The Train-Bridge Coupled Vibration Analysis of a Long-Span Prestressed Concrete Continuous Beam Bridge under Creep Deformation Effect

Abstract

The track smoothness of high-speed railroads is severely limited to ensure train performance. The concrete continuous girder bridge is deformed to influence the smoothness of the bridge decks and track caused by creep bulge. There is a need to research the impact of beam creep on the dynamical response of bridge and train operation safety. This paper used Midas software to calculate the long-term deformation of large-span prestressed concrete continuous beam bridges under concrete creep and shrinkage. The train-bridge coupled system was established by adopting self-programming software. Thereafter, the large-term deformation results of the main girder are considered as track irregularity input into the vibration equation of the train-bridge system. The safety of the train operation was evaluated by calculating the dynamic response of the bridge and analyzing the criteria of train running safety. It was shown that the indicators of the large-span bridge are within the allowable code values under all working conditions. The bridge deformation under creep has an impact on the displacement and acceleration response of the bridge when a high-speed train passes through. There is no noticeable impact of creep deformation on the operational performance of trains. Nevertheless, the criteria for assessing the safety of trains’ operation, such as derailment factor, wheel load differences, lateral wheel forces, and vehicle acceleration, have been increased.

1. Introduction

An increasing number of continuous beam bridges are currently adopted as high-speed railway bridges to cross rivers. Compared with other bridge types, the continuous girder has numerous advantages, such as high structural stiffness, fewer beam joints, small fold angles, and high track smoothness [1]. In high-speed railways, the trains are fully enclosed operation with high requirements for track smoothness and strict restrictions on plane parameters of the line [2]. The smoothness of the rail will be influenced by the creep deformation of the prestressed concrete bridges. After laying the ballastless track, the deformation due to the concrete creep camber is tightly constrained to avoid significant dynamic reactions of the car-bridge system. [3]. To ensure the smoothness of the track, it is essential to investigate the influences of the creep deformation of the concrete bridges on train running performance.

In a report by the American Concrete Institute in 1972, the main creep mechanisms are divided into four parts [4,5,6]. (1) Under the joint action of stress and adsorbed water layer, concrete generates stickiness deformation due to the sliding or shearing of the cement gel. (2) The volume of concrete is reduced under pressure due to the loss of water adsorbed in the structure. (3) Hysteresis elastic strain caused by cement cementitious restraining effect on the structures’ elastic deformation. (4) The phenomenon of microcracking, crystallization damage, and recrystallization occur locally in concrete, and this process produces permanent deformation. Determining the creep coefficient and calculating the shrinkage strain are critical issues in investigating the creep of concrete structures. In engineering, a known shrinkage creep prediction model is generally used to speculate on the unknown shrinkage creep of concrete structures [7].

The study of vehicle-bridge system interactions goes back 100 years. In the early days, most researchers used approximate methods to build simple analytical models of vehicle-bridge systems due to computational power limitations [8]. With the widespread use of the computer, there have been more in-depth studies on the interaction between vehicles and bridges. Manfred Zacher modeled a complex train-bridge coupled system using finite element software [9]. Bogaert investigated the displacement and acceleration of an arch bridge during the simplified train model passing through [10]. Wanming Zhai and He Xia estimate the dynamic response of a coupled train-track-bridge system utilizing computer simulation software (TTBSIM) [11]. Yongle Li et al. used the self-developed analysis software BANSYS to research the vibration pattern of a suspension bridge in various wind velocity fields [12]. Melo, L.T. presented a numerical vibration prediction program for the coupled vibration of trains and bridges by using ABAQUS and MATLAB. [13]. Hongye Gou studies the vibration of high-speed railroad bridges when trains pass through the bridges by using SIMPACK and ANSYS [14].

