Coupled boundary and finite element analysis of vibration from railway tunnels—a comparison of two- and three-dimensional models

Abstract

The analysis of vibration from railway tunnels is of growing interest as new and higher-speed railways are built under the ground to address the transport problems of growing modern urban areas. Such analysis can be carried out using numerical methods but models and therefore computing times can be large. There is a need to be able to apply very fast calculations that can be used in tunnel design and studies of environmental impacts. Taking advantage of the fact that tunnels often have a two-dimensional geometry in the sense that the cross section is constant along the tunnel axis, it is useful to evaluate the potential uses of two-dimensional models before committing to much more costly three-dimensional approaches. The vibration forces in the track due to the passage of a train are by nature three-dimensional and a complete analysis undoubtedly requires a model of three-dimensional wave propagation. The aim of this paper is to investigate the quality of the information that can be gained from a two-dimensional model of a railway tunnel. The vibration transmission from the tunnel floor to the ground surface is analysed for the frequency range relevant to the perception of whole body vibration (about 4–80 Hz). A coupled finite element and boundary element scheme is applied in both two and three dimensions. Two tunnel designs are considered: a cut-and-cover tunnel for a double track and a single-track tunnel dug with the New Austrian tunnelling method (NATM).

Introduction

In recent years great interest has been shown in reduction of vibration in buildings caused by railway traffic. This is a concern both in the design of new railways and in the modification of existing ones. In densely populated areas, trains often run in tunnels. At these locations the transmitted vibration is an important environmental issue [1] and numerical models are required to complement empirical prediction models [2] and for tunnel structure design [3].

For each of these purposes, two-dimensional models have been used; even if the problem is by nature three-dimensional, see, for example, Ref. [4]. Justification for this is based on the fact that the train may be regarded as a long line of incoherent vibration sources. However, rather than using models to provide predictions of absolute vibration levels, they are more likely to be used to predict the effects of changes made to the tunnel structure [2], [3], and it is assumed that such changes are reasonably predicted using a two-dimensional model.

The main reason that a two-dimensional model is preferred is that the numerical analysis with a three-dimensional model requires far more computing time such that it precludes parametric study. This remains the case even though, recently, so-called ‘two-and-a-half-dimensional models’, which calculate a three-dimensional field from a two-dimensional geometry, have been developed [5], [6].

Here, two tunnel structures are considered. Firstly, a cut-and-cover tunnel with masonry abutment walls is analysed with both a two- and a three-dimensional model. The performance of the two-dimensional model is tested against that of the three-dimensional model in the comparison of different designs of the tunnel floor. Secondly, a tunnel is analysed, which has been built using the New Austrian tunnelling method (NATM) [7]. The changes in the response to either a change in the tunnel depth or the application of a wave impeding block (WIB) under the tunnel floor are compared with both a two- and a three-dimensional model.

The models are based on the combined finite element (FE) and boundary element (BE) methods. Solid FEs or BEs are used to model the tunnel. BEs are used to model the surrounding soil. Here the BE method is superior to the FE method due to its inherent ability to model radiating waves. Computer programs for both two- and three-dimensional elastodynamic coupled structure–soil analysis in the frequency domain have been developed. A detailed description of the FE method for elastodynamic analysis may be found, for example, in the book by Petyt [8]. The BE part of the model is an extension of the theory presented by Domínguez [9], which has been modified to account for open domains and to allow coupling with FEs. Here standard three-noded elements are used in the two-dimensional analysis and nine-noded quadrilateral elements are applied in the three-dimensional analysis.