Multilayered Half-space Research Articles

The Possibility of Detrimental Effects on Soil–Structure Interaction in Seismic Design Due to a Shift in System Frequency

Soil–structure interaction (SSI) leads to a modification in the dynamic properties of structure, but due to the complexity of analysis, it is traditionally assumed in seismic designs that the structure is fixed-supported on the ground, which brings about potential risks to the seismic performances of structure. The study works on the possibility of SSI having detrimental effects by comparing the dynamic responses of the SSI system to a fixed-base structure, and presents charts for an evaluation of the system frequency of SSI for the purpose of engineering practice. In order to reveal the physical nature, the SSI model is reduced to its simplest form, consisting of a SDOF oscillator, a three-dimensional rectangular foundation, and a multi-layered half-space. The energy dissipation in the soil is achieved by foundation impedances and the substructure method. Previously, the foundation impedances are usually acquired by two-dimensional or axisymmetric three-dimensional models in uniform half-space to avoid the high cost of the more realistic, fully 3D models, while a high-precision indirect boundary element method is employed, combined with the non-singular Green’s functions of distributed loads to calculate the foundation impedances. Although SSI dampens the peak amplitude of structure response in the frequency domain, case studies on four buildings’ responses to 42 earthquakes in the time history show a possibility of 15–20% that SSI amplifies the dynamic responses of structures, such as the maximum and the mean values in the time history, depending on the properties of the structures and the site, as well as the frequency component of incident waves.

Semi-analytical modelling of realistic tire-pavement contact: an analysis of pavement surface cracking damage under critical driving conditions

This paper aims to evaluate the surface cracking phenomenon on a flexible pavement subjected to diverse vehicle running conditions. Top-down cracking initiation is addressed through a continuum damage framework coupled with a semi-analytical model of tire-pavement rolling contact. The acceleration, braking and cornering scenarios were selected to represent the effect of combining normal and tangential loads. The differences between these loading cases are accurately computed by simulating the realistic contact stresses on the pavement surface through semi-analytical modelling. The computational cost of this modelling technique is reduced, allowing it to be used in industrial applications. It is assumed that tires and pavement are linear elastic multilayered half-spaces. The critical locations of the cracks are identified for each loading scenario and the cracking directions are discussed. This study emphasises the influence of driving conditions on premature pavement cracking which is not typically considered in the current pavement design guides. It is shown that a standard axle load may cause about 9 times more surface damage on a circular intersection compared to a straight-line pavement section.

A time-domain SGFD-FK hybrid method for 2D teleseismic elastic wave modeling and inversion

AbstractFull waveform inversion (FWI) has proved to be a reliable tool for high-resolution imaging of lithospheric structures at various depths down to the upper mantle. However, when the size of the model is large, the computational burden is significative and applications are restricted to low frequencies. To tackle this issue, we developed a new 2D time-domain hybrid method to simulate high-frequency teleseismic body waves propagating through a local heterogeneous elastic Earth model: The frequency-wavenumber (FK) integration method is coupled with the staggered grid finite difference method (SGFD). The FK method is used to compute the wavefield due to obliquely incident plane P and SV waves in a 1D multilayered half space that excites a local heterogeneous region. Inside this region, the velocity-stress staggered grid FD method (SGFD) is used to accurately deal with wave propagation in heterogeneous media. Spurious waves that might be generated at the boundaries of the local region are avoided using convolutional perfectly matched layers (CPML). This new hybrid method inherits the low-memory requirements of the FK method and the accuracy, efficiency and easy implementation of the SGFD. The new hybrid method is benchmarked against the analytical FK method for some canonical models and shows good agreement with analytical solutions. Subsequently, our modeling tool is incorporated into a full waveform inversion algorithm adapted for teleseismic configurations to invert the incident P wave and its coda. The inversion is carried out using a gradient approach that is efficiently implemented via the adjoint-state method. The results suggest that our hybrid method and FWI algorithm represent a valuable tool for 2D forward and inverse regional applications using teleseismic data sets.

