Plane SV Waves Research Articles

Plane SV-waves scattering by seawater convex surface topography and underlying lined tunnel in a saturated half-space

Analytical solutions of boundary-valued problems for aboveground topography and underground buried structures, especially for sites with overlying seawater, have always been challenging. Based on the region decomposition technique (RDT) and Hankel integral transform method (HITM), SV-waves scattering by the convex circular topography with seawater and underlying tunnel in a saturated half-space is solved. The primary problem is tackling additional scattered waves introduced by the water–soil interface and convex topography, while ensuring boundary conditions are all met. To this end, scattering wave potential functions in cylindrical coordinates are first transformed into the Hankel integral form in Cartesian coordinates. This greatly facilitates the straightforward application of free boundary conditions at the soil–seawater flat interface, rather than making a large circular arc approximate assumption, which, in turn, avoids the resultant accumulating errors. Secondly, scattering problem to be solved is mathematically and analytically formulated as solving a 5 × 5 linear equation system comprised of wave functions inside the convex topography and lining tunnel while satisfying all physical continuous boundary conditions. This is the main innovation and resulting consequence from the theoretical derivation process. Finally, validation of numerical examples between this paper, existing studies and theoretical analysis further demonstrates the reliability and general adaptability of solutions. Parametric studies of the incident SV waves are conducted to investigate the spatially varying seismic response. The effects of SV-waves incident angles and frequencies, tunnel burial depth, tunnel lining thickness, saturated soil porosity, and seawater depth are investigated and analyzed in detail. This study provides a feasible scheme of investigating the dynamic response of complex submarine sites, which is of great significance for both theoretical value and engineering purposes.

Formulation and Performance of Multi-Transmitting Formula with Spectral Element Method in 2D Ground Motion Simulations Under Plane-Wave Incidence: SV Wave Problem

ABSTRACT To simulate the ground motions of 2D arbitrary sites subjected to plane SV waves, we put forward a numerical method combining the multi-transmitting formula (MTF) and spectral element method (SEM). The investigation results demonstrate a high simulation accuracy of the presented method. Most importantly, the MTF exhibits a notably superior stability performance in the SEM than in traditional finite-element method (FEM). Without any additional treatment, high-frequency instability does not occur when controlling the Courant number value C and the ratio between the MTF parameter S and C; the upper limit value of this ratio is significantly smaller than that of the SH-wave problem. In addition, a preliminary stability criterion for the MTF in SEM is given based on the homogeneous half-space model. For low-frequency stability, the occurrence time of instability is considerably lagged when no interventions are adopted. Moreover, the low-frequency instability can be readily eliminated by using smaller modification parameter in the SEM compared with the FEM.

Dynamic response of shallow-buried circular composite lining tunnel with imperfect interface under SV waves

In this paper, the dynamic response of lined shallow-buried composite lining tunnel with imperfect interface in semi-infinite elastic medium under plane SV waves is studied, and the spring model is introduced to describe the imperfect interface relationship among the surrounding rock, the isolation layer, the primary support layer and main support layer. By using the wave function expansion method, the analytical solutions of displacements and stresses are presented. The interacting effects of the interface properties and the buried depth under different dimensionless frequencies are discussed. The results show that the peak values of the dynamic stress in the linings gradually shift to the upper part of the tunnel with the increase of dimensionless frequency, and the maximum dynamic stress in the linings also tends to decrease. The dynamic stress around the main and primary support layers increases when the interfaces are weakly bonded, and the effect of transverse springs is greater than that of normal springs. The tunnel with diminutive buried depth has a larger dynamic stress under the low-frequency SV waves. Reducing the stiffness in the isolation layer will lead to the increase of dynamic stress in the surrounding rock, while the dynamic stress in the main and primary support will be weakened.

