Study on the subgrade deformation under high-speed train loading and water–soil interaction

Abstract

It is important to study the subgrade characteristics of high-speed railways in consideration of the water–soil coupling dynamic problem, especially when high-speed trains operate in rainy regions. This study develops a nonlinear water–soil interaction dynamic model of slab track coupling with subgrade under high-speed train loading based on vehicle–track coupling dynamics. By using this model, the basic dynamic characteristics, including water–soil interaction and without water induced by the high-speed train loading, are studied. The main factors-the permeability coefficient and the porosity-influencing the subgrade deformation are investigated. The developed model can characterize the soil dynamic behaviour more realistically, especially when considering the influence of water-rich soil.

Appendix

Equations and Symbols:

$$\begin{aligned} {\begin{array}{l} K_\mathrm{t} =0.5G/(R\Delta l), \\ K_\mathrm{n} =G/(R\Delta l), \\ C_\mathrm{t} =\rho C_\mathrm{s} \Delta l, \\ C_\mathrm{n} =\rho C_\mathrm{p} \Delta l, \\ \end{array}} \end{aligned}$$

(1)

where $K_\mathrm{t}$ and $K_\mathrm{n}$ are, respectively, the tangent and normal stiffness, $C_\mathrm{t}$ and $C_\mathrm{n}$ are, respectively, the tangent and normal damping, and G is the shear modulus. R is the equivalent length between the source and the boundary, and $\Delta l$ is the smallest mesh size (0.15 m). $C_\mathrm{s}$ is the shear wave velocity, and $C_\mathrm{p}$ is the press wave velocity. $C_\mathrm{s}$ and $C_\mathrm{p}$ are calculated by using Eq. (2) as

$$\begin{aligned} {\begin{array}{l} C_\mathrm{s} =\sqrt{\frac{G}{\rho }}, \\ C_\mathrm{p} =\sqrt{\frac{(4/3G+K)}{\rho }}, \\ \end{array}} \end{aligned}$$

(2)

where G, K and $\rho$ are, respectively, shear modulus, bulk modulus, and density of the remaining part under the subgrade bed. These parameters can be seen or calculated from Table 2.

$$\begin{aligned} {\begin{array}{l} F_\mathrm{n}^{(t+\Delta t)}=F_\mathrm{n}^{(t)}-k_\mathrm{n} \Delta u_\mathrm{n}^{(t+1/2\Delta )}L, \\ F_\mathrm{s}^{(t+\Delta t)}=F_\mathrm{s}^{(t)}-k_\mathrm{s} \Delta u_\mathrm{s}^{(t+1/2\Delta )}L. \\ \end{array}} \end{aligned}$$

(3)

Transport Law:

The fluid transport is described by Darcy’s law. For a homogeneous, isotropic soil and constant water density, this law is given in the form:

$$\begin{aligned} q_i =-k\hat{{k}}(s)[p-\rho _f x_j g_j ]_{,l}, \end{aligned}$$

(4)

where $q_{i}$ is the specific discharge vector, p is pore pressure, k is the absolute mobility coefficients (permeability) of the soil, $\hat{{k}}(s)$ is the relative mobility coefficient, which is a function of fluid saturation, s (in FLAC, $\hat{{k}}(s)=s^{2}\left( {3-2s} \right) )$, $\rho _\mathrm{f}$ is the fluid density, and $g_{{j}}$, ${j} = 1, 2$ are the two components of the gravity vector.

Balance Laws:

For small deformations, the fluid mass balance can be expressed as

$$\begin{aligned} -q_{i,i} =\frac{\partial \zeta }{\partial t}, \end{aligned}$$

(5)

where $\zeta$ is the variation of water volume per unit volume of soil due to diffusive fluid mass transport, as introduced by Biot (1956). Note that fluid is compressible, which is reflected by the fluid modulus $K_\mathrm{f} =\Delta p/\left( {\Delta V_\mathrm{f} /V_\mathrm{f} } \right)$.