Indentation over a transversely isotropic, poroelastic, and layered half-space

Poroelastic materials are common in nature and have applications in many engineering fields. In this paper, we derive a general solution of indentation over a multilayered half-space consisting of transversely isotropic and poroelastic materials. The rigid disc-shaped indenter is subjected to a vertical force of Heaviside time-variation. The solution is expressed in terms of the recently introduced powerful Fourier-Bessel series (FBS) system of vector functions combined with the unconditionally stable dual-variable and position method for dealing with layering. Since the problem is a mixed boundary-value one, the Green's functions due to a vertical ring-load are first derived which are then utilized in the integral least-square formulation to derive the solution. In terms of the FBS method, the expansion coefficients, which are further called Love numbers, are discrete, and therefore can be pre-calculated and used repeatedly for different field points on the surface. As such, the solution based on the new FBS method is more efficient and accurate than previous integral-transform methods. This new FBS method is particularly attractive when dealing with mixed boundary-value problems where time-variation is further involved. Numerical examples are conducted to validate the accuracy of the proposed solution and to demonstrate the effects of material layering, geometry, and hydraulic boundary conditions on the contact performance of the material system.

Dynamic Response of Layered Media to Impact Loading (on the Example of Highways)

Introduction. The main way to assess the load-bearing capacity of road structures (which are layered media consisting of materials of different stiffness) is through impact testing. The analysis of the results of these tests is reduced to the consideration of only peak values of vertical displacements without taking into account the change of these parameters in the frequency and time range. At the same time, it is the study of amplitude-frequency characteristics of displacements that can allow solving a number of urgent problems related, for example, to bringing the actual values of vertical displacements to the values under the design modes of operation of the structure.The purpose of the study is to analyze the dynamic response of layered media (road structures) of automobile roads to impact loading, aimed at bringing the actual parameters of the dynamic response to the design values regulated by the requirements of domestic regulatory documents on road design.Materials and Methods. Modern shock loading facilities allow to reproduce the load with a pulse time of 0.03 s, while the design values of elastic moduli are calculated for 0.1 s. To overcome this contradiction, the authors realized an exact solution of the dynamic Lame equation for a multilayered half-space and constructed a series of amplitude-frequency characteristics of displacements, from the processing of which a simplified functional dependence was derived.Results. The methodology of construction of amplitude-frequency characteristics of displacements on the surface of road pavement under impact impact has been developed, which finds its practical application. On the basis of the exact solution of the dynamic Lame equation by means of statistical processing, a simplified regression dependence for vertical displacements is established. An example of calculation of the elastic moduli of the road structure layers taking into account their reduction depending on the time of load application is presented.Discussion and Conclusion. Within the framework of the study, for the first time, a methodology for bringing the actual values of vertical displacements to the values under the design modes of operation of a road structure, based entirely on the computational analysis of its dynamic VAT, in particular, the amplitude-frequency characteristic of deformation under unsteady impact, has been obtained. The developed approach is able to significantly increase the reliability of diagnostics of highway condition, as well as to ensure the unity of measurements and results obtained during the design and diagnostics of roads.

Multilayered elastic medium reinforced with interfacial thin film: A theoretical model for geogrid reinforced HIR asphalt pavement

Aiming at modeling the reinforcement behavior of geogrid reinforced HIR asphalt pavement structure accurately and efficiently, a mathematical formulation using the Generalized Kelvin Solution (GKS) based method for multi-layer three-dimensional elastic system with the consideration of elastic membrane without thickness is put forward in this thesis to simulate the geogrid reinforcement for HIR pavement. Solutions in transform and physical domain, solutions for z = 0, and specific solutions of surface displacements are determined. Four special cases including homogeneous half space with buried and surface thin film reinforcement, Boussinesq's solution, unreinforced multilayered half space are introduced to validate the solution. Moreover, the mechanical effect of geogrid on the deformation, stress state and fatigue life of the pavement structure is investigated using the proposed model and solution. The geogrid stiffness has significant effect on the vertical and radial strain. After the reinforcement, both the radial stress and hoop stress near the interface reduce to a very low level, the stress concentration is effectively alleviated. Meanwhile, geogrid reinforcement plays a positive role on enhancing the fatigue damage resistance of pavement stricture Our proposed method provides a novel way to model reinforcement mechanism of the geogrid on pavement structure, which is helpful to engineering practice.