Analytical model for a circular lined tunnel embedded in the layered soil under seismic waves

AbstractIn this study, based on the wave function expansion method and exact stiffness method (ESM), an analytical model for a circular lined tunnel embedded in the layered soil under seismic waves is established. For convenience, the wave field in the layered‐soil‐tunnel (LST) system is divided into three parts, that is, the free wave field and scattering wave field in the layered soil, as well as the transmitted wave field in the lining. The free wave field in the layered soil is determined by the ESM for the layered soil. The scattering wave field in the layered soil is further decomposed into three parts, namely, the principal and complementary wave fields in the soil layer with the tunnel (the tunnel layer) as well as the modified wave field occurring in the other soil layers. These scattering wave fields are represented by the corresponding principal, complementary, and modified wave functions, respectively. The principal wave functions associated with the direct cylindrical waves emanated from the tunnel are represented by the Hankel functions. The complementary and modified wave functions correspond to the additional responses of the tunnel layer and the other soil layers due to the direct cylindrical waves from the tunnel, and can be obtained by expanding direct cylindrical waves into plane waves together with the application of the ESM. The transmitted wave field in the lining is simply represented by two kinds of Bessel functions. With the aforementioned wave field representations, the boundary value problem for the LST system under seismic waves is formulated. Based on the proposed analytical model, some numerical results and corresponding analyses are presented. Presented numerical results indicate that the dynamic stress concentration factor (DSCF) associated with the stiffer tunnel layer is usually larger than that associated with the softer tunnel layer for the incidence of plane P waves with not very low frequencies, while the circumstance for the incidence of plane SV waves shows an opposite tendency.

SV-wave scattering by an underwater circular cylindrical canyon in a saturated poroelastic half-space

To study the effects of underwater terrain on plane seismic waves (P-SV wave) propagation, the theoretical solution of ground displacement, strain, rocking, and energy for the underwater terrain is derived based on Biot’s theory. Numerical results in terms of incident angle and Poisson’s ratio are illustrated for various porosities and degrees of solid frame stiffness. The results show that the response of a saturated-fluid porous soil under overlying water is controlled by the solid frame stiffness and Poisson’s ratio. Considering the effect of complex site amplification, the scattering theoretical solution of plane SV wave by a circular-arc canyon with overlying water is first derived based on the free-wave field above obtained. The effects of the incident frequencies, the incident angles, and the circular arc canyon on the ground motion of the soil-water interface are investigated parametrically. In particular, the effect of overlying water on ground motion is studied, and the results indicate that the existence of an overlying water layer absorbs the energy of a seismic wave to a certain extent. Meanwhile, the accuracy of the results is also verified by reducing the complex site to that without an overlying water layer and no canyon. Finally, based on the theoretical solution derived above, the coherence functions and the underground variable seismic motion of the site also are simulated, respectively. Compared with ground motions, the numerical results show that underground motions are smaller, which indicates that the canyon has a certain amplification effect on the seismic motions.

Scattering of Plane Waves by Cylindrical Cavity in Unsaturated Poroelastic Medium

The scattering of elastic waves by underground cavities is an active research topic for its broad applications in various fields, such as earthquake engineering, the blast resistance of underground structures, geophysical exploration, etc. In most previous studies, the sounding medium was treated as an ideal elastic medium or a saturated poroelastic medium. The understanding of the scattering of elastic waves by cavities in unsaturated porous media is limited. In this study, the scattering of plane P1 waves and SV waves by a cavity with a permeable surface in an infinite unsaturated porous medium is solved by the wave function expansion method. The dynamic stress concentration at the cavity surface is investigated by taking P1 wave incidence, for example. Numerical results illustrate that the scattering of plane waves around the cavity strongly depends on the frequency of the incident waves, and the saturation, Poisson’s ratio, and porosity of the surrounding medium.

