Vehicle–Bridge Coupling Vibration Analysis of a Highway Pile–Slab Bridge Based on the Contact Constraint Method

Abstract

To investigate the impact of vehicle load on highway pile–slab bridges, the contact constraint method is employed to treat the vehicle and the bridge as two independent subsystems. Through the formulation of point-to-surface contact and constraint equations, a vehicle–bridge coupling vibration analysis is performed, incorporating the effects of bridge deck roughness. The finite element method is utilized to construct the pile–slab bridge model, while the five-axis heavy vehicle model is developed based on the multi-rigid-body dynamics method. The analysis and computational results of the model reveal the effects of pier height, vehicle number, and the friction coefficient on the dynamic response of the pile–slab bridge. The results indicate that pier height significantly influences the dynamic response, and the appropriate pier height should be carefully determined during the design phase. The vertical displacement impact coefficient surpasses the design value derived from the specification, highlighting the need to consider the vehicle’s impact on the bridge. Furthermore, vehicle number and the friction coefficient significantly affect the longitudinal dynamic response and transverse acceleration response of the pile–slab bridge.

1. Introduction

In the construction of expressways in flat areas, the limited availability of land resources can make excavation difficult in terms of roadbed filling. Therefore, a prefabricated highway pile–slab bridge structure is proposed as a solution to this issue [1]. Compared to conventional bridges, pile–slab bridges are lighter in overall weight, with the live load effect in a single span comprising a larger proportion of the total dynamic response. During the structural scheme demonstration, the impact of vehicle load on bridge structures has garnered significant attention [2].

In recent years, the vibration behavior of bridges under external loads has garnered significant attention, particularly in the domain of vehicle–bridge coupling vibrations [3,4,5,6,7]. Deng et al. [8] employed a typical three-axis vehicle model to investigate vehicle–bridge coupling vibrations during braking, analyzing the effects of braking position, deceleration rate, and initial speed on the impact coefficient. Zhu et al. [9] proposed an algorithm that integrates the pseudo-excitation method with adaptive Gauss quadrature to address the random response of the vehicle–bridge system. Through comparison with other studies, the effectiveness and accuracy of the algorithm in solving the random response of the vehicle–bridge system are validated.

Jiang et al. [10] developed a vehicle–bridge-wind coupling system model, accounting for the interaction forces between vehicles and bridges. This model incorporates the vehicle dynamics, bridge structural response, and wind load and is used to compute the bridge’s response under complex loading conditions. Based on Timoshenko beam theory, Wang et al. [11] simplified the vehicle as a spring-damper-mass system and derived the coupling equations with the bridge through the interaction force. The dynamic response of continuous girder bridges with varying spans and cross-sections is determined using the modal superposition method. Stoura et al. [12] employed the Dynamic Partition Method (DPM) to reformulate the vehicle–bridge coupling problem as a system of second-order ordinary differential equations (ODEs). By introducing auxiliary contacts to manage the contact nodes, it was demonstrated that DPM can substantially reduce computational costs. Guo et al. [13] proposed a vehicle–bridge coupling model to predict the dynamic response of long-span curved girder bridges, employing the Newmark-β method to solve the system’s vibration response. The effects of deck roughness, vehicle speed, and load mass on the impact coefficient were systematically analyzed. Zhang et al. [14] developed a dynamic mathematical model for the coupling system of a biaxial vehicle and a modular bridge expansion joint, analyzing the effects of vehicle position and speed on tire dynamic load and center beam vibration displacement. Wang et al. [15,16] employed a vehicle–bridge coupling vibration analysis method that incorporated vehicle jumping to assess the dynamic performance of an in-service curved chord truss bridge. They evaluated the bridge both before and after reinforcement, examining the reinforcement effect and the applicability of various indices.

Zhao et al. [17] calculated the impact of bridge deck roughness on the dynamic response of key bridge components under multi-vehicle traffic flow using an iterative solution method. Liu et al. [18] developed a model based on the integrated simulation platform and interface contact algorithm to describe the vehicle jumping process. The study considers the jump height and vehicle speed, analyzing the impact of the jump on wheel contact force and bridge vibration displacement response. González et al. [19] analyzed the influence of the mass, stiffness, and damping matrices of the vehicle–bridge coupling system on its response using finite element modeling. They also examined the relationship between bridge damage and the forced vibration mode. Li et al. [20] developed a program using Intel Visual Fortran for vehicle–bridge coupling vibration analysis of highway bridges. The program’s reliability was validated through experimental tests. The study investigated the effects of lane position, vehicle–bridge mass ratio, and bridge support configuration on the dynamic response.

