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Aerodynamic loads and bridge responses under train passage: case study of an overpass steel boxgirder cablestayed bridge
Advances in Bridge Engineering volume 3, Article number: 1 (2022)
Abstract
Increased train speeds have led to significant traininduced wind problems. This paper describes a numerical study of the aerodynamic loads produced when a train passes under a bridge and the bridge dynamic response based on a steel boxgirder cablestayed bridge. The results indicate that the wind pressure on the bottom plate and the maximum lift on the beam sections increase as the train speed increases or as the clearance under the bridge decreases. Changes in the intersection angle are found to have a greater impact on the 1/4span and 3/4span regions. The effects of the various influencing factors are basically the same in the construction and operation stages. As the train speed increases or the clearance under the bridge decreases, the bridge responses exhibit an upward trend. As the intersection angle decreases, the vibration displacement tends to increase, but the effects on the vibration acceleration and maximum counterforce of the temporary pier are not consistent. Compared with the construction stage, the dynamic responses of the main girder are smaller in the operation stage due to the larger overall rigidity and the support constraints. In the parameter ranges investigated in this study, the absolute values of maximum vertical bridge vibration displacement and acceleration decrease from 4 to 16 mm and from 85 to 400 mm/s^{2}, respectively, in the construction stage to 1–5 mm and 42–174 mm/s^{2} in the operation stage. Moreover, the counterforce falling on the temporary pier after the main girder turns does not exceed the 2000kN positive force or 300kN negative force limits. In the operation stage, the bridge vibration responses will not affect the comfort of pedestrians.
1 Introduction
In recent years, there have been rapid developments in highspeed trains. With increases in train speeds, the interaction between the trains and the surrounding air has become more significant. Thus, aerodynamics are now a crucial problem for highspeed trains (Schetz 2001).
Current research on the aerodynamic forces created by trains mainly focuses on three methods: scale model testing (e.g., wind tunnel tests, dynamic model tests), real train tests, and numerical simulations. Baker and Brockie (1991) conducted wind tunnel model experiments on aerodynamic resistance using 1:20, 1:40, and 1:76 scale train models, and compared the modelscale results with fullscale vehicles. It was found that smaller model scales produced a greater error in the measurement results. Willemsen (1997) found better agreement when the data extrapolated from highReynoldsnumber wind tunnel tests were compared with fullscale results. Hemida and Krajnović (2009) numerically simulated the flow field around a train using the large eddy simulation (LES) framework, and studied the influence of different wind deflection angles and head shapes on the flow structure and train aerodynamics. An experimental study using a train–wind dynamic model was conducted by Bell et al. (2015), who reported that the peak values of the train wind curves in the nearweak region of two and threecar train models are different, showing distinct wake structure features. Soper et al. (2017) carried out fullscale measurements of the air flow characteristics and track characteristics under a highspeed railway in Britain, and analyzed the dynamic relationship between the undertrain airflow, track, and ballast. Niu et al. (2018) adopted the delayed detached eddy simulation with improved wallmodeling capability to simulate the unsteady aerodynamic performance of a tapered train head with lengths of 8 m and 12 m, and verified the correctness of the numerical simulations through wind tunnel tests. Jiang et al. (2021) studied the aerodynamic performance of a train model through computational fluid dynamics (CFD) and detached eddy simulations, and verified the numerical method by wind tunnel tests.