2D seismic response of shallow sandy basins subjected to obliquely incident waves

Under the effects of the near-field earthquakes, the change in the incident angle of the incoming wave changes the symmetric response of basins to asymmetric. This paper investigates the 2D response of shallow trapezoidal sandy basins to incident plane SV waves. To attain this goal, a trapezoidal basin with 100 m depth, as the representative of shallow sedimentary basins, with an inclined bedrock angle of 10° was selected. These values fall inside of the ranges of the shallow basin depth and bedrock angle of several real basins. To make the results useful in engineering affairs, three different sand types belonging to the different class of soils, with respect to the soil classification codes, were selected. The constructed models were subjected to the collection of 24 earthquakes with different PGA's of 0.1, 0.2, 0.3 and 0.4 g using a fully nonlinear method. The results showed that the change in the incident angle of the motions not only affects the response at two basin edges but also influences the response of the central part. Among different sandy basins, the behavior of the basin with medium dense sand is less affected by the change in the wave incident angle. For basin with loose sand, while greater maximum spectral amplifications happen at the backward direction of the basin, the forward direction is less affected by the change in the wave incident angle. A different pattern was seen for basin with dense sand. Greater maximum spectral amplifications were seen at the forward direction of the basin. Regarding the effect of motion incident angle on the amplification frequency of shallow basins, it was seen that in case of the obliquely incident waves the spectral amplification curves of the points close to the edges have more than one (generally two) maximum points. Besides, unlike the vertical incident wave case that the SAF curves tend to almost one curve at the distant points from edges, no convergence among the SAF curves of the surface points was seen for the obliquely incident wave condition.

Scattering of plane SV-waves by a circular lined tunnel in an undersea saturated half-space

Considering the fluid-soil coupling of saturated soil and seawater-saturated soil-structure dynamic interaction, the Hankel function integral transformation method is employed herein to obtain an analytical solution to the scattering and diffraction problem posed by the plane SV-waves in an undersea lined tunnel. This study is based on the porous medium wave function theory of saturated soil proposed by Biot and the ideal fluid medium wave theory. The impact of factors, such as the incident angle, frequency, water depth, buried depth of tunnel, porosity, and Poisson's ratio of the saturated soil, on the horizontal and vertical displacement at the seawater–saturated soil interface, and on the pore water pressure and total circumferential stress at the saturated soil–tunnel interface, are computed and analyzed.

Dynamic Analysis of a Deeply Buried Tunnel Influenced by a Newly-built Adjacent Cavity with a Special Emphasis on the Minimum Seismically Safe Tunnel Distance

Contemporary life streams, more often than ever, impose the necessity for construction of new underground structures in the vicinity of existing tunnels, with an aim to accommodate transportation systems and utility networks. A previously uninvestigated case, in which a newly-constructed tunnel opening is closely positioned behind an existing tunnel, referred to as the tunnel–cavity configuration, has been considered in this study. An exact analytical solution is derived considering a pair of parallel circular cylindrical structures of infinite length, with the horizontal alignment, embedded in a boundless homogeneous, isotropic, elastic medium and excited by time-harmonic plane SV-waves under the plane-strain conditions. The Helmholtz decomposition theorem, the wave functions expansion method, the translational addition theorem for bi-cylindrical coordinates, and the pertinent boundary conditions are jointly employed in order to develop a closed-form solution of the corresponding boundary value problem. The primary goal of the present study is to examine the increase in dynamic stresses at an existing tunnel structure due to the presence of a closely driven unlined cavity, as well as in a localized region around the tunnel (at the position of the cavity in close proximity), under incident SV-waves. A new quantity called dynamic stress alteration factor is introduced and the aspect of the minimum seismically safe distance between the two structures is particularly considered.

