# Dynamic Responses of Train-Symmetry-Bridge System Considering Concrete Creep and the Creep-Induced Track Irregularity

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Random Vibration Theory Based on KLE-PEM

_{n}(θ) is represented as a set of unrelated random variables that follow a standard normal distribution. λ

_{n}and φ

_{n}(x) are represented as eigenvalues and eigenfunctions, respectively, and the value of M determines the precision of the random field representation.

_{z}

_{2}represents the variance of Y.

## 3. Expression of Track Irregularity Based on KLE

## 4. Dynamic Response Analysis of Uncertain Systems

#### 4.1. Validation of KLE-PEM Method

#### 4.2. Comparison of Train Certainty Analysis and Uncertainty Analysis

^{2}, while the maximum acceleration for deterministic analysis is 0.29 m/s

^{2}, which is 41.4% higher. From Figure 3b, it can be observed that there is a difference in the trend of qualitative analysis and uncertainty analysis for the acceleration of the vehicle body above the bogie. The maximum vertical acceleration for uncertainty analysis is 0.61 m/s

^{2}, while the maximum acceleration for certainty analysis is 0.48 m/s

^{2}, which is 27.1% higher.

#### 4.3. Response of Trains

^{2}. In Figure 4b, the acceleration of the vehicle above the rear bogie increases with the increase of creep camber, reaching a speed of 300 km/s and a camber of 22.5 mm, the maximum vertical acceleration of the vehicle body reaches 1.37 m/s

^{2}. According to the Chinese standard for operational comfort, it is recommended that the vertical acceleration of the vehicle body should not exceed the limit of 0.13 g (=1.275 m/s

^{2}). From Figure 4b, it can be seen that the train running speed is 250 km/h, and the corresponding creep camber amplitude for an acceleration of 1.275 m/s

^{2}is 19 mm. Overall, stochastic analysis can be used to effectively calculate the relationship between creep and response, which is important for ensuring traffic safety.

