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In order to study the random vibration performance of trains running on continuous beam bridge with vertical track irregularity, a time-domain framework of random analysis on train-bridge coupling system is established. The vertical rail irregularity is regarded as a random process. A multibody mass-spring-damper model is employed to represent a moving railway car and the bridge system is simulated by finite elements. By introducing the pseudo excitation algorithm into the train-bridge interaction dynamic system, expressions of the mean value, standard deviation, and power spectral density of the nonstationary random dynamic responses of bridge and vehicles are derived. Monte-Carlo simulations are implemented to validate the presented method. A comprehensive analysis of the train-bridge coupling system with vertical track irregularity is conducted focusing on the effect of the randomness of the vertical rail irregularity on the dynamic behavior of the running train and the three-span continuous concrete bridge. Moreover, stochastic characteristics of the indicator for assessing the safety and the riding quality of the railway cars running on continuous beam bridge are carried out, which may be a useful reference in the dynamic design of the bridge.

The passage of moving trains leads to the dynamic amplification of a bridge; meanwhile, the motion of a bridge deck affects the kinetic characteristic of trains. The study of train-bridge coupling system commenced from the mid-19th century [

The key issue for determining the statistical characteristics of random dynamic responses of the railway bridge system, when high-speed trains are passing through, is the complexity of the coupling system. For instance, nonlinearity exists at the contacts between the wheels and the rail, and the implementation of the theories of nonstationary random vibrations to the coupling system is difficult. A few researchers [

Monte-Carlo method has been adopted in random dynamic analysis of the train-bridge coupling system. Using Monte-Carlo simulations, many samples of the track irregularity are obtained and then analyses according to different parameters of bridges and trains are performed [

In addition to performing Monte-Carlo simulations, other approaches have also been developed to analyse stochastic dynamic problems. Jin [

Many of the investigations on the stochastic characteristic of the bridge are on the simply supported beam. The integrity of the deck of multispan continuous beam is better than that of simple supported girder, so the continuous beam is widely used in Chinese high-speed railway bridge. However, most of the research on the dynamic response of the multispan continuous beam bridge is deterministic [

The remaining parts of this paper are organised as follows. Section

In the train-bridge coupling system, a multibody model is used to represent a moving railway car and a spatial beam element is adopted to simulate the bridge system as shown in Figure

Model of train-bridge coupling system.

In this study, the railway car is modelled by a multibody connected by the primary and secondary springs with 6 degrees of freedom as shown in Figure

A spatial beam element model is applied to simulate the bridge system. Equations of motion of the bridge system can then be expressed as

The interaction between the train and bridge can be calculated through the compatibility condition of the displacement and equilibrium equations of the wheel/rail forces (see Figures

Compatibility condition of the displacement.

Wheel/rail forces.

In Figure

In Figure

After converting the single wheel load vector (

Equation (

If the two pseudo variables are obtained, the cross-power spectral density can be taken as

In this paper, vertical track irregularity is considered as random progress with zero mean. The kinetic equation of the coupled system can be expressed as follows:

Equation (

Substituting (

Furthermore,

Therefore, the responses vector induced by the pseudo load vector as indicated in (

The problems presented in the above section can be solved by applying the numerical integration and programming to the following analysis process:

Average value can be obtained by using (

PSD and SD of the dynamic responses of the train-bridge system are obtained as shown in the flowchart (Figure

Flowchart of the dynamic responses of the train-bridge system.

When normal contact conditions between the wheels and the rail are satisfied, the safety of the running vehicles can be guaranteed. However, under the worst load combination, the derailment or overturn accident may happen. Several parameters can be used to evaluate the safety of the trains, such as derailment coefficient, wheel unloading rate, and lateral swing force. In this paper, we choose the wheel unloading rate as the safety index of the railway cars. Riding quality can show the influence of the vibration of the train on the comfort of the passengers. In ISO2631, Janeway and Jacking comfort curves and Sperling index are usually employed to evaluate the riding quality of the running railway cars. In this study, Sperling index is used to assess the passengers’ riding comfort.

When a train is moving with high speed, the dynamic wheel weight will decrease during the upward movement. Meanwhile, the wheelset may derail owing to the lateral relative displacement between the bridge deck and the wheelset in spite of small lateral wheel-rail interaction force [

Sperling index of the comfort can be expressed as [^{2});

In this section, a single span simple supported bridge is adopted to confirm the rationality of the method presented in Section ^{10} N/m^{2}; Poisson’s ratio is 0.2; ratio of damping is 0.02; area is 7.5 m^{2}; inertia moment in ^{4}, 9 m^{4}, and 70.158 m^{4}, respectively.

The German low-interference track irregularity spectrum is applied; that is,^{−7};

Wavelength coverage of the rail irregularity is from 1 m to 80 m. Eccentricity of the line equals 2.5 m; Rayleigh damping is adopted in this study. The railway cars travel about 100 m on the same line as on the bridge before they pass through the bridge and then travel about 100 m again after getting off the bridge. The time step of the integration procedures is 0.0005 s.

The trigonometric series method is adopted [

To verify the results obtained by the presented method, 500, 5000, 10000, and 20000 Monte-Carlo simulations are implemented. Figures

Statistical characteristics of the bridge response in the midpoint. (a) Mean value of the vertical displacement. (b) Mean value of the acceleration. (c) SD of the vertical displacement. (d) SD of the acceleration.

Statistical characteristics of the railway car. (a) Mean value of the vertical displacement of the car body. (b) Mean value of vertical acceleration of the car body. (c) Mean value of the vertical displacement of the front bogie. (d) Mean value of the vertical acceleration of the front bogie. (e) SD of the vertical displacement of the car body. (f) SD of the vertical acceleration of the car body. (g) SD of the vertical displacement of the front bogie. (h) SD of the vertical acceleration of the front bogie.