The method of fundamental solutions for the elastic wave scattering in a double-porosity dual-permeability medium

For materials that have a matrix porosity at the microscopic scale, and an interconnecting fracture system at the mesoscopic scale, the double-porosity dual-permeability model provides a better prediction of seismic wave attenuation than the single porosity model, due to the flow exchange mechanism between these different size pores. The current work offers for the first time a numerical solution tool based on the method of fundamental solutions for problems of 3-D wave scattering and dynamic stress concentration around scatterers, such as cavities and embedded inhomogeneous inclusions, in an infinite double-porosity dual-permeability medium. The method of fundamental solutions is based on the spherical wave potentials representing three compressional waves and one shear wave. The accuracy and stability of the method are tested against other available results, and are demonstrated to be excellent. Wave scattering and dynamic stress concentration problems due to spherical cavity and poroelastic inclusion are then investigated for incident plane P1 and SV waves in the frequency domain. The research shows that the seismic responses are sensitive to the incident frequency, the boundary drainage condition, and material properties such as porosity.

Seismic response of tunnel near fault fracture zone under incident SV waves

This study investigated the impact of a non-causative fault on the dynamic response of a nearby lined tunnel under the incidence of plane SV waves using the indirect boundary element method. The effects of several critical parameters, such as the incident frequency, the inclination degree of the fault, the distance between the fault and the tunnel on the hoop stress of the lined inner and outer walls, were explored intensively. The numerical results indicated that the non-causative fault could significantly change the hoop stress distribution of inner and outer surfaces of the tunnels. In general, for the vertically incident seismic waves, when the tunnel was located in the foot wall (under the fault), the hoop stress within the tunnel was significantly greater than that of the tunnels in the non-fault half space, with an amplification factor of up to 117%. The amplification effect became more pronounced as the fault dip angle increased. However, when the tunnel was located in the hanging wall (above the fault), the non-causative fault could produce a significant shielding effect on the dynamic response of the tunnel under high frequency wave incidence, with the reduction of hoop stress being up to 81%. For low-frequency waves, though, the fault could lead to an increase of the hoop stress of the tunnel of up to 152%. The research results will provide a reference for the seismic design and safety protection of underground structures in non-causative fault sites.

Dynamic Response of Composite Lining Tunnel with Buffer Layer: An Analytical and Experimental Investigation

Composite lining is often designed for the mountainous tunnels in high-intensity earthquake areas. The application of the buffer layer will bring more advantages, while the shock-absorbing mechanism is still unclear currently. In this paper, based on the Fourier-Bessel series expansion method, the dynamic stress concentration factor of composite lining tunnel with buffer layer subjected to plane SV waves in the half-space is obtained. Then, the influence of geometric and mechanical parameters of the buffer layer on composite lining was systematically analyzed. Finally, the correctness of the analytical solutions is verified by series shaking table tests and numerical simulations. Results suggest that the buffer layer can play the role of “redistributing” the seismic load, and it can effectively reduce the dynamic responses of secondary lining but amplify in primary support. There is an optimal interval of the stiffness and thickness for the buffer layer. When the stiffness ratio of the buffer layer to surrounding rock is 1/10 ∼ 1/50 or the ratio of buffer layer thickness to inner diameters of secondary lining is 1/40 ∼ 1/20, the shock-absorbing performance is remarkable. The general damage observations in tests show that the crown, arch springing, and invert of composite lining in case of no buffer layer are prone to cracking under a strong earthquake. The invert of the composite lining is more susceptible to be damaged after adopting the buffer layer. In general, the analytical results were consistent with experimental and numerical results. The above study results may provide theoretical support and experimental data for the seismic design of composite lining tunnels.

Scattering of seismic waves by three-dimensional large-scale hill topography simulated by a fast parallel IBEM

To solve seismic wave scattering by a large-scale three-dimensional (3-D) hill topography, a fast parallel indirect boundary element method (IBEM) is developed by proposing a new construction method for the wave field, modifying the generalized minimum residual (GMRES) algorithm and constructing an OpenMP plus MPI parallel model. The validations of accuracy and efficiency show that this method can solve 3-D seismic response of a large-scale hill topography for broadband waves, and overcome the weakness of large storage and low efficiency of the traditional IBEM. Based on this new algorithm architecture, taking the broadband scattering of plane SV waves by a large-scale Gaussian-shaped hill of thousands-meters height as an example, the influence of several important parameters is investigated, including the incident frequency, the incident angle and the height-width and length-width ratio of the hill. The numerical results illustrate that the amplification effect on the ground motion by a near-hemispherical hill is more significant than the narrow hill. For low-frequency waves, the scattering effect of the higher hill is more pronounced, and there is only a single peak near the top of the hill. However, for high-frequency waves, rapid spatial variation of displacement amplitude appears on the hill surface.

