^{1}

^{1}

^{2}

^{2}

^{3}

^{1}

^{2}

^{3}

Aiming at the existing heavy-haul railway, bridges hardly meet the transportation requirements. Based on the spatial vibration calculation model of the freight train–track–bridge (FTTB) system, the FTTB spatial vibration model under the condition of auxiliary steel beam reinforcement is established. Besides, according to the random analysis method of train derailment energy, coming up with an evaluation method of auxiliary steel beam reinforcement is based on safety and dynamic response, which is used to discuss the train safety and the change law of FTTB system vibration response. The results show that the derailment resistance of the FTTB system is increased by 22.6% after the auxiliary steel beam is reinforced. Compared with the previous speed (115.56 km/h), the speed is 132.73 km/h after the auxiliary steel beam reinforcement; at the same time, the allowable limit speed increases from 92.49 km/h to 106.18 km/h. In addition, the reinforcement of the auxiliary steel beam can not only effectively reduce the lateral vibration response of the FTTB system under the action of empty wagon but also effectively decline the vertical vibration response of the FTTB system under the action of the loaded wagon, which can meet the stability requirement for running at the speed of 90 km/h. In summary, the reinforcement of auxiliary steel beams can improve the running safety of trains, reduce the vibration response of the FTTB system, and meet the requirements of operation stability.

Bridges are one of the infrastructures of heavy-haul railways as well as an important guarantee for safety train operation. Simply supported concrete T-beam bridge is a common bridge type of heavy-haul railway in China [

Aiming at the traffic safety problems caused by the insufficient bearing capacity and stiffness of the existing bridge, domestic and aboard scholars have carried out a series of reinforcement research for the bridge structure and put forward corresponding reinforcement measures [

In this paper, based on the space vibration calculation model of freight train track bridge system (FTTB system) [

FTTB system is an integrated system that contains freight train, track, and bridge. The connection condition of wheel rail relative displacement [

In formulae (

There is a train running on the heavy-haul railway bridge which is composed of 1 locomotive +

Vehicle unit displacement mode.

Displacement mode | Roll | Pitch | Yaw | |||
---|---|---|---|---|---|---|

Car body | √ | √ | √ | √ | √ | √ |

Front bogie | √ | √ | √ | √ | √ | √ |

Rear bogie | √ | √ | √ | √ | √ | √ |

Each wheelset | — | √ | √ | — | — | — |

According to the above displacement mode, the spatial vibration potential energy

Simultaneously, along the beam span direction of heavy-haul railway, each adjacent diaphragm is divided into a beam segment element. In this way, the beam with effective length _{1}, _{2}, _{4}, _{5}, _{6}, and _{7}; the lateral and vertical damping coefficients are _{1}, _{2}, _{4}, _{5}, _{6}, and _{7}. Assuming that the displacement at the top of the pier is equal to that at the end of the girder, the pier is simulated by beam element and directly consolidated with the ground without considering the pile foundation effect. According to the number of piers and section characteristics, the pier is divided into 1 element. Thus, the spatial vibration calculation model of track bridge system is formed, as shown in Figure

Spatial vibration model of track and bridge system. (a) Main view. (b) Side view.

Based on the above assumption, each beam element is discretized into a finite element model with 50 degrees of freedom, and the node displacement of the element is shown in equation (

In formulae (

According to the node displacement of the element in equation (

Furthermore, by superposing the abovementioned freight train spatial vibration potential energy and the track–bridge system spatial vibration potential energy, the total spatial vibration potential energy of the FTTB system can be obtained as shown in the following equation:

According to the principle of constant value of the total potential energy of elastic system dynamics [

Using the “set in right position” rule [

In formula (

Auxiliary steel beam reinforcement is generally achieved in two ways [

Structural diagram of auxiliary steel beam.

According to the characteristics of the auxiliary steel beam in Figure

The stress–strain relationship of structure is shown in the following equation:

According to the principle of equivalent component strength, the equivalent calculation formula for component strength can be derived as shown in the following equation:

In formula (

Similarly, according to the principle of equivalent bending stiffness of components, the equivalent calculation formula for bending stiffness can be listed as shown in the following equation:

In formula (

According to the above principles, the auxiliary steel beam can be equivalent to a concrete beam, and the equivalent section characteristics are shown in the following equations:

In formulae (

When calculating the transverse moment of inertia of the whole steel–concrete section, the height of the auxiliary steel beam is kept unchanged, and the width of the auxiliary steel beam is equivalent to the height of the auxiliary steel beam according to the ratio of the elastic modulus of the concrete and the steel. Similarly, the vertical moment of inertia is calculated by changing the height of the auxiliary steel beam according to the ratio of elastic modulus. Therefore, taking the equivalent section parameters of auxiliary steel beam and the original

In this paper, the concrete grade of

Cross-sectional characteristics of T-beam before and after reinforcement of auxiliary steel beam.