The literature review reveals that current research on vehicle–bridge coupling vibration predominantly relies on numerical simulations. The focus is gradually shifting towards developing more realistic and accurate simulation methods, as well as more representative vehicle–bridge coupling models. Due to the lack of specialized commercial software, some researchers opt to study vehicle–bridge coupling vibration through self-programming. However, self-programming is not only complex but also lacks versatility. A new trend has emerged to develop vehicle–bridge coupling vibration analysis methods based on existing general finite element software platforms [21,22,23,24,25,26,27]. This method significantly reduces programming difficulty and researchers’ workload. The contact constraint method [28], a vehicle–bridge coupling vibration analysis method that accounts for bridge deck roughness, offers several advantages. It does not require extensive shape function interpolation and is well-suited for complex bridge finite element models, making it widely recognized by researchers.

Although significant progress has been made in research on vehicle–bridge coupling vibration, the influence of vehicle–bridge friction on structural response has not yet been considered. In certain structures with substantial live load effects, vehicle–bridge friction has emerged as a factor that cannot be overlooked. Furthermore, the pier height significantly impacts structural performance. If too low, the structure becomes overly rigid with poor damping; if too high, it results in excessive flexibility and difficulties in controlling deformation. Therefore, it is essential to explore a reasonable range for pier height. In this paper, the contact constraint method is employed to study the vehicle–bridge coupling vibration of a highway pile–slab bridge. The influence of pier height, vehicle number, and the friction coefficient between the vehicle and bridge on the vibration response is analyzed, providing theoretical support for the structural optimization of pile–slab bridges.

2. Establishment of a Finite Element Model for Highway Pile–Slab Bridge

2.1. Overview of the Project

In this study, the standard span of the bridge is 8 m, and the standard adjacent bridge length is 15 × 8 m. The main beam consists of a C40 reinforced concrete prefabricated structure, while the transverse bridge comprises four precast slabs, each 2.9 m wide, connected by three longitudinal wet joints, resulting in a total width of 12.81 m. In the longitudinal direction, a 0.4 m middle cross beam is positioned atop the continuous pier, while a 0.4 m end cross beam is placed on the transition pier, with a transverse joint width of 0.4 m at the pier top. The substructure utilizes prestressed high-strength concrete pipe piles. Each transverse row of the bridge comprises four pipe piles spaced at 3.303 m. PRC-I 500 AB and PHC 500 AB pipe piles are employed. The pipe piles have a diameter of 500 mm and a wall thickness of 100 mm and are constructed using C80 concrete. The continuous pier is pier-girder consolidation, while the transition pier is connected via rectangular skateboard rubber bearings. Figure 1 illustrates the side span and secondary side span of the pile–slab bridge.

2.2. Overall Static Analysis of the Spatial Beam Gird Model

The spatial beam gird model of the pile–slab bridge is established, as illustrated in Figure 2. In the model, the bridge deck and the pipe pile are simulated by the beam element, and the rigid connection is realized by setting the rigid arm between the pile top and the bridge deck. The transverse connection between the main beams is realized by the virtual beam. The pipe piles beneath the soil layer are also simulated using beam elements. Consolidation constraints are applied to the nodes at the bottom of the piles, while soil springs are introduced at other nodes to simulate pile–soil interactions.

The static calculation results for the side beam and middle beam of the pile–slab bridge spatial beam gird model are compared, as shown in Figure 3. The maximum deflection and the largest positive bending moment occur in the side span middle of the side beam, where the positive bending moment due to vehicle load constitutes 77% of the total effect. The maximum negative bending moment and the largest shear force are observed at the fulcrum of the secondary side pier of the middle beam. The negative bending moment due to vehicle load constitutes 58%, while the shear force accounts for 73%. The maximum bending moment of the pipe pile occurs at the top of the secondary side pier of the side beam, with the bending moment caused by uniform temperature constituting 53%. The live load effect of bending moment and shear force accounts for a large proportion of the total effect, indicating that excessive live load can significantly elevate the structural safety risk.

