Dynamic analysis of train and bridge in crosswinds based on a coupled wind-train-track-bridge model

Abstract

The dynamic performance and safety of both the train and the bridge are great concerns for long-span bridges subjected to crosswinds. For railway bridges, there is track laid on the bridge deck. The wind, train, track, and bridge subsystems form a complicated coupling system. This paper presents the dynamic responses of both the train and the bridge under crosswinds based on a wind-train-track-bridge coupled model, in which the vibration effect of the track structure involved in the coupling system is especially considered. For studying the dynamic responses of both the train and the bridge, models of railway vehicle dynamics are used for the train and a three-dimensional finite element model is used for the bridge. The train-track interaction and the track-bridge interaction are respectively considered. The wind action on the train and the bridge consists of a steady-state force and an unsteady-state force. Based on an in-situ test, the proposed numerical wind-train-track-bridge model involving the vibration effect of the track is validated. Then, the application of this proposed model on a long-span cable-stayed bridge on the Shanghai-Suzhou-Nantong HSR (high-speed railway) is presented as a case study. The typical running train and track currently serving in the Chinese railway transportation system are modeled in this study. Excitations from track irregularities are also taken into account. Finally, with the proposed wind-train-track-bridge coupled model, the dynamic performance of both the train and the bridge subjected to crosswinds are investigated.

Keywords

dynamic analysis railway vehicle long-span bridge track vibration crosswinds wind-train-track-bridge model

Introduction

The dynamic behavior and safety of both the train and the bridge in crosswinds are concerned increasingly. In recent years, the spans of the suspension and cable-stayed bridges have been challenging new limits (Zhang et al., 2015). More large-scale bridges are constructed in the costal and canyon areas which makes the bridge and the vehicle more vulnerable to wind actions. Meanwhile, the massive scale of the HSR construction leads to a great increase in the operation speed of trains. The highly raised train speed, to some extent, adds to the risk of fatal accidents of trains under extreme windy environment.

To ensure the working performance of long-span bridges and the running safety of vehicles subjected to crosswinds, it is of great importance to study the dynamic interaction between the crosswind, moving vehicle, and bridge structure (Cai and Chen 2004; Cai et al., 2015; Li et al., 2005). Han et al. (2017) investigated the effect of the aerodynamic interference between the vehicle and the bridge on dynamic responses of the coupled system. Wang and Xu (2015) explored the shielding effect of the bridge tower on the behavior and safety of the vehicle by using a coupled vehicle-bridge-wind system subjected to varying wind loads. Similar critical conditions that can cause sudden change of wind forces acting on the vehicle, for example, when the vehicle drives through a tunnel, windbreak or encountering another vehicle, were also investigated (Bao et al., 2019; Xue et al., 2020; Yang et al., 2020a). He et al. (2020) analyzed the effects on the train-bridge vibrations of non-stationary winds characterized by a transient wind duration and maximum wind speed.

For railway bridges, there is track laid on the bridge deck, which is essentially different from highway bridges. When the train passes over the railway bridge, the train, track, and bridge mutually influence each other. The track structure, as the link between the train and the bridge, plays an important role in the coupled train-bridge system. Extensive studies have been carried out on the train-track coupled dynamics (Paixão et al., 2015; Torstensson et al., 2011; Yang et al., 2020b) and the train-track-bridge coupled dynamics (Antolín et al., 2013; Zhu et al., 2017; Zhai et al., 2019). Zhai et al. (2013a) presented a systematical framework to investigate the high-speed train-track-bridge dynamic interaction including a theoretical model and numerical simulations and an onsite experiment was implemented to validate the calculated results in their companion work (Zhai et al., 2013b). Xu and Zhai (2020) formulated a train-track stochastic analysis model considering the influence of system spatial variability on the train-track interaction. Chen (2020) discussed the relationship between the track stiffness and the dynamic performance of the train-track-bridge system.

However, after the wind was introduced, relatively few studies were found on the wind-train-track coupled dynamics (Xu and Ding 2006). Zhai et al. (2015) proposed a numerical model for analyzing the wind-train-track interaction to investigate the dynamic behavior of a high-speed train running on the track in crosswinds. Heleno et al. (2021) implemented a parametric study from perspectives of variability of train properties on the running safety against crosswinds with a train-track-wind interaction system.

It can be found that the wind, train, track, and bridge components are essentially coupled with each other. Taking advantages of previous groundwork on the wind-vehicle-bridge, train-track-bridge, and wind-train-track coupled dynamics, this study systematically integrates these four components and develops a more realistic wind-train-track-bridge interactive model. This model, compared to the conventional wind-vehicle-bridge coupled model, especially includes the vibration effect of the track structure. Firstly, models of railway vehicle dynamics are used for trains and the track while a three-dimensional finite element model is developed for the bridge. The train-track interaction and the track-bridge interaction are respectively considered. The wind action on the train-bridge system is composed of a steady-state force and an unsteady-state force. Then, according to the dynamic characteristics of each sub-model, a combined explicit-implicit integration method is used to solve the equation of motion for the whole system. Moreover, based on an onsite experiment, the proposed model is validated. Finally, a series of numerical simulations are conducted to demonstrate the effect of crosswind on the dynamic performance of both the train and the bridge.