Reinforcement state | Cross moment of inertia along ^{4}) | Cross moment of inertia along ^{4}) |
---|---|---|

Before reinforcement | 1.1104 | 0.1708 |

After reinforcement | 1.3881 | 0.2778 |

To verify the reliability of the model, the calculation condition in [_{1} = 2.9 × 10^{7} N/m, _{2} = 1.1 × 10^{8} N/m, _{4} = 4.0 × 10^{6} N/m, _{5} = 4.5 × 10^{7} N/m, _{6} = 2.1 × 10^{9} N/m, and _{7} = 2.5 × 10^{11} N/m. The vertical dynamic deflection and lateral amplitude of the fourth span of the auxiliary steel beam before and after reinforcement are obtained through calculation, and the results are shown in Table

The calculated and experimental values of vertical dynamic deflection and transverse amplitude in the middle span of the beams before and after reinforcement.

Reinforced state | Vertical dynamic deflection (mm) | Beam transverse amplitude (mm) | ||
---|---|---|---|---|

Calculated value | Test value [ | Calculated value | Test value [ | |

Before reinforcement | 12.38 | 11.98 | 1.14 | 1.24 |

After reinforcement | 11.47 | 11.02 | 0.93 | 0.94 |

Table

Vertical dynamic deflection-time history curves of auxiliary steel beam before and after reinforcement.

Transverse amplitude-time history curves of auxiliary steel beam before and after reinforcement.

Derailment coefficient and wheel load reduction rate are traditional safety indicators for evaluating train operation [

In the criterion of lateral vibration stability of FTTB system,

Relationship between

Figure _{0} km/h to _{r} km/h, that is,

Then, the calculation formula of energy increment can be listed, respectively, according to the graph of

In formulae (

Formula (

At the same time,

In formula (

At this point, we can determine

Based on the above ideas, the spatial vibration response of FTTB system under normal driving condition is calculated with the actual speed

Evaluation method about running safety and dynamic response of auxiliary steel beam reinforcement.

According to derailment accident and theoretical calculation results, empty vehicles are more likely to derail [

When calculating the lateral vibration stability of FTTB system, the speed of 50 km/h is taken as the starting point. Considering calculation and test error, the lateral vibration stability of FTTB system is checked and calculated once for every 10 km/h increase in vehicle speed. According to the evaluation method in Figure

Relationship

Calculation results of lateral vibration stability of FTTB system before and after the auxiliary steel beams reinforcement.

^{2}) | ^{2}) | Before reinforcement | After reinforcement | |||||||
---|---|---|---|---|---|---|---|---|---|---|

^{2}) | ^{2}) | Whether stable | ^{2}) | ^{2}) | Whether stable | |||||

140 | 302 | 23 | — | — | — | — | 380 | 15 | ＜0 | No |

130 | 279 | 26 | — | — | — | — | 365 | 29 | ＞0 | Yes |

120 | 253 | 59 | 310 | 55 | ＜0 | No | 336 | 66 | ＞0 | Yes |

110 | 194 | 31 | 255 | 36 | ＞0 | Yes | 270 | 33 | ＞0 | Yes |

100 | 163 | 18 | 219 | 19 | ＞0 | Yes | 237 | 25 | ＞0 | Yes |

90 | 145 | 19 | 200 | 23 | ＞0 | Yes | 212 | 22 | ＞0 | Yes |

80 | 126 | 36.6 | 177 | 40 | ＞0 | Yes | 190 | 43 | ＞0 | Yes |

70 | 89.4 | 20.9 | 137 | 24 | ＞0 | Yes | 147 | 22 | ＞0 | Yes |

60 | 68.5 | 17.0 | 113 | 21 | ＞0 | Yes | 125 | 23 | ＞0 | Yes |

50 | 51.5 | — | 92 | — | — | — | 102 | — | — | — |

As shown in Figure ^{2} and the maximum value of ^{2}. The antiderailment ability of FTTB system increased by 22.6% after the reinforcement of auxiliary steel beam.

Table

Taking

Vibration responses calculation results of FTTB system under the action of empty and loaded wagon before and after reinforcement.

