Layered Half-space Research Articles

Numerical investigation of railway subgrade stiffening: Critical speed and free-field vibrations

For a train speed close to the speed of elastic waves in the soil, often referred to as “critical speed”, largely elevated vibration responses occur. This can be a practical problem for soft soil sites, where the phenomenon may cause excessive vibrations in the track and also at distances far from the track. To ensure the running safety of the train, the long-term quality of the track and to reduce the vibrations in the surroundings, such effects must be avoided. An effective counter-measure is to increase the stiffness of the soil underneath the track, thereby increasing the critical velocity.In this paper, a 2.5D finite element model is used for studying the critical velocity phenomenon and its mitigation through soil stiffening, for a ballasted track on a layered half-space with very soft soil. Soil improvement under the track, in the shape of a solid block or as various number of panels, with varying depth and stiffness is considered. The effect of the soil improvement is evaluated both in terms of the maximum rail and free-field displacements. It is shown that a shallow soil stiffening increases the critical velocity and reduces the rail and free-field response for load speeds near the shear wave velocity of the soft top soil layer. It is also demonstrated that a deep soil stiffening, by use of panels along the track direction, increases the critical velocity further, and may also be efficient in reducing the response for load speeds near the shear wave speed of the underlying half-space.

Mitigation of railway-induced vibrations by using periodic wave impeding barriers

Railway-induced vibrations can cause significant environmental issues. This paper proposes an efficient analytical method to investigate the mitigation of railway-induced vibrations by using periodic barriers in a layered half-space. The general solutions for layered ground and multiple inclusions are derived by using the potential decomposition and multiple scattering theory. The conversion equation between cylindrical and exponential functions and the addition theorem are introduced to achieve the transformation between plane and cylindrical wave functions and the translation between cylindrical wave functions. Combined with the transfer matrix method, the fundamental solution for the soil-inclusion dynamic interaction in a layered half-space is derived. The railway train and track are subsequently coupled to the ground-inclusion system. Numerical results demonstrate that the phononic crystal effect induced by the periodic distribution of barriers improves the mitigation efficiency at high frequencies. The increase in the number, size, and stiffness of barriers can give a higher mitigation efficiency in a wider frequency range. The mitigation efficiency of periodic barriers can be guaranteed when their depth is shorter than half the Rayleigh wavelength in the considered frequency range. Owing to the scattering of waves at layer interfaces, the periodic barriers beneath the track have a higher efficiency than those located next to the track, which does not appear in the homogeneous half-space. The performance of periodic barriers is significantly affected by the soil stiffness of the upper shallow layer, while it is less affected by the soil stiffness of the bottom stiffer layer.

Effects of adjacent paleochannel sedimentary on the seismic response of structure with incident SH-wave

This paper combines the indirect boundary element method and the idea of “region-matching technique” to study the impact of adjacent paleochannel on structures in a layered half-space with incident SH-wave. It decomposes the model into paleochannel closed region, layered half-space open region, and rigid foundation-elastic structure. Wave field in the closed region only have scattered wave, and wave field in the open region includes scattered and free field. The scattered and free field are simulated by the dynamic Green's function and the direct stiffness method, respectively. Then, response of soil-structure system is solved by employing the boundary continuity conditions and dynamic equilibrium of structure. After verification of the method, seismic response of structure under the influence of adjacent paleochannel shape, distance between the paleochannel and structure, and shear wave velocity ratio of the paleochannel were analyzed. Results show: All three paleochannel shapes (ellipse, trapezoid, triangle) are adverse to the structure compared to the case of no paleochannel, and ellipse is the most unfavorable shape; Effect of paleochannel should be taken into account when paleochannel distance L/a<100; Higher shear wave velocity ratios result in larger peak of structural displacement; Structural response is more amplified in oblique incidence than vertical incidence.

