Effects of high-speed train excitations on low-pylon cable-stayed bridge vibration considering vehicle-bridge coupling

Abstract

The train operations on railways will inevitably induce cross-railway bridge vibrations, and it is necessary to examine the effects of train-induced vibrations on the cross-railway vehicle-bridge coupling system. In this context, this study focuses on a practical low-pylon cable-stayed bridge to analyze the behavior of the vehicle-bridge coupling system under train passage. By employing theoretical analysis, vibration experiments, and numerical simulations, a comprehensive study is conducted to analyze the vibration response of the vehicle-bridge coupling system under train-induced vibrations. The results demonstrate that vertical vibration acceleration levels at different measurement points on the bridge deck are significantly more pronounced than the horizontal vibration levels. Train passage conditions beneath the bridge exert a noteworthy influence on the vibration acceleration levels and the frequency distribution range of the bridge and vehicle. The train speed, axle load, and driving direction each have varying degrees of impact on the vibration displacement and acceleration of the vehicle-bridge coupling system, emphasizing the importance of considering these factors.

Keywords

Train-induced vibration low-pylon cable-stayed bridge vehicle-bridge coupling system acceleration response vibration test

Introduction

With the rapid development of the economy, there is an increasing demand for road bridges over railway tracks, in which the railway traffic inevitably affects the vibration level of the over-across vehicle-bridge coupling system.^1–3 Factors such as train axle load, speed, and driving direction affect the magnitude of environmental vibrations from railways. Consequently, the effects of railway environmental vibrations beneath bridges on the vibration properties of the vehicle-bridge system are intricate. Therefore, it is necessary to research the vibration response of the vehicle-bridge coupling system over the railway line under the influence of railway environmental vibrations.

Scholars have conducted theoretical research, simulation, and field testing on the railway environment vibration source system and have conducted extensive research on the vibration characteristics of railways and vibration reduction measures.^4–7 Zhou et al.⁸ considered the dynamic interaction of adjacent tunnels, treated the vehicle as a multi-body system, used the Euler beam to simulate the track plate, assumed the soil as a saturated porous medium, and established the corresponding 3D coupled analytical model. Li et al.⁹ considered the coupled losses of soil and building structures, proposing a hybrid method combining Bayesian neural networks and impedance models to predict internal building vibrations. Colaço et al.¹⁰ conducted extensive experimental analysis on building structures near the railway line at the Carregado site in Portugal. Based on a comprehensive study of theory and experiments. Tao et al.¹¹ utilized the Monte Carlo method to quantify the uncertainty in the propagation of ground vibration caused by train-induced ground vibration resulting from spatial variations in soil properties and changes in train speed. He et al.¹² introduced a new two-dimensional semi-analytical approach to compute the ground vibration generated by tunnels in a homogeneous half-space with irregular surfaces. Wang et al.¹³ suggested the implementation of multiple suspension devices with dynamic vibration absorption properties to mitigate the lateral vibrations of high-speed train carriages, aiming to enhance their operability. Also, the vibration issues caused by high-speed train operations in buildings, including environmental vibration and noise generation mechanisms, the propagation laws and attenuation characteristics of vibrations, and prevention and control measures for environmental vibration problems, have received extensive attention.^14–17 Wang et al.¹⁸ investigated the key characteristics of ground vibrations generated by high-speed railways and subways utilizing the substructure method based on the 2.5D FEM-MFS (finite element method-the method of fundamental solutions). They conducted a detailed examination of the vertical vibration behavior of buildings induced by train operations. Furthermore, Sun et al.¹⁹ assessed the viability of novel metamaterial-based Vibration Suppression Stories in mitigating structural vibrations caused by trains from the perspective of vibration propagation. Wang et al.²⁰ introduced a methodology to detect the unstable state of hunting motion in freight trains by implementing the Hunting Coefficient index within the wayside hunting detection system (WHDS). Moreover, Hu et al.²¹ presented a predictive model for vibrational analysis on overhead track structures of subway vehicle segments utilizing a Back-Propagation neural network. Liu et al.²² proposed a deep learning-based technique for effectively identifying train-induced vibration periods to improve the accuracy of vibration assessment.

