Weighted residual method for diffraction of plane P-waves in a 2-D elastic half-space Ⅲ: on an almost circular arbitrary-shaped alluvial valley

Scattering and diffraction of elastic in-plane P-and SV-waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong motion seismologists for over forty years. The case of out-ofplane SH-waves on the same elastic canyon that is semicircular in shape on the half-space surface is the first such problem that was solved by analytic closed-form solutions over forty years ago by Trifunac. The corresponding case of in-plane P-and SV-waves on the same circular canyon is a much more complicated problem because the in-plane P-and SV-scattered waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by Lee and Liu. This paper uses their technique of defining these stress-free scattered waves, which Brandow and Lee previously used to solve the problem of the scattering and diffraction of these in-plane waves on an almost-circular surface canyon that is arbitrary in shape, to the study of the scattering and diffraction of these in-plane waves on an almost circular arbitrary-shaped alluvial valley.     

Scattering and diffraction of plane P-waves in a 2-D elastic half-space Ⅱ:shallow arbitrary shaped canyon

Scattering and Diffraction of elastic in-plane P- and SV- waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong-motion seismologists for over forty years. The case of out-of-plane SH waves on the same elastic canyon that is semi-circular in shape on the half-space surface is the first such problem that was solved by analytic closed form solutions over forty years ago by Trifunac. The corresponding case of in-plane P- and SV-waves on the same circular canyon is a much more complicated problem because, the in-plane P- and SV- scattered waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by the author in the work of Lee and Liu. This paper uses the technique of Lee and Liu of defining these stress-free scattered waves to solve the problem of the scattered and diffraction of these in-plane waves on an almost-circular surface canyon that is arbitrary in shape.     

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.     

A note on three-dimensional scattering and diffraction by a hemispherical canyon–I: Vertically incident plane P-wave

The three-dimensional scattering by a hemi-spherical canyon in an elastic half-space subjected to seismic plane and spherical waves has long been a challenging boundary-value problem. It has been studied by earthquake engineers and strong-motion seismologists to understand the amplification effects caused by surface topography. The scattered and diffracted waves will, in all cases, consist of both longitudinal (P-) and shear (S-) shear waves. Together, at the half-space surface, these waves are not orthogonal over the infinite plane boundary of the half-space. Thus, to simultaneously satisfy both zero normal and shear stresses on the plane boundary numerical approximation of the geometry and/or wave functions were required, or in some cases, relaxed (disregarded). This paper re-examines this boundary-value problem from the applied mathematics point of view, and aims to redefine the proper form of the orthogonal spherical-wave functions for both the longitudinal and shear 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 will be easy to satisfy.     

Zero-stress,cylindrical wave functions around a circular underground tunnel in a flat,elastic half-space: Incident P-waves

The zero-stress boundary conditions at the surface of the half-space in the presence of surface and sub-surface cavities for in-plane, incident cylindrical P- and SV-waves have always posed challenging problems. The outgoing cylindrical P- and SV-waves can be represented by Hankel functions of radial distance coupled with the sine and cosine functions of angle. Together, at the half-space surface the P- and SV-wave functions are not orthogonal over the semi-infinite radial distance from 0 to infinity. Thus, to simultaneously satisfy the zero in-plane, normal, and shear stresses, an approximation of the geometry is often made. This paper presents an analytical formulation of the boundary-valued problem, where the Hankel wave functions are expressed in integral form, changing the representation from cylindrical to rectangular coordinates, so that the zero-stress boundary conditions at the half-space surface can be applied in a more straightforward way.     

半球形凹陷饱和土半空间对入射平面SV波三维散射问题的解析解

以Biot饱和多孔介质动力学理论为基础,利用Fourier-Bessel级展开法,给出了饱和多孔介质半空间中半球形凹陷地形对入射平面SV波散射问题的解析解.利用这一解析解数值计算地表位移幅值,分析入射角、入射波频率对地表位移幅值的影响,并与现存的单相介质情况下三维半球形凹陷问题解析解进行对比得出如下结论：(1)本文两相介质模型与单相介质模型有很大区别;(2)入射角、入射频率对场地表面位移分量有重要影响;(3)随着入射角、入射频率的增大,地表位移分布变得更复杂,且放大效应比单相介质模拟结果更加显著.     

Scattering of plane harmonic SH,SV, P and rayleigh waves by non-axisymmetric three-dimensional canyons: A wave function expansion approach

Scattering of elastic waves by three-dimensional canyons embedded within an elastic half-space is investigated by using a wave function expansion technique. The geometry of the canyon is assumed to be non-axisymmetric. The canyon is subjected to incident plane Rayleigh waves and oblique incident SH, SV and P waves. The unknown scattered wavefield is expressed in terms of spherical wave functions which satisfy the equations of motion and radiation conditions at infinity, but they do not satisfy stress-free boundary conditions at the half-space surface. The boundary conditions are imposed locally in the least-squares sense at several points on the surface of the canyon and the half-space. Through a comparative study the validity and limitations of two-dimensional approximations (antiplane strain and plane strain models) have been examined. It is shown that scattering of waves by three-dimensional canyons may cause substantial change in the surface displacement patterns in comparison to the two-dimensional models. These results emphasize the need for three-dimensional modelling of realistic problems of interest in strong ground motion seismology and earthquake engineering.     