A Unified Analogy-Based Computation Methodology From Elasticity to Electromagnetic-Chemical-Thermal Fields and a Concept of Multifield Sensing

AbstractThis paper reports a unified analogy-based computation methodology, together with a concept of multifield, multifunctional sensing, from elasticity to electromagnetic-chemical-thermal fields, via utilizing the similarities of mechanical-electromagnetic-chemical-thermal (MEMCT) field variables, governing equations, and the material properties pertaining to each individual field. Two equivalences are systemized, which are the field-formulation equivalence and surface-value equivalence. Due to similarity, a number of thermal, electromagnetic, or chemical solutions can be obtained from the direct degeneration of existing mechanical solutions by making specified equivalences of 2G↔k0↔ϖ0↔μ0↔β0 with G for shear modulus, k0 for heat conductivity, ϖ0 for dielectric permittivity, μ0 for magnetic permeability, and β0 for chemical diffusivity, as well as by setting Poisson’s ratio ν → 0.5. These specified equivalences enable quick solutions to other fields directly from mechanics formulations, such as those in the forms of the Galerkin vectors and Papkovich-Neuber potentials, and field coupling, by means of analogy. Several examples are given, one is used to demonstrate that the field solutions of a layered half-space with imperfect thermal, electromagnetic, or chemical interfaces can be readily obtained from the elastic solutions involving interfacial imperfections via the obtained formulation equivalence. A set of simple equations are derived to relate surface behaviors of different fields via the obtained surface-value equivalence, on which a concept of multifield sensing is proposed.

Dynamic interaction analysis of structure-water-soil-rock systems under obliquely incident seismic waves for layered soils

A numerical model for the structure-water-soil (layered)-rock (SWLR) system is proposed in this paper to evaluate the system dynamic response, in which the fluid-solid coupling effect is considered by the interface conditions and the wave radiation effects of far-field domain are simulated by artificial boundary conditions. Moreover, this model covers four prime respects, including the discretization of interface conditions, the absorbing boundary conditions, the equivalent earthquake loading input and the general finite element equation. It can be solved by a standard time integration algorithm such as implicit Newmark's method. Furtherly, the effectiveness of the proposed model is verified by three numerical examples. Apart from these, the dynamic stiffness matrix method (DSM) of fluid and solid adopted in present study to acquire the equivalent earthquake loading effectively accomplishes the oblique incidence of the wave in layered half-space with arbitrary angles. Based on the proposed model, the effects of dynamic water-structure interaction (WSI) and soil-water interaction (SWI) on the seismic responses of offshore structure are investigated, which is conducive to the simplified analysis of dynamic fluid-solid interaction and the design of ocean engineering.

Effect of Initial Stress on an SH Wave in a Monoclinic Layer over a Heterogeneous Monoclinic Half-Space

Considering the propagation of an SH wave at a corrugated interface between a monoclinic layer and heterogeneous half-space in the presence of initial stress. The inhomogeneity in the half-space is the causation of an exponential function of depth. Whittaker’s function is employed to find the half-space solution. The dispersion relation has been established in closed form. The special cases are discussed, and the classical Love wave equation is one of the special cases. The influence of nonhomogeneity parameter, coupling parameter, and depth of irregularity on the phase velocity was studied.

A three-dimensional indirect boundary integral equation method for the scattering of seismic waves in a poroelastic layered half-space

A new indirect boundary integral equation method (IBIEM) is developed to solve the seismic wave scattering problem in a three-dimensional fluid-saturated layered half-space. Based on Biot's theory, the Green's functions of inclined circular loads in porous elastic layered half-space are firstly deduced. According to the single-layer potential theory, when establishing the scattered wave field, the uniform surface loads and fluid sources are located directly on the boundary surfaces of the scatterers, so as to avoid determining the optimal location of fictitious wave sources surface. At the same time, the radiation condition of wave in semi-infinite layered media can be accurately realized by using the dynamic Green's function of concentrated load, and the computational memory can be greatly reduced. The scattering of seismic waves by a canyon topography in saturated layered half-space is examined in detail. The numerical results indicate that with the increase of the porosity of the overlying soil, the amplitude of surface displacement can be amplified by 1∼6 times, and the amplification effect is more significant near the corner of the canyon, which can be attributed to the superposition of wave diffraction effect and resonance amplification effect of saturated soil layer.