Reflection of plane waves on generalized thermoelastic medium under effect of temperature dependent properties and initial stress with three-phase-lag model

The present paper investigates the reflection coefficient of thermoelastic plane waves at the free surface of an elastic half-space influenced by initial stress and temperature-dependent properties in the presence of thermal loading bounded by the three-phase-lag (3PHL) thermoelastic model. It revealed three plane waves, i.e. (thermal wave (T-wave), P-wave, and SV-wave). It provided the numerical solution and presented the effect of initial stress and temperature-dependent properties of the reflection coefficient graphically. Moreover, the paper utilized the altered reflected elastic waves because of initial stress and temperature-dependence to identify the more accurate nature of waves reflection, especially seismic waves.

Scattering of elastic waves by a 3-D inclusion in a poroelastic half space

Using the indirect boundary element method (IBEM) and Biot's theory, surface displacement associated with the field scattered by a 3-D inclusion in a two-phase poroelastic half space under plane SV waves has been studied. According to single-layered potential theory, scattering waves are constructed by the introduction of uniform loads and fluid sources distributed over the surface of the half space and on the interface between the inclusion and the half space. The magnitudes of these loads and sources are obtained by establishing a boundary integral equation based on continuity boundary conditions. The accuracy of this method is verified by comparing the results of degradation with existing results of cavities in elastic media. The scattering of plane waves due to a 3-D inclusion is investigated from several aspects. Numerical results show that the dynamic coupling effect between the solid frame and pore water and the permeability condition of the poroelastic soil boundary result in a generally larger surface displacement amplitude for dry soil than poroelastic soil when the porosity is n = 0.3. By contrast, the surface displacement amplitude of poroelastic soil is larger than that of dry soil when the porosity is n = 0.36. As the depth of burial decreases and as the inclusion becomes wider and thicker, the scattering effect increases. The peak value of the displacement spectrum curve usually appears near the incident frequency η = 1.

Theoretical study of the dynamic response of a circular lined tunnel with an imperfect interface subjected to incident SV-waves

Theoretical analysis is used to predict the dynamic response of a circular lined tunnel with an imperfect interface to plane SV-waves. The linear spring model is introduced to depict the imperfect interface between the rock mass and the lining. The expressions for the stress and displacement of the rock mass and the lining are then derived based on the wave function expansion method and the linear spring model. The dynamic stress concentration factor (DSCF) of the rock mass and lining is systematically discussed in two cases by studying its basic parameters, including the dimensionless wavenumber, the interface spring constants, and the lining thickness. The results show that the imperfect interface has a prominent effect on dynamic stress concentration. The maximum values of DSCF generated by SV-waves in the rock mass and the lining are larger than those generated by the P-waves. Additionally, the thick lining structure and tight adhesion of the lining to the rock mass are conducive to tunnel structure stability. It is suggested that the support should be strengthened at the position of the dynamic stress concentration, and the lining and the rock mass should be combined as closely as possible during the construction process.

Seismic Interaction between a Lined Tunnel and a Hill under Plane SV Waves by IBEM

