Cumulative ground deformation induced by train operations

2.1. Train-track dynamic model

The axial load of the train passing through the uneven track including static load and dynamic load. The static load is generated by the self-weight of the train, and the dynamic load is formed by the contact of the wheel and rail when the train is running. In order to calculate the dynamic response between the train and the track, a train-track interaction dynamic model is established, which takes into account the interaction between the train vehicle, the rails, the underlying underlay and the fasteners, sleepers, track beds and foundations. It includes the vehicle submodel and the track structure submodel, as shown in Fig. 1.

Fig. 1Train-track dynamic model

A train consists of eight cars, each of which includes 10 degrees of freedom. The train is simulated to run on the track at a constant speed, and the relative displacement between the trains can be ignored. According to the Lagrange equation of motion, the dynamic equilibrium equation of the train model can be established as Eq. (1):

where ${\left[M\right]}_c$, ${\left[K\right]}_c$ and ${\left[C\right]}_c$ are mass matrix, stiffness matrix and damping matrix of train, respectively. ${\left\{\ddot{\mathbf{q}}\right\}}_c$, ${\left\{\dot{\mathbf{q}}\right\}}_c$ and ${\left\{\mathbf{q}\right\}}_c$ are acceleration, velocity and displacement vector of train. ${\left\{F\right\}}_c$ is the wheel and track interaction force [6]. The train dynamic parameters are shown in Table 1.

The entire track structure system simulation is based on the finite element method and is set up as a three-layer structure model including rails, sleepers, ballast and subgrade, as shown in Fig. 1. The degree of freedom of the track system is set on the same plane as the degree of freedom of the vehicle system. In order to reduce the size of the finite element stiffness matrix, the steel rail is assumed to be a discrete finite length Euler beam, ignoring the influence of shear force and moment of inertia, and an adjacent analysis unit is formed by the adjacent two sleepers and ballast supporting the rail, as shown in Fig. 2. The track structure dynamic parameters are shown in Table 2.

Based on the Hamilton principle, the dynamic equation of the orbit model is as follows:

where ${\left[M\right]}_1$, ${\left[K\right]}_1$ and ${\left[C\right]}_1$ are mass matrix, stiffness matrix and damping matrix of track structure, respectively. ${\left\{\ddot{\mathbf{q}}\right\}}_1$, ${\left\{\dot{\mathbf{q}}\right\}}_1$ and ${\left\{\mathbf{q}\right\}}_1$ are acceleration, velocity and displacement vector of track structure. ${\left\{F\right\}}_1$ is the wheel and track interaction force [6].

Fig. 2Track structure unit

The wheel-rail contact force acts on the wheel-rail contact point, and the forces acting on the wheel pair and the rail are equal in magnitude and opposite in direction. The wheel-rail contact force expression is as follows:

where $F_0$ is train static load. $q_{wi}$ and $q_{ri}$ are vertical displacement of the wheel at the wheel-rail contact and the vertical displacement of the track, respectively. $h$ represents the irregularity of the track surface at the wheel-rail contact.

Track irregularity is the main reason of dynamic response caused by train operation, and it is the excitation source of wheel-rail system. Considering that the established train-track dynamic model only considers the vertical dynamic response, the track irregularity is based on the spectrum used by Lombaert [7]. The simulation is as follows:

where $k_{\mathrm{1,0}}=$ 1 rad/s, $S\left(k_{\mathrm{1,0}}\right)=\mathrm{}$1×10^-8 m³, $w=$ 3.5.

2.2. Track-ground finite element model

The track-ground model was based on finite element to build. The track structure model consists of rails, sleepers, ballasts. The track adopts 60 kg/m, U75V hot-rolled steel rail, and the distance between two rails is 1.435 m. The relevant physical property parameters are shown in Table 3.

For the ground model, according to the detailed geological investigation, the physical and mechanical parameters of the main soil layers are shown in the table below.

For the train load, this paper simulates moving train loads by applying wheel-rail contact force which obtain from train-track dynamic model to the rail surface. The train speed is 20 km/h. For the boundary condition, by coupling the finite element and the infinite element to simulate infinite ground. It has wide applicability in the problem of simulating the infinite domain without losing the accuracy of calculation. The track-ground model is shown in Fig. 3.

Fig. 3Track-ground finite element model

2.3. Accumulated deformation of soft soil foundation induced by dynamic train load

The cumulative deformation of the ground under the moving train dynamic load can be calculated based on the empirical formula which proposed by Chai and Miura [8]. The formula was established by the soil cumulative plastic strain results based on the indoor dynamic triaxial test, which is defined as:

where ${\epsilon }_p$ is the cumulative plastic strain, $N$ is the number of cyclic loadings, $q_d$ is the dynamic deviatoric stress, $q_s$ is the initial static deviatoric stress, and $q_f$ is the static strength of the soil. When the soil is undrained, $q_f=\text{2}{C}_u$, $C_u$ can be obtained by shear test.

The specific analysis process is: a) dynamic analysis of the three-dimensional track-ground finite element model. The dynamic moving load is the wheel-rail contact force obtained by the train-track dynamic model. b) The dynamic analysis is carried out to obtain the dynamic response of the soil after one single train pass-by. The maximum dynamic deviator stress $q_d$ of the soil can be obtained. c) The static analysis of the model can obtain the initial static deviator stress $q_s$ of the soil. d) Use the empirical formula Eq. (5) to calculate the plastic cumulative strain of the ground under any vehicle load times, and calculate the cumulative deformation of the ground.