Wind-train-track-bridge coupled model

The wind-train-track-bridge coupled model is comprised of four sub-models, namely, the wind, train, track, and bridge sub-models, which are connected by virtue of the wheel-rail interaction, track-bridge interaction, and wind forces acting on the train and the bridge, respectively, as shown in Figure 1.

Figure 1.

Wind-train-track-bridge coupled model.

Vehicle-track coupled model

A high-speed train consists of a set of several identical 4-axle vehicles. Each vehicle is a multi-body system and is discretized into seven components, i.e., one car body, two bogie frames and four wheelsets, which are connected through suspension systems between them. To simplify the modeling but with sufficient accuracy, the process of establishing the vehicle model involves assumptions that each component in a vehicle is a rigid body, neglecting their elastic deformation during the vibration, and the train runs over the bridge at a constant speed so that the displacement of the component in the longitudinal direction is not considered. Thus each rigid component has 5 DOFs, namely the lateral, floating, rolling, pitching, and yawing motion, respectively. Each individual vehicle has 35 DOFs in total. The connections between the bogie and the wheelset as well as between the car body and the bogie are named as the primary and secondary suspension system, respectively, which are characterized by two linear springs and two viscous dashpots in either the horizontal direction or the vertical direction. Figure 2 illustrates the various components of the vehicle-track coupled model, in which the vehicle model and the track model are interacted with each other through the wheel-rail interface.

Figure 2.

Vehicle-track coupled model. (a) Side view (b) End view, (c) Plan view.

For bridges that carry railways, the track is laid on the bridge deck and the forces from wheels of the train are transmitted to the bridge deck through the track. In previous studies, the track on the whole was directly counted as a dead load on the deck, namely only the mass of the track was considered regardless of its vibrations. However, under the effect of the moving train, the track gauge is dynamically varying and increases over time, which evidently will produce a severe effect on the driving stability of the train and the ride comfort of passengers. Moreover, as the driving speed increases, these unique vibration characteristics of the track structure involved in the whole train-bridge coupled system become more obvious and cannot be ignored. Therefore, in order to more precisely evaluate the running performance of the train and to prevent it from derailment, the vibration effect of the track in the train-bridge system should be synthetically considered.

With the rapid development of modern railway transportation system, different types of track structures are becoming widely used to satisfy different running demands. Two main track types are now serving on the Chinese railway lines, namely the ballasted track and the non-ballasted track. The former type is used on the selected bridge in this study, and the model is shown in Figure 2. It consists of two rails, a fastening system, sleepers, and ballast. The two parallel rails are modeled as continuous Bernoulli-Euler beams supported by a three-layer foundation with discrete fasteners, sleepers and ballast included, considering horizontal, vertical, and torsional displacement. The fastening system is idealized by vertical and lateral springs and dampers to represent its dynamic property. The sleeper is simplified as rigid body considering horizontal (i.e., lateral), vertical, and torsional motion, and is connected to upper rails and lower ballast through the represented fasteners. Vertical and lateral springs and dampers are also used between the sleepers and the ballast to represent their elastic and damping properties. According to the spacing between two adjacent sleepers, the ballast is discrete into a number of mass blocks only considering the vertical displacement.

The double-block non-ballasted track system is composed of two rails and a fastening system. The rails are connected with the sub-structure simply through the fastening system which is modeled as vertical and lateral springs and dampers, as shown in Figure 3.

Figure 3.

Double-block non-ballasted track model.

As has been noted, the vehicle and the track model interact with each other through the wheel-rail interface. In the conventional train-bridge coupled dynamics, it is assumed that the wheels of the train and rails are rigid bodies without considering their elastic deformation, and the wheels consistently remain tight contact with the rails without detachment at any position. As the train driving speed increases, the track vibrates more significantly. Under some specific conditions, the wheels of the train may instantaneously detach from the rail, possibly from one side wheel of the wheelset or even both sides of the wheelset (Zhai et al., 2013a). This negatively influences the running safety of the train on the track. Therefore, it is necessary to set up a more realistic wheel-rail interaction model between the train and the track to describe the interface between them, then more accurately determining the wheel-rail interaction force. In this study, a dynamic wheel-rail model is established in which the elastic performance of both the wheels and rails at the contact point is considered, and instantaneous detachment between them is also allowed.