The traininduced pressure will increase rapidly with an increase in the train running speed. The airflow surrounding adjacent structures can be as severe as that in a wind storm. Therefore, it is necessary to predict the aerodynamic load on structures close to the track to aid in their design. Gerhardt and Krüger (1998) investigated the wind and traininduced air movements in train stations and discussed the means of controlling them to prevent excessive air infiltration into the stations. Baker et al. (2014) studied the transient aerodynamic pressures and forces on trackside and overhead structures due to passing trains. Formulas for the aerodynamic loading on trackside structures, which may be useful in future revision of standards, were then derived from the experimental data. Yang et al. (2015) used the FLUENT software and sliding grid technology to investigate how the beam width and beam depth influence the wind pressure on the beam surface when highspeed trains pass under a bridge. Lü et al. (2018) conducted an experimental study on the aerodynamic load characteristics of noise reduction barriers in terms of train speed, distance between the train and sound barrier, and train type. Rocchi et al. (2018) carried out a fullscale experimental test on Italian highspeed trains that obtained experimental results in different environments, and comprehensively compared the fluctuating wind pressure acting on the sound barrier for three different types of highspeed trains operating in the open air. Liu et al. (2019) measured the pressure characteristics of a tunnel when highspeed trains pass through, and studied the effects of train speed, train length, and the intersection of two trains on the peak tunnel wall pressure. Xiong et al. (2020) conducted a field test on the pressure changes caused by a CRH380A train passing the bridge sound barrier. They obtained the fluctuating pressure time history curve and wind pressure peak, and further analyzed the influence of train speed, operating track, measuring point position, marshalling lengths, and ambient wind speed on the peak wind pressure. Zheng et al. (2020) studied the aerodynamic pressure waves produced by highspeed trains passing through a semiclosed sound barrier by establishing CFD and finite element (FE) models.
To date, the aerodynamic characteristics and generation mechanism of traininduced pressure have been intensively studied. However, the effects of the train running speed under a bridge and the relative positions of the bridge and the train on the aerodynamic characteristics and bridge dynamic response require further research. In this study, the aerodynamic characteristics and dynamic response of a steel boxgirder cablestayed bridge are studied in the case of a highspeed train passing underneath. The intersection angle between the bridge and the existing railway is about 66°, and there are 10 railway lines at the intersection. The bridge span layout is (40 + 188 + 55) m, as shown in Fig. 1. The reliability of the numerical simulations is initially verified using existing experimental data. The numerical and FE analysis models of traininduced wind are then established for the case of a highspeed train passes underneath the bridge, and the aerodynamic characteristics and variation pattern are investigated in terms of the train speed, clearance under the bridge, and train–bridge intersection angle. Finally, the dynamic responses of the bridge are studied during the construction and operation stages.
2 Numerical simulation strategy
Turbulence can be simulated based on CFD technology by either direct or indirect numerical simulations. The direct numerical simulation (DNS) method is relatively accurate due to its direct solution to the turbulence control equation (Piller et al. 2002; Wissink 2003), but it is seldom used in engineering because of its high computational requirements. The indirect numerical simulation method approximates or simplifies the turbulence to a certain extent, without directly solving the pulsation characteristics. Indirect approaches include LES, Reynoldsaveraged Navier–Stokes (RANS) equations, and statistical averaging. The RANS method, which is widely used in practical engineering, averages the transient control equations over time and expresses variables in terms of timeaveraged and fluctuation components, which has high accuracy and avoids a large number of calculations (RolletMiet et al. 1999). The twoequation RANS model is the most important turbulence model because it can predict the given turbulence characteristics without knowing the turbulence structure in advance. Twoequation models include the k–ω turbulence model (Bouras et al. 2018; Devolder et al. 2018), standard k–ε turbulence model, and renormalization group k–ε turbulence model (Liu et al. 2017), among others (Baker et al. 2014). The standard k–ε turbulence model (Launder and Spalding 1972) has become widely used because of its efficiency, accuracy, and adaptability.
2.1 Basic theory
The standard k–ε turbulence model is based on the concept of eddy viscosity. In the turbulence model, partial differential equations for k and ε are solved, where k is the turbulent kinetic energy and ε is the dissipation of k (Olsen 2000). The motion equations corresponding to k and ε can be expressed as:
where G_{b} and G_{3ε} are related to buoyancy (which can be ignored in this study), ρ is the fluid density (taken to be 1.225 kg/m^{3}), u_{i} is the velocity vector, t is the time variable, x_{i} is the space tensor, μ is the molecular viscosity, μ_{t} is the viscosity coefficient, and G_{k} is the increased turbulent kinetic energy. σ_{k} and σ_{ε} are the turbulent Prandtl numbers for k and ε, and C_{1ε} and C_{2ε} are model coefficients: C_{1ε} = 1.44, C_{2ε} = 1.92, σ_{k} = 1.0, σ_{ε} = 1.3 (Zheng et al. 2020).