Most studies on coupled train-bridge systems focus on external loads such as wind and waves and earthquakes or concentrate on the effect of parameters such as train speed and vehicle weight on the dynamic response of car-bridge system. However, the prestressed concrete continuous girder bridge will undergo shrinkage creep deformation in the long-term process due to concrete characteristics. This long-term deformation can lead to rail deformation affecting the safety and comfort of train operation. This paper investigates the effect of creep camber on the dynamic response of a train bridge system by using a prestressed concrete continuous girder bridge with a main span of 108 m as the research object. The accuracy of the creep model was confirmed by comparing the Midas calculation results with the test results. The creep deformation of the main girder was calculated and entered into the train-bridge system equations as the track irregularity. The interaction between the train and bridge was calculated, and the train running safety was analyzed under the effect of creep deformation.

2. Numerical Analysis Model

2.1. Concrete Creep Deformation

It is critical to determine the creep coefficient and shrinkage strain in studying the shrinkage creep of concrete structures. These parameters are impossible to obtain ultimately through experiments. It is more efficient to use the existing shrinkage creep prediction model to speculate on the long-term effects of concrete shrinkage creep.

Based on the “Design Code for Railway Bridges and Culverts” issued by China, the CEB-FIP (2010) model is used to calculate the creep coefficient of concrete structure [15]. The definition of the coefficient is as follows.

$$\phi \left(t,\tau \right)=\frac{{\epsilon }_c(t,\tau )E_{28}}{\sigma (\tau )}$$

(1)

where $\tau$ is loading age, and $\sigma (\tau )$ is the uniaxial constant stress applied to the concrete at time $\tau$. $\phi (t,\tau )$ is concrete creep coefficient from time $\tau$ to time t. E₂₈ is the modulus of elasticity of concrete with a curing time of 28 days. ${\epsilon }_C(t,\tau )$ is the creep strain generated between time $\tau$ to time t. The creep strain is determined by the relative humidity of the environment, the concrete composition and the geometric parameters of the structure.

2.2. Vehicle-Bridge Coupled Model

According to the theory proposed by Academician Qingyuan Zeng and Professor Xiangrong Guo of Central South University, in this paper, high-speed trains, tracks and bridges are treated as a total system of interactions [16,17,18]. The kinetic equations of the car-bridge coupling system are established using Dahlberg’s principle. The equation is solved by using the set-in-right-position rule and the principle of Total Potential Energy with Stationary Value in Elastic System Dynamics [19,20,21].

2.2.1. Train Spatial Vibration Equation

A CRH (China Railway High Speed Train) carriage can be simplified to four wheels, two bogies and one body which means seven rigid bodies. The topology diagram of the carriage is shown in Figure 1. The vehicle at its center point will have five DOFs, i.e., lateral, rolling, yawing, pitching and vertical displacements, described as $\{v_c\}={\{X_c\hspace{1em}{Y}_c\hspace{1em}{\psi }_{c\hspace{1em}}{\phi }_c\hspace{1em}{\theta }_c\}}^{\mathrm{T}}$ [22]. The train’s bogie has four DOFs at the center of gravity, i.e., lateral, roll, yaw and vertical movement. $\{v_{bq}\}={\{X_{bq}\hspace{1em}{Y}_{bq}\hspace{1em}{\psi }_{bq\hspace{1em}}{\theta }_{bq}\}}^{\mathrm{T}}$ were designated as four DOF of front bogie, and another one denoted as $\{v_{bh}\}={\{X_{bh}\hspace{1em}{Y}_{bh}\hspace{1em}{\psi }_{bh\hspace{1em}}{\theta }_{bh}\}}^{\mathrm{T}}$. Only two DOFs are included for the ith wheelset on the bogie: lateral and rolling displacements, which are denoted as $\{v_{si}\}={\{X_{si}\hspace{1em}{\psi }_{bh}\}}^{\mathrm{T}}$. A simplified model of the train is shown in Figure 1, and their definitions are concluded in

Table 1. The parameters of the train model.

