle and the MBEJ is not considered in this type of model.
is will lead to too large deviations in the solution results of this type of model in the high-frequency vibration region. erefore, the coupled effect between vehicles and MBEJ needs to be considered [12].
To make the simulation results more accurate, Wu et al. [13] established a numerical model of vehicle-steel boxgirder based on rigid wheel model to verify that the existence of expansion joints significantly increases the impact effect of vehicles on the bridge. Sun and Zhang [14] applied a single-DOF vehicle model to the finite element model of a vehicle-MBEJ coupled system to analyze the vibration characteristics of the MBEJ. e results showed that the vehicle loads will cause the significant dynamic response of the MBEJ. Deng et al. [15] proved that the damage condition of expansion joints has a significant impact on the bridge deck by establishing a vehicle-bridge coupled model in which the expansion joint was defined as a road-surface profile according to the damage degree of the expansion joint. Friedl and Mangerig [16] and Mangerig and Friedl [17] proposed a vehicle-expansion joint coupled system model based on a plane tire model to analyze the dynamic response of the coupled system when the vehicle passed through the expansion joint. e model considered the nonuniform pressure distribution in the contact area between the tire and the center beam. Ding et al. [18,19] proposed a distributed spring-damper element for dynamic simulation of vehicles passing through expansion joints, in which the contact patch between the tire and the expansion joint was considered. However, the existing mathematical models do not consider the change in the tire contact patch length, which will cause the impact of tire load in the calculation process to be overestimated, resulting in the limitations of the use of models. erefore, it is necessary to establish a more precise model for the contact relationship between the tire and the expansion joint in the coupled system model.
In this paper, a dynamic mathematical model of the vehicle-MBEJ coupled system by introducing a flexible roller tire model is proposed. is model considers the excitation characteristics of the tire load. e validity of this model has been verified in our published paper [20]. Based on the dynamic mathematical model, the vibration characteristics of the coupled system is simulated, and the influence of the parameters on the impact factors of the tire load and the center beam are analyzed. Figure 1 shows an example of a vehicle-MBEJ coupled system. x is the width direction of the bridge deck, y is the vertical upward direction, z is the axial direction of the bridge deck, and the subscripts C, Z, and H denote the gravity center of the vehicle body, center beam, and crossbeam, respectively. e MBEJ in the model consists of 5 center beams and 10 crossbeams. Center beams are perpendicular to the driving lane, while crossbeams are laid parallel to the driving lane under the center beams. Center beams and the crossbeams are connected by sliding bearings, and crossbeams and the bridge are connected by bearings installed in displacement box. A 2-axle heavy vehicle is taken as an example.

e Vehicle Dynamic Model.
e 7-DOF vehicle model [21] is proved to be efficient to simulate 2-axle heavy vehicle. e 7-DOF vehicle models are the vertical displacement, the roll and pitch angles of the vehicle body, and the four vertical displacements of the wheels. Based on the D'Alembert principle, the differential equation of the vehicle system is expressed as where M V , C V , and K V are the mass matrix, stiffness matrix, and damping matrix of the vehicle system, respectively, A V , V V , and X V are the acceleration vector, velocity vector, and displacement vector of the vehicle system, respectively, and R V is the force vector of the vehicle system. e specific explanations of these parameters can be found in [21]. e parameters of the 2-axle heavy vehicle adopted in simulation are listed in Table 1.

2.2.
e MBEJ Dynamic Model. e center beam and crossbeam are slender beams, of which the cross-section dimensions are far less than the lengthwise dimensions. erefore, the vibration equations of the center beam and crossbeam can be obtained by Euler beam theory, and the calculation accuracy is proved to be acceptable [11]. Denoting the vertical displacement of the ith center beam as y Zi (x, t), the vibration equation is given as follows.