The formulas for calculating μ_{t} and G_{k} are (Markatos 1986):
where S is the modulus of the average strain rate tensor, \(S=\sqrt{2{S}_{ij}{S}_{ij}}\). The strain rate tensor \({S}_{ij}=\frac{1}{2}\left(\frac{\partial {U}_i}{\partial {x}_j}\frac{\partial {U}_j}{\partial {x}_i}\right)\), where U_{i} and U_{j} are the doubleaveraged velocity components in the x_{i} and x_{j} directions, respectively. The model constant C_{μ} = 0.09.
2.2 Model validation
To verify the reliability of the FLUENT numerical simulation results, an aerodynamic model was established to simulate a footbridge previously studied by Yang et al. (2015), and the measured data were compared with the simulation results. The footbridge is located at the centerline of the railway station and has an overall length, clear width, and clear construction height of 82 m, 15 m, and 2.2 m, respectively. The structural form is a steel structure with a reinforced concrete composite bridge deck. A photograph of the bridge is shown in Fig. 2a.
The train type considered in this study is the CRH380, which has a streamlined appearance. On the premise of ensuring the calculation accuracy, the bogie, wheel rail, and pantograph structures were ignored, and only the aerodynamic shape of the train was retained and smoothed (Zheng et al. 2020; Yang et al. 2015). Numerical simulations of the traininduced wind pressure were carried out on a CRH380 train with eight carriages, as shown in Fig. 3.
Considering the influence range of the traininduced wind pressure, only the main span beam above the train was considered in establishing the aerodynamic model. To avoid the influence of the flow field boundary on the calculation results, the final calculation domain size was taken as 720.0 × 168.0 × 78.6 m. The initial position of the train was 108 m away from the beam flange. The top and surroundings of the fluid domain adopted periodic boundary conditions. The ground, train body, and bridge surface adopted nonslip wall boundary conditions. Data exchange between the dynamic grid and the static grid was realized through the interface. The final volume grid consisted of approximately 3.63 million elements, as shown in Fig. 2b.
The measuring points on the bridge surface were located at the bottom of the windward surface, vertically perpendicular to the train operation direction, with a vertical distance of 4.5 m from the top of the train. A comparison of the time history between the simulated and measured wind pressures at the measuring points is shown in Fig. 2c. The simulated and measured train wind time histories are basically consistent, but the simulation results are relatively smooth. This is because the RANS method only gives the average pressure, rather than the fluctuating pressure. Nevertheless, it is acceptable since the measured results obtained by Yang et al. (2015) also showed that the lowfrequency component dominated while the highfrequency pulsation component was small. The absolute differences between the extrema of the simulated “head wave” and “tail wave” and the measured values are less than 10 Pa and 3 Pa, respectively, and the simulation error is about 10%, which is within an acceptable range.
3 Aerodynamic modeling
This section introduces the train model and modeling parameters as described in Section 2. The computational domain size is 720.0 × 208.0 × 47.6 m, as shown in Fig. 4. To guarantee the numerical accuracy and efficiency, the layering mesh technique is applied. The simulation domain is divided into two parts: a stationary domain including the overpass bridge and the surrounding flow field, and a moving domain including the train. The contact face between the moving and stationary domains is defined as an interface to exchange data. When the train moves forward, its front part compresses grids and the rear part stretches the grids, resulting in grid reconstruction in the dynamic domain. The grid height is 0.5 m in dynamic domain, which decides the time of grids creating and destroying. The grid destroying coefficient is 0.2 and the grid splitting coefficient is 0.4. The train is initially 108 m away from the girder flange.
