^{1}

^{1}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

With the fast development of rail transit, the environmental vibration problems caused by subways have received increasing attention. A 3D finite element model was built in this study to investigate the ground vibrations induced by the moving load operating in the parallel twin tunnels. Compared to the model consisting of a single tunnel that was commonly adopted in the past studies, a pair of tunnels is considered and the surrounding medium of the tunnels is taken as a saturated porous medium. The governing equations of the 3D finite element method modeling of the saturated poroelastic soil have been derived according to Biot’s theory. Computed results showed that the dynamic response of the twin-tunnel model is greater than that of the single tunnel model. And the spacing between two tunnels, tunnel buried depth, and load moving speed are the essential parameters to determine the dynamic response of the tunnel and soil.

Recently, concerns about the environmental vibrations induced by the rail transit have increased substantially. When the subways run close to the existing infrastructures, vibrations are transmitted to the infrastructure through the ground, which can cause annoyance to inhabitants or result in malfunction interruption to sensitive equipment. Due to the high groundwater table in southeastern area of China, e.g., Shanghai, Zhejiang, Jiangsu, and Guangdong, subways in these areas are running in the saturated soil. Besides it is a common practice to construct underground railway lines in pairs. The vibrations caused by twin tunnels cannot be simplified to be the sum of those caused by two single tunnels. Therefore, an investigation on ground vibrations induced by subways in the context of twin tunnels embedding in a saturated ground is necessary and desirable.

The vibrations caused by subways have been investigated by many scholars using both analytical and computational methods. Metrikine et al. [

Compared with the analytical method, the computational method, e.g., finite element method (FEM), stands out for its easy handling of the material heterogeneity and complex/irregular geometry. Balendra et al. [

However, the wave field generated by the moving load is three-dimensional in nature, which restricts the applicability of 2D computational models. Gardien and Stuit et al. [

As mentioned above, the current research focuses on the environmental vibrations problems generated by subways running in a single tunnel. The effect of the neighboring tunnel is not considered although it is a common practice that underground railway lines are constructed in pairs. The vibration caused by two metro tunnels is different to the sum of two single tunnels since the neighboring tunnel can impede and screen the vibration waves generated by the operating tunnel. To the best of the authors’ knowledge, Kuo et al. [

The ground in the above-referred papers using numerical approach is generally modeled as single-phase elastic medium in which the ground water is not considered. Existing research by Yuan et al. [

This paper focuses on the coupled vibrations of the liner, the twin-parallel tunnels, and the saturated ground using the finite element method. A moving point load is applied to the invert to represent the subway excitation. The dynamics of the surrounding saturated soil are governed by Biot’s poroelastic theory. To suppress spurious reflections at the truncation boundaries, an artificial boundary condition named the multi-transmitting formula (MTF) that is proposed by the Liao and Wong [

According to Biot’s theory [

After manipulation, the pore pressure

There are two reasons for keeping

The same interpolation function is used for both the soil-skeleton displacement

Following the standard Galerkin procedure, the governing equations of the finite element formulation representing the saturated poroelastic soil medium can be derived as

The expressions of the elements in the mass matrix, the damping matrix, the stiffness matrix, and the force vector are given in the appendix.

Due to the semi-infinite nature of the ground, the ground vibration analysis using the finite element method requires artificial boundary conditions to make the computational domain finite. Many artificial boundary conditions have been proposed for absorbing/damping the outgoing waves for both the elastic medium and the saturated poroelastic medium. For a detailed review, one is referred to the paper by Shi et al. [

The MTF extrapolates displacement on the artificial boundary at time

where

Since the computational points (

The operators

Correspondingly, the operators

According to Shi et al. [

To verify the accuracy of the proposed finite element formulation and the effectiveness of the MTF boundary condition, the transient response due to an edge pressure applied radially at a cavity surface is computed and compared to the analytical solution given by T. Senjuntichai [

Due to the symmetry, a 1/4 model is established for the cavity and the surrounding saturated soil medium. Symmetric boundary conditions are imposed for the left and bottom boundaries, while MTF boundary conditions are applied to the top and right boundaries to account for the infinite extension of the soil medium. The mesh in the

A cavity embedded in an infinite saturated poroelastic medium: (a) model setup and mesh grid; (b) edge pressure.