Dynamic response of three-dimensional rigid and flexible foundations on layered soils with local inhomogeneities

The dynamic response of rigid and flexible three-dimensional (3D) foundations of arbitrary shape, placed on a homogeneous or layered halfspace, which may contain a finite, longitudinally invariant structure or inhomogeneity, is determined numerically. A 2.5D coupled Integral Transform Method (ITM) - Finite Element Method (FEM) approach is used to compute the dynamic compliance at the surface of the stratified soil with inclusion, whereas 3D ITM fundamental solutions are applied in case of a homogeneous or layered subgrade. The foundation is modelled by 3D finite elements in both cases and coupled to the underlying ground by enforcing the compatibility conditions at the interface. Compliance functions at the soil foundation interface are presented and compared with existing solutions for verification as well as displacement distributions within the underlying ground are illustrated. The influence of the foundation stiffness on the total response as well as on the frequency dependent power transmission into the soil due to different load types is studied. Finally, the method is applied to evaluate the influence of a tunnel and a stiff cylindrical inclusion on the dynamic response of the foundation, thereby demonstrating the importance of taking into account the Structure-Soil-Structure-Interaction (SSSI).

A 2.5D coupled FEM–SBM methodology for soil–structure dynamic interaction problems

In this paper, a novel numerical methodology to deal with longitudinally invariant soil–structure interaction problems is proposed. The methodology uses the finite element method to model the structure and the singular boundary method to model the wave propagation in the soil, both formulated in the wavenumber–frequency domain. The versatility presented by the finite element method combined with the simplicity and computational efficiency of the singular boundary method results in a friendly, robust and accurate novel methodology to address the soil–structure interaction problems. The accuracy of the proposed method is assessed by comparing it against 2.5D FEM–MFS and 2.5D FEM–BEM approaches for two cases: a thin cylindrical shell and a star-like beam structure, both embedded in a full-space medium. Moreover, the computational efficiency of the proposed method is evaluated against 2.5D FEM–MFS and 2.5D FEM–BEM methods. In the final step, the applicability of the proposed method is studied through an example of a railway tunnel embedded in a layered half-space. The results presented in this work exhibit the advantages of the novel approach in modelling simplicity, numerical efficiency and robustness with respect to previous methodologies.

A scaled boundary finite element method procedure for arch dam-water-foundation rock interaction in complex layered half-space

In the seismic analysis of arch dams, the heterogeneity of the foundation rock is always not included with the use of assumption of a homogeneous half-space. This study is devoted to the analysis of dam-water-foundation rock interaction considering a complex layered half-space. For this purpose, a novel scaled boundary finite element method (SBFEM) for the analysis of 3D dam-foundation interaction is developed. The novel SBFEM can satisfy the radiation damping rigorously of a complex layered half-space, which overcomes the limit of scaling requirements and broadens the application of the original SBFEM. In addition, two efficient models in the framework of SBFEM have been established in the early research, respectively, to solve hydrodynamic analysis of dam-water interaction and free-field motion of an infinite canyon. Thus, based on the substructure method, a SBFEM procedure and the corresponding program, named SBFEM-LEACD, is developed to solve the earthquake analysis of arch dams considering a complex layered half-space. The response of an arch dam at the design stage in southwest China, where the foundation rock is idealized as three kinds of model: homogeneous half-space, horizontal layered half-space and inclined half-space, is analyzed to demonstrate the effect of the complex layered half-space.

A novel scaled boundary finite element method for dynamic impedance of an arch dam foundation in a complex layered half-space

This study is devoted to the evaluation of dynamic impedance of the arch dam foundation embedded in a complex layered half-space. For this purpose, a modified scaled boundary finite element method (SBFEM) is developed for modeling the far field. In order to overcome the limits of scaling requirements of the original SBFEM, a scaled boundary transformation based on a scaling surface is developed, which can model the geometry of an infinitely long canyon exactly. Besides, the complex material portions of half-space can also be easily and exactly simulated using the modified scaled boundary transformation. Corresponding to the novel scaled boundary transformation, the governing equations of the modified SBFEM in terms of the displacement and dynamic stiffness are derived based on the frame work of Hamiltonian system. Comparison with existing solutions for a foundation supported on a semi-circular canyon in a homogeneous half-space confirms the accuracy and efficiency of the proposed approach. The effects of horizontal layers, shape of cross-section and the curved axis of canyon are investigated. The results show that the layers of half-space may have a significant effect on the dynamic stiffness of the canyon foundation.