Ground vibrations induced by the passage of double-line trains in underground tunnels

Subway systems consist of double lines to transport passengers in the outbound and inbound directions. The existing prediction models for subway-induced vibrations ignored the superimposed effect caused by the successive passage of double-line trains. This study develops a computationally efficient method to simulate ground vibrations during the simultaneous passage of double-line trains in underground tunnels. An analytical model is applied to calculate the transfer function of a track-tunnel-ground system in a multi-layered half-space. Two alternative designs of tunnels in a double-line subway system are considered, which are a double-track tunnel and twin single-track tunnels. By introducing the relation between two different moving frames of reference, a fully coupled solution for the train-track-tunnel-ground system that considers the dynamic interaction between the double-line trains is proposed. The numerical results indicate that the simultaneous passage of double-line trains induces a significant superimposed effect of ground vibrations. The superimposed effect can increase the vertical velocity in the one-third octave spectrum by up to 5 dB, and the running RMS value by up to 40% under the same track roughness. The ground vibrations from one double-track tunnel are more than 30% higher than those from two single-track tunnels due to the passage of the single-line train and double-line trains.

Simulation of thermoelastic wave propagation in 3-D multilayered half-space media

SUMMARY Thermal phenomena are common in both the generation and propagation of seismic waves. To investigate the propagation characteristics of thermoelastic waves generated by point sources in a 3-D multilayered half-space, herein, we develop a method that can solve problems involving coupled mechanical and thermal motions. The method first transforms the governing equations into two sets of first-order ordinary differential equations with a unified form in the frequency–wavenumber domain and then solves them by a generalized reflection and transmission coefficient method. To verify the correctness of our method and the corresponding computer program, we compare our solutions with analytical solutions. We further apply the method to different multilayered models, and the numerical results clearly demonstrate the reflection and transmission phenomena of thermoelastic waves within the interlayer. Furthermore, the generated thermal and seismic signals are synchronous, and the two P waves predicted by generalized thermoelastic theory can be converted into each other. The records on the ground show that thermal motions influence our observation of direct waves, especially in the presence of high thermal conductivity subsurface media, it can significantly impact the propagation of both body and surface waves, making surface seismic observations more complex than those predicted by elastic theory.

Dynamic response of the multilayered half-space medium due to the spatially periodic harmonic moving load

This study theoretically investigated the dynamic responses of a multilayered half-space subjected to a spatially periodic harmonic moving load using the direct stiffness method. First, a periodic model of the multilayered half-space was established. Based on the generalised modal functions and Fourier transform, the direct stiffness matrices relating displacements and tractions along layer interfaces for both the interior layers and semi-infinite region were derived. A global direct stiffness matrix was established by assembling the direct stiffness matrix for each layer. Then, the accuracy and feasibility of the proposed direct stiffness method were verified by comparing the results in the homogeneous and multilayered half-spaces with the literature. Finally, parametric analyses were conducted to investigate the effects of wavenumber in one periodicity length, ground layering, load speed, and load frequency on the dynamic responses. It is concluded that the characteristics of the displacement responses in the horizontal direction differ from those in the vertical direction in both time and frequency domains. The increase in wavenumber, load speed, and load frequency decreased the response amplitudes, increased the characteristic frequencies, and broadened the critical frequency bandwidths. The increase in the layer number broadens the critical frequency bandwidths, but it has no impact on the characteristic frequencies. The moving and Doppler effects cause variations in the characteristic frequencies and critical frequency bandwidths. The results of the multilayered half-space under spatially periodic harmonic moving load can be utilised as fundamental solutions in the periodic boundary element formulations, which will be the focus of future work.