The soil properties and the load parameters are the same as those given by Senjuntichai [

Parameters of the edge pressure and the saturated poroelastic medium.

Parameter | Symbol | Value |
---|---|---|

Young’s modulus | | 5.33 × 10^{7} Pa |

Poisson’s ratio | | 0.33 |

Density of the bulk material | | 2000 kg/m^{3} |

Density of the pore fluid | | 1000 kg/m^{3} |

Porosity | | 0.4 |

Biot parameters | | 0.98 |

Biot parameters | | 4.0 × 10^{8} Pa |

Additional mass density | | 0 kg/m^{3} |

Darcy permeability coefficient | | 4.91 × 10^{−3} m/s |

Triangular pulse load | | 2.00 × 10^{7} Pa |

Radius of tunnel | | 1.0 m |

Comparison of present results with the solution proposed by Senjundichai [

In this section, coupled vibrations of the liner, the twin tunnels, and the saturated ground are investigated using the developed finite element along with the developed MTF boundary condition. Parametric studies have been performed for the burying depth of tunnel, the tunnel spacing, the load velocity, and the soil permeability.

The computation model consisting of two identical circular tunnels that are parallel to each other and embedded in a three-dimensional saturated ground is shown in Figure

Schematic of the 3D computation model.

Side view

Cross section view

Four observation points are chosen to record the responses caused by the moving load: points A and B are located at the liner bottom of the tunnel with the load applied and the other tunnel, respectively; points C and D are located on the ground surface right between the twin tunnels and above the center of the tunnel loaded, respectively. All observation points are located at the middle of the model along the tunnel axial direction (i.e.,

The bottom surface of the model is fixed to consider the underlying hard stratum. The top surface is free and set as permeable. The MTF boundary condition is applied to the remaining 4 side surfaces to account for the infinite extension of the saturated ground. For a reliable modeling of wave propagations, the element sizes of the ground should meet the requirement

Parameters of the liner, the saturated ground, and the moving load are summarized in Table ^{−5} m and 2.0 × 10^{−5} m/s, respectively.

Parameters of the liner, the saturated ground, and the moving load.

Saturated Soil Ground | Width a (m) | Height b (m) | Length l (m) | Biot parameters | |

42 | 24 | 120 | 0.99 | ||

Spacing | Burying depth | Porosity | Poisson’s ratio | ||

9–24 | 6–18 | 0.286 | 0.4 | ||

Young’s modulus | Darcy permeability coefficient | Shear wave velocity | Biot parameters | ||

| | (m/s) | | ||

2.23 × 10^{8} | 1.0 × 10^{−5}–1.0 × 10^{−2} | 191.1 | 5.74 × 10^{9} | ||

Density of the bulk material ^{3}) | Density of the pore fluid ^{3}) | ||||

2.178 × 10^{3} | 1.0 × 10^{3} | ||||

| |||||

Concrete Liner | Young’s modulus | Poisson’s ratio | Density of the bulk material | outer radius | Thickness |

| | ^{3}) | | | |

2.50 × 10^{10} | 0.2 | 2.400 × 10^{3} | 3.0 | 0.3 | |

| |||||

Moving Load | Point load value (N) | Load velocity (m/s) | |||

2.0 × 10^{4} | 19.11–191.1 |

In this section, a single tunnel model and a model with both tunnels loaded are compared with the present twin-tunnel model. The cross sections of the two models are presented in Figures

Cross sections of single tunnel model and the model with both tunnels loaded.

Single tunnel model

Model with both tunnels loaded

To investigate the influence of interactions between the two tunnels on the vibration-prediction results, the comparison of dynamic responses between single tunnel model and the present model is shown in Figures

Comparison of dynamic responses between a single-tunnel model and a twin-tunnel model.