2.3. Multi-Scale Finite Element Model

This paper focuses on detailed modeling of the side span and secondary side span. The aim is to accurately analyze the dynamic response of the most critical part of the pile–slab bridge under vehicle loads. The beam slab and pipe pile are modeled using the Solid185 element. This element offers high versatility and precise three-dimensional simulation capabilities, which help analyze complex structures and handle issues like large deformation and stress stiffening. The beam slab and pipe pile are rigidly connected through common nodes. Since other spans are not the focus of this study, the Shell181 element is used for beam slab sections to improve computational efficiency. This element accounts for both bending and membrane stiffness. It is suitable for simulating thin to medium-thickness shell structures and can handle nonlinear analysis with large rotations. The pipe pile above the soil layer is modeled using the Beam188 element, which reflects the slenderness ratio of the pile. The PRC pipe pile beneath the soil layer is also modeled with the Beam188 element, which also considers the pile–soil interaction and improves calculation efficiency.

The grid type and density in the model significantly affect both the calculation accuracy and efficiency. A hexahedral mesh is used for the most critical parts to ensure computational stability and higher accuracy, with a mesh density of 0.1 m. A tetrahedral mesh is used for the other parts. To improve computational efficiency, the mesh size is set to 0.2 m.

The connection between the solid, shell, and beam elements uses the MPC (multi-point constraint) contact method [29]. Vertical displacement constraints are applied at the bearing of the solid element. Consolidation constraints are applied at the bottom nodes of the pipe pile beneath the soil layer, while soil springs simulate the pile–soil interaction at the other nodes. An approach bridge model is created to simulate vehicles entering and leaving the bridge. Forced displacement is applied to each node of the vehicle model at each time step to simulate its motion on the bridge. The displacement is calculated using the uniform variable motion formula, $S=v_{0}t-1/2a{(t-t_0)}^2$, where $S$ is the longitudinal horizontal displacement applied to the vehicle model, and $t_0$ is the initial duration of uniform motion. The multi-scale finite element model of the pile–slab bridge is shown in Figure 4.

The secondary load is converted into mass, and the natural vibration characteristics of the pile–slab bridge are determined using the Block Lanczos method. The first seven natural vibration characteristics of the pile–slab bridge are presented in

Table 1. The first seven natural vibration characteristics of the pile–slab bridge.

Mode Number	Solid Model Frequency (Hz)	Spatial Beam Gird Model Frequency (Hz)	Mode Shape Description
1	0.728	0.71	First-order antisymmetric transverse vibration
2	0.809	0.75	First-order symmetrical transverse vibration
3	0.811	0.76	First-order longitudinal vibration
4	2.773	2.25	Second-order symmetric transverse vibration
5	6.981	4.58	Second-order antisymmetric transverse vibration
6	11.208	7.14	First-order symmetric vertical bending vibration
7	11.280	9.70	First-order antisymmetric vertical bending vibration

.

As shown in Table 1, compared to the spatial beam gird model, the solid model accounts for the actual size effect of the structure, demonstrating greater structural stiffness and a more significant impact on higher-order frequencies. However, generally, the deviation in the first three frequencies between the two models is small, and the first five vibration modes are consistent, indicating that the established pile–slab bridge model accurately reflects the dynamic characteristics of the bridge. The first three frequencies of the pile–slab bridge are low, indicating weak transverse and longitudinal stiffness, which makes the bridge susceptible to transverse and longitudinal vibrations.

3. Analysis Methods for Vehicle–Bridge Coupling Vibration

3.1. Equation of Vehicle–Bridge Coupling Vibration

Using the multi-rigid-body dynamics method and the finite element method, the vehicle and the bridge are modeled as multi-node finite element models. The vibration equation of the vehicle is derived using the generalized virtual work principle and finite element theory, while the vibration equation of the bridge is obtained using finite element theory. The vehicle and bridge vibration equations are coupled by the interaction force at their contact node, forming the vehicle–bridge coupling vibration equation, which is represented by the following formula:

$$\begin{array}{c}{M}_v\ddot{z}+C_v\dot{z}+K_{v}z=F_{bv}+F_{vg}\\ M_b\ddot{y}+C_b\dot{y}+K_{b}y=F_{vb}\end{array}$$

