Underground tunnels are important infrastructures due to diverse applications in civil engineering. The dynamic behavior of the underground tunnels when exposed to seismic waves or the passage of high-speedmoving trains is of particular interest. Amplifications of displacements and stress concentrations may occur due to wave scattering and wave interference. Engineers concern about the vibration stability of moving trains as so-called anomalous Doppler waves may be generated when trains move at high speeds; the corresponding energy is fed into the vibration of the vehicle. Therefore, the traintrack- soil interaction should be properly considered to predict when the vehicle vibration becomes unstable. This thesis aims to present a semi-analytical solution for the response of a half-space with an embedded tunnel subject to seismic waves and to analyse the vibration stability of high-speed trainsmoving through that underground tunnel. Previous studies indicate that the method of conformal mapping is a promising analytical method to solve the two-dimensional (2D) wave scattering problem due to its computational efficiency and accuracy. Thus, the first objective of this thesis is to extend the method of conformal mapping to three-dimensional (3D) case and systematically evaluate its performance. Results reveal that inaccurate results maybe obtained, particularly at high frequencies. This observation motivates the second objective of this thesis, which focuses on verifying the accuracy of the specific application of the method of conformal mapping in which thewaves scattered fromthe half-space surface are represented by cylindricalwaves that originate from an image source of a priori unknown intensity. To this end, a simpler 2D model is considered, involving a cylindrical cavity embedded in an elastic half-space subject to a harmonic anti-plane shear wave. The performance of the indirect Boundary Element Method (indirect BEM) is evaluated too for this model in view of the choice of the appropriate solution method for the second type of dynamic problem considered in this thesis. For this second type of dynamic problem, due to the identified inaccuracies at high frequencies for the 3D problem, the indirect BEM is utilised to investigate the stability of vibrations of an oscillator moving at high speeds through a tunnel embedded in soft soil, which is the third objective of this thesis… @en

This paper presents a semi-analytical solution for the 3D problem of a cylindrical tunnel embedded in an elastic half-space subject to plane harmonic compressional and shear waves. Both the tunnel and soil are modelled as an elastic continuum. Conformal mapping is employed to transform the original physical domain with boundary surfaces of two different types onto an image domain with surfaces of the same type, which makes the problem easier to solve. The total wave field in the half-space consists of incident and reflected (from the half-space surface) plane waves, as well as directly and secondary scattered cylindrical waves, while the total wave field in the tunnel consists of refracted cylindrical waves. The secondary scattered waves, generated when the cylindrical waves directly scattered from the tunnel meet the half-space surface, are represented by cylindrical waves that originate from an image source, which is in line with the spirit of the method of images. The unknown amplitude coefficients of the cylindrical waves are determined from the boundary and continuity conditions of the tunnel–soil system by projecting those onto the set of circumferential modes, which results in a set of algebraic equations. Results show that the present method converges for a small number of circumferential modes. We observe very good agreement between the obtained results and those in literature. In a systematic evaluation, we demonstrate that the method works well for the frequency band of seismic waves, as well as for the complete considered ranges of the tunnel/soil stiffness ratio, the embedded depth of the tunnel, the vertical incident angle and the tunnel thickness. However, the results obtained for a moderate tunnel–soil stiffness contrast under the incident compressional wave are inaccurate when Hankel functions are used to represent the cylindrical waves in the tunnel, which is due to the refracted shear waves in the tunnel transitioning from propagating to evanescent (in the 3D case). These inaccuracies can be perfectly overcome by representing the waves in the tunnel by Bessel functions. We also find that the present method generally works better for the incident compressional wave than for the incident shear wave, as the condition number of the matrix (related to the mentioned algebraic equations) is often larger in the latter case. In view of engineering practice, we conclude that the tunnel is safer when the surrounding soil is stiffer, the tunnel is thicker and the vertical incident angle is larger. Finally, the present method, which is in general fast, elegant and accurate, can be used in preliminary design so as to avoid pronounced resonances, and to assess stress distributions and ground vibrations.@en

An analytical solution for the scattering of harmonic P1 and SV waves in a poroelastic half-plane with a shallow lined tunnel is obtained using the plane complex theory in elastodynamics. In light of the wave function expansions, the wave fields of the poroelastic medium and the liner with unknown coefficients are obtained based on Biot's theory and Helmholtz decomposition. Complex-valued expressions of the effective stresses, the fluid stress, and the displacements of the poroelastic medium and the liner are expressed by the complex variable function method and the conformal transformation technique. With the boundary conditions and the continuity of the medium-liner interface, the boundary value problem results in a series of algebraic equations. The unknown coefficients in the infinite set of algebraic equations can be solved numerically by truncating the series number. A parametric study for the incident SV waves is performed to investigate dynamic stress concentrations and fluid stress of the medium and the liner. Numerical results show that the embedment depth of the tunnel, the incident angle of the excitations, and the porosity of the medium have considerable influence on the dynamic responses of the medium and the liner. The shielding effect of the tunnel on the incident SV waves is obvious. For the big embedment depth of the tunnel, the scattered waves contribute little to the displacements and dynamic stress concentration of the medium and the liner. For a high porosity close to the critical value, the response of the medium-liner system to the incident waves is great.