Rolling contact on a viscoelastic multi-layered half-space

This paper’s purpose is the contact of a rolling body on a viscoelastic multi-layered half-space. Firstly, the contact is formulated with the classical definition of the gap and the pressure states in the contact zone and outside. L layers are considered on a substrate, L being any positive integer. Secondly, the influence coefficients related to an elastic multi-layered half-space are found, using the Papkovich–Neuber potentials. Tractions and displacements continuity is assumed at the interfaces, and then a construction of matrix systems corresponding to a unit pressure and a unit shear imposed at the top surface, leads to the general problem to solve. Further, the solution for any kind of pressure and/or shear distributions at the contact surface is inferred by convolution with the influence coefficients found earlier, using the Fast Fourier Transform (FFT) algorithms. Throughout the steps, an Elastic/Viscoelastic correspondence is used in order to take into account not only the change of behaviour of the half-space during time, but also to superpose the load history. Conjugate Gradient Method (CGM) algorithms are used to solve the variational problem that yields from the contact problem definition. In the present paper, a parametric study is performed to highlight the effects of the elastic modulus and the relaxation time on some relatively simple cases of rolling contact.

An efficient prediction model for vibrations induced by underground railway traffic and experimental validation

This study proposes an efficient prediction model for the vibration generated by underground railways. The proposed model is based on an analytical approach for a tunnel in a horizontally multi-layered half-space, showing the advantage of high computational efficiency and accuracy. Three alternative modelling strategies regarding hypotheses about the cross-section behavior of the slab and the connection condition between the slab and tunnel are introduced. A multi-body vehicle model is subsequently coupled with the track-tunnel-ground system in the moving frame of reference to determine the train-induced vibration. The predicted vibrations on the track, tunnel, and ground surface induced by the passage of trains are compared with the measured vibrations from an underground railway line in London from the literature. The predicted results for the double-beam track model with uniform support and the continuum track model exhibit a good agreement with the site measurements. The cross-section deformation of the track can be ignored and has little influence on the responses of the tunnel and ground. However, the assumption of the one-line connection between the slab and tunnel results in an overestimation of the track vibration, up to 20 dB, and an underestimation of the tunnel and ground vibration, up to 10 dB.

A reflection-transmission matrix method for time-history response analysis of a layered TI saturated site under obliquely incident seismic waves

This paper proposed a reflection-transmission matrix method to solve the time-history response of a layered transversely isotropic (TI) saturated site subjected to obliquely incident seismic waves. Based on Biot's theory of wave propagation in a TI poroelastic medium, the displacement, stress and pore pressure components are derived in the frequency domain using the boundary and continuity conditions of the layered half-space. By employing inverse Fourier transform and convolution theorem, time-domain solution for the site response under earthquake excitation is given. The methodology is implemented by the numerical platform FORTRAN, and is verified through comparison with degenerated solution for a layered isotropic single-phase case. Taking single-layered and multi-layered half-space as the examined models respectively, the time-history response of the stratified TI saturated site under obliquely incident seismic waves are presented and parameters studies are performed. Numerical results illustrate that the TI parameters, earthquake excitation, incident angle, and layering sequence have significant important influence on the seismic responses. The presented solutions can provide effective reference for studying the input ground motion and seismic soil-structure interaction concerning specially the layered TI saturated site.

An Energy Method for Determining the Residual Resource of Nonrigid Road Pavements at the Stage of Operation

The topical problem of assessing the residual resource of nonrigid road pavements of highways is studied. An energy method has been developed to solve this problem. The method is based on comparing the actual total dissipation of energy in the pavement structure determined from field measurements using an FWD shock loading unit and the total energy of dissipation calculated using a dynamic model of the stress-strain state of the road structure. Experimental studies have shown that the area of the dynamic hysteresis loop and, accordingly, the specific energy dissipation depend on the operational state of the pavement. Using a mathematical model of the dynamic stress-strain state of a multilayered half-space, the possibility of constructing dynamic hysteresis loops for the design traffic load is substantiated. It is shown that this method makes it possible to determine the design energy resource of the pavement, as well as its gamma-percentage residual resource and residual service life.