The computational grid is generated hierarchically. Since the overhead bridge surface is the area of concern, a finer structured grid is used to define these areas, while for the rest a coarser grid is adopted. For the calculation mesh of the moving domain, faces covering the train body are created and meshed first. The volume around the train is then meshed. Note that not only hexahedral, but also tetrahedral elements are applied for the air volume around the train because of the complex geometry of the train. The final number of elements is approximately 4.80 million. Grid independence test results show that the average pressure on the monitoring region obtained by further refining the grid has limited changes (< 5%). Thus the current model can ensure the validity of the calculated results.
It is worth mentioning that this paper does not aim at accurately modeling the flow features around the bridge girder. And, as a matter of fact, for the studied bridge girder with such a complex configuration, one can hardly obtain a meshindependent solution regarding the flow features since it is very sensitive to the mesh size (Bruno et al. 2012). Nevertheless, from the engineering practical perspective, the modeling methodology itself is generally accepted. The innovation of this study is believed to provide insights regarding the anticipated aerodynamic loads and dynamic responses of the overpass bridge. The results will be also used in deciding particulars during the bridge design stage.
Figure 5 shows a schematic diagram of the main span beam section and the block division of L1. To obtain a more detailed response condition of the beam, the main span is divided into 19 beam sections at 5m intervals along the longitudinal bridge direction on the left and right, which are denoted as L1–L19 and R1–R19, respectively. The train is located in the middle of the main span, as shown in Fig. 5a. “Windward side” refers to the surface on the same side as the initial position of the train, with the beam centerline as the boundary, while “Leeward side” refers to the surface on the opposite side; “adi1” corresponds to the first bottom plate on the windward side (see Fig. 5b).
The aerodynamic model illustrated in Fig. 4 is used to analyze the wind pressure characteristics of the beam surface when a highspeed train passes under the bridge. The train speed is 350 km/h, the clearance under the bridge is 7.25 m, and the intersection angle between the train and the bridge is 90°. Figure 6 shows wind pressure nephograms at the beam surface when the train is at different positions.
As shown in Fig. 6, when the train passes under the bridge, the surrounding flow field presents a typical threedimensional distribution, with positive and negative pressure peaks spreading from the head and tail of the train, forming an ellipsoidal wind pressure distribution on the beam surface. For the different positions illustrated in the figure, the wind pressure on the beam surface alternates in the form “positive–negative–negative–positive”. Before the nose of the head carriage reaches the beam body, the beam is basically only subjected to positive pressure. After the head of the train has entered the beam body, the beam body is subjected to both positive and negative pressure. After the nose of the head carriage has moved out of the beam body, the beam body is basically only subjected to negative pressure. The aerodynamic action of the train’s tail exhibits the opposite pattern. The absolute values of the extreme positive and negative wind pressures are roughly equal when the nose of the head carriage and the tail of the train reach the centerline of the beam. The extreme values of wind pressure when the head and tail of the train nose reach the centerline of the beam are 302.5 Pa and − 294.7 Pa, respectively. It is clear that the predicted wind pressure magnitudes and variation characteristics in this study are judged comparable to related field measurements (Yang et al. 2015) and CFD simulations (Yan et al. 2014).
4 Factors influencing the aerodynamic loads
The aerodynamic model is now used to examine the influence of train speed, bridge clearance, and the intersection between train and bridge on the wind pressure on the girder surface and the lift force on the girder section.
4.1 Train speed
To obtain the wind pressure at different train speeds, the wind pressure of the bottom plate, “adi1”, is fitted at different train speeds. The fitting formula P = av^{b} is adopted, where P is the pressure extremum. The fitting relationship is described in Table 1.
To investigate the bridge dynamic responses under wind, the lift force, drag force, and torsion moment can be introduced to evaluate the loads applied on the bridge girder (Yan et al. 2014). As an example, the lift force applied on the beam section is discussed herein. Figure 7a and b plot the aerodynamic curves at different train speeds. The time history of each beam section is basically consistent at different train speeds. As the train speed increases, the extremum increases rapidly. The time differences between the head and tail waves and the time differences between the positive and negative pressure extrema of the head and tail waves become significantly shorter, and the shortening time decreases as the train speed increases. The attenuation trend of the extreme lift force along the longitudinal bridge direction at different train speeds is basically unchanged, and decreases as the distance increases. A faster train speed produces a more rapid decay in the lift force along the longitudinal bridge direction.