(1)

where $M_v$, $C_v$ and $K_v$ represent the matrices of mass, damping, and stiffness of the vehicle, respectively; $M_b$, $C_b$ and $K_b$ represent the matrices of mass, damping, and stiffness of the bridge, respectively; $z$, $\dot{z}$ and $\ddot{z}$ denote the vectors of displacement, velocity and acceleration of the vehicle system nodes, respectively; $y$, $\dot{y}$ and $\ddot{y}$ denote the vectors of displacement, velocity and acceleration of the bridge system nodes, respectively; $F_{vg}$ is the node load column vector of the vehicle system due to the vehicle’s gravity; $F_{bv}$ and $F_{vb}$ are the load vectors for the vehicle and bridge system nodes, respectively, derived by distributing the contact force at the vehicle–bridge interface through the shape function. The contact force is a function of the vehicle displacement at the vehicle–bridge contact node, bridge displacement and bridge deck roughness [30].

3.2. Vehicle–Bridge Coupling Vibration Model

The vehicle model employs a three-dimensional multi-degree-of-freedom space model, where the vehicle body, frame, wheel, and suspension system are represented by mass, spring, damper, and rigid beam elements, respectively.

As illustrated in Figure 5, the contact constraint method treats the vehicle and the bridge as two independent subsystems, coupling them through force balance and displacement coordination at the contact nodes. This method is based on the displacement contact method [31], where a coincident node is established at the wheel bottom, and the Conta175 element is employed to generate the contact node. Simultaneously, the Targe170 target element is defined on the bridge deck unit over which the vehicle moves to establish point-to-surface contact. A constraint equation is then formulated between the contact node and the wheel bottom node, with its constant term representing the roughness of the bridge deck, simulating the impact of bridge deck roughness during vehicle motion.

In the absence of any separation between the wheel and the bridge deck, the displacement coordination relationship between the vehicle and the pile–slab bridge is represented by the following formula:

$$Y_{vi}-Y_{bi}=R_{vi}$$

(2)

$Y_{vi}$ represents the vertical displacement of the vehicle wheel bottom node, while $Y_{bi}$ denotes the vertical displacement of the bridge at the contact node of the wheel bottom. Additionally, $R_{vi}$ refers to the bridge deck roughness at the same contact node.

The harmonic superposition method simulates the response of complex signals, and time-domain samples of bridge deck roughness are obtained through inverse Fourier transform to generate A, B, and C-grade bridge deck roughness samples, as shown in Figure 6.

3.3. Analytical Process

The process of simulating vehicle–bridge coupling vibration for the pile–slab bridge, based on the contact constraint method, is outlined as follows:

(1)

The finite element model of the bridge is constructed, and approach bridge models of specified lengths are established at both ends of the bridge model.

(2)

Define the parameters of the vehicle and the lane and construct a multi-rigid-body model for the vehicle in space.

(3)

Generate bridge deck roughness samples using a programming language and input these samples into the table array of the finite element software.

(4)

Point-to-surface contact is established between the contact node and the bridge deck unit.

(5)

In the solution layer, the analysis type is defined as transient analysis, and the constraint equation is established between the wheel bottom node and the contact node. The constant term of the constraint equation is set to zero, and the bridge deck roughness is not considered. The effects of structural gravity, secondary pavement, and uniform temperature are incorporated. The static analysis of the vehicle–bridge coupling model is carried out by closing the time integration in the first two load steps. In the first load step, the load is applied, and initial conditions are established, while in the second load step, the initial velocity of the structure is eliminated.

(6)

In subsequent load steps, enable the time integration, define the appropriate integration time step and the number of sub-steps of the load step, specify the load as the slope load, and define the mass damping coefficient. For each load step, apply the displacement constraint along the driving direction to all nodes and contact nodes of the vehicle model. Update the constraint equation with its constant term, which is defined as the bridge deck roughness value at the corresponding location.

(7)

Repeat step 6 until the vehicle has completely traversed the bridge.

(8)

Complete the solution and proceed to time–history post-processing analysis for the results.

4. Analysis of the Influencing Factors Affecting Vehicle–Bridge Coupling Vibration in Pile–Slab Bridges

The typical five-axle heavy vehicle model, combined with the multi-scale model above, is used to analyze vehicle–bridge coupling vibration [32]. This model is a mass-spring-damping system commonly used in vehicle–bridge coupling vibration analysis. The vehicle body is simplified as a mass point at the center of mass with rotational inertia, and the wheel is modeled as a mass point with only mass. Both are simulated using the Mass21 element. The interaction between the wheel center of mass and wheel bottom, as well as the suspension system between the frame and wheel, is simplified as a spring-damper system simulated using the Combin14 element. The frame is simulated using the Beam4 element. The front and rear body connections are simulated using the Combin14 spring element. The vehicle model is shown in Figure 7, with parameter values provided in

Table 2. Calculation parameters of the five-axle heavy vehicle model.