Item	Notation
Mass of car body, bogie, wheelset	Mc, Mg, Ms
One half of the longitudinal distance between the front and rear bogie centers of gravity	L
Half the length of the wheelbase	L1
Distance from the center of gravity of the vehicle to the central transverse spring	h1
Distance from the center of gravity of the frame to the central transverse spring	h2
Distance from the center of gravity of the frame to the transverse spring of the axle box	h3
Distance from the center of gravity of the wheelset to the transverse spring of the axle box	h4
Stiffness of the suspension systems between the car body and the bogie	Kux, Kuy, Kuz
Stiffness of the suspension systems between the bogie and wheelset	Kdx, Kdy, Kdz
Damping of the suspension systems between the car body and the bogie	Cux, Cuy, Cuz
Damping coefficient between frame and wheel pair	Cdx, Cdy, Cdz

.

The relative position of the train to the bridge is constantly changing as the train travels over the bridge. Based on the two fundamental theories above, it is possible to derive the stiffness, mass, damping matrix ([KV], [MV], [CV]) and load vector quantity {P} of the train [19,20]. Based on the set-in-right-position rule of matrix formation, the overall stiffness matrix [K], mass matrix [M], damping matrix [C] and load array {P} of the train-bridge system at any moment t can be calculated by combining the train and bridge dynamic matrices.

It can be deduced from D’Alembert’s principle for the total potential energy of the jth carriage.

$${\Pi }_{vj}=V_{mj}+V_{Fj}+U_{Ej}+U_{Ewj}+V_{gj}$$

(2)

The total vibration potential energy of j train on the bridge at t time as:

$$\begin{array}{c}{\prod }_v={\displaystyle \sum _{j=1}^N{\prod }_{vj}={\displaystyle \sum _{j=1}^N[V_{msj}+V_{mgj}+V_{mcj}+V_{Fu1j}+V_{Fu2j}+V_{Fu3j}+V_{Fd1j}}}+V_{Fd2j}+\\ V_{Fd3j}+V_{Fdj}+U_{Eu1j}+U_{Eu2j}+U_{Eu3j}+U_{Ed1j}+U_{Ed2j}+U_{Ed3j}+U_{E\omega j}+V_{gj}]\end{array}$$

(3)

where V_mj is the work of inertial forces on car body, bogie and wheelset. V_Fj is the work of damping suspension systems. U_Ej is the energy of deformation at damping suspension systems. U_ej is the strain energy at damping suspension systems. U_Ewj is the gravitational potential energy at wheelsets. V_gj is the gravitational potential energy at car body.

2.2.2. Bridge Spatial Vibration Model

The spatial vibration of the box girder bridge is studied by the FE method of beam segment element [23]. The main assumptions in the calculation are as follows:

(1)

The tapered edge sections and reinforcement are not considered in the limit element model.

(2)

The deflection on both sides at each corner of the box section is assumed to be equal to $\gamma /2$.

(3)

The dynamic modulus of elasticity of concrete is set at 115–120% the static modulus with elasticity.

The box girder section has a torsion center k and shape center c, and the coordinate origin is set at the torsion center as depicted in Figure 2. There are four DOFs in the space of the box girder section, i.e., transverse displacement U, vertical displacement V, torsion angle φ, distortion angle γ [24]. The displacement at any point in the cross-section of the beam can be separated into the transverse displacement u (z, t), the vertical displacement v (z, t), the torsion angle φ (z, t) and the distortion angle γ (z, t) of the cross-section.

Calculating the damping matrix of a bridge accurately is very difficult. According to the empirical data of bridge damping, this paper determines the damping matrix’s ap-proximation by the bridge structure’s overall energy consumption during the vibration process [19].

The beam stiffness matrix [K] can be derived from the strain energy of each element, diaphragm plate and partition wall. A linear combination of the mass matrix [M] and the stiffness matrix [K] is used to represent the damping matrix of the beam structure.

$$\left[C\right]=\alpha \left[M\right]+\beta \left[K\right]$$

(4)

where $\alpha$ and $\beta$ are parameters with the following expressions:

$$\left\{\begin{array}{c}\alpha =\frac{2\left({\xi }_i{\omega }_j-{\xi }_j{\omega }_i\right)}{{\omega }_j^2-{\omega }_i^2}{\omega }_i{\omega }_j\\ \beta =\frac{2\left({\xi }_j{\omega }_j-{\xi }_i{\omega }_i\right)}{{\omega }_j^2-{\omega }_i^2}\end{array}\right.$$

(5)

where ${\omega }_i$ and ${\xi }_i$ denote the i-order natural frequency and the i-order damping ratio, respectively. ${\omega }_j$ and ${\xi }_j$ denote the j-order natural frequency and the j-order damping ratio, respectively.

