2.5D prediction of soil vibrations due to railway loads by the isogeometric analysis with scaled boundary

https://doi.org/10.1016/j.enganabound.2021.10.012Get rights and content

Abstract

The 2.5D approach is an efficient tool for the dynamic analysis of half-space with longitudinally invariant properties. In this paper, a newly formulated 2.5D version of the NURBS-based isogeometric analysis (IGA) and the scaled boundary IGA (SBIGA) is proposed for analyzing the bounded and unbounded domains, respectively, of a soil-tunnel system subjected to moving railway loads. The bounded domain is discretized by IGA, since the NURBS used can exactly represent the variations in geometry and materials with negligible errors. The unbounded domain is discretized by SBIGA, which automatically satisfies the Sommerfeld radiation condition and can be easily connected with the bounded IGA domain. The 2.5D approach differs from the 2D approach in that both the in-plane and out-of-plane displacements of the profile of the soil-tunnel system are included in analysis. By applying the Fourier transformation, the IGA and SBIGA equations of the 2.5D version in the frequency-wavenumber domain are derived individually and then assembled at the interface of the bounded and unbounded domains. Two examples are prepared for the ground and underground trains moving at subsonic and transonic speeds. Both single and multi moving loads are considered. It was demonstrated that the wave propagation characteristics including the Mach cones at transonic speeds can be precisely predicted. The results in time and frequency domains have been compared with existing ones. It was shown that the proposed method is accurate and reliable for application to a wide range of soil-tunnel systems.

Introduction

In recent years, highspeed railways and subways have emerged as efficient tools of transportation for inter- and intra-city passengers, respectively, for their economy, punctuality, and environmental friendliness. They can ease greatly the traffic pressure, while promoting the economic development of the areas along the route. However, railway lines inevitably run close to or through residential or industrial areas. The vibrations induced by moving trains may affect the living quality of residents and the production quality of precision factories along the lines [1].

Recently, due to the residents’ increasing environmental concern, the vibration of moving trains, either on the surface or underground, has become a hot subject of research in railway engineering. It is realized that the foundation of a railway track, e.g., ballast, sleepers, fasteners, etc., may affect the local vibration transmission of a moving train. Concerning the train-induced wave propagation, focus has often been placed on the dynamic response on a half-space to loads moving on the surface or inside the half-space via tunnels. Methods that have been adopted by researchers for predicting the ground-borne vibrations including those of the analytical or semi-analytical [2], [3], [4], [5], [6], [7], empirical [8], [9], [10], and numerical nature [11], [12], [13].

The early analytical methods to study soil vibrations induced by surface trains were simplified to a model of moving harmonic point loads on a homogeneous half-space surface [2,3]. Later, Sheng, et al. [4] introduced an influencing factor to account for the effect of railway tracks. For the analytical methods of soil vibration induced by underground trains, Forrest and Hunt [5] proposed the Pipe-in-Pipe (PiP) model for predicting soil vibrations from a tunnel in a homogeneous full-space, which was later extended to a tunnel in a multi-layered half-space by Hussein et al. [6] Recently, He et al. [7] used the wave decomposition method to calculate the effect of twin tunnels in a multi-layered half-space. Analytical methods have the advantages of accuracy, readiness and high computational efficiency, but are limited by the assumptions adopted.

The reliability of empirical methods is highly dependent on the accuracy of the measurements and the level of similarity to the locations covered by the database. To cope with the irregularity in geometry and materials, numerical methods have often been adopted, which is enhanced by the availability of high-performance computers and the advancements in computational techniques.

Balendra et al. [14] used a two-dimensional (2D) finite element model to predict the ground-bourne vibrations in a subway building by moving trains. Other 2D analyses also exist in the literature. However, the results predicted by 2D models are known to be good in the qualitative, but not in the quantitative sense, since no account was taken of the wave propagation characteristics along the track due to moving trains. Subsequently, some sophisticated three-dimensional (3D) models have also been adopted, such as Luco et al. [15], Seo et al. [16], Galvín and Romero [17]. Although the 3D models can simulate the wave propagation behavior to a certain extent, their computational costs are prohibitively high.