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This paper investigates the instability of vertical vibrations of an object moving uniformly through a tunnel embedded in soft soil. Using the indirect Boundary Element Method in the frequency domain, the equivalent dynamic stiffness of the tunnel-soil system at the point of contact with the moving object, modelled as a mass-spring system or as the limiting case of a single mass, is computed numerically. Using the equivalent stiffness, the original 2.5D model is reduced to an equivalent discrete model, whose parameters depend on the vibration frequency and the object's velocity. The critical velocity beyond which the instability of the object vibration may occur is found, and it is the same for both the oscillator and the single mass. This critical velocity turns out to be much larger than the operational velocity of high-speed trains and ultra-high-speed transportation vehicles. This means that the model adopted in this paper does not predict the vibrations of Maglev and Hyperloop vehicles to become unstable. Furthermore, the critical velocity for resonance of the system is found to be slightly smaller than the velocity of Rayleigh waves, which is very similar to that for the model of a half-space with a regular track placed on top (with damping). However, for that model, the critical velocity for instability is only slightly larger than the critical velocity for resonance (of the undamped system), while for the current model the critical velocity for instability is much larger than the critical velocity for resonance due to the large stiffness of the tunnel and the radiation damping of the waves excited in the tunnel. A parametric study shows that the thickness and material damping ratio of the tunnel, the stiffness of the soil and the burial depth have a stabilising effect, while the damping of the soil may have a slightly destabilising effect (i.e., lower critical velocity for instability). In order to investigate the instability of the moving object for velocities larger than the identified critical velocity for instability, we employ the D-decomposition method and find instability domains in the space of system parameters. In addition, the dependency of the critical mass and stiffness on the velocity is found. We conclude that the higher the velocity, the smaller the mass of the object should be to ensure stability (single mass case); moreover, the higher the velocity, the larger the stiffness of the spring should be when a spring is added (oscillator case). Finally, in view of the stability assessment of Maglev and Hyperloop vehicles, the approach presented in this paper can be applied to more advanced models with more points of contact between the moving object and the tunnel, which resembles reality even better.

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A wave function expansion method for out-of-plane scattering of SH waves by two symmetrical circular cavities in two bonded exponentially graded half spaces is presented by using the plane elastic complex variable theory. The medium is composed of a semi-infinite homogeneous space with a circular cavity and a bonded exponentially graded half-space with a symmetrical circular cavity along the boundary interface. In terms of Helmholtz decomposition to the wave propagation equation, the stresses and displacements of the medium are expressed by the displacement potentials. The scattered waves by the boundary interface are assumed to transmit from the images of the origins of the respective two cavities. A conformal mapping function is introduced to convert the physical plane of two bonded semi-infinite spaces into two jointed concentric annular regions. The boundary value problem is formulated as a series of infinite algebraic equations. The convergence of the present solution is examined by investigating the variations of the solution results with the truncation of the series number. Parametric study shows that the interface displacements are almost independent to the inhomogeneous coefficient βR of the right medium. The amplification of the interface displacement becomes great for high incident frequency and with the decreasing of the distance-to-radius ratio h/R. The distribution of stress concentration of the right cavity depends significantly on the inhomogeneous coefficient βR, the distance-radius ratio h/R, the incident angle and frequency of the excitations.

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This paper addresses the dynamic response of an infinitely long cylindrical structure embedded in an elastic half-space. The structure has a circular cross-section and its axis is parallel to the half-space surface. Excitation can be incident body waves or forces applied on the surface of the half-space and/or the structure. The model can be used to assess the integrity of structures when acted upon by seismic waves, to predict ground-borne vibration due to circulation of vehicles, and to infer about the safety of vehicles during earthquake events. Because the half-space and the structure surfaces possess different symmetries, the solution is not straightforward. In order to circumvent this difficulty, the physical domain is conformally mapped onto an auxiliary domain with a cylindrical symmetry, in which the free surface of the half-space and the surface of the structure are located at concentric cylindrical surfaces. The solution of the original boundary value problem is finally obtained by solving a set of algebraic equations. Truncation of the summation over circumferential modes is needed in the numerical implementation. Convergence tests, validations and comparisons of stresses and motions for twoand three-dimensional cases are presented and discussed as well as the advantages and disadvantages of the proposed method. Additionally, the effect of the presence of the tunnel is analysed by considering a limiting case of the half-space with just a cylindrical cavity of the same radius as the outer radius of the tunnel. @en