2.2.3. Vehicle-Bridge Coupling Vibration Equation

In analyzing the vitality equation of the bridge system, the train and the bridge are considered a whole system. The total potential energy of the train-bridge system at t seconds is equal to the sum of the potential energy of the train, girder and piers [19].

$$\Pi ={\displaystyle \sum _{i=1}^n{\Pi }_{vi}+{\Pi }_{bi}+{\Pi }_{pi}}$$

(6)

where ${\Pi }_{vi}$,${\Pi }_{bi}$ and ${\Pi }_{pi}$ denote, respectively, the potential energy of i train, i bridge, i piers. The dynamic matrix of the prestressed concrete continuous girder bridge-train system at time t can be obtained [19,20].

$$[M]\{\ddot{\delta }\}+[C]\{\dot{\delta }\}+[K]\{\delta \}=\{P\}$$

(7)

where $\{\ddot{\delta }\}$, $\{\dot{\delta }\}$, $\{\delta \}$ express, separately, the train-bridge array of acceleration, velocity and displacement.

The train-bridge displacement array $\{\delta \}$ can be divided into k known parameters and n unknown parameters, i.e., Equation (7) can be represented by Equation (8).

$$\left[\begin{array}{cc}{M}_{kk}& M_{kn}\\ M_{nk}& M_{nn}\end{array}\right]\left\{\left.\begin{array}{c}{\ddot{\delta }}_k\\ {\ddot{\delta }}_n\end{array}\right]\right.+\left[\begin{array}{cc}{C}_{kk}& C_{kn}\\ C_{nk}& C_{nn}\end{array}\right]\left\{\left.\begin{array}{c}{\dot{\delta }}_k\\ {\dot{\delta }}_n\end{array}\right]\right.+\left[\begin{array}{cc}{K}_{kk}& K_{kn}\\ K_{nk}& K_{nn}\end{array}\right]\left\{\left.\begin{array}{c}{\delta }_k\\ {\delta }_n\end{array}\right]\right.=\left\{\begin{array}{c}{P}_k\\ P_n\end{array}\right\}$$

(8)

The function of track irregularity and creep irregularity can be input in Equation (8) to calculate the system spatial vibration response. Equation (8) can be written as:

$$[M_{nn}]\{{\ddot{\delta }}_n\}+[C_{nn}]\{{\dot{\delta }}_n\}+[K_{nn}]\{{\delta }_n\}=\{P_n\}-[M_{nk}]\{{\ddot{\delta }}_k\}-[C_{nk}]\{{\dot{\delta }}_k\}-[K_{nk}]\{{\delta }_k\}$$

(9)

$$[M_{kk}]\{{\ddot{\delta }}_k\}+[C_{kk}]\{{\dot{\delta }}_k\}+[K_{kk}]\{{\delta }_k\}+[M_{kn}]\{{\ddot{\delta }}_n\}+[C_{kn}]\{{\dot{\delta }}_n\}+[K_{kn}]\{{\delta }_n\}=\{P_k\}$$

(10)

where $\{{\ddot{\delta }}_k\}$, $\{{\dot{\delta }}_k\}$, $\{{\delta }_k\}$ is the excitation for transverse vibrations in the train-bridge system.

Typically, the irregularity of the track is entered into the coupled vibration equations as a function of power spectral density [25]. In this paper, the creep amplitude of the main girder is considered as track irregularity and is introduced into the car-bridge system for dynamic calculations. The complex nonlinear dynamical equations above are generally solved by a time domain stepwise integration method [26].