Recently, research on vehicle-bridge coupling vibration and train-bridge interaction vibration has received extensive attention.^23–29 Zhu J et al.³⁰ evaluated the driving safety of moving vehicles on coastal bridges and the structural safety of the bridges, and assessed the ride comfort of vehicles using the ISO 2631-1 standard. Gou et al.³¹ investigated the influence of bridge pier settlement and beam creep deflection, two typical long-term deformations of the bridge, on the vibration of high-speed trains. Wu et al.³² constructed a comprehensive analysis framework for the ice-wind-vehicle-bridge interaction and investigated the vibration response of this coupled system. The study of coupling vibration of train–vehicle–bridge system is introduced.^33,34 Xiang et al.³⁵ established the model of coupled vibration of the train-vehicle-bridge system considering train-induced winds. Lei et al.³⁶ developed an analytical framework to simulate the complex wind-road vehicle-train-bridge system. They proposed a scalable multiple-time-step algorithm to overcome the computational efficiency issue of traditional single-time-step algorithms for solving the complex wind-road vehicle-train-bridge system interaction. Wang et al.³⁷ utilized a multi-objective multi-parameter optimization approach by integrating the improved niche genetic algorithm into the numerical simulation procedure. This optimization significantly enhanced the vehicle’s vibration reduction capabilities and effectively managed equipment vibration within a reasonable range. Similarly, Song et al.³⁸ examined the impact of crosswind-induced bridge vibrations on vehicles within the wind-vehicle-bridge system framework. By formulating the fundamental equations of the vehicle model, they derived three critical speed formulas for wind-induced vehicle incidents. Furthermore, Zhang et al.³⁹ investigated the aerodynamic coefficients of three typical highway vehicles under two scenarios: traveling on a twin-box girder bridge deck and a flat surface, using the overlapping dynamics mesh technology in computational fluid dynamics simulations. Li et al.⁴⁰ conducted tests on a truck-bridge deck coupled model under vortex-induced vibrations inside a large wind tunnel, exploring the impact of turbulence on the average aerodynamic forces, peak aerodynamic forces, and acceleration responses of trucks on the vortex-induced vibration bridge deck. The research on the train-vehicle-bridge system mainly focuses on road-rail bridges with similar upper and lower bridge spans. The low-pylon cable-stayed bridge^41,42 is a new type of bridge between continuous beam bridges and cable-stayed bridges, and the research on the mechanism of the vehicle-bridge coupling is limited. Hence, investigating the vibration response from vehicle-bridge coupling of low-pylon cable-stayed bridges under the influence of train loads contributes to advancing its promotion and structural design.

As mentioned above, most studies have focused mainly on the vibration characteristics of single vehicle-bridge coupling or railway environments. In addition, some studies have concentrated on the horizontal vibration effects of wind, earthquakes, and ship impacts on vehicle-bridge coupling systems. In contrast, vertical vibration is the predominant sustained vibration in railway environments. With an increasing number of cross-railway bridges, the vibration effects of railway environments on the vehicle-bridge coupling systems of cross-railway bridges are becoming more prevalent. Despite this, the research in this specific area remains limited. In this context, this study conducts vibration testing and experimental research on the vehicle-bridge coupling system during the passage of trains in a practical low-pylon cable-stayed bridge project over-across two railway lines. Examine the vibration response characteristics of the vehicle-bridge coupling system under various train and vehicle driving conditions. Furthermore, a three-dimensional coupling model is established to analyze the vibration response of the vehicle-bridge coupling system under the influence of railway environmental vibration, specifically exploring the effects of train speed, axle load, and driving mode on the acceleration of the vehicle-bridge coupling system. The research findings provide valuable technical support for studying vehicle-bridge coupling vibration in cross-railway bridges.

Theoretical background

Interaction between vehicle and bridge

The vehicle body, suspension, and wheels are considered as rigid bodies with six degrees of freedom in three-dimensional space. The dynamic equation of the vehicle can be expressed as follows:

[M_{v}] {{\ddot{Z}}_{v}} + [C_{v}] {{\dot{Z}}_{v}} + [K_{v}] {Z_{v}} = {F_{v}}

(1)

where [M_v], [K_v], and [C_v] represent the mass, stiffness, and damping matrixes of the vehicle, respectively.

{Z_{v}}

{{\dot{Z}}_{v}}

, and

{{\ddot{Z}}_{v}}

denote the displacement, velocity, and acceleration matrixes of the vehicle, respectively. {F_v} is the external load matrix of the vehicle wheel.

Bridge model

The bridge model is discretized using finite element analysis to establish the relationship between displacement and external loads for each unit. This equation is solved through step-by-step integration to obtain the dynamic response results of the structure. The dynamic equilibrium equation is:

[M_{b}] {{\ddot{δ}}_{b}} + [C_{b}] {{\dot{δ}}_{b}} + [K_{b}] {δ_{b}} = {F_{b}}

(2)

where [M_b], [C_b], and [K_b] are the mass, damping, and stiffness matrixes of the bridge, respectively. {

δ_{b}

}, {

{\dot{δ}}_{b}

},{

{\ddot{δ}}_{b}

}represent the displacement, velocity, and acceleration matrixes of the bridge, respectively. {F_b} denotes the external load matrix.