This paper investigates the dynamic interaction between a lined tunnel and a hill under plane SV waves using the indirect boundary element method (IBEM), with the displacement and stress characteristics of the system presented in frequency domain. The IBEM has several unique advantages such as reducing calculation dimension, automatically satisfying the infinite radiation condition, etc. The numerical results indicated that the dynamic response of the tunnel–hill system is strongly dependent on incident wave characteristics, geometrical and material properties of the lined tunnel, as well as the topography of the hill. For a dimension ratio between the hill and tunnel of less than 10.0, the lined tunnel has large amplification or deamplification effect on the dynamic response of the hill. Correspondingly, the hill also greatly amplifies the displacement and stress concentration of the tunnel especially in the lower-frequency range, due to the complicated interference effect among the reflected waves and diffracted waves induced by the tunnel and hill. Also demonstrated is that the displacement and stress amplitude spectrums highly depend on the incident frequency and the space location, and there exist multiple peaks and troughs in the spectrum curve with the peaks usually appearing in the low-frequency range. Thus, for the seismic safety assessment of a hill slope or hill tunnel in practice, the dynamic interaction within the tunnel–hill system should be taken into consideration.

IBIEM Analysis of Dynamic Response of a Shallowly Buried Lined Tunnel Based on Viscous‐Slip Interface Model

A viscous‐slip interface model is proposed to simulate the contact state between a tunnel lining structure and the surrounding rock. The boundary integral equation method is adopted to solve the scattering of the plane SV wave by a tunnel lining in an elastic half‐space. We place special emphasis on the dynamic stress concentration of the lining and the amplification effect on the surface displacement near the tunnel. Scattered waves in the lining and half‐space are constructed using the fictitious wave sources close to the lining surfaces based on Green’s functions of cylindrical expansion and the shear wave source. The magnitudes of the fictitious wave sources are determined by viscous‐slip boundary conditions, and then the total response is obtained by superposition of the free and scattered fields. The slip stiffness and viscosity coefficients at the lining‐surrounding rock interface have a significant influence on the dynamic stress distribution and the nearby surface displacement response in the tunnel lining. Their influence is controlled by the incident wave frequency and angle. The hoop stress increases gradually in the inner wall of the lining as sliding stiffness increases under a low‐frequency incident wave. In the high‐frequency resonance frequency band, where incident wave frequency is consistent with the natural frequency of the soil column above the tunnel, the dynamic stress concentration effect is more significant when it is smaller. The dynamic stress concentration factor inside the lining decreases gradually as the viscosity coefficient increases. The spatial distribution and the displacement amplitudes of surface displacement near the tunnel change as incident wave frequency and angle increase. The effective dynamic analysis of the underground structure under an actual strong dynamic load should consider the slip effect at the lining‐surrounding rock interface.

2D dynamic structure-soil-structure interaction for twin buildings in layered half-space II: Incident SV-waves

A two-dimensional model for interaction through the soil of two identical vibrating structures, supported by rigid foundations embedded in a layered half-space, is presented for in-plane motions, and incident plane SV-wave. The formulation is based on the Indirect Boundary Element Method (IBEM). A similar model for incident SH-waves was presented in a companion paper. The formulation and results show higher level of coupling than for anti-plane motions. The vertical motions are coupled with the rocking and horizontal motions even for small motions, which is not the case for a stand-alone structure. At close distances, building relative response may be reduced by as much as 30% compared to the stand-alone structure, while otherwise it may be amplified, by as much as 50% for vertically incident SV-wave. For obliquely incident SV-wave, the building relative response of the illuminated structure may increase by 25%, while that of the structure behind would be reduced. Such phenomenon is consistent with earthquake observations in clusters of full-scale identical buildings. The effect of the other structure on the system frequency is small, within 3% for the cases considered.

In-plane soil-structure interaction excited by incident plane SV waves

In-plane soil-structure-interaction of a shear wall on a rigid circular foundation embedded in a linear elastic half-space, excited by an incident plane SV wave, is solved by relaxing the zero-stress boundary conditions on the ground surface. An approximate solution of this model was previously presented for the cylindrical approximation of the ground surface. This paper formally neglects such boundary conditions, and as a result, the complexity of the formulation is significantly reduced. Solutions for foundation and building motions in frequency space are compared to the previous results and it is shown that the present approximate solution provides excellent and almost identical results. The objective of this paper is to provide further insight on the role and the effects of surface boundary conditions for this type of problem.