Parameter	Value	Parameter	Value
Quality of vehicle body 1/kg	2276.5	Suspension damping/(kN·s·m−1)	10~53
Quality of vehicle body 2/kg	45,246	Tire damping/(kN·s·m−1)	3
Pitching moment of inertia for vehicle body 1/(kg·m2)	20,196	Spacing of vehicle body/m	2.5
Pitching moment of inertia for vehicle body 2/(kg·m2)	285,900	Wheel spacing/m	1.8
Rollover moment of inertia for vehicle body 1/(kg·m2)	2189.2	Axle spacing/m	1.4~7
Rollover moment of inertia for vehicle body 2/(kg·m2)	43,512	Spacing between vehicle body centroid, axle, and hanging shaft/m	1~4.5
Quality of axle suspension/kg	700~1000	Height difference between vehicle body centroid and hanging shaft/m	0.1~1
Suspension stiffness/(kN·m−1)	300~1250	Height difference between vehicle body centroid and wheel centroid/m	1
Tire stiffness/(kN·m−1)	1500~3000	Height difference between wheel centroid and wheel bottom/m	1.2

.

To study the extreme and most unfavorable conditions during the operation of the pile–slab bridge, this paper applies a maximum uniform temperature rise of 34 °C and arranges three lanes laterally with a five-axle heavy vehicle, as shown in Figure 8.

Since the maximum deflection and the largest positive bending moment occur in the side span middle of the side beam, measuring point B1 is located at the bottom of the longitudinal wet joint in this region. Moreover, the displacement of the end bearing of the multi-span pile–slab bridge is considerable along the longitudinal direction, which could lead to bearing damage. Consequently, measuring point B2 is placed at the bottom of the right end bearing of the middle beam slab. Simultaneously, a rectangular skateboard rubber bearing is employed at the transition pier, with a vertical displacement constraint applied in the model. As shown in Table 1, the transverse stiffness of the multi-span pile–slab bridge is low, making it prone to transverse vibrations due to the influence of vehicle loads. Therefore, measuring point T1 is located at the lower flange of the left and right bearings of the end of the side beam slab. According to the “General Specification for Design of Highway Bridges and Culverts” in China, the impact coefficient of the vehicle load in this study is calculated to be 0.05, based on the structural fundamental frequency [33].

4.1. Analysis of the Effect of Pier Height

A row of vehicles passes through the entire bridge at a constant speed of 120 km/h. The friction coefficient between the vehicle and the bridge is 0.02, while the pier heights are 5.4, 8.4, and 11.4 m, respectively. The bridge deck has a roughness grade of C. The structural weight, second-stage pavement, and uniform temperature effect are also considered. The dynamic response of the pile–slab bridge at different pier heights is presented in Figure 9 and Figure 10, with the acceleration response shown in Figure 11.

As shown in Figure 9, when the front wheel of the vehicle reaches the top of the second row of side piers, the vertical displacement at the side span middle of the beam slab reaches its maximum value. As the pier height increases, the displacement impact coefficient rises by 0.24%, while the maximum dynamic displacement and maximum acceleration decrease by 19.83% and 40.13%, respectively. This suggests that increasing the pier height is an effective means to control the vertical dynamic response of the side span in the middle of the beam slab. The impact coefficient of vertical displacement is more than 17.33% higher than that calculated by the specification. As shown in Figure 10, as the pier height increases, the maximum value of the longitudinal dynamic displacement at the right bearing of the beam slab increases by 6.41%. The amplitude of the dynamic displacement first decreases by 12.50%, then increases by 114.28%. However, the maximum acceleration decreases by 61.33%. This indicates that controlling the pier height between 5.4 and 8.4 m helps optimize the longitudinal dynamic response of the pile–slab bridge, utilizing the structure’s flexibility to reduce vibration. As shown in Figure 11, the transverse acceleration response at the left and right bearings of the beam slab gradually decreases as the pier height increases, with a change range of 43.71% to 86.88%. This indicates that increasing the pier height reduces the transverse stiffness of the structure, making the transverse direction of the bridge more flexible, thereby significantly reducing the transverse vibration of the beam slab.