The wheel-rail interaction specifically refers to the geometry relationship and the interaction force between them. The wheel-rail geometry relationship mainly refers to the spatial position of the wheels and rails. The wheel-rail interaction force includes the wheel-rail normal contact force calculated based on the nonlinear Hertzian elastic contact theory, and the tangential wheel-rail creep force calculated by using the Kalker’s linear creep theory, details of which can be found in Zhai et al. (2013a).

Track-bridge coupled model

To have a good understanding of the dynamic performance of the bridge, a premise is to have knowledge of the bridge modal characteristics, i.e., the natural frequencies and corresponding vibration modes of the bridge. Modal analysis is performed for the bridge through a three-dimensional finite element model with different types of finite elements. Commonly, trusses, piers, towers, and beams of the bridge are modeled by using beam elements while cables of the bridge are modeled by using truss element. The connection between the beam and the pier is characterized by the main-subordinate node according to the real constraint condition between them. The displacement of the bridge deck at any section is identified in terms of lateral, vertical, and torsional displacement with respect to the centroid (or shear center) of the deck cross section. The structural damping of the bridge is modeled using the Rayleigh damping assumption. Thus, the damping matrix of the bridge is a linear combination of the mass and stiffness matrix of the bridge with two structural damping ratios (Li et al., 2005).

Different track structures on the bridge will affect the track-bridge interaction force. For instance, the ballasted track interacts with the bridge through the sleeper’s lateral vibration and the ballast’s vertical vibration, while the non-ballasted slab track connects to the bridge via the vertical and lateral constraint relations to the slab on the deck. For a variety of track structures, approaches for developing the track-bridge relationship are generally the same except the calculation of the track-bridge interaction force. Generally speaking, the interaction between the track and the bridge is discretized into a series of point-to-point interactions which are connected by linear springs and dampers. Therefore, as long as the vibration displacement and velocity of both the track and the bridge are determined, the interaction force between them can be obtained. Then, this interaction force is respectively used in the track’s equation of motion as the support reaction force on the track and in the bridge’s equation of motion as the excitation force on the bridge. Details of the track-bridge interaction can be referred to Zhai et al. (2013a).

Wind forces on train-bridge system

The wind forces acting on the bridge deck consist of three components, i.e., the static force, the buffeting force, and the self-excited force. The static wind force is due to the mean wind component in the flow, and usually expressed by three components, that is, the drag force $D_{st}^{B}$ , the lift force $L_{st}^{B}$ , and the moment $M_{st}^{B}$ as follows (Simiu and Scanlan, 1996):

D_{st}^{B} = \frac{1}{2} ρ U^{2} H C_{D} (α)

(1a)

L_{st}^{B} = \frac{1}{2} ρ U^{2} B C_{L} (α)

(1b)

M_{st}^{B} = \frac{1}{2} ρ U^{2} B^{2} C_{M} (α)

(1c)

where the subscript “

st

” refers to the static wind component while the superscript “

B

” refers to the bridge;

ρ

is the air density;

U

is the mean wind speed;

B

is the width of the deck along the mean wind flow;

H

is the height of the deck normal to the main stream direction;

α

is the angle of attack; and

C_{D} (α)

C_{L} (α)

, and

C_{M} (α)

are the static coefficients (or tri-component coefficients) of drag, lift, and moment, respectively, which usually vary with the angle of attack, and can be determined by routine section model tests or computational fluid dynamics (CFD) technique.

The buffeting force is induced by the fluctuation wind, and based on the quasi-steady theory, the buffeting drag, lift, and moment force for a unit deck length can be expressed as (Scanlan and Jones, 1990):

D_{bu}^{B} (t) = \frac{1}{2} ρ U^{2} B [2 C_{D} (α) \frac{u (t)}{U} γ_{D} + C_{D}^{'} (α) \frac{w (t)}{U} γ_{D}^{'}]

(2a)

L_{bu}^{B} (t) = \frac{1}{2} ρ U^{2} B {2 C_{L} (α) \frac{u (t)}{U} γ_{L} + [C_{L}^{'} (α) + C_{D} (α)] \frac{w (t)}{U} γ_{L}^{'}}

(2b)

M_{bu}^{B} (t) = \frac{1}{2} ρ U^{2} B^{2} [2 C_{M} (α) \frac{u (t)}{U} γ_{M} + C_{M}^{'} (α) \frac{w (t)}{U} γ_{M}^{'}]

(2c)

where the subscript “

bu

” refers to the buffeting wind component;

u (t)

and

w (t)

are the transverse (i.e., along wind) and vertical fluctuation wind velocity component, respectively;

C_{D}^{'} (α) = \partial C_{D} (α) / \partial α

C_{L}^{'} (α) = \partial C_{L} (α) / \partial α

, and

C_{M}^{'} (α) = \partial C_{M} (α) / \partial α

;

γ_{i}

and

γ_{i}^{'} (i = D, L, M)

are the aerodynamic admittance functions which present the unsteady characteristics of wind.