## 5. Conclusions

- (1)
- By comparing with the Monte Carlo method, the accuracy of the KLE-PEM method was verified, and the KLE-PEM method has higher computational efficiency.
- (2)
- There is a significant difference between the deterministic dynamic response analysis and the uncertain dynamic response analysis of the train track bridge system. The uncertainty dynamic analysis of the train track bridge system can more accurately and effectively reflect the actual situation of the train during operation.
- (3)
- The vertical acceleration of the vehicle body above the front bogie of the train increases with the increase of train operating speed; The vertical acceleration and wheel load reduction rate of the vehicle body above the front and rear bogies increase with the increase of the upper creep camber; Based on the safety and comfort indicators of train operation, the creep upper arch limit of a 48 m + 80 m + 48 m continuous beam bridge is given as 19 mm.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Jiang, L.Z.; Liu, X.; Xiang, P.; Zhou, W.B. Train-bridge system dynamics analysis with uncertain parameters based on new point estimate method. Eng. Struct.
**2019**, 199, 109454. [Google Scholar] [CrossRef] - Trong-Phuoc, H.; Hwang, C.-L.; Limongan, A.H. The long-term creep and shrinkage behaviors of green concrete designed for bridge girder using a densified mixture design algorithm. Cem. Concr. Compos.
**2018**, 87, 79–88. [Google Scholar] - Au, F.T.K.; Si, X.T. Accurate time-dependent analysis of concrete bridges considering concrete creep, concrete shrinkage and cable relaxation. Eng. Struct.
**2011**, 33, 118–126. [Google Scholar] [CrossRef] - Xiang, P.; Huang, W.; Jiang, L.Z.; Lu, D.G.; Liu, X.; Zhang, Q. Investigations on the influence of prestressed concrete creep on train-track-bridge system. Constr. Build. Mater.
**2021**, 293, 123504. [Google Scholar] [CrossRef] - Zhao, H.; Wei, B.; Jiang, L.; Xiang, P. Seismic running safety assessment for stochastic vibration of train–bridge coupled system. Arch. Civ. Mech. Eng.
**2022**, 22, 180. [Google Scholar] [CrossRef] - Jiang, L.Z.; Liu, C.; Peng, L.X.; Yan, J.W.; Xiang, P. Dynamic Analysis of Multi-layer Beam Structure of Rail Track System under a Moving Load Based on Mode Decomposition. J. Vib. Eng. Technol.
**2021**, 9, 1463–1481. [Google Scholar] [CrossRef] - Nakov, D.; Markovski, G.; Arangjelovski, T.; Mark, P. Experimental and Analytical Analysis of Creep of Steel Fibre Reinforced Concrete. Period. Polytech.-Civ. Eng.
**2018**, 62, 226–231. [Google Scholar] [CrossRef] - Chen, Z.; Zhai, W.; Cai, C.; Sun, Y. Safety threshold of high-speed railway pier settlement based on train-track-bridge dynamic interaction. Sci. China-Technol. Sci.
**2015**, 58, 202–210. [Google Scholar] [CrossRef] - Rahmanzadeh, S.; Tariverdilo, S. Evaluating Applicability of ASTM C 928 Approach in Assessing Adequacy of Patch Repair of Bridge Piers. Int. J. Eng.
**2020**, 33, 2455–2463. [Google Scholar] - Xiong, Z.; Chen, J.; Liu, C.; Li, J.; Li, W. Bridge’s Overall Structural Scheme Analysis in High Seismic Risk Permafrost Regions. Civ. Eng. J.
**2022**, 8, 1316–1327. [Google Scholar] [CrossRef] - Chan, Y.W.S.; Wang, H.P.; Xiang, P. Optical Fiber Sensors for Monitoring Railway Infrastructures: A Review towards Smart Concept. Symmetry
**2021**, 13, 2251. [Google Scholar] [CrossRef] - Wang, X.; Luo, F.; Ye, A. A holistic framework for seismic analysis of extended pile-shaft-supported bridges against different extents of liquefaction and lateral spreading. Soil Dyn. Earthq. Eng.
**2023**, 170, 107914. [Google Scholar] [CrossRef] - Wang, X.; Ji, B.; Ye, A. Seismic Behavior of Pile-Group-Supported Bridges in Liquefiable Soils with Crusts Subjected to Potential Scour: Insights from Shake-Table Tests. J. Geotech. Geoenviron. Eng.
**2020**, 146, 04020030. [Google Scholar] [CrossRef] - Wang, X.W.; Ye, A.J.; Ji, B.H. Fragility-based sensitivity analysis on the seismic performance of pile-group-supported bridges in liquefiable ground undergoing scour potentials. Eng. Struct.
**2019**, 198, 109427. [Google Scholar] [CrossRef] - Wang, X.W.; Ye, A.J.; Shang, Y.; Zhou, L.X. Shake-table investigation of scoured RC pile-group-supported bridges in liquefiable and nonliquefiable soils. Earthq. Eng. Struct. Dyn.
**2019**, 48, 1217–1237. [Google Scholar] [CrossRef] - Wang, X.; Ye, A.; Shafieezadeh, A.; Padgett, J.E. Fractional order optimal intensity measures for probabilistic seismic demand modeling of extended pile-shaft-supported bridges in liquefiable and laterally spreading ground. Soil Dyn. Earthq. Eng.
**2019**, 120, 301–315. [Google Scholar] [CrossRef] - Wang, X.; Shafieezadeh, A.; Ye, A. Optimal EDPs for Post-Earthquake Damage Assessment of Extended Pile-Shaft-Supported Bridges Subjected to Transverse Spreading. Earthq. Spectra
**2019**, 35, 1367–1396. [Google Scholar] [CrossRef] - Xia, Q.; Xiang, P.; Jiang, L.Z.; Yan, J.W.; Peng, L.X. Bending and free vibration and analysis of laminated plates on Winkler foundations based on meshless layerwise theory. Mech. Adv. Mater. Struct.
**2022**, 29, 6168–6187. [Google Scholar] [CrossRef] - Yang, D.; Chen, H.; Meng, Z.; Chen, G. Random Vibration and Dynamic Reliability Analyses for Nonlinear MDOF Systems under Additive Excitations via DPIM. J. Eng. Mech.
**2021**, 147, 04021117. [Google Scholar] [CrossRef] - Huo, H.; Zhou, Z.; Chen, G.H.; Yang, D.X. Exact benchmark solutions of random vibration responses for thin-walled orthotropic cylindrical shells. Int. J. Mech. Sci.
**2021**, 207, 106644. [Google Scholar] [CrossRef] - Liu, X.; Jiang, L.Z.; Xiang, P.; Zhou, W.B.; Lai, Z.P.; Feng, Y.L. Stochastic finite element method based on point estimate and Karhunen-Loeve expansion. Arch. Appl. Mech.
**2021**, 91, 1257–1271. [Google Scholar] [CrossRef] - Chen, G.; Zhou, J.; Yang, D. Benchmark solutions of stationary random vibration for rectangular thin plate based on discrete analytical method. Probabilistic Eng. Mech.
**2017**, 50, 17–24. [Google Scholar] [CrossRef] - Liu, X.; Xiang, P.; Jiang, L.; Lai, Z.; Zhou, T.; Chen, Y. Stochastic Analysis of Train-Bridge System Using the Karhunen-Loeve Expansion and the Point Estimate Method. Int. J. Struct. Stab. Dyn.
**2020**, 20, 2050025. [Google Scholar] [CrossRef] - Zhao, H.; Wei, B.; Jiang, L.; Xiang, P.; Zhang, X.; Ma, H.; Xu, S.; Wang, L.; Wu, H.; Xie, X. A velocity-related running safety assessment index in seismic design for railway bridge. Mech. Syst. Signal Process.
**2023**, 198, 110305. [Google Scholar] [CrossRef] - Xiang, P.; Ma, H.; Zhao, H.; Jiang, L.; Xu, S.; Liu, X. Safety analysis of train-track-bridge coupled braking system under earthquake. Structures
**2023**, 53, 1519–1529. [Google Scholar] [CrossRef] - Xiang, P.; Wei, M.; Sun, M.; Li, Q.; Jiang, L.; Liu, X.; Ren, J. Creep Effect on the Dynamic Response of Train-Track-Continuous Bridge System. Int. J. Struct. Stab. Dyn.
**2021**, 21, 2150139. [Google Scholar] [CrossRef]

