In the realm of subway shield tunnel operations, the impact of tunnel settlement on the operational performance of subway vehicles is a crucial concern. This study introduces an advanced analytical model to investigate rail geometric deformations caused by settlement within a vehicle-track-tunnel coupled system. The model integrates the geometric deformations of the track, attributed to settlement, as track irregularities. A novel “cyclic model” algorithm was employed to enhance computational efficiency without compromising on precision, a claim that was rigorously validated. The model’s capability extends to analyzing the time-history responses of vehicles traversing settlement-affected areas. The research primarily focuses on how settlement wavelength, amplitude, and vehicle speed influence operational performance. Key findings indicate that an increase in settlement wavelength can improve vehicle performance, whereas a rise in amplitude can degrade it. The study also establishes settlement thresholds, based on vehicle operation comfort and safety. These insights are pivotal for maintaining and enhancing the safety and efficiency of subway systems, providing a valuable framework for urban infrastructure management and long-term maintenance strategies in metropolitan transit systems.

As urban rail transit operations densify and operational speeds continue to improve, the safety and stability requisites for these systems are becoming increasingly stringent. Currently, shield tunneling, with its distinctive advantages, has progressively become the principal technical method for underground space development. Factors such as falling groundwater levels leading to segment subsidence, groundwater-triggered segment floatation in water-rich strata, and soil deposition-induced segment subsidence, can impact the smoothness of subway lines. This may result in a gamut of issues, including degradation of train running smoothness, comfort, and safety [

In light of the issue of subway tunnel settlement, numerous researchers have conducted studies. Guo et al. [

Track smoothness significantly affects the operational performance of vehicles, and this subject has been the focus of extensive research [

To gauge the influence of subgrade differential settlement on the dynamic performance of the rail-plate track system, Guo et al. [

Sun et al. [

The main contributions of the paper are as follows:

An analytical equation for rail deformation caused by shield tunneling settlement or uplift is proposed.

An efficient calculation model for vehicle-track-shield tunnel segment-soil dynamic system is proposed.

The impact of different wavelengths and amplitudes of settlement and on vehicle operation performance are discussed.

The paper will elaborate on the following aspects: first, the analytical equation for rail deformation caused by the settlement of shield tunnel segments is derived; second, the efficient calculation model for vehicle-track-shield tunnel segment-soil dynamic system is introduced; finally, the effects of settlement wavelength, amplitude, and vehicle speed on operational performance are discussed.

To expediently determine track irregularity prompted by tunnel settlement, an analytical model of settlement-track irregularity was devised. The resolution of the model was premised on the following assumptions: (1) The mortar layer beneath the track plate is treated as a Winkler elastic foundation; (2) No coupling relationship exists between the deformation of the shield segment and the track.

According to the relevant principles of material mechanics, the vertical upward displacement _{r,i} of the rail at the force position of the

_{r,i} denotes _{r,i} denotes the distance between _{f} denotes the distance of the two adjacent fastener; _{r} denotes elastic modulus of rail; _{r} denotes total length of rail; _{r} denotes the cross section moment of inertia of the rail.

The corresponding rail displacement at each fastener position can be obtained from the above formula, which is now integrated into a matrix form and expressed as follows:_{r} denotes the rail deformation matrix of _{r,i}, _{r} denotes the fastener force matrix;

The vehicle model was conceptualized as a multi-rigid-body-spring-damping system, wherein each vehicle consisted of a car body, two bogies, and four wheel-sets, all of which were deemed rigid bodies. The suspension system linking the car body and bogie (secondary suspension), and the one between the bogie and wheel set (primary suspension), were depicted through spring stiffness viscous damping, as illustrated in

_{v} is the mass matrix of vehicle, and it can be written as:_{vi} denotes ^{th} mass matrix of vehicle, and it can be represented as:

_{v} denotes stiffness of vehicle, and it can be represented as:_{vi} denotes stiffness matrix of ^{th} carriage, and it can be represented as:

The soil and shield segments can be represented using the finite element software ABAQUS, as illustrated in

Both the rail and the track are modeled using Euler beam elements. The fastener connecting the rail and the track is simulated via a spring-viscous damper, while the connection between the rail plate and the shield segment is also represented through a spring-viscous damper.