To explore the relationship between the lift force extremum and train speed, curve fitting is applied to the head wave lift force positive extremum of each beam section and the train speed. Examples for beams L1, L3, and L5 are shown in Fig. 7c. The functional relation between the positive lift force extremum F and train speed v is expressed as F = av^{b}, where a and b are fitting parameters and v is given in km/h. The correlation coefficient R^{2} is close to 1, indicating that the fitting results are fairly good. The power exponent is close to 2, indicating that the relationship between the lift force and speed is quadratic.
Furthermore, to study the functional relationship between the lift force extrema of the beam sections and the distance from the track centerline, we now introduce the lift force extremum coefficient C_{F}. The formula for the positive lift force extremum can be expressed as F = 0.5C_{F}ρv^{2}, where the air density ρ is 1.225 kg/m^{3}.
Figure 7d shows the attenuation of C_{F} along the longitudinal bridge direction. C_{F} and the distance from the track centerline have been fitted by a power exponent according to the functional expression C_{F} = ae^{bx}, where a and b are the fitting parameters and x is the distance from the track centerline in meters. R^{2} is close to 1, which means the fitting results are fairly good. Therefore, C_{F} has a power exponential relationship with the distance from the track centerline.
4.2 Clearance under bridge
To obtain the wind pressure at different clearances under the bridge, the wind pressure of the bottom plate “adi1” is fitted at different bridge clearances. The fitting formula is P = ah^{b}, and the fitting relationship is described in Table 2.
Figure 8a and b plot the aerodynamic curves at different clearances under the bridge. The results reveal that, at different clearance levels, the lift force time history of each beam section is basically consistent, and the occurrence times of the extrema are largely unchanged. However, as the clearance under the bridge decreases, the extreme values increase sharply, and smaller clearance levels produce greater increases in amplitude. The attenuation trend of the extreme lift force along the longitudinal bridge direction is broadly consistent. A smaller clearance under the bridge results in more rapid decay in the lift force extremum along the longitudinal bridge direction.
Figure 8c shows the fitting curve of the positive lift force extremum for the head waves at beams L1, L3, and L5 with respect to the clearance under the bridge. The positive extremum of the head wave lift force has a negative power exponential relationship with the clearance under the bridge, and the functional relation can be expressed as F = ah^{b}, where a and b are fitting parameters and h is the clearance under the bridge in meters. R^{2} is close to 1, which means the fitting results are fairly good. The power exponent b decreases from − 1.3129 for beam L1 to − 0.8479 for beam L5. Therefore, there is a negative power exponential relationship between the lift force extremum of the beam section and the clearance under the bridge, but the exponents are different. Farther away from the track centerline, the absolute value of the exponent becomes smaller.
4.3 Intersection angle
Figure 9 shows the aerodynamic curves at different intersection angles. The time history curves of the lift force on each beam section at different intersection angles exhibit little variation. Along the longitudinal bridge direction, the lift force extrema at both the side and midspan beams are hardly affected by changes in the intersection angle, while those at the 1/4span and 3/4span beams are more strongly affected. This is because, as the intersection angle decreases, the distance between the midspan beam and the track centerline changes little, and so the angle change has little impact. For both sidespan beams, although the distance from the track centerline changes more, shifting the angle within the range 15–90° means that the side span is still far from the track centerline, and is therefore outside the main influence range of traininduced wind. For the 1/4span and 3/4span beam sections, the distances from the track centerline change significantly and these regions are within the main influence range of traininduced wind, so the lift force extremum changes considerably. At smaller intersection angles, the lift force extremum exhibits greater changes with the intersection angle.