**Figure 3.**Comparison of Deterministic and Uncertain Analysis of Vehicle Acceleration: (

**a**) Position above the front bogie (

**b**) Position above the rear bogie.

**Figure 4.**Vertical acceleration of vehicle: (

**a**) Position above the front bogie (

**b**) Position above the rear bogie.

**Table 1.**Point Gaussian Hermite integration abscissa and weight coefficients [23].

Point | 1 | 2 | 3 |
---|---|---|---|

${x}_{GH,l}$ | −1.22474 | 0 | 1.22474 |

${\omega}_{GH,l}$ | 0.29541 | 1.18146 | 0.29541 |

Parameters | Units | Numerical Value |
---|---|---|

Elastic modulus of bridge | N/m^{2} | 3.45 × 10^{10} |

Elastic modulus of track plate | N/m^{2} | 2.06 × 10^{11} |

Cross section mass moment of inertia | m^{4} | 12.744 |

Poisson’s ratio | — | 0.2 |

Mass per unit length | kg/m | 2.972 × 10^{4} |

Beam unit length | m | 0.64 |

Damping ratio | — | 0.05 |

First natural frequency | Hz | 2.25 |

Second natural frequency | Hz | 4.81 |

Third natural frequency | Hz | 5.65 |

Fourth natural frequency | Hz | 5.93 |

Parameter | Speed (km/h) | ||
---|---|---|---|

250 | 300 | 350 | |

Safety indicators | 21.1 mm | 21.3 mm | 21.5 mm |

Comfort indicators | 19.0 mm | 21.6 mm | 21.8 mm |

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## Share and Cite

**MDPI and ACS Style**

Li, W.; Ma, H.; Wei, M.; Xiang, P.; Tang, F.; Gao, B.; Zhou, Q.
Dynamic Responses of Train-Symmetry-Bridge System Considering Concrete Creep and the Creep-Induced Track Irregularity. *Symmetry* **2023**, *15*, 1846.
https://doi.org/10.3390/sym15101846

**AMA Style**

Li W, Ma H, Wei M, Xiang P, Tang F, Gao B, Zhou Q.
Dynamic Responses of Train-Symmetry-Bridge System Considering Concrete Creep and the Creep-Induced Track Irregularity. *Symmetry*. 2023; 15(10):1846.
https://doi.org/10.3390/sym15101846

**Chicago/Turabian Style**

Li, Wenfeng, Hongkai Ma, Minglong Wei, Ping Xiang, Fang Tang, Binwei Gao, and Qishi Zhou.
2023. "Dynamic Responses of Train-Symmetry-Bridge System Considering Concrete Creep and the Creep-Induced Track Irregularity" *Symmetry* 15, no. 10: 1846.
https://doi.org/10.3390/sym15101846