Dynamic response of an embedded flexible foundation of general shape in a transversely isotropic and multilayered half-space

A new semi-analytical method is developed for solving the dynamic response of an embedded flexible foundation of general shape in a transversely isotropic and multilayered half-space. It is based on the hybrid Green's functions (GFs) and finite-element method (FEM) via the modified subtraction method. The GFs in the multilayered half-space are derived by virtue of the dual variable and position (DVP) method whilst the FEM is utilized to discretize the domain of the embedded foundation. Making use of the modified subtraction method, the GFs and FEM formulations are combined to derive the dynamic impedance matrix of the embedded foundation. After verifying the accuracy and stability of the proposed algorithm, numerical studies are carried out to investigate the effect of the foundation modulus, half-space anisotropy, and layering on the dynamic impedance matrix of the embedded foundation. Results show that these factors could significantly affect the dynamic impedance matrix of the foundation, which could provide reference for practical engineering design.

An efficient three-dimensional method for the prediction of building vibrations from underground railway networks

This paper proposes a new efficient three-dimensional (3D) method to predict the vertical building vibration generated by trains running in tunnels in a multi-layered half-space. The tunnel-ground-building dynamic interaction involving in the proposed method is handled using the sub-structuring approach. The multi-story building is considered as rectangular floors supported by a distribution of columns. Each building floor is divided into several sub-plates. The dynamic stiffness matrix of a single sub-plate derived by the dynamic stiffness method is assembled to obtain the global one of each building floor. The free edge condition of the building floor is satisfied in this solution. The train-track-tunnel-ground dynamic interaction is modelled by a fully-coupled analytical method. The proposed method shows sufficient efficiency and accuracy, making it a suitable tool for predicting large-scale vibrations and performing parametric studies. The responses of a four-story building generated by the passage of trains in a tunnel are analyzed. The building vibrations are significantly affected by the dynamic properties of the building and the building-ground dynamic interaction. The dynamic interaction between the shallow foundations of buildings has a considerable influence on the building vibrations in the low-frequency range. The isolation effectiveness of the floating slab track and the base-isolation of buildings is highly dependent on the isolation frequency.

The near-surface velocity reversal and its detection via unsupervised machine learning

In land seismic data processing, picking the first arrivals and imaging the near-surface velocity structures are important tasks. However, in many areas, the near-surface weathering layer includes high-velocity reversals, causing the first arrivals to exhibit shingling effects, which are difficult for picking at the far offset. We have used an acoustic full-waveform modeling method in a multilayered half-space to simulate first arrivals with the velocity reversal. Numerical tests indicate that under certain conditions, shingling occurs if the seismic wave propagates through a thin velocity reversal layer embedded in the shallow structures. Detection of shingling is essential for the selection of valid near-surface imaging solutions, such as first-arrival refraction, or waveform solutions for the appropriate areas. We find that an automated detection scheme that uses unsupervised machine learning can help identify the velocity reversal. We test the method on synthetic and real data, and the testing shows that the automated detection result matches our visual judgment well. After the automated detection, appropriate inversion approaches can be applied to corresponding areas.