Vehicle-bridge interaction model

The solution to the vibration response of the vehicle traveling the bridge process can be obtained by iteratively applying displacement coordination conditions and equivalent force formula.

(1) Displacement compatibility conditions

Assuming that the vehicle’s wheels always keep tight contact with the bridge surface while driving. Considering the unevenness of the bridge surface, the vertical displacement at the contact point between the vehicle and the bridge is determined by both the vertical displacement of the bridge and the unevenness of the bridge surface. The displacement can be expressed as follows:

Z_{g} = Z_{b} (x, t) + r (x)

(3)

where Z_b (x,t) represents the vertical displacement of the bridge, r(x) represents the unevenness of the bridge surface, and its value is only associated with the position of the wheels.

At this moment, the vertical velocity of the contact point of the vehicle-bridge can be expressed as:

{\dot{Z}}_{g} = \frac{\partial_{Z_{b}}}{\partial_{t}} + \frac{\partial_{Z_{b}}}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial r}{\partial x} \frac{\partial x}{\partial t} = {\dot{Z}}_{b} + v Z_{b}^{'} + v r^{'}

(4)

where v represents the velocity of the vehicle,

Z_{b}^{'}

is the partial derivative of the vertical displacement of the bridge with respect to its longitudinal direction, and

r^{'}

denotes the partial derivative of the vertical unevenness of the bridge deck concerning its longitudinal direction. When both the vehicle speed and the vertical displacement of the bridge are small, the additional term

v Z_{b}^{'}

can be negligible.

(2) External loads of the vehicle system.

External loads on vehicle systems are induced by bridge vibrations, primarily associated with the vertical displacement and velocity of the bridge oscillations. The computation of each non-zero component in the external vehicle load vector can be determined as follows:

f = C_{w} \cdot {\dot{Z}}_{g} + K_{w} \cdot Z_{g}

(5)

where f is the external load vector of the vehicle system, C_w and K_w are the damping and stiffness of the wheel contacting the bridge surface.

Thus, the excitation force resulting from bridge vibration on each wheel can be calculated using the above equation (5). These excitation forces are independent and do not correlate with one another. Ultimately, the external load vector of the vehicle system can be assembled by following the order of nodal degrees of freedom for each wheel.

(3) External loads of the bridge system.

The bridge is subjected to vertical forces from the wheels of vehicles moving. These forces include the vehicle’s weight supported by the wheels and the excitation force generated by the elastic and damping properties of the wheels, and it is given as:

P_{b} = - M g - K_{w} Z_{g} - C_{w} {\dot{Z}}_{g}

(6)

where M represents the wheels bear the vehicle mass.

When the force from the wheels is applied to the central section of the bridge element, it becomes essential to utilize the shape function N to translate this force into nodal equivalent loads. Subsequently, these loads are then systematically integrated into the bridge structure. At this stage, the calculation formula for the shape function and the external load vector of the bridge system are given as:

[N] = [1 - \frac{x}{l} - x + \frac{2 x^{2}}{l} - \frac{x^{3}}{l^{2}} 1 - \frac{3 x^{2}}{l^{2}} + \frac{2 x^{3}}{l^{3}} \frac{x}{l} \frac{x^{2}}{l} - \frac{x^{3}}{l^{2}} \frac{3 x^{2}}{l^{2}} - \frac{2 x^{3}}{l^{3}}]

(7)

{F_{b}} = [N] {P_{b}}

(8)

Structural damping matrix

Rayleigh damping is a commonly used approximation method that assumes the damping matrix of a structure can be treated as a linear combination of the mass matrix and the stiffness matrix, simplifying the damping determination. The formula is as follows:

[C] = - α [M] + β [K]

(9)

where α and β denote the proportional coefficients of the mass and stiffness matrixes, respectively. They are determined by the natural frequencies and damping ratios of any two modes of the structure and can be expressed as:

α = \frac{2 (ζ_{i} ω_{j} - ζ_{j} ω_{i})}{ω_{j}^{2} - ω_{i}^{2}} ω_{i} ω_{j}

(10)

β = \frac{2 (ζ_{j} ω_{j} - ζ_{i} ω_{i})}{ω_{j}^{2} - ω_{i}^{2}}

(11)

where ω_i and ω_j represent the natural frequencies of the i-th and j-th modes of the structure,

ζ_{i}

and

ζ_{j}

represent the damping ratios of these two mode shapes. Generally, the damping ratio for concrete bridge structures is typically 0.05.