On the fact that the railway track is a long structure, one may assume the track to be invariant or uniform along its longitudinal direction. With this, the 2.5D approach was proposed by Yang and Hung [18] as an alternative for treating the 3D wave propagation problems. The 2.5D approach is featured by the fact that only the 2D profile of the railway track is needed in the finite element simulation, but it enables us to consider the effect of wave propagation along the track or train-moving direction. The computational efficiency of the 2.5D approach is comparable to but slightly lower than the 2D approach, since it requires both the in-plane and out-of-plane displacements of each nodal point of the profile to be included in analysis. Using the finite elements (for bounded domain) and infinite elements (for unbounded domain), Yang and co-workers have analyzed various soil vibration problems due to moving train loads, including the effects of rail roughness and incident earthquakes [19], [20], [21], [22].

The 2.5D approach has evolved over the years. Jean et al. [23] adopted the same 2.5D concept, but used the boundary element (BE) method for the problem, which was later extended to include the finite element (FE) domain [24]. Along these lines, Sheng et al. used the FE-BE model to predict the ground vibrations caused by moving trains [25,26]. Jin et al. [27] used a similar model for simulating the subway tunnels and compared their results with field measurements. François et al. [28] used the finite element-perfectly matched layer (FE-PML) model to study soil vibrations caused by ground trains. Lopes et al. [29,30] used the FE-PML model to study the vibrations induced by trains moving through the tunnels. Amado-Mendes et al. [31] used the method of fundamental solutions (MFS) to replace the PML in their 2.5D MFS-FEM formulation. Yaseri et al. [32] presented a 2.5D approach by combining the FEM with scaled boundary finite element method (SBFEM) for ground vibration analysis. Zhou et al. [33] used the 2.5D FE-BE model to study the dynamic response of a segmented tunnel in saturated soils. Recently, Ma et al. [34] used the 2.5D FEM-PML model to model the wave propagation induced by curvilinearly moving loads. Liravi et al. [35] used the FEM-BEM to represent the FE domain, resulting in the so-called 2.5D FEM-BEM-MFS scheme.

As can be seen from the above review, almost all of the methods adopted for simulating the half-space are “joined” methods, with the bounded domain mainly modeled by FEs. They only differ in the modeling of the unbounded domain by different approaches to cope with non-deflective outgoing waves.

In the vibration analysis of a half-space, however, one may inevitably encounter the inclusion of structures such as tunnels, above-ground buildings, or others. In this regard, geometry modeling errors may occur when using the FEs to simulate the inclusions with curved shapes or sharp corners. In addition, errors may also be brought in when using the interpolation functions, such as polynomials, to approximate the displacement field of each FE. To reduce such errors, a large number of FEs are often required for regions with abrupt changes in geometry, which renders the FE simulation not a very efficient scheme.

In computer-aided design, non-uniform rational B-spline (NURBS) is an efficient tool for geometry rendering, since it can accurately represent arbitrary free shapes in a compact form. Based on NURBS, Hughes et al. [36] proposed the isogeometric analysis (IGA) for both geometry modeling and field variable interpretation, particularly by using the NURBS basis functions to replace the conventional interpolation functions such as Lagrange polynomials. High level of accuracy can be achieved by the NURBS when used to simulate both the geometric model and displacement field. The concept of IGA has revolutionized the analysis procedure. It has been proved to be an effective alternative to the conventional FEM.

The NURBS with high continuity can be applied to higher-order differential equations, including plate and shell problems. Moreover, the NURBS function can offer any degree of continuity by adjusting the multiplicity of knots, which can hardly be achieved by the conventional FEM. Due to these attractive advantages, the NURBS-based IGA has been applied to a variety of problems such as plates [37], cohesive elements [38], fluid-structure interaction [39], optimization [40], composite plates [41], curved beams [42], and so on.