Shock and Vibration
where E Z is the elasticity modulus of the center beam, I Z is the section inertia of the center beam, F Hji is the supporting force from the jth crossbeam to the ith center beam, F TLki (t), F TRki (t) are the contact forces between the wheels of the kth axle and the ith center beam with subscripts TL and TR representing the left and right wheels of the vehicle, NJ is the total number of crossbeams, x j is the x coordinate of the jth crossbeam central axis, NK is the total number of axles, and x Lk , x Rk are the x coordinates of the left and right wheels on the k th axle. δ is the Kirchhoff function. e supporting force F Hji (t) can be described as where k 1 , c 1 are the spring stiffness and damping coefficient of the center beam bearing, y Hj (z i , t) is the displacement of the jth crossbeam at the coordinate z i , and z i is the z coordinate of the ith center beam central axis.
Similarly, the vertical displacement of jth crossbeam y Hj (z, t) is expressed as Center beam

Crossbeam
Bridge deck  where E H is the elasticity modulus of the crossbeam, NI is the total number of center beams, F Iij (t) is the supporting force from the ith center beam to the jth crossbeam, F 0hj (t) is the supporting force from bridge deck to the jth crossbeam, h is the bearing in the displacement box at the end of the crossbeam, and z 0h is the z coordinate of the bearing. e supporting force F Iij (t) can be described as For the supporting forces received by the crossbeam, we can obtain where k 2 , c 2 are the spring stiffness and damping coefficient of the crossbeam bearing. e structural parameters of the MBEJ adopted in simulation are listed in Table 2.

Contact Relationship between the Tire and the MBEJ.
Due to the discontinuity of the top surface of the MBEJ, it is necessary to consider the local separation between the bottom of the tire and the MBEJ at the gap [19]. In order to accurately represent the contact relationship between the tire and the MBEJ, a tire model with the profile function of MBEJ's top surfaces added to the revised flexible roller contact tire model [22,23] is proposed [20], as shown in Figure 2.
e compression deformations of the contact points between the kth tire and the MBEJ in the model are expressed as where y 0 (z k + z k ′ ) is the profile function of MBEJ's top surfaces, z k is the z-direction displacement of the kth wheel axis, z k ′ is the local z coordinate in the contact patch of the kth tire, −r 0 ≤ z k ′ ≤ r 0 , r 0 is the tire free radius, and y Tsk is the static load-deflection of the kth tire. e profile function y 0 (z k + z k ′ ) is given as where B Z , H Z are the width of the center beam top surface and the height of the center beam, respectively, and B is the gap width.
en, the contact force between the kth tire and the MBEJ F Tk (t) and the contact force between the kth tire and the ith center beam are given as where k Tk (z k ′ ), c Tk (z k ′ ) are spring stiffness and damping coefficient of the kth tire vertical distributing, and a ki is the contact patch length between the kth tire and the ith center beam. e tire vertical distributing spring stiffness k Tk and damping coefficient c Tk can be obtained by a simplified formula [24], expressed as where a k is the half of contact patch length between the kth tire and contact surface. e a k can be defined as where f is the load-deflection of the tire tread. When radial tire is under load, the belt will have rigid displacement [25]; i.e., the wheel axis deviates from the tire center. Let f b denote the rigid displacement, the tire loaddeflection of the wheel axis Δy can be expressed as where r is the tire rolling radius.

Numerical Integration Method.
With regard to the vehicle-MBEJ coupled system, the dynamic equation can be described as where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the system; the subscripts V and E represent the vehicle system and expansion joint system; X, V, and A represent the generalized displacement, velocity, and acceleration matrix; R is the generalized force matrix.
e simple fast explicit integration method [26] is used to solve (13). e initial conditions are listed as The i th center beam r 0

Shock and Vibration 5
A test of a four-axle vehicle passing an MBEJ has been completed to verify the accuracy of the vehicle-expansion joint coupled dynamics mathematical model. It shows that the dynamic model simulation results of the vertical velocity at the center beam test points match well with the test results. e deviations of the center beam maximum sinking displacement between the simulation results and the test results are less than 10%. e test method and conclusions have been published in [21] in 2021.

