Two-dimensional scattering of in-plane body waves due to a discontinuity in bedrock

The direct boundary integral equation technique is used to study in-plane surface amplification of in-plane seismic body waves for the case of an inhomogeneity in a bedrock half-space. In the studied soil configuration, a soil layer rests on a rock half-space which includes a rock inclusion. The rock inclusion considered is a semi-infinite horizontal rock layer in which its upper boundary borders the soil layer. Materials in the soil–rock configuration are considered viscoelastic except for the section of the rock half-space below the level of the rock inclusion which is considered elastic. A parametric study is performed to determine controlling factors for surface displacement due to in-plane body waves. The study investigates varying the stiffness and the thickness of the rock inclusion for a range of frequencies and wave incidence angles. Anti-plane waves for this type of soil-rock configuration have been addressed in a previous article by Heymsfield (Earthquake Engng. Struct. Dyn. 28 : 841–855 (1999)). Copyright © 1999 John Wiley & Sons, Ltd.     

Two-dimensional scattering of SH waves in a soil layer underlain with a sloping bedrock

A two-dimensional analysis is applied to examine the effect that a sloping bedrock half-space has on the amplification of an anti-plane shear wave. The direct boundary integral equation method is used for the two-dimensional analysis. The particular soil–rock configuration investigated includes a homogeneous soil layer underlain by a sloping rock half-space. The rock half-space dips for a horizontal distance L and then becomes horizontal so that the overlying soil layer has a thickness H that remains constant from this point to infinity. The materials in the soil–rock configuration are considered viscoelastic except in the rock half-space below soil layer thickness H, which is considered elastic. This limitation in damping is due to the correction used for the truncation of the half-space boundary. Four cases are used to study the relationship between rock slope and surface displacement, vertical, 1:2, 1:4, 1:8. Surface displacements are determined for each of these cases for half-space incidence angles of 90, 75, and 60°. To allow for applicability to a wide range of problems, results are determined as a function of dimensionless parameters. In addition, solutions from a one-dimensional analysis are compared with the results of the two-dimensional analysis to establish limits outside of which a one-dimensional analysis suffices.     

层状半空间中洞室对平面SH波的放大作用

利用间接边界元法,求解了弹性层状半空间中洞室对入射平面SH波的放大作用问题,并以基岩上单一土层中洞室对入射平面SH波的放大作用为例进行了数值计算分析。本文模型的特点之一是考虑了层状场地的动力特性,因而更接近于实际工程;特点之二是计算精度非常高。研究表明,层状半空间中洞室对波的放大作用与均匀半空间中情况有着本质的差别;层状半空间中洞室附近地表动力响应由土层动力特性和洞室对波的散射二者共同决定。土层动力特性不仅影响洞室附近地表位移的幅值,还会影响地表位移的频谱。在土层的前几阶共振频率附近,随着基岩与土层剪切波速比的增大,土层的影响随之增大,而随着土层厚度的增加,土层的影响随之减小,并逐渐趋于均匀半空间情况。     

弹性层状半空间中凸起地形对入射平面SH波的放大作用

对Wolf理论进行拓展,使之可解决凸起地形对波的散射问题,进而利用间接边界元法,求解了弹性层状半空间中凸起地形对入射平面SH波的放大作用问题。本文模型的显著特点之一是考虑了层状半空间的动力特性以及层状半空间和凸起地形的阻尼;特点之二是计算精度高。文中以基岩上单一土层中半圆凸起地形对入射平面SH波的放大作用为例进行了数值计算分析。研究表明,基岩上单一土层中凸起地形对入射平面SH波放大作用和均匀半空间中凸起地形有着本质的差别;土层动力特性不仅影响凸起地形地表位移的幅值,还会影响地表位移的频谱;阻尼会显著降低凸起地形对高频波的放大作用。     

Rayleigh modes in anisotropic,heterogeneous poroelastic layers

The generalized Rayleigh type surface waves are studied in a multilayered medium consisting of anisotropic poroelastic solid layered stack beneath a fluid layer and overlying a heterogeneous elastic solid half-space. The heterogeneity, considered, is of vertical type. The interface between solid layer and half-space is treated as an imperfect interface and suitable boundary conditions are applied thereat. The technique of transfer matrix is used to obtain the dispersion equation in compact and convenient form. Numerical results are obtained for particular models. The effects of anisotropy and heterogeneity on the surface waves speed are discussed.     