The self-excited force is due to the aeroelastic interaction between the wind and bridge motion. According to the impulse response function model proposed by Lin and Yang (1983), the self-excited force can be expressed as convolution integrals between bridge motion and impulse response functions. For example, the self-excited lift force is expressed as:

L_{se}^{B} (t) = L_{p} (t) + L_{h} (t) + L_{φ} (t)

= \int_{- \infty}^{t} f_{L p} (t - τ) p (τ) d τ + \int_{- \infty}^{t} f_{L h} (t - τ) h (τ) d τ + \int_{- \infty}^{t} f_{L φ} (t - τ) φ (τ) d τ

(3)

where the subscript “

se

” indicates the self-excited wind force;

p

h

, and

φ

are the lateral, vertical, and torsional displacement, respectively;

f_{L p}

f_{L h}

, and

f_{L φ}

are the impulse response functions, which can be determined by the flutter derivatives of the deck measured from wind tunnel experiments and the rational function approximation approach (Chen et al., 2000; Chen and Kareem, 2001). For example, the self-excited lift force induced by the vertical displacement of the deck can be expressed as:

L_{h} (t) = ρ U^{2} B {C_{1} h (t) + C_{2} \frac{B}{U} \dot{h} (t) + \sum_{k = 3}^{n} C_{k} \int_{- \infty}^{t} \exp [- \frac{d_{k} U}{B} (t - τ)] \dot{h} (τ) d τ}

(4)

where $C_{k}$ and $d_{k} (k = 3, \dots, n)$ are frequency independent coefficients of rational functions, which can be obtained by linear and nonlinear least-square fitting methods based on the measured flutter derivatives at the reduced frequencies; the value $n$ indicates the accuracy of this approximation; and the term of convolution integral can be calculated by the recursion algorithm.

The terms in equation (4) are respectively the structural displacement related aerodynamic stiffness term, velocity related aerodynamic damping term, and time-history related nonlinear convolution term. The self-excited drag and moment force can be similarly obtained.

The wind forces acting on the moving train in the wind-train-track-bridge coupled model consist of the static force and buffeting force and are in the same form of that on the bridge deck under crosswinds. For the sake of brevity, the derivations of wind forces on the train are referred to Li et al. (2005).

The self-excited force is resulted from the interaction between moving flexible structure and fluid. Due to the blunt cross section and small width of the train, the aerodynamic interaction between wind and train motion is weak such that the self-excited force acting on the train can be neglected. Additionally, owing to the small windward area of bogies and wheelsets of the train, the wind forces acting on them are not considered, that is, only wind forces acting on the car body of the train are included.

Numerical simulation

As mentioned earlier, the wind-train-track-bridge interaction model is comprised of a train sub-model, a track sub-model, a bridge sub-model, and wind field. The train, track, and bridge dynamically interact with each other by virtue of the wheel-rail interaction as well as the track-bridge interaction. The wind force on the bridge deck includes three components, i.e., the drag, lift, and moment force. Each component contains a steady-state force due to the mean wind, a buffeting force due to the fluctuation wind, and a self-excited force due to the aeroelastic interaction between the wind and bridge motion. Wind forces acting on the moving train only consider the steady-state force and the buffeting force, omitting its self-excited loads.

Therefore, the equations of motion for the coupled train-track-bridge system under crosswinds can be expressed as:

M_{V} {\ddot{X}}_{V} + C_{V} {\dot{X}}_{V} + K_{V} X_{V} = F_{VT} + F_{st}^{V} + F_{bu}^{V}

(5a)

M_{T} {\ddot{X}}_{T} + C_{T} {\dot{X}}_{T} + K_{T} X_{T} = F_{TV} + F_{TB}

(5b)

M_{B} {\ddot{X}}_{B} + C_{B} {\dot{X}}_{B} + K_{B} X_{B} = F_{BT} + F_{st}^{B} + F_{bu}^{B} + F_{se}^{B}

(5c)

where

M_{V}

M_{T}

, and

M_{B}

are the mass matrix of train vehicle, track, and bridge, respectively;

C_{V}

C_{T}

, and

C_{B}

are the damping matrix of train vehicle, track, and bridge, respectively;

K_{V}

K_{T}

, and

K_{B}

are the stiffness matrix of train vehicle, track, and bridge, respectively;

X_{V}

{\dot{X}}_{V}

, and

{\ddot{X}}_{V}

denote the vectors of the displacement, velocity, and acceleration of the train vehicle, respectively;

X_{T}

{\dot{X}}_{T}

, and

{\ddot{X}}_{T}

denote the vectors of the displacement, velocity, and acceleration of the track, respectively;

X_{B}

{\dot{X}}_{B}

, and

{\ddot{X}}_{B}

denote the vectors of the displacement, velocity, and acceleration of the bridge deck, respectively;

F_{VT}

is the vector of the wheel-rail interaction force between the vehicle and the track;

F_{st}^{V}

and

F_{bu}^{V}

are vectors of the two components of wind forces acting on the vehicle, i.e., the static wind force and the buffeting force, respectively;

F_{TV}

is the vector of the wheel-rail interaction force between the track and the vehicle;

F_{TB}

is the vector of the track-bridge interaction force;

F_{BT}

is the vector of the bridge-track interaction force; and

F_{st}^{B}

F_{bu}^{B}

, and

F_{se}^{B}

are vectors of the three components of wind forces acting on the bridge, i.e., the static wind force, the buffeting force, and the self-excited force, respectively.

