Analytical layer element with a circular cavity and its application in predicting ground vibrations from surface and underground moving sources

Abstract

The ground-borne vibrations caused by surface and underground moving trains have been recognized as an important environmental issue. There are good reasons to develop prediction models to compute the ground-borne vibration from moving sources. Underground sub-surface structures linked with infrastructure construction are always present in cities like cavities, tunnels, pipes and subways. When these inclusions exist in the ground, the solution to the wave field in a soil layer with inclusions is required for modelling the vibrations from moving sources on the ground surface or inside the inclusion. One of the simplest inclusions in the ground can be an empty infinitely-long circular cavity. In this paper the concept of analytical layer element with a circular cavity is proposed by using the wave function method aiming at providing a powerful tool to compute the vibrations from surface and underground moving sources. A complete solution to the wave field in a soil layer of finite thickness with a circular cavity is given, and the layer element can be easily assembled with other soil layers (with or without inclusions) by the formulations commonly used for a multi-layered ground. Moving sources can be located on the ground surface or inside the cavity. Two application examples are provided to show the capability of the proposed layer element: one is a theoretical investigation on the influence of a cavity/tunnel on ground vibrations due to a surface moving load. The other is an example to calculate ground vibrations from underground moving trains.

Introduction

The study of mechanical response of a multi-layered ground has received considerable attention, in particular in the context of seismology, transportation and engineering since the multi-layered ground represents a closer approximation to natural soil profiles, which are normally stratified in character. With the restriction to a linear problem, integral transformations are used to formulate the element stiffness matrix of each soil layer and half-space. The global stiffness matrix for multiple soil layers can be established through the global stiffness method (Kausel and Roësset, 1981, Schevenels et al., 2004), thin layer method (TLM) (Lysmer and Waas, 1972, Wass and Hanks, 1972, Kausel and Peek, 1982, Tassoulas and Kausel, 1983) and transmission and reflection matrix method (TRM) (De Barros and Luco, 1994). Analytical solution to the wave field in a soil layer and half-space allows for the solutions of various important problems, such as steady/transient wave propagation due to a forced excitation, the dispersion and attenuation analysis of surface waves and site amplification due to incident seismic waves (Mesgouez and Lefeuve-Mesgouez, 2009, Zhao et al., 2016, Kausel, 2018).

In most exiting studies, the above-mentioned approaches are adopted to model a multi-layered ground without any inclusions (Rajapakse and Senjuntichai, 1995, Senjuntichai and Rajapakse, 1995, Beskou and Theodorakopoulos, 2011, Han et al., 2020). Underground sub-surface structures linked with infrastructure construction are always present in cities like cavities, tunnels, pipes and subways. The addition of an inclusion into a soil layer significantly changes the wave filed due to the scattering effect of the inclusion. This call for a need for good analytical method to be developed to calculate the mechanical response in the soil layer with inclusions. One of the simplest inclusions in the ground can be an empty infinitely-long circular cavity. Biot, 1952, Boström and Burden, 1982 investigate the surface waves along a circular cavity surrounded by soils of infinite extent (full-space geometry) for rotationally symmetric case and arbitrary azimuthal order respectively. Gregory, 1967, Gregory, 1970) proposes an expansion theorem to the problem of wave propagations in a two-dimensional (2D) elastic half-plane with a circular cavity. In Gregory’s work it has been proved that the elastic potentials are expansible in a series form, which automatically satisfies the governing equations, the boundary conditions and the radiation conditions at infinity. The prescribed boundary conditions at the scattering surfaces are shown to lead to an infinite system of equations for the expansion coefficients. In this paper the concept of three-dimensional (3D) analytical layer element with a circular cavity is generalized by using the wave function method aiming at providing a powerful tool to formulate the wave field in a soil layer with a circular cavity. The proposed methodology facilitates the numerical investigation of ground vibrations from surface and underground moving sources such as busy road traffics and underground railways. Two application examples are provided to show the capability of the layer element for modelling a cavity/tunnel embedded in a multi-layered half-space.

In the followings a brief literature review is made on the topic of ground vibrations from surface and underground moving sources. The increasing urban population is leading to the exploitation of building sites close to sources of ground-borne vibration, such as busy roads and underground railways. For the sources moving on the ground surface, the solution to the wave field in a homogenous soil layer (without any inclusion) provides a prerequisite to calculate the mechanical response in the ground. The earliest research on the topic of ground vibrations from surface moving loads can be traced back to the work by Eason (1965), in which the dynamic response of an elastic half-space subjected to a constant moving load is investigated. Later Fryba, 1972, Alabi, 1992, Krylov, 1995, Yang and Hung, 2001 study a similar problem with the focus on the development of numerical algorithms to solve the problem and the resonance effect of the moving speed. To include the stiffness of the track structure and pavement, some improved models (Dieterman and Metrikine, 1996, Takemiya, 2003, Karlström and Boström, 2006) are proposed for the calculation of vibrations from moving loads running on the ground surface. The above models mentioned so far assume the ground to be a homogeneous/multi-layered half-space without any inclusions. In urban areas underground sub-surface structures linked with infrastructure construction are present, such as cavities, tunnels, pipes and subways. To the authors’ knowledge few studies have considered the influence of the existence of an underground cavity/tunnel on the ground vibration generated by road traffics. As the first application example of the proposed layer element, the influence of an underground cavity/tunnel on the mechanical response in the ground due to a surface moving source is investigated theoretically.