3. The Creep Deformation Calculation of Continuous Beam Bridge

3.1. Engineering Background and FE Model

In this article, a long-span prestressed concrete continuous beam bridge is taken from Wuhan-Guangzhou High-Speed Railway in China. The superstructure of the bridge is single-cell box girders with variable cross sections and material of C55 concrete. There are two parallel train lines on the bridge, which are designed for a speed of 350 km/h. The bridge is a six-span prestressed concrete continuous bridge with a span arrangement of 64 m + 4 × 125 m + 64 m, as seen in Figure 3. The maximum beam height is 8.7 m, the minimum height is 5.2 m, and the soffit varies according to a circular curve (R = 467.125 m). Piers are round-ended hollow piers made of C35 concrete.

3.2. Comparison of Theoretical and Experimental Values of Creep Deformation

Midas Civil, a commercial structural analysis and design software in the bridge field, was used to calculate the effect of shrinkage creep on the deflection of the main beam. The deflections at 60 and 180 days after the completion of track laying are calculated according to the CEB-FIP (2010) model. The calculated results are then compared with the test results of creep in the main beam. Figure 4 shows the comparison between the calculated and theoretical values of bridge creep at 90 and 180 days. The tendency is relatively consistent between the calculated results and the test results for the concrete continuous girder bridge. Due to the effect of ambient temperature variation on the deflection of the main beam, the calculated results are smaller than the test results. The accuracy of the bridge’s creep model was demonstrated by comparing the test results with the measured results.

The creep deformation is calculated at different time periods along the beam according to this method, as shown in Figure 5. The large deformation of the main girders will occur within three years after the building of the bridge due to shrinkage creep. After three years, the growth rate of deflection in the main beam gradually decreases with the development of shrinkage creep. For the long-span continuous beam bridge, the camber deformation reaches the maximum at ten years. In that period, the upper arch of the bridge side span reached 1.6 mm and the main span deflected up to 8 mm.

4. Effects of Concrete Creep on Train-Bridge Systems

The smoothness of the bridge deck and track can be affected due to the creep deformation of the concrete continuous beam bridge. There are strict limits on the creep values for prestressed concrete bridges in the Chinese Railway Bridge Code. The creep deformation of the bridge is input into the train-bridge coupling system as track irregularity using self-programmed software. The effect of creep deformation of large-span prestressed concrete continuous girder bridges on the capacity of high-speed trains was calculated using this method.

4.1. The Natural Vibration Characteristics of Bridge

The structural stiffness of the bridge can be reflected by its natural frequency and mode of vibration, which is very important for the seismic design. The finite element model of continuous beam in this paper is built by the software programmed by Professor Guo of Central South University. The finite segment method is used to model the main girder section, and the elastic modulus E and Poisson’s ratio μ are taken according to the current bridge code. The piers are modeled by the space beam element, and the m-method is used to consider the joint pile-soil action. The natural vibration characteristics of the continuous beam bridge are analyzed by building the dynamical matrix as shown in

Table 2. The natural vibration characteristics of continuous beam bridge.

Mode Frequency (Hz)	Finite Element Model	Vibration Characteristics
1st Mode 0.4070 Hz		Longitudinal drifting of main beam and pier
2nd Mode 0.8740 Hz		Main beam and pier bendingin lateral
3rd Mode 1.010 Hz		Anti-symmetrical bending of main beam in vertical

.

4.2. Bridge Dynamic Response Analysis under Creep Effect and Train Load

The dynamic response of a train passing through a bridge at different speeds was calculated based on the CRH train model. The train model is composed of sixteen train groups in the form of twelve motor cars and four trailer cars. The track irregularity spectrum of German high-speed lines was adopted in the train-bridge system. The train formation and working conditions are shown in

Table 3. Introduction to the train model.

Train Model	Abbreviations	Train Formation	Velocity (km/h)	Working Condition
China Railway High-Speed (China)	CRH	(M 1 + M + T 1 + M + M + T + M + M) × 2	250, 275, 300, 325, 350	double line train

¹ The M is motor car, and the T is trailer car.

.