Impact Effect Evaluation
In the study of a vibration system, the impact factor is often used to evaluate the impact effect [8][9][10][14][15][16][17][18][19][20]. In the vibration analysis of the vehicle-MBEJ coupled system, the impact of the dynamic tire loads and the impact of center beam vibration displacements are both very important. erefore, two different impacts are adopted in this paper. In order to evaluate the impact effect of the dynamic tire loads, the dimensionless impact factor μ of the tire load is defined as [18][19][20] where F Tmax is the maximum tire load, and F Ts is the static tire load. en, in order to evaluate the impact effect of the center beam vibration displacements, the dimensionless impact factor IM of the center beam vibration displacements is defined as [8][9][10][11][14][15][16][17] IM � y max − y s y s , where y max is the maximum displacement of the center beam due to the vehicle traveling at designated speed, y s is the static displacement of the center beam due to a vehicle axle stopping directly above the center beam or the maximum displacement due to the vehicle traveling at low speed [9], and the velocity of 10 km/h [10] is adopted for y s in the following study. e impact factor μ can be applied to the design of the bridge local structure, and the impact factor IM can be applied to the design of expansion joints. In most cases, the values of these two factors are equal, but in the vehicleexpansion joint coupled system, we study them separately. e impact factors specified by some bridge design codes vary from country to country. e impact factors specified in Chinese bridge code [27] and European bridge code [28] are 0.3, and that in American bridge code [29] is 0.75.

Dynamic Response of Vehicle.
Before the simulation, we set the basic simulation conditions, where v � 80 km/h, B � 60 mm, k1 � 60 kN/m, and the center line of the left rear wheel contact patch passes through the midpoint of the center beam. e parameters in the following simulations are modified based on the basic conditions. e response of the vehicle wheels when the vehicle passes through the MBEJ is simulated by using the basic simulation conditions, as shown in Figures 3 and 4. Figure 3 shows that both the vibration displacement waveforms and the vibration acceleration waveforms of the left and right tires are basically the same. Due to the large wheelbase, the interaction between the front and rear wheels can be ignored. Figure 4 shows that the dynamic loads of the left and right tires basically agree with each other, and the maximum loads of the front and rear wheels occur at the end of the bridge deck behind the MBEJ, which is consistent with the conclusion of Ding et al. [19]. e tire load impact factors μs of the left front and rear wheels are 0.23 and 0.24, respectively. Figure 5 shows that the vibration displacement response of the MBEJ center beams' midpoints during the vehicle crossing though an MBEJ. Since the rear axle of the vehicle is heavier than the front axle, the maximum displacement of the center beam caused by the crossing rear wheels is greater than that caused by the crossing front wheels. e interaction of the center beam vibrations caused by the front and rear wheels can be ignored due to the large wheelbase. erefore, the actions of the front and rear axles in a vehicle-MBEJ coupled system can be studied separately. e maximum displacement increases from 1st center beam to 5th center beam, which is consistent with the change of tire load on middle beams as shown in Figure 4. e simulation result at the midpoint of center beams is shown in Table 3. It shows that the impact factor IM of each center beam displacement by the crossing front wheels is greater than that of the rear wheels. e impact factors IMs of the 1st center beam displacement by the crossing front and rear wheels are the largest, which are 0.42 and 0.26, respectively. e impact factor IM of the center beam displacement is larger than the impact factor μ of the tire load. e waveform of the dynamic tire loads acting on each center beam is approximately a sine wave, as shown in Figure 6, which verifies the assumption of tire load in [3,[5][6][7].

Parameters Affecting the Impact Factor μ of the Tire Load.
In this part, the effects of the vehicle position, vehicle velocity, gap width, and spring stiffness of center beam bearing on the impact factor μ of the tire load are analyzed. When studying the effect of the vehicle position, the vehicle is considered to pass through an MBEJ from eight positions as shown in Figure 7. In positions 1, 3, 5, and 7, the left rear wheel of the vehicle passes directly above the crossbeams, and in other positions, the left rear wheel of the vehicle passes over the midpoints of the center beams span. e calculation results of the tire load impact factors μs are shown in Figures 8-11.

Shock and Vibration
with the vehicle velocity, and the maximum tire load impact factors μs of the left front and rear wheels are 0.33 and 0.34, respectively, which are greater than 0.3 but less than 0.75. e gap width of the MBEJ will affect the tire load amplitude on a center beam and the time in which the wheel passes through a center beam. As the gap width increases, the maximum tire load acting on a center beam increases, causing the maximum displacement of the center beam to increase, which eventually leads to an increase in the impact of the tire load. erefore, the tire load impact factors μs of the vehicle wheels rise with the increase of gap width, as shown in Figure 10. e maximum tire load impact factors μs of the left front and rear wheels are 0.34, which are greater than 0.3 but less than 0.75. Figure 11 shows that the tire load impact factors μs of the vehicle wheels decrease slowly with the increase of the center    beam bearing's spring stiffness. is is because as the spring stiffness of the center beam bearing increases, the maximum sinking displacement of the center beam gradually decreases, resulting in a decrease in the axle displacement. However, when the spring stiffness of the center beam bearing increases to more than 80 kN/mm, this decreasing trend becomes slower. erefore, in the design of the MBEJ, the center beam bearing's spring stiffness of 80 kN/mm can be a preferred recommended value.