4.4 Simplified aerodynamic load model
As a guide for engineering practice, Fig. 10 presents a simplified diagram of the wind pressure curve at bottom plate “adi1”. In the figure, l_{head} and l_{tail} are the train running distances between the positive and negative extrema of head wave and tail wave, respectively, P_{head}(+) and P_{head}(‐) are the positive and negative pressure extrema of the head wave, respectively, P_{tail}(+) and P_{tail}(‐) are the positive and negative pressure extrema of the tail wave, respectively, K_{head} is the slope between the positive and negative extrema of the head wave, and K_{before head}, K_{after head} are the slopes before the positive extremum and after the negative extremum of the head wave, respectively. When calculating K_{before head} and K_{after head}, the slope of the connecting line between the wind pressure extremum and the 1/3 wind pressure extremum is taken as the wind pressure slope before and after extrema. K_{tail}, K_{before tail}, and K_{after tail} are defined similarly.
The train considered in this study is the CRH380 type. The wind pressure extrema P_{head}(+), P_{head}(‐), P_{tail}(+), and P_{tail}(‐) can be calculated by the above fitting formulas. The wind pressure models of the bottom plate at different train speeds and different clearances under the bridge are now studied.
The parameters at different train speeds are listed in Table 3(a). The clearance under the bridge is 7.25 m and the intersection angle is 90°. As the train speed changes, the parameters of the wind pressure curve exhibit little variation.
The parameters at different clearances under the bridge are given in Table 3(b). The train speed is 350 km/h and the intersection angle is 90°. There is a big difference in l_{head} and l_{tail}, which is mainly caused by the pressure wave pattern generated by the train. In the direction of train height, the distance between the positive and negative extrema increases with distance from the train.
The generalized wind pressure curve model shown in Fig. 10a and the corresponding parameter values listed in Table 3 can be jointly regarded as a “load spectrum”. The “load spectrum” is useful for other bridges with similar scenarios. Nevertheless, when other conditions change drastically (e.g., girder section, train type), the specific parameter values should be carefully checked.
5 Bridge responses in different stages
5.1 Evaluation indexes
The FE model of the bridge was established using the MIDAS/Civil software and the dynamic response of the bridge was analyzed in the construction and operation stages. The stay cable was simulated by a bar element, and the tower, girder, and beam were simulated by beam elements. The structure was discretized based on the theoretical completed bridge alignment, and the calculation diagram for the cantilever stage before closure and the operation stage were formed according to the erection process, as shown in Fig. 11. The number of elements and nodes in the cantilever stage were 191 and 221, respectively, and the number of elements and nodes in the operation stage were 243 and 388, respectively. The first five natural vibration modes and frequencies are shown in Fig. 12. Based on the aerodynamic loads aforementioned (e.g., Figs. 7a, 8a, and 9a), the timevarying nodal forces were applied on the FE model. Note that the nodal forces include three components, i.e., lift force, drag force, and torsion moment. The direct integration method was used based on a time step of 0.001 s. A damping ratio of 1% was adopted for the dynamic response calculation.
The rotation construction of the bridge was planned to be carried out during the “skylight” time. After the rotation had been completed, the main span was supported on the temporary pier. The position of the closure segment is shown in Fig. 13. Before closure, the temporary pier was supported at the diaphragm. To avoid the temporary pier from being subjected to large forces and adversely affecting the steel beams, the total counterforce of the temporary pier should not exceed 2000 kN. Calculations indicate that the negative counterforce limit is 300 kN.
As the bridge is equipped with sidewalks, the beam vibration caused by traininduced aerodynamic action may reduce the comfort of pedestrians. Therefore, it is necessary to research the bridge comfort level in the operation stage. In this paper, the German EN03 standard (Butz et al. 2008) is adopted to evaluate the comfort, and the evaluation indexes are presented in Table 4. Due to the special working condition in which the beam falls on the temporary pier and is temporarily not closed, this section analyzes the vibration displacement and acceleration of the cablestayed bridge during the cantilever state (conservative consideration) and the operation stage, as well as the supporting counterforce when the main beam falls on the temporary pier (with vertical constraints) and the comfort during the operation stage under the traininduced aerodynamic force. The specific evaluation indexes of each stage are listed in Table 5.