The dynamic equation of the rail-shield cyclic tunnel model obtained is shown as follows:_{ts} and _{ts} are the mass matrix and stiffness matrix of rail-shield circulation tunnel model, _{ts} is the damping matrix, and it can be obtained by Rayleigh damping; _{ts}, _{ts}, _{ts} and _{ts} denotes the acceleration, velocity and displacement vector of rail-shield circulation tunnel system; _{ts} is the load array acting on the rail-shield cyclic tunnel system.

The vehicle and rail-shield cyclic tunnel model can be coupled according to the wheel-rail contact. According to

The explicit integration method for gradually solving the above equation system [

The selected example is the section from Changle Binhai Express Airport Station to the middle wind shaft, as shown in

This soil mass is divided into five distinct layers: silty sand, muddy sand, and three layers of silty clay, each with different attributes, starting from the top. The silty sand layer has a height of 5 m, and the muddy sand layer is 10 m high. The three layers of silty clay, each with unique properties, have respective heights of 15, 5, and 5 m. The segments’ outer diameter is 8.3 m, while the inner diameter is 7.5 m.

Layer | Density (g/cm^{3}) |
Elastic modulus (kPa) | Poisson’s ratio |
---|---|---|---|

Silty fine sand | 1.8 | 7000 | 0.32 |

Silty fine sand | 1.78 | 5000 | 0.32 |

Silty clay 1 | 1.89 | 3900 | 0.32 |

Silty clay 2 | 1.91 | 6600 | 0.35 |

Silty clay 3 | 1.89 | 4000 | 0.32 |

Segment | 2.4 | 3.2e7 | 0.2 |

2_{1}/m |
2_{2}/m |
_{V}/m |
_{c}/kg |
_{t}/kg |
_{w}/kg |
---|---|---|---|---|---|

2.2 | 12.5 | 19.0 | 48800 | 2310 | 2080 |

The vehicle-track-tunnel coupled system model is intricate, involving multiple layers of system dynamics. To validate the accuracy of our system dynamic model’s calculation results, we compared them to the dynamic response results provided by Xia et al. [

Due to the ongoing construction of the tunnel, the current geometric deformation mainly relies on numerical calculations and settlement measurements, without involving track irregularities. The initial track irregularity in the manuscript was simulated using the German low interference spectrum, and the irregularity caused by settlement was obtained through assumptions. The German low interference spectrum can be written as:

_{c} equals to 0.8246 rad/m, _{v} equals to 1.080 × 10^{−6} m^{2}·rad/m, Ω_{r} equals to 0.0206 rad/m. The trigonometric series method is used to simulate the track irregularity samples.

Tunnel settlement reasons are multifaceted, leading to various forms of uneven settlement. Theoretically, this inhomogeneous sedimentation can be broadly classified into two types: “sudden” and “gradual”. To simulate the impact of gradual tunnel settlement on vehicle running performance, we used a concave full-wave cosine curve [

After obtaining the tunnel settlement values, the geometric deformation of the rail can be calculated using the analytical method described in

In addition to the deformation induced by tunnel settlement, the track also possesses initial irregularities. These two factors were superimposed to create a compound irregularity. This compound irregularity was then input into the vehicle-track-tunnel dynamic simulation model to analyze the dynamic response of the vehicle.

The paper delves into the dynamic response of vehicles impacted by tunnel settlement through a time history response analysis. This analysis was based on a vehicle running at a calculated speed of 100 km/h, with a tunnel settlement wavelength (S) of 40 m and an amplitude (A) of 150 mm.

Under regular operating conditions, the vehicle’s body oscillates due to the influence of track irregularities. Upon passing through the tunnel’s settlement area, the vehicle body’s acceleration experiences abrupt changes. Once the vehicle exits the tunnel’s settlement area, the body’s acceleration diminishes and gradually returns to its normal operating state.