There is a requirement for the creep upwarp value of ballastless track deck girders, i.e., the maximum value is 20 mm according to the code for design of high-speed railway (TB1062-2014). The results of creep amplitude show that the maximum value of concrete creep up-arch is reached after ten years. Considering the different train speeds and different amplitudes of beam creep, the impact coefficients of the bridge are calculated, respectively, as follows.

Figure 6 shows the relationship between bridge impact factor and train speed, this coefficient increases with the rising speed. When the train speed is greater than 300 km/h operation, the impact factor of the bridge is increased affected by the shrinkage creep of the concrete. The displacement and acceleration responses of the beam mid-span were calculated in order to investigate further the impact of creep convexity on the bridge, as shown in Figure 7, Figure 8 and Figure 9.

Figure 7a indicates that the response of vertical displacement at mid-span is enhanced with increasing train speed, and the increase is unremarkable from 275 to 300 km/h. With the increase in train speed, the lateral displacement response of the mid-span decreases, as shown in Figure 8a. At the mid-span position of the main beam, both lateral and vertical accelerations in the mid-span reach their maximum values with a vehicle velocity of 325 km/h. Under the ten-year creep, there is a greater amplitude of displacement and acceleration than without creep in the mid-span. The upward climbing of the main beam has little effect on the lateral dynamic response of the top of the pier, as seen in Figure 9. The comprehensive analysis shows that there is relatively less influence of creep camber on the dynamic responses of the bridge.

4.3. The Criteria for Train Running Safety under the Effect of Creep Camber

Although the influence of creep is relatively minor on the bridge, it is necessary to analyze the impact of creep camber on the train-bridge system by the standard of train operation safety. These criteria include derailment factor, unloading factor, wheel/rain lateral force, car acceleration and Sperling comfort index in the High-Speed Rail Design Code (HSR Code) [27,28,29]. The criteria indexes for train running safety of the vehicle are calculated, combining the different train speeds and the beam creep effect.

4.3.1. The Derailment Factor

The derailment factor is an indicator to assess the security of the vehicle wheels against derailment and is defined as Q₁/P₁ in Equation (11) [30,31]. As the lateral force Q₁ increases, the wheel tread will gradually lift if the vertical force P₁ cannot prevent the wheel from climbing up.

$$\frac{Q_1}{P_1}=\frac{tg{\alpha }_1-{\mu }_1}{1+{\mu }_{1}tg{\alpha }_1}$$

(11)

where Q₁ is lateral force. P₁ is vertical force. ${\alpha }_1$ is wheel tread angle. ${\mu }_1$ is the friction coefficient between flange and rail. The limit value of Q₁/P₁ is 0.8 in the HSR code. The CRH trains are composed of motor (M) cars and trailers (T) cars, whose derailment factors are calculated under different train speeds and creep effect in Figure 10.

The derailment factors of M and T cars tend to increase significantly with the increase in speeds in the range of 250–300 km/h. The derailment coefficient of motor vehicles decreases with the increase in speed after the train speed reaches 300 km/h. In contrast, the derailment coefficient of the trailer cars increases more significantly. The train derailment factor under the ten-year creep is more significant than the train derailment factor with no creep. It can be concluded that there is an impact of creep on the derailment factor for the train.

4.3.2. Offload Factor

The wheel load decreases due to the wheel vibrating upside during the vehicle running on the bridge. The train can be derailed even with small lateral forces if the wheel load is reduced to a certain value [32]. Therefore, the offload factor is used as a measure of safety against derailment. The expression of the unloading factor is shown in Equation (12).

$$\left\{\begin{array}{c}\begin{array}{cc}\Delta P/P=0.65\hfill & \left(\mathrm{Frist} \mathrm{limit}\right)\hfill \end{array}\hfill \\ \begin{array}{cc}\Delta P/P=0.60\hfill & \left(\mathrm{Second} \mathrm{limit}\right)\hfill \end{array}\hfill \end{array}\right.$$

(12)

where $\Delta p$ is the reduction of the wheel load on the unloaded side, and $p$ is the average load of both wheels. The M and T cars’ offload factors are calculated under different train speeds and creep effect as shown in Figure 11.