For the soil vibration problems, the scaled boundary FEM (SBFEM) was first presented by Song and Wolf [43] for the unbounded domain, which can meet the Sommerfeld radiation conditions. It has been shown that the SBFEM is precise and effective for dealing with problems such as infinite domains, anisotropic media, and material inhomogeneity [44].

The conversion of the conventional coordinate system to the scaled boundary coordinate system for the unbounded domain is an essential step for SBFEM, a semi-analytical method that presents the results analytically and numerically in the radial and circumferential directions, respectively. However, as in FEM, the conventional Lagrange polynomials are chosen as the basis functions in SBFEM, which cannot accurately cope with geometries of curved shapes. Inspired by the concept of IGA, the NURBS is also chosen as the basis function of SBFEM. The NURBS-based SBFEM is named the scaled boundary IGA (SBIGA), which works well for the 2D problems including the elastostatics problems [45], fracture mechanics [46] and electrostatic problems [47].

As of now, the IGA has not been extended to the 2.5D formulation. In this paper, the 2.5D version of the IGA will be formulated for the first time and used to predict the soil vibration caused by moving trains. For the soil-tunnel system, the IGA will be used for the bounded domain (to accommodate the irregularity in geometry) and the SBIGA for the unbounded domain (to simulate the radiation damping at infinity). They can be easily be assembled at the interface of two domains, since they are both of the 2.5D version based on the NURBS. The rest of the paper is organized as follows. In Section 2, a brief introduction is given of the NURBS and IGA. In Section 3, a detailed formulation is given of the 2.5D version of NURBS-based IGA, followed by that for the SBIGA. In Section 4, numerical examples are prepared to demonstrate the accuracy and efficiency of the proposed scheme. Particularly, the wave propagation characteristics on the surface caused by single and multi moving loads will be investigated, including those of the Mach cones for the transonic case. Finally, in Section 5, the conclusions are drawn along with future research identified.

Section snippets

B-spline and NURBS

This section provides a brief overview of NURBS. For a more detailed description and implementation, interested readers should be referred to Ref. [36]. NURBS is a generalization of uniform B-spline, characterized by its basis functions, control points and knot vectors. A spline function is a piecewise polynomial function of degree p with its intersection points called knots. The number of knots must be equal to or greater than p +  1. A remarkable feature of the spline functions is that they

2.5D analysis of elastodynamics

A 3D soil-tunnel system that is longitudinally invariant is considered (see Fig. 4), as encountered in the vibration analysis of soils induced by trains moving on the surface or inside a tunnel. To facilitate analysis, the soil-tunnel system is assumed to be invariant or uniform along the longitudinal (z-) axis in terms of geometry and material. Using only the 2D profile (i.e., xy-plane) and assigning three displacement DOFs to each control point, the vibration of the soil-tunnel system along

Numerical verification

This section aims to evaluate the accuracy of the 2.5D version of the IGA-SBIGA procedure developed in previous sections. To facilitate comparison with existing analytical or numerical solutions, two simple examples are prepared to verify the accuracy of the proposed method. In the first example, a viscoelastic half-space under the surface moving loads at various speeds is considered. The results obtained by the proposed approach will be compared with the analytical solutions given by Hung and

Conclusions

In this paper, the 2.5D version of the NURBS-based IGA-SBIGA approach has been formulated for predicting the ground response to moving railway loads. The reliability and accuracy of the proposed approach has been demonstrated in the numerical examples via comparison with previous solutions. Based on the theoretical formulation and results of analysis, the following conclusions are drawn: (1) As an improvement over the conventional FEM, the geometry for the soil-tunnel system is exactly

Declaration of Competing Interest

The authors declare no conflict of interest.

Acknowledgements

This research reported herein is sponsored by the following agencies: Chongqing Science and Technology Commission (Grant No. cstc2020yszx-jscxX0002), Chongqing Science and Technology Commission (Grant No. cstc2018jcyj-yszxX0013), National Natural Science Foundation of China (Grant No. 52078082), Chongqing Science and Technology Commission (Grant No. cstc2019yszx-jcyjX0001), and China State Railway Group Co., Ltd. (Grant No. K2019G036).

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