Wave Scattering of Plane P, SV, and SH Waves by a 3D Alluvial Basin in a Multilayered Half-Space

ABSTRACTA special indirect boundary element method (IBEM) is proposed to investigate the waves scattering of plane P, SV, and SH waves by a 3D alluvial basin embedded in a multilayered half-space. The new IBEM, which uses half-space Green’s functions for uniformly distributed loads acting on an inclined plane as its fundamental solutions, has the merits of (1) excellent capability of dealing with the stratification of the basin and the external half-space, (2) without the problem of singularity due to fictitious distributed loads being directly applied on the real boundaries, and (3) good adaptability to complex models with trapezoidal or triangular elements being used to discretize the boundaries. The validity and accuracy of the new method are verified by comparing its results with those in the literature. To illustrate the general applicability and efficiency of the new method further, 3D alluvial basins of varying shapes, depths, and sedimentary sequences embedded in a single layer overlying a homogeneous half-space are numerically studied. Numerical results show that the basin’s shape, depth, and sedimentary sequence all have significant impact on the ground seismic responses; the incident angle also has noticeable effects on the surface motion, and these effects are more prominent at the observation points along the incident direction of the plane waves; for the case of layered model, the displacement spectral amplification is affected by the eigenmodes of the vibrations of the layers, both inside and outside the basin.

Nonlinear contact force law for spherical indentation of FGM coated elastic substrate: An extension of Hertz's solution

This paper develops a nonlinear contact force law for the indentation of an elastic substrate with thin functionally graded material (FGM) coating by a rigid smooth sphere. The nonlinear contact force law is explicitly expressed in form of extended Hertz's solution with a correction factor, which is related to the coating thickness, radius of the indenter, contact interference and the modulus ratio. The explicit expression of the correction factor for arbitrary coating modulus gradation can be determined statistically with extensive numerical results. For computational efficiency and accuracy, an analytical contact model based on multilayered half space is developed and the associated mixed boundary value problem is converted to a Fredholm integral equation of the second kind. This model discretizes the thin FGM coating into n dissimilar and fully bonded sub-layers. Each sub-layer is a homogeneous elastic seam of finite thickness and constant elastic modulus and Poisson's ratio. Variation of the coating elastic properties along the thickness direction is accurately approximated by the multilayered system. The non-linear contact force laws in closed-form are specifically given for both linear and exponential modulus gradations. The load-displacement relations predicted by these force laws are shown to be in exact agreement with the numerical results from the Fredholm integral equation. Additionally, this nonlinear contact force law for FGM coating is further incorporated into the Greenwood and Williamson model, which provides a feasible and effective way for modeling the contact of FGM coated rough surfaces.

Three-Dimensional Analytical Model for Coupled Track-Tunnel-Soil System in a Multilayered Half-Space

AbstractAn analytical model is presented in this paper to predict the three-dimensional dynamic response of a circular tunnel in an elastic multilayered half-space. By applying the transfer matrix ...

The wave function method for calculation of vibrations from a twin tunnel in a multi-layered half-space

The ground vibrations due to underground moving trains cause disturbances to nearby buildings and their residents, which has been viewed as an important environmental issue. Various numerical or semi-analytical models are used to calculate the vibrations from underground railways, however, most of the prediction models adopt a single tunnel-soil system for modelling vibrations from underground trains and neglect the influence of a neighboring tunnel. In this paper, a semi-analytical wave function method is proposed for modelling vibrations from twin tunnels in a multi-layered viscoelastic and poroelastic half-space. The two tunnels are modelled as two hollow cylinders while the surrounding soil is modelled as a multi-layered half-space containing two circular cavities. The vibrations of the tunnel walls are taken as a superposition of outgoing and regular cylindrical waves. The wave field in the soil layer containing two cavities consists of down-going and up-going plane waves and cylindrical waves propagating outwards from each tunnel. In the wave function method, the transformation and translation of the plane and cylindrical waves make it possible to apply the boundary conditions at the scattering surfaces. Numerical results show that the half-space model predicts higher surface vibrations than a full-space model at equivalent surface locations. When the second tunnel is added near a deeper existing tunnel, a wider range of the free-field response above the two tunnels is significantly changed, however, the response below twin tunnels is less affected. If the separation distance between the two tunnels is smaller than five times the tunnel diameter, a twin tunnel model in a half-space is more suitable for an accurate vibration prediction at the ground surface. The cutoff effect of the rigid base is more visible for shallow bedrock at lower frequencies while the influence of the bedrock is quite limited at higher frequencies.