5.2 Construction stage
For the construction stage, when the track centerline is arranged near the beam end, only the aerodynamic force on one side of the track centerline is exerted on the beam body, and thus the dynamic response of the beam is less than the maximum at this time. According to our calculations, the beam end suffers the maximum vertical displacement when the distance between the track centerline and the beam end is about four beam sections (approximately 20 m). Therefore, in the construction stage, the track centerline is about four beam sections away from the beam end, i.e., it is regarded as the extreme scenario. The scenarios of multiple presence of trains are not considered since they are rare.
5.2.1 Influence of train speed
Figure 14 shows the bridge dynamic response at different train speeds. As the train speed increases, the maximum values of the vibration displacement, acceleration, and counterforce of the temporary pier basically show an upward trend. The different trends for individual speeds are mainly due to the following reasons:

(1)
As higher speeds will enhance the fluctuations in the lift time history curve, the time difference between positive and negative extrema becomes shorter, resulting in a faster unidirectional acceleration time for the beam. Increases in the train speed and the resulting increase in the lift extremum aggravate the vibration of the beam, but the reduction in the unidirectional force action time weakens the beam vibration. Therefore, increasing the train speed may not aggravate the beam vibration.

(2)
The vertical vibration modes mainly include the secondorder natural vibration mode (0.36 Hz) and the fourthorder natural vibration mode (0.68 Hz). As the train speed changes, the disturbance frequency of the traininduced wind also changes. Therefore, the contribution ratio of each natural vibration mode to the response varies at different train speeds, and so an increase in the train speed may not aggravate the beam vibration.
5.2.2 Influence of clearance under bridge
Figure 15 shows the bridge dynamic response at different clearances under the bridge. As the clearance under the bridge decreases, the maximum vibration displacement, acceleration, and counterforce of the temporary pier increase to different degrees, with the rate of increase being greater at smaller clearance levels. For example, as the clearance under the bridge decreases uniformly from 13.25 m to 5.25 m, the maximum vibration displacement of the beam end initially increases by 1.2 mm and eventually increases by 2.6 mm.
5.2.3 Influence of intersecting angle
Figure 16 shows the bridge dynamic response at different intersection angles. In general, a decrease in the intersection angle causes the vibration displacement to increase, but the variations in the vibration acceleration and maximum counterforce of the temporary pier are irregular. This is mainly because, at smaller intersection angles, the influence range of the “positive–negative–negative–positive” pressure wave of the head and tail of the train widens as the train passes along the beam, which intensifies the disturbance to the beam body. However, changes to the fluctuating wind frequency and the inconsistent stress time of different beam sections complicate the beam vibration.
Within the parameter ranges investigated in this paper, the absolute values of the maximum vertical vibration displacement and acceleration of the beam are within 4–16 mm and 85–400 mm/s^{2}, respectively, in the construction stage. The influence of traininduced wind on the steel boxgirder of the closure segment is limited. Moreover, the extreme values of the positive counterforce are less than 105 kN, which is negligible compared with the maximum limit of 2000 kN. The absolute values of the negative counterforce are less than 300 kN, which will not cause the temporary pier to void.
5.3 Operation stage
During the operation stage, the track centerline is arranged at the midspan position. Through comparative analysis, it is found that the maximum bridge responses can be obtained in this scenario. The scenarios of multiple presence of trains are not considered. In the operation stage, the influence of train speed, clearance under the bridge, and train–bridge intersection angle on the vibration displacement and acceleration of the beam are basically the same as in the construction stage. Within the parameter ranges investigated in this paper, the absolute values of the maximum vertical vibration displacement and acceleration of the beam are within 1–5 mm and 42–174 mm/s^{2}, respectively, in the operation stage. Compared with the construction stage, the displacement and acceleration of beam decrease, which is mainly because of the stronger bridge boundary constraints and greater total stiffness.