During the passage of the train across the bridge, the position of the train changes with time. Thus, the coupled equation (5) is essentially time-dependent with time-varying coefficients. As such, the matrices in equation (5) should be updated at each time step after the new position of the train is identified. In this study, according to the dynamic characteristics of each sub-system, a combined explicit-implicit integration method is used to solve the equations of motion of the whole system. To be more specific, for the vehicle and track sub-systems, an explicit integration method is applied to obtain their dynamic performance; while for the bridge sub-system, an implicit Newmark-β method is applied to obtain its dynamic response. Finally, a computer program is developed for the simulation of the proposed wind-train-track-bridge coupled model.

Validation

As has been noted earlier, extensive studies have been carried out on the wind-train-bridge coupling system which, however, does not take into account of the vibration effect of the track structure in the train-bridge system. Therefore, a direct validation of the proposed wind-train-track-bridge coupled model is very difficult due to the lack of related data. Based on an available onsite experiment, this section mainly validate partially the proposed model without considering the wind effect.

The in-situ experiment was carried out on the Beijing-Shanghai HSR, which is a typical high-speed railway line in China. In the experiment, a steel bridge and a CRH3 EMU (electric multiple units) were adopted as the tested bridge and train, respectively. The steel bridge is a five-span continuous steel truss girder with a total length of 728 m (112 m+3×168 m+112 m). The width and height of the cross section of the truss girder is 30 m and 16 m, respectively. The main truss is triangle shaped with vertical web members and the internode spacing is 14 m. The flexible arch structures are constructed at the second, third, and fourth span of the bridge. The design speed is 350 km/h and the configuration of the bridge is shown in Figure 4.

Figure 4.

Configuration of tested bridge. (a) Side view, (b) Cross section view.

Compared in Figure 5 are the measured and calculated results of the train dynamic indices at different train speeds. The measured train speed is from 250 km/h to 350 km/h, while the calculated train speed is from 250 km/h to 375 km/h. It is shown that the calculated results are generally in good agreement with those from the test. The discrepancies are acceptable compared to their practical values.

Figure 5.

Comparison of measured and numerical results of train dynamic responses at different train speeds.

Displayed in Table 1 is the maximum measured and calculated running safety indices of the train when it passes over the bridge at speeds from 250 km/h to 350 km/h. The calculated results include those with and without considering the vibration effect of the track structure. The proposed WTTB model considers the vibration effect of the track structure while the WTB (wind-train-bridge) model does not. It can be found that the WTTB results show a difference with WTB results. Compared with the WTB model results, the WTTB calculation is more consistent with the measured results and the TTBSIM results (Zhai et al., 2013b) that also considers the vibration effect of the track structure. This is because the WTTB model is more realistic and practical.

Table 1.

Comparison between measured and numerical results of train dynamic responses at speed from 250 to 350 km/h.

Running safety indices	$R_{M}$	$R_{TTBSIM}$	$R_{WTTB}$	$R_{WTB}$
Running safety indices	$R_{M}$	$R_{TTBSIM}$	$\| \frac{R_{WTTB} - R_{M}}{R_{M}} \|$ %	$\| \frac{R_{WTB} - R_{M}}{R_{M}} \|$ %
Derailment coefficient	0.19	0.186	0.179	0.164
Derailment coefficient	0.19	0.186	5.8%	13.7%
Wheel-load reduction ratio	0.78	0.702	0.730	0.431
Wheel-load reduction ratio	0.78	0.702	6.4%	44.7%
Wheel-rail lateral force (kN)	24.57	16.83	15.27	13.44
Wheel-rail lateral force (kN)	24.57	16.83	37.9%	45.3%
Vertical sperling index	2.11	1.99	2.01	1.89
Vertical sperling index	2.11	1.99	4.7%	10.4%
Lateral sperling index	2.18	2.14	2.22	2.08
Lateral sperling index	2.18	2.14	1.8%	4.6%

WTTB: wind-train-track-bridge WTB: wind-train-bridge.

$R_{M}$ : Measured result $R_{TTBSIM}$ : TTBSIM result

$R_{WTTB}$ : WTTB result $R_{WTB}$ : WTB result.