In addition to road traffics, the ground vibration from underground moving trains is another type of disturbance to the inhabitants who live near the railways. As metro lines grow rapidly, the vibration pollution induced by underground railways becomes more prominent due to a closer distance between railways and nearby buildings. Regarding the topic of ground vibrations from underground railways, there are lots of numerical models developed for the computation of vibrations from underground, such as the finite element (FE), boundary element (BE) and coupled FE-BE models (Stamos and Beskos, 1995, Gardien and Stuit, 2003, Andersen and Jones, 2006). The main disadvantage of numerical models is the high computational cost. To reduce the computation time, a two-and-a-half dimensional (2.5D) approach is a good choice, which enables obtaining a three-dimensional (3D) solution from a two-dimensional (2D) mesh (Yang and Hung, 2008, Galvín et al., 2010, Lopes et al., 2014, Jin et al., 2018). In above studies the soils are usually assumed to be linear elastic and the wheel-rail forces to be linear contact as well, which may be unrealistic in some cases and instead non-linear behaviors in the railway system should be modelled (Costa et al., 2010). Recently within the framework of 2.5D methodology, Ruiz et al. (2019) develop a novel hybrid vibration prediction model consisting of a tunnel construction model and an elastodynamics model. Using the construction model, the effect of soil stiffness degradation due to larger strains in the soils (in the range of 10^-4-10^-2) during tunnel drilling and lining construction can be taken into account. Then the 2.5D elastodynamics model is used to analyze the effect of tunnel construction on future underground railway vibrations. To include the nonlinear properties of wheel-rail forces, Connolly et al., 2019, López-Mendoza et al., 2020 propose a hybrid time–frequency approach to model the generation of railway vibration caused by singular defects. Since the wheel-rail-defect interaction and soil-structure interaction of buildings are solved in the time domain, the influence of defect type and building type on the commonly used vibration metrics can be easily analyzed. An alternative to numerical approaches is an analytical solution, which can give accurate predictions and provide benchmark results to verify the correctness of other modelling approaches. Also the analytical solution requires less computation time, which meets the expectations of the designers to assess the environmental vibrations quickly. Krylov (1995) proposes the buried load model, where the load on each sleeper is regarded as a point load in the interior of a half-space. Metrikine and Vrouwenvelder (2000) propose the embedded beam model, where the tunnel is viewed as a single Euler-Bernoulli beam embedded in soil layers. Using the separation-of-variable method, Forrest and Hunt (2006) present a pipe-in-pipe (PiP) model consisting of an infinitely long shell tunnel embedded in a full-space with a cylindrical cavity. Hussein et al. (2014) improve the PiP model to incorporate the half-space geometry. Recently, Yuan et al., 2017(a)., Yuan et al., 2017(b). develop an analytical wave function method for modelling vibrations from a tunnel embedded in a homogeneous half-space (a soil layer of infinite thickness), which considers the multiple scattering effects between the tunnel and the free surface exactly. Two tunnels close to each other give a problem that in some ways resembles that with a single tunnel below the ground surface. To investigate the influence of a neighboring tunnel on the ground vibration from underground vibrating sources, Yuan et al., 2019(a)., Yuan et al., 2019(b). propose an analytical wave function method for modelling a twin tunnel in a full- and half-space. The transformation and translation properties between plane and cylindrical waves are employed simultaneously to express the waves radiating outwards from one scattering surface in terms of the coordinates defined at other scattering surfaces.

In the present paper based on the previous work done by Yuan with other collaborators (Yuan et al., 2018), the concept of an analytical layer element with a circular cavity is generalized by using the wave function method. The exact solution to the wave field in a soil layer (of finite thickness) with a circular cavity facilitates the numerical investigation on the mechanical response in a multi-layered ground due to various vibration sources, such as the seismic waves generated by earthquake and moving sources by road and railway traffics. The proposed layer element with a circular cavity can be easily assembled with other soil layers (with or without inclusions) by the formulations commonly used for a stratified ground, such as the global stiffness method, TLM and TRM. In contrast to Yuan et al. (2018), the implementation is extended to allow a multi-layered geometry in this contribution, thereby giving the possibility to simulate a closer approximation to natural soil profiles (normally stratified in character). Meanwhile the application of moving loads and the evaluation of the mechanical response are possible on either the ground surface or the tunnel, and the approach permits an investigation of the ground vibration problem induced by both surface and underground moving sources. For the underground railway-induced vibrations, we further address how to model a complete train-track-tunnel-soil system with the track irregularities taken into account. The novelty of this paper mainly lies in the following two aspects: one is that the previous works consider only stationary/moving point source inside a tunnel with no attempt to account for the vibration source on the ground surface. The other is that the present paper provides the general procedure to model a complete train-track-tunnel-soil system incorporating track irregularities. The wave field in a soil layer (of finite thickness) with a circular cavity is the superimposition of down-going waves from the upper interface of the soil layer, up-going waves from the lower interface of the layer and outgoing waves from the cavity. To demonstrate the capability of the proposed layer element, two typical examples are given: one is a theoretical investigation on the influence of an underground cavity/tunnel on the ground vibration due to a surface moving source; the other is an application example for the calculation of ground vibrations from underground moving trains.