Rotational cone models for a soil layer on flexible rock half-space

The unfolded cone model used for calculating the dynamic response of a disk on the surface of a soil layer resting on flexible rock for translational motion is extended to rotational motion. The method is analogous to that for a layer on rigid rock, the only modification being that the reflection coefficient – α replaces the coefficient of total reflection – 1. The modified value of – α, which, in general, is frequency-dependent, is determined by considering one-dimensional wave propagation along the cone for the first impingement at the layer–rock interface. The low- and high-frequency limits of – α for the rotational motion are the same as for translational motion. As these limits do not depend on frequency, the dynamic analysis using cones can be performed in the familiar time domain. The transfer function constructed by addressing the reflections–refractions at the soil–rock interface and the reflections at the free surface in the unfolded cone model is highly accurate, resulting in the same accuracy of the dynamic response of a disk on a layer resting on flexible rock as that on a homogeneous half-space modelled with a cone.     

Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space

A half-space containing horizontally multilayered regions of different transversely isotropic elastic materials as well as a homogeneous half-space as the lowest layer is considered such that the axes of material symmetries of different layers and the lowest half-space to be as depth-wise. A rigid circular disc rested on the free surface of the whole half-space is considered to be under a forced either vertical or horizontal vibration of constant amplitudes. Because of the involved integral transforms, the mixed boundary value problems due to mixed condition at the surface of the half-space are changed to some dual integral equations, which are reduced to Fredholm integral equations of second kind. With the help of contour integration, the governing Fredholm integral equations are numerically solved. Some numerical evaluations are given for different combinations of transversely isotropic layers to show the effect of degree of anisotropy of different layers on the response of the inhomogeneous half-space.     

层状半空间中周期分布凸起地形对平面SH波的散射

提出了一种新的以层状半空间中周期分布斜线荷载动力格林函数为基本解的间接边界元方法,研究了周期分布凸起地形对平面SH波的散射问题.方法将散射波场分解为凸起内部散射波场和凸起外部散射波场.凸起内部散射波场通过在凸起闭合边界上施加虚拟斜线荷载产生的动力响应来模拟,而凸起外部散射波场则通过在凸起与半空间交界面上施加虚拟周期分布斜线荷载产生的动力响应来模拟.周期分布斜线荷载动力格林函数的引入,使得本文方法仅需针对一个凸起进行边界单元的离散和求解,便可完成问题的求解,避免了通过截断无限边界求解而引入的误差,方法具有较高精度的同时显著降低了求解自由度.文中通过与已有结果的比较,验证了方法的正确性,并以均匀半空间和基岩上单一土层中周期分布凸起为例进行了数值计算分析.研究表明,凸起间距对凸起地形间的动力相互作用有着显著的影响,同时层状半空间中周期分布凸起地形对SH波的散射与均匀半空间情况也有着显著的差别.     

地表下圆形夹塞区出平面散射对地面运动的影响

采用波函数展开方法，提出弹性半空间埋藏的圆形弹性夹塞区对平面ＳＨ波二维散射的封闭级数解答．利用内域型Ｇｒａｆ加法公式，将解答归结为一组无穷代数方程组的求解．通过对无穷方程组的截断计算，得到了解答的数值结果；并通过边值条件的收敛，检验了截断计算的精度．最后给出若干典型算例说明夹塞区产生的散射场对地表运动的复杂影响．     

Saturation effects on horizontal and vertical motions in a layered soil–bedrock system due to inclined SV waves

A study of the effects of pore-water saturation on the horizontal and vertical components of ground motions in a multi-layered soil–bedrock system due to inclined SV waves is presented. Both the soil and the rock are modeled as a partially water-saturated porous medium which is characterized by its degree of saturation, porosity, permeability, and compressibility. An efficient formulation is developed for the computation of the two-dimensional ground motions, which are considered as functions of the angle of incidence, the degree of saturation, the frequency, and the geometry of the system. Numerical results for both the half-space model and the single-layered model indicate that the effect of saturation may be significant, and is dependent on the angle of the incidence. Even a slight decrease of full saturation of the overlying soil may cause appreciable difference in the amplitudes of ground motions in both the horizontal and vertical components and the amplitude ratios between the two components at the ground surface, implying that one may need to carefully take into account the saturation conditions in the interpretation of field observations.     

Scattering of SV waves by a canyon in a fluid-saturated, poroelastic layered half-space, modeled using the indirect boundary element method

The scattering of SV waves by a canyon in a fluid-saturated, poroelastic layered half-space is modeled using the indirect boundary element method in the frequency domain. The free-field responses are calculated to determine the displacements and stresses at the surface of the canyon, and fictitious distributed loads are then applied at the surface of the canyon in the free field to calculate the Green's functions for displacements and stresses. The amplitudes of the fictitious distributed loads are determined from the boundary conditions, and the displacements arising from the waves in the free field and from the fictitious distributed loads are summed to obtain the solution. The effects of fluid saturation, boundary conditions, porosity, and soil layers on the surface displacement amplitudes and phase shifts are discussed, and some useful conclusions are obtained. It is shown that the surface displacement amplitudes due to saturation and boundary conditions, different porosities, or the presence of a soil layer can be very dissimilar, and large phase shifts can be observed. The resulting wavelengths for an undrained saturated poroelastic medium are slightly longer than those for a drained saturated poroelastic medium; and are longer for a drained saturated poroelastic medium than those for a dry poroelastic medium. As porosity increases, the wavelengths become longer; and a layered half-space produces longer wavelengths than a homogeneous half-space.     