The balance of momentum has the form

$$\begin{aligned} \sigma _{ij,j} +\rho g_i =\rho \frac{\mathrm{d}v_i }{\mathrm{d}t}, \end{aligned}$$

(6)

where $\rho =\left( {1-n} \right) \rho _\mathrm{s} +ns\rho _\mathrm{w}$ is the bulk density, and $\rho _\mathrm{s}$ and $\rho _\mathrm{w}$ are the densities of the soil and water, respectively. Note that $\left( {1-n} \right) \rho _\mathrm{s}$ corresponds to the dry density of the matrix.

Constitutive Laws:

In the FLAC, changes in the variation of fluid content are related to changes in pore pressure p, saturation s, and mechanical volumetric strains $\varepsilon$. The response equation for the pore fluid is formulated as

$$\begin{aligned} \frac{1}{M}\frac{\partial p}{\partial t}+\frac{n}{s}\frac{\partial s}{\partial t}=\frac{1}{s}\frac{\partial \zeta }{\partial t}-\alpha \frac{\partial \varepsilon }{\partial t}, \end{aligned}$$

(7)

where M is the Biot modulus [$\hbox {N/m}^{2}$], n is the porosity, and $\alpha$ is the Biot coefficient.

The constitutive response for the soil has the form

$$\begin{aligned} \sigma _{ij}^{{\prime }}+\alpha \frac{\partial p}{\partial t}\delta _{ij} =H(\sigma _{ij} ^{{\prime }},\xi _{ij} ,\kappa ), \end{aligned}$$

(8)

where $\sigma _{ij} ^{{\prime }}$ is the effective stress rate, H is the functional form of the constitutive law (Mohr–Coulomb model in this study), $\kappa$ is a history parameter, $\delta _{{ij}}$ is the Kronecker delta, and $\xi _{{ij}}$ is the strain rate, which can be transformed into displacement (velocity, acceleration) according to compatibility equations.

In the FLAC numerical approach, the flow domain is discretized into quadrilateral zones defined by four nodes. (The same discretization is used for mechanical concerns, when applicable.) Both pore pressure and saturation are assumed to be nodal variables. Internally, each zone is subdivided into a triangle, in which the pore pressure and the saturation are assumed to vary linearly. The numerical scheme relies on a finite difference nodal formulation of the fluid continuity equation. The formulation can be paralleled to the mechanical constant stress formulation that leads to the nodal form of Newton’s law. It is obtained by substituting the pore pressure, specific discharge vector, and pore-pressure gradient for velocity vector, stress tensor, and strain-rate tensors, respectively. The resulting system of ordinary differential equations is solved using an explicit mode of discretization in time.

$$\begin{aligned} F_{\sup ,i} (t)=k_{\sup } Z_{\sup ,i} (t)+c_{\sup } \dot{Z}_{\sup ,i} (t), \end{aligned}$$

(9)

where $F_{\sup }$ is the discrete rail-supporting force, $k_{\sup }$ and $c_{\sup }$ are, respectively, the supporting stiffness and damping, and $Z_{\sup }$ and $\dot{Z}_{\sup }$ are, respectively, the displacement and the velocity.

$$\begin{aligned} F_{\sup } (t)=\sum _{-n}^n {k_i \cdot F_{\sup ,i} (t),(-n\le i\le n)}, \end{aligned}$$

(10)

where $k_{{i}}$ is the reduction factor. For a slab track, according to the reduction analysis by using the ANSYS, $k_{0}$ is 1, $k_{\pm 1}=0.286$, $k_{\pm 2}=0.0038$, $k_{\pm 3}=0.00055$, $k_{\pm 4}=0.00031$, and so on. The fastener number within a vehicle length is ($\hbox {2}n+1$) (${n}=10$).

$$\begin{aligned} z=A+\frac{B-A}{1+(k/k_0 )^{p}}, \end{aligned}$$

(11)

where $k_{0}=0.01145$, $p=2.1297$, $A =0.56756$, and $B =0.50654$ for the subgrade top.

$$\begin{aligned} \alpha =(z_{i,\max } -z_{i,\min } )/z_{i,\mathrm{mid}}, \end{aligned}$$

(12)

where $\alpha$ is the permeability coefficient influence factor, $z_{\max }$, $z_{\min }$, and $z_{\mathrm{mid}}$ are the maximum, minimum, and middle deformation in different depths, respectively. $i=1, 2, 3$ corresponds to the changing permeability coefficient in the first layer, the second layer, and the third layer, respectively.