According to the definition of comfort in the German EN03 standard, the vertical acceleration of the beam during single line operation is less than 0.50 m/s^{2}, which is classified as “very comfortable.” Therefore, within the parameter ranges investigated in this paper, the beam vibration caused by traininduced aerodynamic action will not affect pedestrian comfort.
6 Conclusions
Based on FE numerical simulations of a steel boxgirder cablestayed bridge, this paper has investigated the aerodynamic characteristics of highspeed trains passing through a bridge. The influence of the train speed, clearance under the bridge, and train–bridge intersection angle was considered, and the bridge dynamic response during construction and operation stage was studied. The main conclusions can be summarized as follows:

(1)
The wind pressure on the bottom plate and the beam lift extremum have a quadratic relationship with the train speed and a negative power exponential relationship with the clearance under the bridge. At different intersection angles, the lift amplitude at both the sidespan beam and the midspan beam section is less affected than at the 1/4span and 3/4span beams.

(2)
The bridge dynamic responses in the construction and operation stages are basically the same under the different influencing factors. As the train speed increases and the clearance under the bridge decreases, the vibration displacement, acceleration, and maximum counterforce of the temporary pier basically exhibit an upward trend. As the intersection angle decreases, the vibration displacement generally increases, but the variations in the vibration acceleration and the maximum counterforce of the temporary pier do not exhibit clear trends. A simplified aerodynamic load model has been developed to guide engineering practice, and wind pressure models of the bottom plate at different train speeds and different clearances under the bridge were summarized.

(3)
Within the parameter ranges investigated in this paper, the absolute values of the maximum vertical vibration displacement and acceleration of the beam were found to be within 4–16 mm and 85–400 mm/s^{2}, respectively, in the construction stage. The influence of traininduced wind on the steel boxgirder of the closure segment was found to be limited. During the operation stage, the absolute values of the maximum vertical vibration displacement and acceleration of the beam were within 1–5 mm and 42–174 mm/s^{2}, respectively. Compared with the construction stage, the dynamic response of the beam was significantly decreased by the larger overall stiffness and the constraints of the support.

(4)
Within the parameter ranges investigated in this paper, the absolute values of the positive counterforce of the main girder falling on the temporary pier after turning were less than 105 kN, which is negligible compared with the maximum limit of 2000 kN. The absolute values of the negative counterforce were less than 300 kN, which will not cause the temporary pier to void.

(5)
Within the parameter ranges investigated in this paper, the beam vertical acceleration during the operation stage was less than 0.50 m/s^{2}, which is classified as “very comfortable.” Therefore, the beam vibration caused by traininduced aerodynamic action will not have a significant impact on pedestrian comfort.
The main results of this study are obtained based on CFD simulations. Nevertheless, the presented results themselves are believed to be conceptually true in terms of their prediction insights concerning the overall performance of the specific case from an engineering perspective. The instrumentation plan, loading scheme, the anticipated wind pressures and structural responses in the upcoming field test will be mainly based on the findings of this article.
Availability of data and materials
The data and materials in current study are available from the corresponding author on reasonable request.
Abbreviations
 LES:

Large eddy simulation
 CFD:

Computational fluid dynamics
 FE:

Finite element
 DNS:

Direct numerical simulation
 RANS:

Reynoldsaveraged Navier–Stokes
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This work was supported financially by the National Nature Science Foundation of China (Grant Nos. 51778534, 51978580).
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ZL: Formal analysis, Investigation, Writing – original draft. XZ: Conceptualization, Supervision, Writing – review & editing, Funding acquisition. TC: Validation. YL: Data curation. BL: Software. All authors read and approved the final manuscript.
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Liu, Z., Zhang, X., Chen, T. et al. Aerodynamic loads and bridge responses under train passage: case study of an overpass steel boxgirder cablestayed bridge. ABEN 3, 1 (2022). https://doi.org/10.1186/s4325102100051w
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DOI: https://doi.org/10.1186/s4325102100051w