Scattering of plane waves by a 3D canyon in a transversely isotropic fluid-saturated layered half-space

Scattering of plane waves by a three-dimensional (3D) irregularity in a transversely isotropic (TI) fluid-saturated layered half-space is investigated for the first time. The derived exact stiffness matrix is applied to solve the dynamic response of free fields. The scattered fields, which arise from the presence of canyon irregularity, are described by applying dynamic Green's functions for uniformly distributed loads and pore pressure acting on an inclined plane, that is, the special indirect boundary element method (IBEM). The accuracy of the proposed method is verified via the comparison between the calculated results and those of published literature. By taking a semi-spherical canyon cut in a homogeneous, single-layered, and three-layered TI saturated half-space as examples, the influences of material anisotropy, boundary drainage condition, and sedimentary sequence on dynamic response are investigated. The results show that the material anisotropy and boundary drainage conditions all have a significant impact on the seismic responses, which is highly dependent on the waves type, incident angle, and frequency. For the layered medium, the displacement amplification is affected by the soil layer properties and sedimentary sequence.

BEM analysis of protection of rock caverns from cracking by water infilling and rock softening

This paper studies the elastic behavior of near ground cavern infilled with water in a layered rock mass by applying axisymmetric boundary element method (BEM). The results of two case studies are presented to quantitatively show the effect of water infilling on the elastic stress and deformation of layered isotropic halfspaces with a spherical cavern. The cavern water pressure has a significant effect to the normal and shear stress distributions on the cavern surface. Moreover, the effect of water softening on the elastic modulus of the rock mass is considerable to the displacements on both the ground and cavern surfaces. However, this effect is insignificant for all induced stresses on the cavern surface. Hence, the full infilling of water in the rock cavern can protect the surrounding rock from cracking or fracturing because water softening can also increase the rock mass toughness.

Practical technique for analysing free-field response of horizontally layered viscoelastic half-space excited by vertically travelling seismic waves

A practical technique for directly calculating the response of a viscoelastic layered half-space excited by vertically travelling seismic waves in the time domain was developed. Based on the relationship between the superposition of the upward and downward waves incident on the soil interface and the superposition of the upward and downward waves emitted from the interface, the governing equations were derived to prevent the cumbersome recording and synthesis of the upward and downward waves. In addition, a technique to decompose seismic waves was applied to precisely model the hysteretic damping of soil in the time domain. The proposed method is simple, practical, and easy to implement in programs, and it is an alternative approach for computing the response of the viscoelastic layered half-space. Several examples, including the viscoelastic layered half-space response generated by incident harmonic and seismic waves, are presented to demonstrate the preciseness and versatility of the proposed method.

Full waveform inversion to estimate the material properties of a layered half-space

Estimations of material properties of layered infinite media have been considered in a variety of problems, including geophysical subsurface imaging, geotechnical site investigations, and nondestructive tests. The full waveform inversion (FWI) approach can be employed because it is versatile when used to estimate the properties of complicated systems. A rigorous formulation of the FWI approach used in conjunction with the thin-layer method, which is accurate and efficient for wave-propagation in a layered medium, is developed to estimate the material parameters of a layered half-space. We measure the vertical displacements of a layered half-space subjected to a harmonic vertical disk load on its surface. The responses are compared with those calculated using the estimated material properties of the considered medium. The estimated responses are obtained using a thin-layer model of the system. The thin-layer model is composed of mid-point integrated finite elements, the thickness of which can be complex, and perfectly matched discrete layers, which are appropriate for the representation of a layered half-space. An objective function, which is defined as the L2-norm of the difference between the measured and estimated responses, is minimized by means of gradient-based optimization. The gradient of the objective function is derived in terms of the derivatives of the eigenvectors and eigenvalues of the thin-layer representation of the considered medium. We then estimate the material properties by solving the optimization problem. The proposed approach is applied to various examples to estimate the profiles of shear-wave velocities of layered half-spaces. We demonstrate that the developed FWI strategy can estimate the material properties of layered half-spaces with satisfactory accuracy.