Train load excitation

The factors such as speed impact and track roughness should be considered when simulating train loads. The representative formula for simulating high-speed train loads is as follows:

P (t) = k_{1} k_{2} (P_{0} + P_{1} \sin ω_{1} t + P_{2} \sin ω_{2} t + P_{3} \sin ω_{3} t)

(12)

where P₁, P₂, and P₃ represent vibration loads corresponding to typical values of calculation parameters for three control conditions of track roughness. P₀ denotes the static wheel load of a single side of the vehicle. k₁ is the superposition coefficient of adjacent wheel-rail forces, typically ranging from 1.2 to 1.7. k₂ is the dispersion coefficient of the steel rail, typically ranging from 0.6 to 0.9. The vibration load can be expressed as follows:

P_{i} = M_{0} a_{i} ω_{i}^{2} (i = 1, 2, 3)

(13)

where M₀ represents the undercarriage mass of the train,

a_{i}

is the typical rise.

ω_{i}

is the vibration frequency, which can be expressed as:

ω_{i} = 2 π v / L_{i} (i = 1, 2, 3)

(14)

where

v

represents the velocity of the train, and

L_{i}

represents the typical wavelength.

This study analyzes the load exerted by the CRH1A high-speed train that is presently in operation on the Chinese railway system. The specific simulated load parameters are as follows: axle load of 17 t, P₀ = 85 kN, M₀ = 750 kg, k₁ = 1.2, k₂ = 0.6, v = 60 m/s, L₁ = 10 m, a₁ = 0.0035 m, L₂ = 2 m, a₂ = 0.0004 m, L₃ = 0.5 m, a₃ = 0.00,008 m.

Experimental analysis of vehicle-bridge under train loads

Sensor layout of the bridge and vehicle

Wanbaolu Bridge is a cable-stayed bridge with a double tower and an overall length of 340 m. Figure 1 shows the span arrangement of the bridge; the main span is 160 m, and each side span is 95 m. The main bridge features a prestressed concrete box girder with a variable cross-section, while the towers are constructed using concrete casting, and the bridge piers are slab pier designs. Besides, the main bridge and towers are constructed using C55 concrete, with the bridge piers made of C50 concrete. The stay cables are arranged using a single-cable surface double-row method, with the intersection of the stay cables and the towers vertically spaced 1 m and a horizontal cable spacing of 4 m. A total of 44 pairs of stay cables, totaling 88 cables, are utilized across the entire bridge.

Figure 1.

Details of the bridge structure tests.

The distribution of vibration measurement points on the bridge deck and within the test vehicle is also illustrated in Figure 1. Measurement location A1 is situated on the bridge deck at the position of the right tower, while point A2 is located at the 1/4 position on the right side of the mid-span, and point A3 is positioned at the mid-span of the central span. Figure 2 shows that point A4 is situated at the co-driver’s foot position inside the test vehicle, and point A5 is located on the right rear seat of the test vehicle. The key parameters of the vehicle are listed in Table 1.

Figure 2.

Sensor layout of the bridge and vehicle.

Table 1.

Parameters of the test vehicle.

Parameters	Mass (kN)	Sizes (mm)	Wheel radius (mm)	Wheel width (mm)
Values	20	4800 × 1900 × 1500	320	215

The accelerometer sensor chosen is the TST126 vibration sensor, illustrated in Figure 3(a). This sensor is suitable for ultra-low or low-frequency vibration testing, with a sensitivity of 30 V/g, a range of 0.2 g, and a resolution of 0.3 ug. The accelerometer sensor data is collected using the TZT3828 E dynamic and static signal acquisition instrument, with the sampling frequency ranging from 1 Hz to 1 kHz, as shown in Figure 3(b). Figure 3(c) is the LD-98 radar speedometer to determine the train’s running speed.

Figure 3.

The details of the test instruments.

Parameters of test cases

Collect on-site test data to determine the axle load information of trains passing beneath the bridge. The type of the train is CRH1A with 17 t weight, and the information of three random test cases is listed in Table 2.

Table 2.

The train and vehicle information in the field tests.

Tests	Vehicle	Vehicle speed (km/h)	Train	Train speed (km/h)
Case A-1	BYD Qin PLUS DM-i	40	CRH1A	161
Case B-1	BYD Qin PLUS DM-i	60	CRH1A	164
Case C-1	BYD Qin PLUS DM-i	80	CRH1A	164

Test results under train loads

Figures 4–6 are the measured acceleration responses and corresponding Fourier spectra of the bridge and vehicle in case A-1. The vibration acceleration of the bridge deck is apparent when the vehicle and train traverse the bridge simultaneously at speeds of 40 km/h and 160 km/h, respectively. Vertical (Z-direction) acceleration is more pronounced than horizontal vibrations. At measurement point A1, vibration response frequencies are uniformly distributed within 0-100 Hz, while A2 and A3 demonstrate a concentration within 0-50 Hz. Within the vehicle, the dominant vibration frequencies of A4 and A5 are concentrated within 0-40 Hz. Particularly, the vibration level at the rear seat measurement point A5 is higher than at point A4.