Displayed in Table 2 is the maximum bridge displacement and acceleration responses at each mid-span with train speeds from 250 km/h to 375 km/h. The WTTB results show a satisfactory agreement with the verified TTBSIM ones.

Table 2.

Maximum dynamic responses of bridge at each mid-span within the speed from 250 to 375 km/h.

Span	Middle point displacement (mm)				Middle point acceleration (m/s²)
	TTBSIM result		WTTB result		TTBSIM result		WTTB result
	Vertical	Lateral	Vertical	Lateral	Vertical	Lateral	Vertical	Lateral
1^st span	5.638	0.341	5.678	0.327	1.581	0.167	2.168	0.115
2^nd span	5.350	0.348	5.421	0.367	2.139	0.103	2.484	0.128
3^rd span	5.315	0.312	5.374	0.316	1.017	0.083	1.092	0.089
4^th span	5.324	0.289	5.395	0.305	1.411	0.122	1.414	0.097
5^th span	5.342	0.292	5.575	0.306	0.944	0.127	1.302	0.124

In comparison with the results from TTBSIM, WTB, and field measurement, it can be concluded that the WTTB numerical model can take a relatively full consideration of the dynamic participation of each part in the coupled system. To some extent, it can better analyze the dynamic performance of the bridge and evaluate the running safety of trains in crosswinds as well as provide a reference for engineering applications.

Description of case study

Bridge and train prototype

A long-span cable-stayed bridge which carries a six-lane highway at the upper level of the deck and two Chinese grade-I trunk lines and two intercity passenger dedicated railways at the lower level of the deck on the Shanghai-Suzhou-Nantong HSR is chosen as an engineering example. The main configuration of the bridge is shown in Figure 6. This long-span cable-stayed bridge has an overall length of 2296 m with a main span of 1092 m, two side approach spans of 462 m each, and two end spans of 140 m each. The height of the vase-shaped tower is 325 m.

Figure 6.

Bridge prototype (unit: m). (a) Layout of bridge spans, (b) Cross section.

The bridge deck is a steel truss girder with three main trusses. The orthotropic plate and the steel box girder are constructed for the highway and railway bridge deck, respectively. The width of the cross section is 35 m with a distance of 17.5 m between the two adjacent main truss planes, which is triangle-shaped with vertical web members, 16 m in height and 14 m in internode length. The deck is 74.5 m above water. A three-dimensional finite element method was adopted to analyze the vibration characteristics of the bridge and the main vibration modes are displayed in Table 3.

Table 3.

Main vibration modes of the bridge.

Mode no	Natural frequency (Hz)	Mode type
1	0.077	1^st symmetric lateral mode
3	0.187	1^st symmetric vertical mode
4	0.221	1^st asymmetric lateral mode
5	0.265	1^st asymmetric vertical mode
12	0.493	1^st symmetric torsional mode

The train vehicle used in this study is a high-speed EMU train assembled with two identical marshals. Each marshal is composed of 8 cars (M+T+M+M+M+M+T+M), where M denotes the motor car and T the trailer car. The static axle loads of the motor car and the trailer car are 160 kN and 146 kN, respectively. Main parameters of the train vehicle can be referred to Xia et al. (2012). The designed speed of the train is 250 km/h, therefore, the German high-speed low-disturb track spectrum was transformed into the time domain and added as the track irregularity excitation source.

Generation of fluctuation wind

Since there is no field measurement data about the wind environment at the bridge site, the wind spectrum is selected and simulated to obtain the time-history of wind velocities. According to the properties of both the structural type and vibration modes of the long-span cable-stayed bridge and considering the correlation characteristics of natural wind, the three-dimensional wind velocity field is simplified into many one-dimensional wind velocity fields along the pylons and the deck (Li et al., 2004). The bridge deck considers the lateral and vertical wind velocity components while the pylon considers the lateral and longitudinal wind velocity components. The auto-spectral density functions based on the Chinese code are selected to simulate the lateral and vertical turbulent components of wind, while the longitudinal turbulent component is modeled according to Kaimal’s equation as expressed in equation (6):

\frac{n S_{u} (ω)}{u_{*}^{2}} = \frac{200 f}{{(1 + 50 f)}^{5 / 3}}

(6a)

\frac{n S_{ν} (ω)}{u_{*}^{2}} = \frac{15 f}{{(1 + 9.5 f)}^{5 / 3}}

(6b)

\frac{n S_{w} (ω)}{u_{*}^{2}} = \frac{6 f}{{(1 + 4 f)}^{2}}

(6c)

where

S_{u} (ω)

S_{ν} (ω)

, and

S_{w} (ω)

are the wind spectra density function in the lateral (along wind), longitudinal (along bridge), and vertical direction, respectively;

n

is the frequency;

u_{*}^{2}

is the shear velocity of the wind; and

f

is the dimensionless normalized frequency.