Maximum displacement at the observation points for different tunnel spacing

Maximum displacement at the observation points for different burying depth

The comparison between the single tunnel model and the present model is shown in Figure

Comparison on maximum displacement at observation points between a single-tunnel model and a twin-tunnel model.

Practically, both metro tunnels are always loaded at the same time due to the high operation frequency of the subways. The dynamic responses of the model with both tunnel inverts loaded are studied in this section.

In contrast to the result of the present model, the dynamic responses of point C and point D due to two moving point loads are larger in terms of the maximum displacement. In Figure

Comparison of dynamic responses between the present model and a model with both tunnels loaded.

Maximum displacement at the observation points for different tunnel spacing

Maximum displacement at the observation points for different burying depth

The comparison of the displacements and the vibration velocity between the present model and the model with both tunnels loaded is shown in Figures

Comparison of maximum displacement at observation points between a model containing one load and a model containing two loads.

Comparison of maximum vibration velocity at observation points between a model containing one load and a model containing two loads.

To illustrate the necessity in introducing the saturated ground model, the dynamic responses of the twin tunnel embedded in a single-phase ground are compared to those of the saturated ground model and presented in Figure

Comparisons of dynamic response at observation points between a saturated ground and an elastic ground.

Maximum vibration velocity at observation points for different burying depth

Maximum displacement at observation points for different load velocity

Maximum vibration velocity at observation points for different load velocity

The influences of load velocity on the dynamic responses of the liner-tunnel-ground system are investigated in this section. Six different velocities

It is observed from Figure

Maximum vertical displacements at observation points for different load velocities.

In this section, the influence of the tunnel spacing on the dynamic responses is investigated by considering 6 different twin tunnel distances, i.e.,

Maximum vertical displacements at the observation points for different tunnel spacing.

The displacement at point A is significantly larger than those at other points, since point A is traversed directly by the point load. When the tunnel spacing increases, the displacements at points B and C decrease gradually. However, the displacements at points A and D remain almost the same. This observation is within expectation because the positions of points A and D remain the same while points B and C are shifted away from the load when the spacing increases. The displacements at points B and C vary greatly when the tunnel spacing

In this section, parametric study is conducted for 5 different burying depths of the twin tunnel, i.e.,

The variation of the maximum vertical displacements at the observation points with respect to the burying depth is presented in Figure

Vertical maximum displacements for different burying depth of tunnels.

In this paper, a 3D twin-tunnel model is presented to evaluate the dynamic responses of saturated half-space induced by subways. The effects of spacing between two tunnels, tunnel buried depth, load moving speed, and soil permeability on the vibration response are investigated. Based on the derivation and numerical examples presented above, the following conclusions can be drawn:

The existence of the other tunnel has influences on the vibration caused by the moving subway. And when the twin tunnels are both loaded, the dynamic responses cannot be calculated by simply summing up the results of a single tunnel model. Therefore, it is necessary to consider a twin-tunnel model when predicting the dynamic responses induced by subways for all load velocities. The dynamic responses of the single-phase ground are larger than those in the saturated ground which denotes that using a single phase model to predict the vibrations is more conservative.

The influence of the spacing between two tunnels on the dynamic response of the tunnel without the load applied is greater than that on the ground surface. With the increase of the spacing

The subways moving at a low speed will generate a smaller dynamic response in the saturated half-space than moving at a higher speed. And when the load velocity approaches the Rayleigh wave velocity of the ground, which is around

The expressions of the elements in the mass matrix, the damping matrix, the stiffness matrix, and the force vector are presented as follows:

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by the National Key R&D Program of China (Grant No. 2016YFC0800200), the Projects of International Cooperation and Exchanges NSFC (Grant No. 51620105008), National Science Foundation for Young Scientists of China (Grant No. 51608482), and National Natural Science Foundation of China (Grant No. 51478424).