Figure 4.

Triaxial acceleration response of bridge deck in case A-1.

Figure 5.

Frequency spectrums of the triaxial acceleration response on bridge deck.

Figure 6.

Time- and frequency-domain of acceleration responses in the vehicle.

Figures 7–9 illustrates the acceleration responses and the corresponding Fourier spectrums from the measurement points on the bridge and vehicle, which were exposed to vibrations from passing trains and vehicles in case B-1. The findings demonstrate that as the vehicle passes the bridge at 60 km/h, the vertical vibration level of the bridge surpasses that in the horizontal direction under the combined effect of the vehicle and train. Analysis of the Fourier spectra shows that the dominant vibration frequency at measurement point A1 predominantly falls within 40-60 Hz. Additionally, there is a noticeable trend of the vibration frequencies at other measurement points shifting towards higher frequencies within the dominant frequency range. Furthermore, the response amplitudes and frequencies of A4 and A5 points inside the vehicle coincide with the trend observed at the measurement points of the bridge deck, where dominant frequencies are concentrated in 0-40 Hz.

Figure 7.

Triaxial acceleration response of bridge deck in case B-1.

Figure 8.

Frequency spectrums of the triaxial acceleration response on bridge deck in case B-1.

Figure 9.

Time- and frequency-domain of the vehicle acceleration responses in case B-1.

Figures 10–12 depict the vibration acceleration and Fourier spectrum of the bridge and the vehicle under operating condition C-1. The vibration acceleration is more pronounced in all three directions of the bridge when the vehicle with a speed of 80 km/h passes, in contrast to vehicles traveling at 40 km/h and 60 km/h. The frequencies of the measured acceleration responses at each measuring point exhibit an even distribution within 0-100 Hz, without any concentration of vibration response frequencies in specific intervals as observed in operating conditions A-1 and B-1. The results suggest that the proportion of high-frequency responses gradually increases with higher vehicle speeds. Additionally, the vibration spectrum reveals that higher vehicle speeds correspond to increased amplitude in the vibration response. Notably, there is no significant difference in the vibration response between the front and rear measuring points, with the vibration frequency response consistently concentrated within 0-40 Hz.

Figure 10.

Triaxial acceleration response of bridge deck in case C-1.

Figure 11.

Frequency spectrums of the triaxial acceleration response on bridge deck in case C-1.

Figure 12.

Time- and frequency-domain of the vehicle acceleration responses in case C-1.

Model establishment and validation

The vehicle FE model

Figure 13 depicts the vehicle FE mode using ANSYS APDL. Tables 3 and 4 contain the specific parameters of the elements and material properties of the vehicle. The model integrates factors such as geometric complexity, material properties, stress concentration, computational efficiency, and convergence analysis. For models with geometric complexity that have areas of stress concentration, contact areas, or complex loading, finer meshes are used to capture details. The division size for the vehicle body is set at 200 mm, the axle elements at 100 mm, and the wheel elements are divided into 30 equal parts.

Figure 13.

FE model of the vehicle.

Table 3.

Selected elements for the vehicle components.

Component	Body	Suspension	Wheel	Contact pairs	Mass
Element	SHELL63	COMBIN14	SOLID45/SOLID185	CONTA173	MASS21

Table 4.

Material properties of the vehicle components.

Component	Body	Suspension	Wheel rubber	Wheel hub
Elasticity modulus (MPa)	2.1 × 10⁵	2.1 × 10⁵	1.0 × 10⁵	2.1 × 10⁵
Poisson ratio	0.3	0.3	0.4497	0.3

The bridge FE model

Figure 14 illustrates the comprehensive FE model of the railway-bridge-vehicle system. In this model, the bridge piers are modeled using SOLID45 element, and the tower is represented using BEAM188 beam element. Due to the varying cross-section and complex internal structure of the main girder in the cable-stayed bridge, accurately modeling it with a single solid element poses challenges. Therefore, the uniform cross-section part of the main girder is simulated using SOLID45 element, while the variable cross-section part utilizes BEAM188 element, with a rigid connection between the two elements. Additionally, the cable is simulated using LINK10 element with unidirectional stress characteristics, and tension force is simulated by applying initial strain.

Figure 14.

FE model of the bridge and railway model.

The mesh size significant impacts the computational results of the vibration problem caused by the operation of the simulated train. Acceptable computational results can be obtained when the distance between the load position and the model boundary exceeds half of the maximum wavelength of the soil, and the equation is given as:

L > λ_{\max} = \frac{C_{s}}{2 f_{\min}}

(15)

where

C_{s}

represents the shear wave velocity of the soil,

f_{\min}

denotes the lower limit value of concern frequency. According to the soil parameters of the test site, the maximum shear wave velocity is 810 m/s. Furthermore, this study focuses on the frequency range 0-80 Hz, with a minimum frequency of 20 Hz, and the corresponding half of the maximum wavelength of the soil is 20.25 m. Therefore, the model dimensions of the soil are set as 96 m × 192 m × 21 m.