Dynamic response of an alluvial valley consists of three types of soil

Dynamic response of an alluvial valley consisting of three different types of soil was studied. In the two-dimensional model, the alluvial valley was assumed to have half-cylindrical shape. The alluvial valley contained three different types of soil with different shapes. The valley was surrounded by a half-space. All of the soil types and the half-space were assumed to be isotropic, homogeneous and linear elastic. The half-space was excited by simple harmonic SH-waves radiated from a strike-slip fault. The fault was assumed to have circular-arc shape in the mathematical model. The governing equations were solved by using wave function expansion method where boundary conditions were relatively simple. In the regions where boundary conditions were more complex, finite difference method was employed. Consequently, the wave propagation problem was solved semi-analytically. The obtained numerical results showed that surface displacement amplitudes are significantly affected by the material properties of the different soil types of the alluvial valley. It was also observed that the shapes of the soil types in the alluvial valley played a considerable role in surface displacements.     

Two-dimensional scattering and diffraction of P- and SV-waves around a semi-circular canyon in an elastic half-space: An analytic solution via a stress-free wave function

A well-defined boundary-valued problem of wave scattering and diffraction in elastic half-space should have closed-form analytic solutions. This two-dimensional (2-D) scattering around a semi-circular canyon in elastic half-space subjected to seismic plane and cylindrical waves has long been a challenging boundary-value problem. In all cases, the diffracted waves will consist of both longitudinal (P-) and shear (S-) rotational waves. Together at the half-space surface, these in-plane longitudinal P- and shear SV-waves are not orthogonal over the infinite half-space flat-plane boundary. Thus, to simultaneously satisfy both the zero normal and shear stresses at the flat-plane boundary, some approximation of the geometry and/or wave functions often has to be made, or in some cases, relaxed (disregarded). This paper re-examines this two-dimensional (2-D) boundary-value problem from an applied mathematics points of view and redefines the proper form of the orthogonal cylindrical-wave functions for both the longitudinal P- and shear SV-waves so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite-canyon surface are then comparatively less complicated to solve. This is now a closed-form analytic solution of the 2-D boundary-valued problem satisfying the half-space zero-stress boundary conditions exactly.     

地下夹塞断面形状随机性对平面SH波散射的影响

采用波函数展开法及边界离散的方法给出了任意断面形状的地下夹塞对平面SH波散射的半解析解; 利用蒙特卡罗方法随机模拟产生了30组夹塞断面样本, 通过对该样本在平面SH波入射下地表位移幅值的统计分析, 研究了夹塞断面形状随机性对平面SH波散射的影响. 结果表明, 夹塞断面形状随机性对平面SH波的散射具有重要影响. 以圆形夹塞为例, 当断面半径的变异系数为0.1时, 地表位移响应幅值变异系数可达0.71. 随着入射频率的升高, 变异系数逐渐增大; 随着夹塞刚度的降低, 变异系数逐渐增大; 随着夹塞埋深的增加, 变异系数逐渐减小.      

MODAL ANALYSIS OF SOFT-SOIL SITES INCLUDING RADIATION DAMPING

For the one-dimensional analysis of soft-soil layers on an elastic half-space, a general form of analytical solution is developed for converting radiation damping due to energy leaking back to the half-space into equivalent modal damping, allowing the modal analysis technique to be extended to a site where radiation damping has to be accounted for. Closed-form solutions for equivalent modal damping ratios and effective modal participation factors are developed for a single layer with a shear wave velocity distribution varying from constant to linearly increasing with depth. Compact and recursive forms of solutions for equivalent modal damping ratios are developed for a system with an arbitrary number of homogeneous layers on an elastic half-space. Comparisons with numerical solutions show that the modal solutions are accurate. The nominal frequency of a site, i.e. the inverse of four times the total shear wave travel time through the layers, is an important parameter for estimating the high mode frequencies. A parameter study shows that for the same impedance ratio of the bottom layer to the elastic half-space, a system of soil layers with an increasing soil rigidity with depth has, in general, larger peak modal amplifications at the ground surface than does a single homogeneous layer on an elastic half-space, while a system with a decreasing soil rigidity with depth has smaller modal peak amplifications. © 1997 by John Wiley & Sons, Ltd.     