The comparison of the robustness of the dynamic responses and the computation efficiency between Monte-Carlo method and the presented method (denoted by SRCTB) is listed in Table

Comparison of robustness and computational efficiency.

Method | Computational time (h) | Robustness | Efficiency | |
---|---|---|---|---|

Bridge responses | Vehicle response | |||

Monte-Carlo |
0.9 | Fair | Poor | Fair |

Monte-Carlo |
9.8 | Fair | Poor | Poor |

Monte-Carlo |
19.4 | Excellent | Fair | Poor |

Monte-Carlo |
27.8 | Excellent | Excellent | Poor |

SRCTB | 0.5 | Excellent | Excellent | Excellent |

In this section, a three-span nonuniform prestressed continuous beam bridge along the Chinese high-speed railway line is chosen to demonstrate the random properties of the train-bridge coupling system. Vehicles adopted are ICE high-speed train [

Three-span continuous beam bridge.

The cross section of the beam is a single cell box girder as shown in Figure ^{10} N/m^{2} and Poisson’s ratio is 0.2.

Cross section of the beam.

The wavelength coverage of the vertical rail irregularity is from 1 m to 80 m; the vehicle velocity is 275 km/h; eccentricity of the rail line equals 2.5 m; Rayleigh damping is adopted; the trains travel about 100 m on the same line as on the bridge before they pass through the bridge, and they travel about 100 m again after getting off the bridge.

As shown in Figure

Displacement of the midpoint in the side span. (a) Mean value. (b) SD.

From Figures

Acceleration of the midpoint in the side span. (a) Mean value. (b) SD.

Dynamic responses of the midpoint in the middle span. (a) Mean value and SD of the displacement. (b) (a) Mean value and SD of the acceleration.

As shown in Figures

As shown in Figure

PSD of the bridge displacement of (a) midpoint of the first span, (b) midpoint of the middle span, and (c) midpoint of the third span.

From Figure

PSD of the bridge acceleration in the midpoint of (a) first span, (b) middle span, and (c) third span.

The PSDs have different varying rules with each other as shown in Figures

From Figures

Displacement of the first railway car. (a) Mean value. (b) Standard deviation.

Displacement of the car bodies. (a) Mean value. (b) Standard deviation.

Displacement of the bogies. (a) Mean value. (b) Standard deviation.

Acceleration of the vehicles. (a) Mean value. (b) Standard deviation.

The mean and SD of the displacement as to the four car bodies vary with time in almost the same pattern. When all the four railway cars are on the bridge deck and the third vehicle passes around 0.4 times of the middle span length, the SD of the displacement about the car bodies reach their maximum value (Figure

Position corresponding to the maximum SD values.

Vehicle number | Position of the vehicle |
---|---|

1 | 0.6 times of the middle span length |

2 | 0.6 times of the middle span length |

3 | 0.4 times of the middle span length |

4 | 0.1 times of the middle span length |

Note: the distance is counted from the starting point of the middle span.

Figures

As shown in Figure

PSD of the displacement of the car bodies in the centre of the (a) first car body, (b) second car body, (c) third car body, and (d) fourth car body.

As for the vertical wheel/rail force of the first railway car, the mean values of four wheelsets are around their static wheel load, that is, 156.8 kN (motor car) or 143.08 kN (trail car). As shown in Figures

Vertical force and wheel load reduction rate as for number 1 railway car. (a) Mean value of the vertical wheel/rail force. (b) SD of the vertical wheel/rail force. (c) Mean value of the wheel load reduction rate. (d) SD of the wheel load reduction rate.

As shown in Table

Sperling index of the vehicles.

Vehicle number | Car body | Bogie | ||
---|---|---|---|---|

Mean | SD | Mean | SD | |

1 | 0.751 | 0.858 | 1.081 | 1.763 |

2 | 0.655 | 0.823 | 1.014 | 1.948 |

3 | 0.659 | 0.823 | 1.033 | 1.938 |

4 | 0.682 | 0.869 | 1.014 | 1.795 |

Random vibration analysis of train-bridge coupling system is implemented by extending the pseudo excitation method (PEM). Expressions of the mean value, standard deviation, and power spectral density of the nonstationary random dynamic responses of bridge and vehicles are derived. Evaluation indicators for assessing the safety and riding quality of the railway cars are developed. Results produced by the redeveloped PEM agree with those generated by the Monte-Carlo simulation method, but the presented method requires less computational efforts.

By taking a high-railway bridge with three-span continuous beam as a case study, the whole histories of an ICE3 train running on the bridge subjected to vertical random rail irregularity are investigated. The following observations and conclusions may be drawn from the present numerical study:

The location of vehicle affects the bridge responses.

The randomness of the dynamic responses of the midpoint in the middle span is different from that in the side span. The vertical displacement of the midpoint is mainly affected by the certain load, that is, the wheel load.

Nonstationary features can be found from the PSD of the responses of the bridge.

Randomness exists in the dynamic responses of the vehicles.

The value of the vertical wheel/rail force and wheel load reduction is mainly affected by the static wheel load.

The mean value and the standard deviation of the vertical Sperling index of the car body are smaller than those of the bogie due to the vibration suppression caused by the bogie. The Sperling index has high-level dispersal, which should be considered in the analysis and design of the coupling system.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the National Natural Science Foundation of China (Grant no. 51308470), Fundamental Research Funds for the Central Universities (Grant no. SWJTU12CX062), and Australian Research Council through Discovery Projects.