Regarding the issue of the mesh size, more accurate results are achieved when the unit length is set to be ∆x = λ_smin/12. Also, a calculation accuracy guarantee (excluding the excitation source at 0.5λ_smin) can be obtained if ∆x = λ_smin/6. The frequency calculation formula for the wave velocity is given as:

f = \frac{v}{λ}

(16)

where

v

represents the wave velocity, and λ represents the wavelength.

In this study, the soil’s minimum vibration wavelength is 80 Hz, corresponding to a minimum shear wave velocity of 310 m/s. According to equation (15), the maximum mesh size is calculated to be less than 0.65 m; thus the excitation source region is set to 0.6 m. Moreover, the time step should be no greater than half of the vibration period, which is determined to be 0.005 s.

To simulate the soil in a semi-infinite space outside the FE and minimize the influence of boundary reflection waves, the equivalent three-dimensional viscoelastic boundary proposed by Gu et al.⁴³ is adopted. The material properties of the extended boundary elements, that is, equivalent elastic modulus, equivalent shear modulus, and Poisson’s ratio, are expressed as:

\tilde{ν} = {\begin{cases} \frac{α - 2}{2 (α - 1)} & α \geq 2 \\ 0 & α < 2 \end{cases}, α = α_{N} / α_{T}

(17)

\tilde{E} = α_{N} h \frac{G}{R} \frac{(1 + \tilde{ν}) (1 - 2 \tilde{ν})}{1 - \tilde{ν}}

(18)

\tilde{G} = α_{T} h \frac{G}{R}

(19)

where

α_{T}

and

α_{N}

represent the tangential and normal correction factors, respectively. h is the thickness of the equivalent unit, R denotes the vertical distance from the wave source to the artificial boundary, and G represents the shear modulus of the medium.

Development of vehicle-bridge interaction

In this study, the vehicle-bridge coupling system is established using the contact element CONTA173, as illustrated in Figure 15. The wheels are then coupled with the bridge deck nodes via the contact elements to simulate the interaction between the wheels and the bridge. The transient dynamics analysis module is chosen, and parameters such as time step, iterations, and convergence criteria are set to uphold the precision and stability of the calculation outcomes. Figure 16 illustrates the comprehensive FE model, which contains the vehicle, bridge and track-soil systems. The three subsystems are assembled into the comprehensive model capable of transmitting loads and ensuring consistency in displacement and deformation. By adjusting the parameters of vehicle body mass and axle quantity, the vibration responses of vehicles with different axle loads can be simulated with flexibility. Varying the load increments at critical nodes allows the simulation of forces acting on the wheels at varying vehicle speeds. The bridge system comprehensively considers self-weight, structural design. Various levels of bridge deck roughness serve as simulation parameters for modeling vehicle-bridge coupling vibrations across different road surface conditions. Within the railway environment, vibrations resulting from different train operating conditions are replicated by adjusting the magnitude and duration of wheelset loads.

Figure 15.

Schematic diagram of axle coupling finite element model.

Figure 16.

Detail structure of the integrated FE model.

Validation of the comprehensive FE model

The measured and simulated peak accelerations were compared to validate the effectiveness of the comprehensive model. Figure 17(a) compares the peak accelerations in three directions at each measurement point on the bridge deck, obtained from simulation and measured when a vehicle passes the bridge at 60 km/h speed. Furthermore, Figure 17(b) compares simulated and measured peak accelerations in three directions at each measuring point for a vehicle speed of 60 km/h on the bridge deck and a train speed of 160 km/h passing under the bridge. The results demonstrate a close match between the measured and simulated peak values, indicating that the comprehensive FE model can accurately capture the coupling vibration of the vehicle-bridge system and conduct precise vibration analysis under train excitations. Additionally, the simulated vibration values at each measurement point generally slightly exceed the measured values. Notably, the difference between measured and simulated peak values at the top of the piers surpasses one-fourth of the span position under both operational conditions, with a minor difference observed at the mid-span position of the bridge.

Figure 17.

Comparison of acceleration peak values for field measurement and FE analysis.

Figure 18 compares acceleration spectra at different measurement points on the bridge in simulated data and experimental measurements, corresponding to the vehicle and train speeds of 60 km/h and 160 km/h, respectively. The findings demonstrate a close alignment in the vehicle-bridge coupling system between the measured and simulated vibration spectra in the frequency domain, with high-frequency components predominantly falling within 0-40 Hz. The changes in the vibration measurement results agree with the simulation results, further validating the effectiveness of the established finite element overall model.