Parameters Affecting the Impact Factor IM of the Canter Beam Displacement.
To evaluate the effects of the vehicle position, vehicle velocity, gap width, and spring stiffness of center beam bearing on the impact factor IM of the center beam displacement, a series of simulations are performed, and the calculation results are shown in Figures 12-15. Figure 12 illustrates the effect of vehicle position on the impact factors IMs of the canter beam displacement. Similarly to impact factor μ of the tire load (see in Figure 8), the        Figure 13 shows that the impact factor IM of each center beam displacement fluctuates with the increase of vehicle velocity when the vehicle velocity is less than 60 km/h. However, when the vehicle velocity is greater than 60 km/h, the impact factor IM of each center beam displacement increases rapidly. e influence of vehicle velocity on the impact factor IM is complex [15], which may be related to the contact time between the wheel and center beam. When the vehicle passes the MBEJ at 120 km/h, the factor IM of 1st center beam displacement by the crossing front and rear wheels is the largest, which is 0.55 and 0.42, respectively. Both are greater than 0.3 but less than 0.75. Figure 14 shows that the impact factors IMs of all center beams displacement except 3rd beam first decrease and then increase with the increase of the gap width, which is due to the influence of the gap width on the contact time and tire load between the wheel and a single center beam. e impact factor IM of the 3rd center beam fluctuates with the increase of the gap width. When the gap width is 80 mm, the impact factor IM of 1st center beam displacement by the crossing front and rear wheels reaches maximums of 0.66 and 0.49. Both are greater than 0.3 but less than 0.75. e effect of the center beam bearing's spring stiffness on the impact factor IM of the center beam displacement is shown in Figure 15. e impact factor IM of each center beam displacement decreases with the increase of the center beam bearing's spring stiffness. However, the decreasing trend slows down when the spring stiffness of the center beam bearing increases to greater than 80 kN/mm. Similar results were obtained by Li et al. [10].

Conclusions
In this paper, a dynamic mathematical model of a vehicleexpansion joint coupled system is established with a flexible roller tire model. e numerical simulations are carried out with different system parameters to study the dynamic response of the coupled system. Two impact factors μ and IM are defined to evaluate the dynamic impact effects of the tire load and the center beam displacement. e following conclusions can be drawn from the studies: (1) For the 2-axle vehicle, the interaction between the front and rear wheels can be ignored when the vehicle passes through the MBEJ, so they can be studied separately. e impact factor μ increases with the vehicle velocity and the gap width. is implies that limiting the vehicle velocity contributes to improving the vehicle safety, especially in winter with a big gap.
(2) For the bridge expansion joint, the tire loads on the edge beam and the concrete at the end of the bridge deck are the largest. e strength of the connection between the MBEJ and the bridge deck should be considered during the design of an MBEJ. e impact effects of the vehicle and the expansion joint can be improved when the vehicle wheel passes directly above the crossbeam. erefore, the position design of the crossbeams should take into account the distribution of tire road imprints during the design of an MBEJ. However, the relationships of the impact factor IM with the speed and the gap width are not monotonous functions. When the vehicle speed is 60 km/h or the gap width is 40 mm, the impact factor IM will decrease suddenly. e increase of the center beam bearing's spring stiffness can reduce the impact of both vehicle and expansion joint.
(3) e calculated impact factor IM is greater than the impact factor μ. erefore, a distinction should be made when using impact factors for static calculations. e maximum impact factor μ can reach 0.34, and the maximum impact factor IM can reach 0.58. Both impact factors may exceed the values specified by the Chinese or European bridge codes.
e simulation results will provide useful guidance for the design and maintenance of expansion joint and bridge. However, the interaction between the parameters and the influence of different types of vehicles on the vehicle-expansion joint coupled system have yet to be achieved. ese issues remain potential topics of future work.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.