RS | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

E | L | E | L | E | L | E | L | E | L | E | L | E | L | E | L | ||

60 | A | 2.12 | 0.88 | 3.07 | 11.74 | 26.37 | 23.25 | 2.77 | 11.14 | 0.57 | 0.49 | 1.54 | 0.98 | 4.11 | 3.45 | 2.39 | 3.18 |

B | 1.73 | 0.55 | 2.78 | 10.52 | 22.87 | 21.71 | 2.58 | 10.63 | 0.48 | 0.43 | 1.37 | 0.86 | 3.72 | 3.09 | 2.14 | 2.92 | |

70 | A | 2.20 | 1.15 | 3.25 | 12.38 | 30.23 | 26.16 | 2.83 | 11.81 | 0.62 | 0.59 | 1.62 | 1.07 | 4.32 | 3.75 | 2.46 | 3.37 |

B | 1.81 | 0.93 | 2.98 | 11.48 | 27.32 | 24.76 | 2.68 | 11.18 | 0.54 | 0.49 | 1.45 | 1.01 | 3.97 | 3.49 | 2.38 | 3.06 | |

80 | A | 2.52 | 1.49 | 3.61 | 13.46 | 37.31 | 29.85 | 2.99 | 12.31 | 0.77 | 0.64 | 1.76 | 1.19 | 4.43 | 3.96 | 2.63 | 3.53 |

B | 2.18 | 1.41 | 3.32 | 11.52 | 34.12 | 28.02 | 2.84 | 11.64 | 0.69 | 0.52 | 1.59 | 1.14 | 4.11 | 3.67 | 2.43 | 3.28 | |

90 | A | 3.15 | 1.72 | 3.71 | 15.14 | 45.83 | 31.81 | 3.27 | 13.14 | 0.85 | 0.70 | 1.93 | 1.29 | 4.54 | 4.13 | 2.79 | 3.64 |

B | 2.77 | 1.64 | 3.49 | 13.53 | 42.51 | 30.27 | 3.11 | 12.63 | 0.78 | 0.57 | 1.77 | 1.23 | 4.23 | 3.86 | 2.62 | 3.39 |

Figure

Figure

Figure

Figure

Figures _{p} both exceeded the limits of 0.65 and 1.0 [

Figures

In order to reflect the time history response changes of the abovementioned indexes before and after reinforcement, the time-history responses with

FTTB system spatial vibration response before and after the reinforcement under the action of empty wagon. (a) Time history curve of

FTTB system spatial vibration response before and after the reinforcement under the action of a loaded train. (a) Time history curve of

Based on the spatial vibration calculation model of FTTB system and the derailment energy random analysis method of train, the spatial vibration calculation model of FTTB system under the condition of auxiliary steel beam reinforcement is established, and then the evaluation method about running safety and dynamic response of auxiliary steel beam reinforcement is presented. The influence of antiderailment capacity, the critical speed, the allowable limit speed, and the spatial vibration response of FTTB system are analyzed. The following conclusions are obtained:

Through comparative analysis, the calculated results of the model are basically consistent with the test results in the literature, which verifies the rationality of the model.

The results show that the antiderailment capacity of FTTB system is improved by 22.6%; before the reinforcement of auxiliary steel beam, the critical speed and allowable limit speed are 115.56 km/h and 92.49 km/h, respectively, while after reinforcement, the critical speed and allowable limit speed are 132.73 km/h and 106.18 km/h, respectively; the critical speed and allowable limit speed of auxiliary steel beam are increased by 14.8%.

Under the action of empty wagon, the mid-span transverse amplitude and vertical dynamic deflection of beam body, lateral displacement and vertical displacement of car body, wheel-load reduction rate, derailment coefficient, and lateral and vertical Sperling stability index increase with the increase of vehicle speed; after the auxiliary steel beam is reinforced, the maximum values of the above indexes are reduced by 12.1%, 5.9%, 13.3%, 4.8%, 8.2%, 8.3%, 6.8%, and 6.1%, respectively. The lateral vibration response of FTTB system under the action of empty wagon is greatly affected by the reinforcement of auxiliary steel beam, which can meet the requirements of running stability at the speed of 90 km/h.

Under the action of loaded wagon, the mid-span transverse amplitude and vertical dynamic deflection of beam body, lateral displacement and vertical displacement of car body, wheel load reduction rate, derailment coefficient, and lateral and vertical Sperling stability index increase with the increase of vehicle speed; after the auxiliary steel beam is reinforced, the above indexes decrease by 4.6%, 10.6%, 4.8%, 3.9%, 18.6%, 4.3%, 6.5%, and 6.7%, respectively. When the speed is 90 km/h, the reinforcement of auxiliary steel beam has great influence on the vertical vibration response of FTTB system under the action of loaded wagon, which can meet the requirements of running stability at the speed of 90 km/h.

The above method and results can provide reference for evaluation and formulation of reinforcement measures with derailment prevention function and meeting the requirements of driving stability.

All data generated or analyzed during this study are included in the published article.

The authors declare that they have no conflicts of interest regarding the publication of this paper.

This research was financially supported by the National Natural Science Foundation of China (51578238 and 52068028), Natural Science Foundation Committee of China and Shenhua Group Corporation Limited (U1261113), Jiangxi Province’s Key Research and Development Plan (20181BBE50013), and Science and Technology of Jiangxi Provincial Education Department (GJJ170392).