Figure 18.

Comparison of vibration spectra of bridge deck measurement points.

Numerical analysis of vehicle-bridge interaction under train loads

Effect of train speed on bridge displacement response

Based on the integrated FE model, this section investigates the influence of the vehicle-bridge coupling vibration on bridge deformation during train passage. The displacement response at the mid-span of the cable-stayed bridge under passing trains with speeds of 150 km/h, 200 km/h, and 250 km/h, respectively, is illustrated in Figure 19 when the vehicles on the bridge deck are set to a speed of 60 km/h. The mid-span displacement of the cable-stayed bridge increases proportionally with the train speed. Specifically, increasing the train speed from 150 km/h to 200 km/h results in a 16.8% increase in the peak value of the mid-span displacement. Furthermore, it is noteworthy that increasing the train speed from 200 km/h to 250 km/h leads to a nearly 70% increase in the displacement peak value. Therefore, it is evident that the increase in train speed beneath the bridge is a significant factor influencing the displacement response of the vehicle-bridge coupling system.

Figure 19.

Mid-span displacement of vehicle-bridge coupled vibration under different train speed.

Effect of train speed on bridge acceleration response

Passing trains beneath the bridge results in unavoidable vibrations in the vehicle-bridge coupling system, affecting the bridge’s safety and the passengers’ comfort. Consequently, it becomes imperative to examine the effect of train speed on the acceleration response of the vehicle-bridge coupling system. In this context, simulations are conducted on the bridge with a vehicle weighing 2 t traveling at 60 km/h, and the statistical analysis recorded acceleration peak values on both the bridge deck and the vehicle, corresponding to a train weighing 17 t at various speeds are shown in Figure 20. It is evident that as the train speed increases, the X-acceleration peak value at measuring point A1 increased by 7.4% and 6.6%, while the peak growth rates of Y-acceleration are 53.8% and 39.4%, and the peak growth rates of Z-acceleration are 63.2% and 37% respectively. Similar to measuring point A2, the peak growth rates of X-acceleration are 66.7% and 34%, the peak growth rates of Y-acceleration are 61.3% and 46.4%, and the peak growth rates of Z-acceleration are 65% and 34.3% respectively. Likewise, at measuring point A3, the peak growth rates of X-acceleration are 23% and 38.1%, the peak growth rates of Y-acceleration are 7.7% and 30.3%, and the peak growth rates of Z-acceleration are 51.9% and 33.3% respectively. Consequently, it can be inferred that the increase in train speed significantly amplifies the peak vibration levels of the vehicle-bridge coupling system.

Figure 20.

Comparison of acceleration peak values at measurement points under different train speed.

Performing a fast Fourier transform on the time-domain acceleration enables acquiring energy distribution information in the frequency domain, which reveals the distribution characteristics of the acceleration induced by train vibration on both the bridge and vehicles. For analysis, the acceleration spectrum is segmented into five equal divisions to examine the main frequency distribution properties of the bridge acceleration under train-induced vibration. Each frequency bandwidth is 20 Hz, with ranges of 0-20 Hz, 20-40 Hz, 40-60 Hz, 60-80 Hz, and 80-100 Hz, respectively. Figure 21 illustrates the proportion of each frequency band at the measuring points under the mentioned train conditions. The results indicate a significant influence of train speed on the frequency range of acceleration response for the bridge. The bridge’s acceleration response shifts towards higher frequencies with the increasing train speed.

Figure 21.

The proportion of each frequency band at the measuring points.

Effect of train axle load on bridge acceleration response

Simulate a train pass beneath the bridge at 200 km/h to examine the effects of trains with varying axle loads on the peak vibration acceleration of both the bridge deck and vehicles. Figure 22 illustrates the time history of acceleration response for the bridge deck and vehicle with the train axle loads of 17t, 20t, and 25t. The results demonstrate varying growth rates in the peak acceleration values of X, Y, and Z directions at measurement points A1, A2, and A3. In detail, the growth rates of X-direction acceleration peak values are 35% and 13.9% at point A1, for Y-direction are 35.4% and 10.7%, and for Z-direction are 46.9% and 5.9%. At point A2, the growth rates of X-direction acceleration peak values are 19.3% and 27.5%, for Y-direction are 18.3% and 27.6%, and for Z-direction are 30.4% and 12.5%. At point A3, the growth rates of X-direction acceleration peak values are 25.8% and 20%, for Y-direction are 29.3% and 25.5%, and for Z-direction are 34.9% and 7.6%. Thus, the peak acceleration of the bridge-vehicle coupling system increases with the increase of train axle load, and the vertical acceleration peak value of the bridge is much higher than the other two horizontal directions. However, Figure 23 indicates that the increase in train axle load does not significantly impact the concentration interval of vibration frequencies at the measurement points.