4.2. Analysis of the Effect of Vehicle Number

The vehicle passes across the entire bridge at a constant speed of 120 km/h. The longitudinal spacing between vehicles is 15 m, with the number of rows being 1 and 2, respectively. The friction coefficient between the vehicle and the bridge is 0.02, the pier height is 8.4 m, and the bridge deck has a roughness grade of C. Additionally, the structural weight, second-stage pavement, and uniform temperature effect are considered. The dynamic response of the pile–slab bridge under varying vehicle numbers is presented in Figure 12 and Figure 13, with the acceleration response shown in Figure 14.

As shown in Figure 12, when the vehicle passes through the side span and secondary side span of the pile–slab bridge, the vertical displacement of the side span middle of the beam slab reaches its peak value, and the shape of the time–history curve of displacement is similar. The maximum single variation in vertical dynamic response is 1.54%, which can be attributed to the relatively short standard span of the pile–slab bridge and its high vertical stiffness. Therefore, when the longitudinal spacing between vehicles is appropriate, the cumulative superimposed effect resulting from an increase in vehicle numbers remains small. The impact coefficient of vertical displacement is more than 17.52% higher than that calculated by the specification. As shown in Figure 13, the increase in the vehicle numbers leads to a significant rise in the maximum acceleration of the longitudinal bridge at the right bearing of the beam slab, with an increase of 312.50%. As shown in Figure 14, as the vehicle numbers increase, the maximum acceleration difference exhibits a cumulative superimposed effect. When the number of vehicle rows increases from 1 to 2, the increase reaches 57.81%, which exacerbates the relative vibration of the beam slab at both ends in the transverse direction of the bridge.

4.3. Analysis of the Effect of the Friction Coefficient Between Vehicle and Bridge

A row of vehicles passes across the entire bridge at a constant speed of 120 km/h. The friction coefficient between the vehicle and the bridge is 0, 0.02, and 0.04. The pier height is 8.4 m, and the bridge deck has a roughness grade of C. Additionally, the structural weight, second-stage pavement, and uniform temperature effect are considered. The dynamic response of the pile–slab bridge under varying friction coefficients is presented in Figure 15 and Figure 16, with the acceleration response shown in Figure 17.

As shown in Figure 15, the maximum single amplitude change in the vertical dynamic response of the side span middle of the beam slab is 0.12%, indicating that it is less sensitive to variations in the friction coefficient between the vehicle and the bridge. The impact coefficient of vertical displacement is more than 17.43% higher than that calculated by the specification. As shown in Figure 16, as the friction coefficient between the vehicle and the bridge increases, the dynamic response of the longitudinal bridge at the right support of the beam slab shows an increasing trend. The change in dynamic displacement amplitude is most significant, with increases of 65.48% and 118.71%, respectively. As shown in Figure 17, when the vehicle passes through the side span and secondary side span of the pile–slab bridge, the transverse acceleration response at the left and right bearings of the beam slab reaches its peak, with no significant change during the middle section of travel. The maximum acceleration difference is highly sensitive to variations in the friction coefficient between the vehicle and the bridge. The single variation range is 27.29% to 35.85%, with the peak value reaching 0.557 m·s⁻² when the friction coefficient is 0.02.

5. Conclusions

(1)

The pier height significantly influences the dynamic response of the pile–slab bridge. Numerical simulation results show that the vertical dynamic response and transverse acceleration response decrease as pier height increases. However, a higher pier may increase the longitudinal dynamic response. Therefore, during the design phase, it is crucial to consider the impact of pier height on various dynamic responses and determine an optimal range.

(2)

For bridge deck roughness grade C, the vertical displacement impact coefficient exceeds the specified value. Consequently, during bridge operation and maintenance, attention should be given to protecting the bridge deck to minimize impact. Additionally, the vehicle numbers and the friction coefficient between the vehicle and bridge significantly affect the longitudinal dynamic response and transverse acceleration response. In practice, this can be managed by coating the bridge deck and controlling lane distribution.

While this study focuses on vehicle–bridge coupling vibration of pile–slab bridges, existing research is limited due to the relatively new nature of this structure and the lack of comparable engineering examples. Future research will further examine the impact of repeated loading on the fatigue life of pile–slab bridges, particularly under complex vehicle loads [34,35,36].