Dynamic interaction between 3-D rigid surface foundations and comparison with the ATC-3 provisions

The dynamic behaviour of a system of three-dimensional, massless, rigid, surface foundations of arbitrary shape perfectly bonded to the elastic half-space is numerically studied with the frequency domain boundary element method. This method employs the dynamic Green's function for the surface of the half-space and this results in a discretization of only the soil-foundation interfaces. In addition, use of isoparametric quadratic quadrilateral boundary elements increases the accuracy of the method, which is confirmed by comparison with other known numerical solutions. Externally applied loads, harmonically varying with time, are considered. The through the soil coupling effect between the foundations as a function of distance and frequency is assessed through extensive parametric studies involving two and four rigid foundations being isolated or interconnected. It is found that the assertion of ATC-3 regulations that omission of coupling effects leads to conservative results is not always correct for all frequencies.     

弹性层状半空间中无限长洞室对斜入射 平面SH波的三维散射(Ⅱ) 数值结果与分析

以基岩上单一土层场地为例, 计算分析了在斜入射平面SH波作用下弹性层状半空间中无限长洞室附近的地表位移. 研究表明, 层状半空间中地下洞室对波的散射与均匀半空间情况存在显著差别. 层状场地由于考虑了场地自身的动力特性, 使得洞室附近地表位移幅值的空间变化更为复杂, 基岩与土层刚度比、 土层厚度对散射效应均有着重要影响. 随着基岩与土层刚度比的增大, 地表位移幅值整体上逐渐增大; 随着土层厚度的增大, 土层对地表位移幅值的影响逐渐减小. 在频域解答的基础上, 给出了层状半空间中洞室对斜入射SH波散射的时域解答, 并以Ricker波为例进行了数值计算.      

Response of a randomly inhomogeneous layered media to harmonic excitations

We present a unified formulation of an analytical method for evaluating the response of a random soil medium to surface or earthquake excitations. Specifically, we are interested in this study to the case of a horizontally stratified layered soil profile. Soil properties, mass density and shear modulus of each layer are modeled as spatial random fields. The soil profile is laying on a homogeneous half-space. Integro-differential equations are formulated and solved using the Laplace transform method. Numerical results are, firstly, obtained in terms of effects of soil properties on distributions of mean dimensionless displacement and stress with depth. Then, the amplification function of the surface layer of a soil profile is formulated and a parametric study is conducted to examine the effects of stochastic variations of mass density and shear modulus on the amplification function. We, lastly, computed the seismic response of a site located in the basin of Metidja in Algeria to base rock accelerations which were recorded at Keddara's station during the May 21, 2003 Boumerdes earthquake, in terms of free surface Fourier spectrum amplitudes and accelerations. Comparison between computed accelerations and recorded ones at Dar Elbeida site has proved the effectiveness of the approach.     

Impedance functions for three-dimensional foundations supported on an infinitely-long canyon of uniform cross-section in a homogeneous half-space

A direct boundary element procedure is presented to determine the impedance matrix for a three-dimensional foundation supported on an infinitely-long canyon of uniform cross-section cut in a homogeneous half-space. The uniform cross-section of the canyon permits analytical integration along the canyon axis leading to a series of two-dimensional boundary problems involving Fourier transforms of the full-space Green's functions. Solution of these two-dimensional boundary problems leads to a dynamic flexibility influence matrix which is inverted to determine the impedance matrix. The accuracy of the procedure is demonstrated by comparison with previous solutions for a surface-supported, square foundation and results obtained by a three-dimensional boundary element method (BEM) for a foundation of finite-width supported on an infinitely-long canyon. Compared with the three-dimensional BEM, the present method requires less computer storage and is more accurate and efficient. The foundation impedance matrix determined by this procedure can be incorporated in a substructure method for earthquake analysis of arch dams.     

Dynamic stiffness of foundation on layered soil half-space using cone frustums

Approximate dynamic-stiffness coefficients of a disk on the surface of a single layer on a half-space may be calculated using cone models. This concept is generalized to the case of a horizontally stratified site consisting of many layers on a homogeneous half-space. After constructing the so-called ‘backbone cone’ determining the radii of the disks at all interfaces, the dynamic-stiffness matrices of the layers (modelled as cone frustums) and the dynamic-stiffness coefficient of the underlying half-space (modelled as a cone) are assembled to that of the site. The dynamic-stiffness matrix of a layer is a complex-valued function of frequency because radiation of energy in the horizontal direction is considered. In this model of the layered half-space the properties of the cone reproduce themselves (cloning). The advantages of using cone models are also present for the layered half-space; in particular, no transformation to the wave-number domain is performed.