Mathematical Problems in Engineering

Volume 2019, Article ID 5962737, 16 pages

https://doi.org/10.1155/2019/5962737

## Influence of the Hydraulic Boundary Condition between the Embankment and Saturated Half-Space on the Train-Induced Ground Vibration

^{1}College of Civil Engineering and Architecture, Zhejiang University of Technology, Hangzhou 310014, China^{2}Research Center of Coastal and Urban Geotechnical Engineering, Department of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310027, China

Correspondence should be addressed to Zhi-gang Cao; nc.ude.ujz@1102gnagihzoac

Received 13 December 2018; Accepted 31 January 2019; Published 11 February 2019

Academic Editor: Gaetano Giunta

Copyright © 2019 Zong-hao Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A three-dimensional analytical model was proposed to investigate the influence of the hydraulic boundary of the ground surface on the train-induced ground vibrations. The ground was simulated as a fully saturated poroelastic half-space and the embankment as a rectangular elastic soil layer with finite width. The rails and the sleepers were modeled as Euler beams and a Kirchhoff plate, respectively. Two hydraulic boundaries of the interface between the embankment and the ground surface were simplified as two limiting cases, namely, the permeable and impermeable cases. The linearized dynamic equations of motion for the fully saturated poroelastic soil and the elastic embankment were solved by Fourier transform and Fourier series. The vertical velocity and pore pressure were firstly calculated in the frequency-wavenumber domain and then transformed into the time-space domain by inverse Fourier transform. The influence of the hydraulic boundary at the ground surface on the train-induced ground vibration was specially investigated. It was found that the hydraulic boundary condition has a significant influence on ground-borne vibrations for the high-speed moving train with high load excitation frequencies.

#### 1. Introduction

The high-speed trains usually cause disturbance to residents and detrimental effects on the buildings nearby. In the past decades, researchers have made a lot of efforts to investigate the dynamic response of the ground generated by moving traffic loads.

Some researchers [1–3] have established several three-dimensional models to study the dynamic response of a homogeneous elastic half-space due to the moving constant load applied on the ground surface. In order to include the track, Metrikine and Dieterman [4] and Kim [5, 6] investigated the dynamic interaction between an Euler beam and the underlying elastic half-space, and it was found that resonance phenomenon occurs when the moving load velocity approaches the critical velocity of the beam-soil model. Sun [7] presented a theoretical three-dimensional model to simulate the vibrations of a rigid pavement resting on an elastic half-space subjected to a moving point load for the subsonic, transonic, and supersonic speed, respectively. Using a simplified formulation of the railway track, namely, a layered beam structure resting on a layered elastic ground, Sheng et al. [8–10] proposed a series of analytical models to study the ground vibrations by solving the wave equations for the Euler beam and the viscoelastic half-space.

All the works mentioned above treated the soil as a single-phase elastic and viscoelastic medium. However, underground water usually exists, which leads to the fact that the soil is saturated with water and becomes a two-phase material. For the two-phase medium, the coupling effects between the soil skeleton and the pore-fluid cannot be neglected when subjected to moving train loads [11]. If the half-space is modeled as a saturated poroelastic medium by using Biot theory [12], the pore-fluid related parameters and hydraulic boundary conditions can be considered. By neglecting the coupling between the soil skeleton and the fluid, Siddharthan et al. [13] solved Biot equations of** u-p** formulation under the plane strain condition, and the displacement and pore pressure responses caused by the moving load with a low velocity were given. Based on Siddharthan work, Theodorakopoulo et al. [14] and Theodorakopoulo [15] solved the complete** u-w-p** equation and presented a two-dimensional model to study the dynamic responses of the saturated soil under rectangular moving loads. Jin [16] established a three-dimensional model to calculate the dynamic response of the saturated soil for the case of a point load moving on the ground surface. By adopting four scalar potential functions and Helmholtz decomposition theorem, Liu et al. [17] studied the dynamic response of saturated soil to a harmonic moving load under conditions of plane strain. Xu et al. [18] researched the vibrations of the infinite Euler beam on a multilayered saturated half-space induced by moving loads on the beam, in which the effects of soil properties and the load velocity were studied. Gao et al. [19, 20] presented a track-ground interaction model to evaluate the train-induced vibration in the saturated ground. The influence of permeability coefficient and shear wave velocity on the attenuation of ground vibration was analyzed in detail. Chahour et al. [21] carried out a spectral analysis of a railway track coupled to the multilayered poroelastic half-space for the case of a moving harmonic load moving on the rails. However, in the above works the stress boundary condition between the beam and the underlying half-space was simplified as uniform stress distribution in the transverse direction, and the displacement compatibility condition is only valid at the center line of the track, which could not model the real dynamic interaction between the railway track and the ground precisely when subjected to high frequency loads. Based on Biot’s theory, Cai et al. [22] and Cao et al. [23] established a track-embankment-soil model with continuous displacement and stress compatibility conditions at the track-ground interface to investigate the ground vibration induced by a moving train.