The coherence function uses Davenport’s form. Since the points where the wind velocity is simulated in this study are uniformly distributed along the deck, the coherence function can be simplified as:

C o h_{j m} (ω) = \exp (- λ \frac{ω r_{j m}}{2 π U (z)})

(7)

where

λ

is the decay factor ranging from 7 to 12;

U (z)

is the mean wind velocity at the height of

z

; and

r_{j m}

is the distance between point

j

and

m

Taking the simulation of the lateral and vertical fluctuation wind of bridge deck as an example, the total length of the bridge is 2,296m with the internode length of 14 m. Thus, the main truss has 164 sections with 165 nodes and the fluctuation wind velocities are simulated at these nodes. The train is also in the wind field before entering the bridge and after leaving the bridge, therefore, 5 more points on the roadbed at each bridge end are considered. A total of 175 points are generated along the deck and the total length of the simulated wind velocity field is 2436 m. For finite element nodes between two successive simulated points, linear interpolation method is applied to determine their wind velocities. The sampling frequency and the duration is 20 Hz and 100 s, respectively. Time-histories of the lateral and vertical wind velocities at the middle point of the main span with a mean wind speed of 20 m/s at the deck level are shown in Figure 7, respectively.

Figure 7.

Generated turbulent wind velocity at mid-span when $U$ =20 m/s. (a) Lateral turbulent wind velocity, (b) Vertical turbulent wind velocity.

Aerodynamic coefficients from wind tunnel tests

When the train runs over the bridge under crosswinds, the aerodynamic characteristics of the train and the bridge will influence each other. Therefore, when there are trains on the bridge, the aerodynamic performance of the bridge is measured taking into account of the aerodynamic effect from trains, and vice versa. With wind tunnel tests, the static coefficients (

C_{D}

C_{L}

, and

C_{M}

) of both the train and the bridge and their first derivatives to the attack angle (

C_{D}^{'}

C_{L}^{'}

, and

C_{M}^{'}

) at

0^{\circ}

attack angle are obtained, as listed in Table 4 (Xiao, 2017).

Table 4.

Aerodynamic coefficients.

	Bridge only	Vehicle only	Bridge-vehicle
	Bridge only	Vehicle only	Bridge	Vehicle
Lift coefficient at 0° attack angle $C_{L}$	0.452	0.166	–0.076	0.014
Drag coefficient at 0° attack angle $C_{D}$	0.895	1.573	0.833	1.049
Pitching coefficient at 0° attack angle $C_{M}$	–0.004	–0.039	0.028	0.067
$\partial C_{L} / \partial α$ at 0° attack angle	8.317	–2.056	6.127	0.029
$\partial C_{D} / \partial α$ at 0° attack angle	−0.659	1.475	–0.410	–3.392
$\partial C_{M} / \partial α$ at 0° attack angle	0.125	–0.246	–0.498	–1.075

Dynamic responses of bridge and train for case study

Bridge dynamic response

Figure 8 depicts the time-histories of lateral and vertical displacement of the bridge girder at the mid-span under different mean wind speeds (0–30 m/s). The train runs over the bridge on the first traffic lane (see Figure 6) at a speed of 250 km/h. It is seen that the maximum lateral and vertical displacement for the bridge at the mid-span are 79.4 cm and 25.1 cm, respectively, when the mean wind speed is 30 m/s.

Figure 8.

Displacement response time-histories of bridge girder.

One can find that the lateral responses of the bridge are more sensitive to the action of crosswind than the vertical ones. The lateral displacement of the bridge under crosswind is far larger than the case without crosswind. In comparison, it seems that the crosswind only has slight effects on the vertical response of the bridge. The reason is probably that the vertical stiffness of the bridge is relatively large, and the interaction force transmitted from the upper train produces main effects on the vertical response of the bridge. It can be concluded that the lateral response of the bridge is mainly influenced by the fluctuation wind, while the vertical response is mainly excited by the loading of the train.

The maximum displacements and accelerations of the bridge versus the mean wind speed (0–30 m/s) are displayed in Figure 9 when the train passes the bridge at a speed of 250 km/h. The maximum displacements and accelerations of the bridge versus the train speed (140–300 km/h) are shown in Figure 10 for a given mean wind speed of 20 m/s. While the displacement and acceleration responses of the bridge girder increase rapidly as the mean wind speed increases as shown in Figure 9, the train speed has smaller influences on the bridge dynamic responses than the wind speed.

Figure 9.

Maximum bridge responses versus mean wind speed.

Figure 10.

Maximum bridge responses versus train speed.

Train dynamic response

Displayed in Figure 11 are the time-histories of the lateral and vertical accelerations of the first car (motor car) in a train with and without crosswind actions. The train speed is 250 km/h. The numerical results show that the maximum lateral acceleration of the first car under crosswind is 0.841 m/s², while the maximum vertical acceleration is 0.916 m/s². Compared to the case without wind action, the dynamic responses of the train under crosswind are significantly larger.