The local response, as seen in

Nadal index and wheel unloading are the most commonly used indexes to measure running safety. The Nadal index can be calculated by the following formula:

The wheel unloading can be calculated by the following formula:

When the settlement amplitude increases, the amplitude of track irregularity correspondingly increases, and the dynamic response of the vehicle will increase. In an effort to thoroughly investigate the effect of tunnel settlement amplitude on vehicle performance, we computed the maximum acceleration of the car body above the front bogie, the maximum wheel unloading, and the maximum Nadal index for settlement wavelengths of 10, 40, and 70 m. The vehicle speed was held constant at 80 km/h. The results of these calculations can be seen in

In order to systematically investigate the effect of tunnel settlement wavelength on vehicle performance, we calculated the maximum acceleration of the car body above the front bogie, the maximum wheel unloading, and the maximum Nadal index for settlement amplitudes of 20, 50, and 80 mm. The results of these calculations are depicted in

Vehicle speed significantly influences the responses. We calculated vehicle responses for settlements with amplitudes of 40 mm and wavelengths of 40, 50, and 60 m, respectively, under conditions of 80–110 km/h. The results of these calculations are illustrated in

To more effectively assess the actual settlement of subway lines, we calculated the vehicle running performance under various conditions, focusing primarily on the car body acceleration, wheel unloading, and Nadal index. The respective limits for car body acceleration, wheel unloading, and the Nadal index are 1.3 m/s², 0.6, and 0.8.

The outcomes of these calculations are presented in

The findings presented in

Settlement wavelength (m) | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
---|---|---|---|---|---|---|---|---|

80 km/h | 0.034 | 0.057 | 0.101 | 0.183 | 0.240 | 0.334 | – | – |

90 km/h | 0.031 | 0.050 | 0.086 | 0.132 | 0.185 | 0.256 | 0.376 | – |

100 km/h | 0.029 | 0.044 | 0.071 | 0.107 | 0.144 | 0.196 | 0.280 | – |

110 km/h | 0.026 | 0.377 | 0.061 | 0.088 | 0.119 | 0.154 | 0.219 | 0.303 |

Settlement wavelength (m) | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
---|---|---|---|---|---|---|---|---|

80 km/h | 0.175 | 0.399 | 0.476 | – | – | – | – | – |

90 km/h | 0.144 | 0.376 | 0.448 | – | – | – | – | – |

100 km/h | 0.121 | 0.335 | 0.388 | 0.638 | – | – | – | – |

110 km/h | 0.103 | 0.294 | 0.382 | 0.584 | 0.785 | – | – | – |

During subway tunnel construction and operation, soil settlement can lead to shield segment sinking. This condition often results in a series of problems, such as segment damage and rail deformation, negatively impacting the comfort and safety of vehicle operation. This study aimed to investigate the effects of tunnel settlement on vehicle running performance. To this end, an analytical mapping model was established to represent the relationship between tunnel settlement and rail deformation. The model’s accuracy was then validated using a case study. Moreover, a cyclic vehicle-track-tunnel coupled system was developed, its accuracy confirmed via literature comparison. The analysis of the vehicle’s dynamic response while traversing the settling area was carried out, followed by a discussion on the influence of different parameters on running performance. The main conclusions are as follows:

(1) The analytical mapping model accurately calculates the rail geometry deformation due to tunnel settlement. Compared to the Finite Element (FE) model, this model simplifies the calculations.

(2) Tunnel settlement affects driving comfort more significantly than it does running safety.

(3) Both the settlement wavelength and amplitude significantly impact the vehicle’s running performance. Increasing the settlement wavelength improves vehicle performance, whereas augmenting the settlement amplitude impairs it.

(4) Through the developed threshold table, rapid assessment of tunnel settlement is possible. The running safety threshold is higher than the comfort threshold.

(5) It is important to note that this study primarily focuses on the impact of track geometric deformation, caused by settlement, on the dynamic running performance. It does not take into account instances of segment breakage.

(6) Shield deformation has an impact on the performance of vehicle operation, and under the action of vehicle loads, it may further deteriorate the deformation of shield tunneling. This is mutual and will be a research focus of future attention.

None.

This study was funded by the Scientific Research Startup Foundation of Fujian University of Technology (GY-Z21067 and GY-Z21026).

The authors confirm contribution to the paper as follows: study conception and design: Gang Niu and Guangwei Zhang; data collection: Guangwei Zhang and Zhaoyang Jin; analysis and interpretation of results: Wei Zhang and Xiang Liu; draft manuscript preparation: Gang Niu, Xiang Liu and Wei Zhang. All authors reviewed the results and approved the final version of the manuscript.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors declare that they have no conflicts of interest to report regarding the present study.