It should be noted that a fully permeable boundary condition is usually assumed at the ground surface or between the track structure and underlying half-space in the above studies. However the hydraulic boundary condition between the track-embankment and ground may deteriorate during the service time due to the breakage of road bed filling particles; thus the contact surface between the track and half-space is actually partially permeable. The partially permeable interface between the embankment and ground and the fully permeable free ground surface leads to a mixed boundary value problem, which is not easy to be solved analytically. To circumvent the difficulty of the mixed boundary value problem, two limiting cases, namely, fully permeable and fully impermeable ground surface, were used to bound the actual hydraulic boundary condition. A study by Zhou et al. [24] showed that the dynamic response of a saturated half-space for the permeable and impermeable ground surface exhibits an obvious difference; however the dynamic interaction between the track system and the ground was not considered in this study. To date, few studies focus on the effects of the hydraulic boundary condition on the train-induced ground vibration. In order to make accurate prediction of train-induced vibrations, the effect of the hydraulic boundary condition on the train-induced ground vibration needs to be investigated.

In the present paper, the effects of the hydraulic boundary condition between the embankment and the saturated half-space on the train-induced ground vibrations are investigated by a three-dimensional semianalytical model. The model consists of rails, sleepers, an embankment, and a half-space which are simulated as Euler beams, a Kirchhoff plate, an elastic layer, and a poroelastic soil medium, respectively. By using Fourier transform and Fourier series techniques, the linearized dynamic equations of motion for the embankment and fully saturated poroelastic half-space were solved in the frequency-wavenumber domain. The critical velocity of the embankment-ground structure under different hydraulic boundary conditions was investigated. The effects of the hydraulic boundary condition on the dynamic responses of the ground generated by the moving trains with different self-excitation frequencies were analyzed.

#### 2. Governing Equations and Solutions

##### 2.1. Governing Equations

A detailed track-embankment-ground model is shown in Figure 1, consisting of a half-space, an embankment, sleepers, and two rails. The embankment is modeled as a rectangular elastic layer with finite width 2 and finite thickness which is resting on a homogeneous poroelastic half-space. The sleeper and rails are simulated as a Kirchhoff plate and Euler-Bernoulli beams. A high-speed train with 5 carriages is used to simulate the moving train loads and the detailed load amplitude and geometric distribution are presented in Figure 2. The drainage condition of interface between the embankment and the ground usually deteriorates during the service period due to the particle breakage and the invasion of fines. Thus, the interface is actually partially impermeable; then the hydraulic boundary of the ground surface becomes a mixed boundary with the partially permeable interface and the permeable free surface. In order to facilitate the analytical solutions of this model, two limiting cases, namely, the permeable and impermeable ground surface, are considered in the present study to bound the effects of the hydraulic boundary at the ground surface.