Figure 22.

Comparison of acceleration peak values at measurement points with different train axle load.

Figure 23.

The proportion of each frequency band at the measuring points with different train axle load.

Effect of train driving direction on bridge acceleration response

The train driving direction varies would be resulting in corresponding variations in the vehicle-bridge coupling vibration. To investigate the impact of different train driving modes on the acceleration response of the vehicle-bridge coupling system, setting a vehicle with a 2 t axle load and a speed of 60 km/h traveling on the bridge, while the train axle load is 17 t. Figure 24 presents the acceleration peak values at the measurement points of the vehicle-bridge coupling system under three operating conditions, that is, single train-single track, double train-syntropy track, and double train-opposite track. The results indicate that the acceleration peak values increased to varying degrees under different operating conditions. Changing the driving direction from the syntropy track to the opposite track had no significant impact on the acceleration level at the measurement points, suggesting that the direction of train travel did not significantly affect the acceleration of the vehicle-bridge coupling system. Also, the acceleration peak value of the vehicle-bridge coupling system can be attributed primarily to the increased number of trains. Furthermore, Figure 25 shows a decreasing trend in the proportion of low-frequency vibration response at each measurement point as the train driving direction changes from the single train-single track to the double train-syntropy track and double train-opposite track. This result further indicates that both train loads influence the vibration frequency of the vehicle-bridge coupling system to some extent.

Figure 24.

Comparison of acceleration peak values at measurement points with different train driving direction.

Figure 25.

The proportion of each frequency band at the measuring points with different train axle load.

Conclusions

This study focuses on a practical low-pylon cable-stayed bridge. It examines the influence of railway environment vibrations on the vibration response of the vehicle-bridge coupled system by integrating on-site tests and FE simulations. Through analyzing the vibration of the vehicle-bridge coupled system under railway environment vibrations in both time-domain and frequency-domain using on-site measured and simulation data, the main conclusions are as follows:

(1) The vertical acceleration response of the bridge deck is more pronounced than the horizontal vibration response. An increase in vehicle speed within the same train speed has a discernible effect on the vibration acceleration of the bridge deck in the vehicle-bridge coupling system. In contrast, the impact on the internal vibration acceleration of the vehicle remains negligible. The vibration response frequencies at bridge measuring points predominantly range between 1 and 100 Hz when subjected to train and vehicle vibration excitation, whereas the frequencies at internal measuring points of the vehicle primarily fall within 0∼50 Hz. Elevating the vehicle’s speed leads to a higher frequency shift in the concentrated interval of bridge deck vibration, yet the effect on the internal vehicle vibration frequency remains insignificant.

(2) This study establishes the comprehensive FE model, which contains the vehicle, bridge, and track-soil systems. By comparing the acceleration response and vibration frequency of the measured data with simulated operating conditions, the results show that the simulated values generally marginally exceed the measured values. The differences between the measured and simulated acceleration peak values at the top of the bridge pier exceed a quarter of the span position. In contrast, it is minimal for the acceleration peak value at the mid-span. The comprehensive model’s effectiveness has been validated, and experimental operating condition simulations can be conducted.

(3) The mid-span displacement of the bridge increases with the increase in train speed. Analysis of vibration response from bridge deck measurement points demonstrates a notable acceleration in bridge deck vibration with increasing train speed. Moreover, higher train speeds reduce the excitatory force period, leading to an elevation in the high-frequency vibration components at the bridge deck measuring points. Comparing the vibration peak value and frequency, it is evident that the impact of increasing train speed on the vibration level of the vehicle-bridge coupling system surpasses that of increasing vehicle speed.

(4) The peak acceleration of vibrations at both the bridge deck and vehicle measurement points increases with the increase in train axle load. Nevertheless, the impact of train axle load on the frequency distribution at the bridge measurement point is minor. Additionally, a linear relationship does not exist between the peak acceleration of vibrations at the bridge measurement point and the train axle load. Compared to the augmentation in train speed, the increase in axle load exerts a more negligible influence on the vehicle-bridge coupling system.

(5) The impact of changes in train composition on the vehicle-bridge coupling system is primarily manifested through variations in the train axle load. Multiple trains, as opposed to a single train, can substantially influence the peak vibration response and the higher vibration frequency of the vehicle-bridge coupling system. On the other hand, differences in the driving direction have a negligible effect on the vibration response distinctions within the vehicle-bridge coupling system.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research was supported by and the Science and Technology Research Project of Education Department of Jiangxi Province (GJJ180300), and the National Natural Science Foundation of China (51688022).

ORCID iD

Zhenwei Zhou

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