There is no significant growth trend in the offload factor of M cars with increasing speed. However, the offload factor of T cars has an increased offload factor in the 300–350 km/h range. For the offload factor with the same condition, the offload factor of the car considering creep irregularity (i.e., ten-year creep) is much greater than that neglecting creep irregularity (i.e., no creep). Thus, it is concluded that the irregularities of creep have an effect on the unloading factor of the car. According to the HSR code, the above offload factors, which are less than the second limit 0.60, satisfy the safety of traffic.

4.3.3. Car Body Acceleration

The measuring points of acceleration are located on the traction beams of the vehicle underframe [33]. The limits for acceleration are set in the China code, i.e., the maximum value of vertical acceleration is 1.3 m/s² and the lateral acceleration is 1.0 m/s². The acceleration of the car body is calculated as shown in Figure 12.

From Figure 12, the acceleration of the body can be seen to increase significantly with vehicle speed and creep effect for M and T cars. Under the same creep irregularity, the vertical dynamic response of the motor car is larger than that of the trailer car, but the lateral dynamic response is smaller. In addition, this evaluation indicates that the T and M train are within the limit at the all-work conditions.

4.3.4. Sperling Comfort Index

The Sperling comfort index is applied in the dynamic analysis of train-bridge systems to assess the operational stability of the vehicles [34]. The sperling comfort index can be calculated as Equations (13) and (14). In the China code, the standard for giving the Sperling comfort index is shown in

Table 4. Criteria of Sperling comfort index in HSR Code.

Vehicle	Criteria of Sperling Comfort Index
	Excellent	Good	Accepted
motor cars	≤2.75	2.75~3.10	3.10~3.45
trailers car	≤2.50	2.50~2.75	2.75~3.00

.

$$W=\sqrt[10]{W_1^{10}+W_2^{10}+\cdots +W_N^{10}}$$

(13)

$$W_i=7.08 \sqrt[10]{\frac{{A_i}^3}{f_i}F(f_i)}$$

(14)

where $F(f_i)$ is frequency weighting function. $f_i$ is the ith frequency with unit of Hz. $A_i$ is the amplitude of the acceleration component with frequency $f_i$.

As can be seen from Figure 13, the Sperling comfort indexes of the train are excellent in the speed range of 250 to 275 km/h. However, the vehicle’s operational stability decreases as the train’s speed increases. At the same vehicle speed, the running stability of the train that considers creep irregularity is worse than that neglecting creep camber.

5. Conclusions

This paper investigates the impact of creep deformation at the main girder on the operational capability of high-speed trains, taking a large-span continuous beam bridge on a high-speed railway as the background. Midas software was used to establish the creep model of this bridge, then the long-term deformation of the bridge was calculated under a constant load. The coupled vibration equations of the train bridge were established using the set-in-right-position rule and the principle of Total Potential Energy with Stationary Value in Elastic System Dynamics. The long-term bridge creep, as a form of track irregularity excitation, is imported into the train-bridge coupling system. Finally, this complex system is solved by a self-designed program to analyze the dynamics index of the train and the bridge. The following conclusions are drawn:

(1)

When long-span continuous beam bridge is completed, the main girders will have large deformation due to the shrinkage and creep of the concrete. After three years, the main beam’s deflection growth rate gradually decreases with shrinkage creep development. For this bridge, the camber deformation reaches the maximum at ten years.

(2)

There is no significant effect of creep deformation on the dynamic responses of the bridge as a high-speed train passes through. The displacement and acceleration responses of key sections are below the limits of the design code under all work conditions. The results show that long-term creep deformation of the bridge has no significant effect on the stiffness of the bridge.

(3)

The creep deformation of the main girder also has no noticeable impact on the train running safety of high-speed trains. However, the criteria for train running safety, for example, the car body acceleration, derailment factor, Sperling comfort index and offload factor, have increased to a certain extent. Under the influence of long-term creep, the deformation will not cause significant problems to the safety of train operations. However, it also has an impact on the comfort of passengers traveling.