Coupled boundary and finite element analysis of vibration from railway tunnels—a comparison of two- and three-dimensional models
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.
Section snippets
Boundary elements for an open domain
In the frequency domain, the equation of motion and the boundary conditions for a viscoelastic two- or three-dimensional body Ω with the surface Γ readHere is the complex amplitude of the displacement field, are the body forces and are the stresses that may be computed from the displacements by the constitutive relation. As indicated in Eq. (2) the displacement amplitude
Analysis of a cut-and-cover tunnel
Three cut-and-cover tunnel structures are analysed. In Section 3.1 the structures are described, and in Section 3.2 the results of a two- and a three-dimensional model are compared.
Analysis of a NATM tunnel
Now, a similar study is carried out for a NATM tunnel. The popularity of this method has increased over the last decades for deep tunnels in soil with medium stiffness, because the construction process is cheaper than boring as no specialized equipment is necessary. NATM tunnels are dug with the use of standard excavators, and the concrete lining is cast in situ [7].
Conclusions
Two- and three-dimensional combined finite and boundary element analyses have been carried out for two railway tunnel structures. The aim has been to investigate, what information can reliably be gained from a two-dimensional model to aid a tunnel design process or an environmental vibration prediction based on ‘correcting’ measured data from another tunnel in similar ground.
In the case of both types of tunnel examined, only small changes in the vibration response are predicted from the
Acknowledgements
Lars Andersen would like to thank the Danish Technical Research Council for financial support via the research project: ‘Damping Mechanisms in Dynamics of Structures and Materials’.
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