Fundamental solution of general time-harmonic loading over a transversely isotropic, elastic and layered half-space: An efficient and accurate approach

Time-harmonic loading over layered elastic half-spaces has applications in various science and engineering fields. While various approaches have been proposed in solving the related boundary-value problems, in this paper, we propose a new approach, which is based on the novel Fourier-Bessel series system of vector functions and the dual variable and position method (DVP). While the DVP method was proposed recently and verified to be computationally stable and efficient, the Fourier-Bessel series system of vector functions is newly introduced. Similar to the cylindrical system of vector functions, the normal (dilatational) and shear (torsional) deformations (waves) can be separated and solved in terms of the LM- and N-types of the new vector function system. The new formulation is coded, and the corresponding algorithm/program is applied to a couple of cases. It is shown that, by comparing previous approaches, this new series system of vector functions is equally accurate, but much more computationally powerful. Since it is substantially time saving in calculation, it is hopeful that this new approach would have broad applications related to transient response and inverse problems in elastodynamics of layered systems.

Coupled FEM/SBFEM investigation on the characteristic analysis of seismic motions of a trapezoidal canyon in a layered half-space

Layered topographies and geologies are known to have considerable effects on the spatially variable seismic motions scattered by a canyon in a layered half-space. In this study, based on a substructure replacement technique, a coupled finite element method and scaled boundary finite element method based on scaling splicing lines (hereafter referred to as coupled FEM/SBFEM) is developed for a scattering field analysis, considering the effect of the layered topographical characteristics. The near field is modelled using the finite element method (FEM), and the far field is modelled using an improved scaled boundary finite element method, which can model the radiation damping of a layered half-space precisely. In the modified version of SBFEM, a novel dimensionless frequency is proposed to facilitate obtaining the time-domain response efficiently using a fast Fourier transformation. Several examples are investigated to demonstrate the versatility and validity of the proposed method. Parametric studies are performed to investigate the characteristics of the ground motions of a trapezoidal canyon in a layered half-space in the frequency and time domains (where the ground motions significantly differ from those of a uniform half-space), and to investigate the effects on ground motions caused by the soil layer thickness, burial depth, and impedance ratio.

A special indirect boundary element method for seismic response of a 3D canyon in a saturated layered half-space subjected to obliquely incident plane waves

A special indirect boundary element method (IBEM) is proposed to deal with the wave scattering of plane P1-, SV- and SH-waves by a 3D canyon cut in a saturated layered half-space. First, the dynamic stiffness matrix method is used to determine the free field dynamic responses. Then, based on Biot's theory, the 3D dynamic Green's functions for uniformly distributed loads and pore pressure acting on an inclined plane in a saturated layered half-space are derived, which is subsequently utilized to construct the scattered fields. Finally, the densities of uniformly distributed loads are determined by introducing boundary conditions, and the total dynamic responses are obtained by superimposing the free and scattered fields. The accuracy of the proposed method is verified via the comparison between the calculated results and those of published literature. The influences of boundary drainage conditions, medium porosity, and soil layering on seismic response are investigated in the frequency domain through parametric analysis. The numerical results show that the influences of boundary drained conditions on dynamic response are apparent, especially for high incident frequency. The effect of porosity on dynamic response seems to more sensitive to undrained conditions. Furthermore, the soil layering will bring an impact on the scattering mechanism of the layered system, appearing difference in the displacement amplitude and its distribution.

Estimation of Shear-Wave Velocities of Layered Half-Space Using Full Waveform Inversion with Genetic Algorithm

Estimation of Shear-Wave Velocities of Layered Half-Space Using Full Waveform Inversion with Genetic Algorithm

What Is Fling Step? Its Theory, Simulation Method, and Applications to Strong Ground Motion near Surface Fault Ruptures

ABSTRACT We present the theory of the fling step and a theoretical method for simulating accurately the near-fault strong motions, and apply it to reproduce various strong-motion records near surface faults. Theoretically, the fling step is the contribution of the static Green’s function in the representation theorem (Hisada and Bielak, 2003), and we show that this theory holds for any seismic velocity structure. We first demonstrate the validity of this theory using theoretical solutions of a circular fault model in a homogeneous full-space. Next, we apply the theory to layered half-spaces, present a theoretical method based on the wavenumber integration method, and introduce various techniques to simulate the near-fault ground motions including fling steps with high accuracy. Finally, we demonstrate the effectiveness of the method by reproducing various strong-motion records near surface fault ruptures and discuss the characteristics of near-fault strong motions including the fling step and the forward directivity pulse. We made all of the software and data used in this article available on the internet.