Figure 11.

Acceleration response time-histories of first car.

Compared in Figure 12 are the maximum lateral and vertical accelerations of the motor car and the trailer car versus the mean wind speed. It is seen that, as the wind speed increases, the lateral responses for both the motor car and the trailer car increase more remarkably than the vertical ones. While the lateral acceleration of the trailer car is relatively larger than that of the motor car, the vertical acceleration is the opposite. This is similar to the bridge dynamic responses discussed above, that is, the lateral response of the vehicle is primarily influenced by the crosswind, while the vertical response is mainly determined by the vehicle loading. The greater weight of the motor car than that of the trailer car exemplifies these remarks.

Figure 12.

Maximum vehicle acceleration responses versus mean wind speed.

According to the Chinese standards (Ministry of Railways of PRC, 2015), the running safety of railway vehicles, as an important issue in railway transportation system, must be paid close attention to especially in windy environment and on long-span bridges. Four important indices are used as a reference for the safety evaluation, namely, the derailment coefficient ( $\leq$ 0.8), the wheel-load reduction ratio ( $\leq$ 0.6), the overturning coefficient ( $\leq$ 0.8), and the wheel-rail lateral force ( $\leq 0.85 (10 + P_{st} / 3)$ ) where $P_{st}$ is the vertical static wheel-rail load. In this case, the static load of the motor car and the trailer car are 160 kN and 146 kN, respectively, leading to an allowable wheel-rail lateral force of 53.83 kN and 49.87 kN, respectively.

Depicted in Figure 13 are the time-histories of the derailment coefficient and the wheel-load reduction ratio of the first car with and without crosswind actions. Figures 14–15 sketch the maximum value of the four safety indices versus the mean wind speed, respectively. It is seen that the crosswind has a strong effect on the train dynamic responses. Specifically, first, all the graphs of the safety coefficients for both the motor car and the trailer car show an increasing trend as the wind speed increases. Second, all safety indices of the trailer car are larger than those of the motor car. This is because the trailer car is lighter than the motor car, thus it is more sensitive to the crosswind action. However, for the wind speed less than 15 m/s, the derailment coefficient, wheel-load reduction ratio, and wheel-rail lateral force of both the motor car and the trailer car are slightly affected by the wind speed. For wind speed beyond 15 m/s, these three indices of the trailer car increase more remarkably than that of the motor car, and the influence of wind is becoming more pronounced. Third, when the wind speed reaches 25 m/s, the wheel-load reduction ratio and the wheel-rail lateral force of both the motor car and the trailer car exceed the allowances discussed above.

Figure 13.

Derailment coefficient and wheel-load reduction ratio time-histories of first car.

Figure 14.

Maximum derailment coefficient and wheel-load reduction ratio versus mean wind speed.

Figure 15.

Maximum overturning coefficient and wheel-rail lateral force versus mean wind speed.

Conclusions

A framework for performing dynamic analysis on the coupled train, track, and bridge systems under crosswinds has been presented. The static wind force, buffeting force, and self-excited force are considered for the bridge deck, while the static wind force and buffeting force are considered for train vehicles. Models of the train, track, and bridge are established, among which the train-track interaction and the track-bridge interaction are presented. To verify the model effectiveness, results from a field experiment, the verified TTBSIM program, and the conventional WTB calculation are adopted and compared, which shows a good sign for the valid application of the proposed model in this study. Furthermore, a case study of a long-span cable-stayed bridge with high-speed running trains is presented. The dynamic responses of both the bridge and the train in crosswinds are investigated. It is concluded that the lateral responses of the bridge and the train are more sensitive to the wind effect, and the trailer car in a train is more vulnerable than the motor car from safety consideration.

It is worthwhile to mention that the proposed wind-train-track-bridge dynamic model, compared to the conventional wind-vehicle-bridge model, especially considers the vibration effect of the track structure in the whole system, which is particularly vital for high-speed railways as has been stated. It is also noted that for a wind-train-track-bridge model, it would be more convincing for the verification by involving both the wind and train loads on the whole dynamic system. However, it is difficult to collect information for such a complicated system and there exist many uncertainties in the field including uncertainties of wind field, track properties, moving train, and bridge properties. Thus, a direct and comprehensive validation of the analysis framework is very challenging, which is also the focus of our future work. Nevertheless, the work in this study has laid a foundation for future investigations on the dynamic performance of the train-track-bridge coupled system in winds.

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: This work is supported by the National Natural Science Foundation of China (U1434205, 52178450), and Zhejiang Provincial Natural Science Foundation of China (LY19E080016).

ORCID iDs

Jinfeng Wu

Xiaozhen Li

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