圆弧形凹陷饱和土场地对平面P波散射问题的解析解

以Biot 饱和多孔介质动力学理论为基础，利用Fourier Bessel级数展开法, 得到饱和多孔介质半空间中圆弧形凹陷地形对平面P波的散射问题的解析解. 数值计算给出地表位移幅值，分析了入射波波长、入射角、圆弧高宽比对地表位移幅值的影响，并与现存的在单相介质情况下得到的结论进行对比.     

A new series solution method for two-dimensional elastic wave scattering along a canyon in half-space

This paper presents a semi-analytical method for studying the two-dimensional problem of elastic wave scattering by surface irregularities in a half-space. The new method makes use of the member of a c-completeness family of wave functions to construct the scattering fields, and then applies equal but opposite tractions to those of the foregoing constructed scattering fields on the horizontal surface of the half-space to produce additional scattering fields. These additional scattering fields are a series of Lamb's solutions. Thus the whole scattering field constructed in the series automatically satisfies the Navier equations, the condition of zero traction on the half-space surface, and the radiation boundary conditions at infinity. Using the traction-free conditions along the canyon surface, the coefficients of the series solutions are determined via a least-squares method. For incident P, SV, and Rayleigh waves, the numerical results are presented for the scattering displacements in the vicinity of a semi-circular canyon in the half-space.     

Dynamic soil-tunnel interaction in layered half-space for incident P-and SV-waves

The dynamic soil-tunnel interaction is studied by the model of a rigid tunnel embedded in layered half-space,which is simplified as a single soil layer on elastic bedrock to the excitation of P- and SV-waves.The indirect boundary element method is used,combined with the Green's function of distributed loads acting on inclined lines.It is shown that the dynamic characteristics of soil-tunnel interaction in layered half-space are different much from that in homogeneous half-space,and that the mechanism of soil-tunnel interaction is also different much from that of soil-foundation-superstructure interaction.For oblique incidence,the tunnel response for in-plane incident SV-waves is completely different from that for incident SH-waves,while the tunnel response for vertically incident SV-wave is very similar to that of vertically incident SH-wave.     

地震波在不规则界面上的散射效应及其某些应用

本文运用数值解法,求解了两类散射问题:(1)在声学近似下,平面P波在半无限介质空间表面上任意形状的三维空腔上的散射;(2)平面SH波在半无限弹性空间中埋藏着任意形状截面的无限长、且平行于地面的弹性柱体上的散射,得到了几种几何形状的物体所引起的散射数值结果。把某些特殊情况下的散射结果与已知的精确解作对比,两者能很好地吻合。     

Diffraction of plane P waves by a canyon of arbitrary shape in poroelastic half-space (Ⅰ): Formulation

This paper presents an indirect boundary integration equation method for diffraction of plane P waves by a two-dimensional canyon of arbitrary shape in poroelastic half-space. The Green's functions of compressional and shear wave sources in poroelastic half-space are derived based on Biot's theory. The scattered waves are constructed using the fictitious wave sources close to the boundary of the canyon, and magnitude of the fictitious wave sources are determined by the boundary conditions. The precision of the method is verified by the satisfaction extent of boundary conditions, the comparison between the degenerated solutions of single-phased half-space and the well-known solutions, and the numerical stability of the method.     

Scattering of plane sh waves by a semi-elliptical canyon

In this paper we analyse the two-dimensional scattering and diffraction of plane SH waves by a semi-elliptical canyon. The exact series solution of the problem, for general angle of incidence of the plane SH waves, has been used to examine the dependence of surface amplifications inside and near the canyon. The nature of ground motion has been found to depend on two key parameters: (a) The angle of incidence. (b) The ratio of the canyon width to the wave length of incident SH waves. For short incident waves surface displacement amplitudes change rapidly from one point to another, while for the long waves and shallow canyons displacement amplitudes display only minor departure from the uniform half-space amplification of 2. For shallow canyons and long incident waves, the angle of incidence introduces only minor changes into the overall behaviour of surface amplitudes. For deep canyons and nearly grazing incidences, a prominent shadow zone is realized behind the canyon.     

饱和层状场地中任意凹陷地形对入射SV波的散射

利用饱和土层的精确动力刚度矩阵和动力格林(Green)函数,采用间接边界元法,在频域内求解了层状饱和场地中任意凹陷地形对入射SV波的散射问题.通过自由场反应分析,求得凹陷地形表面各点的位移和各单元的应力响应;同样计算了虚拟分布荷载的格林影响函数,求得相应的位移和应力响应;根据边界条件确定虚拟分布荷载,将自由场位移响应和...     

Application of the weighted residual method to diffraction by 2-D canyons of arbitrary shape: I. Incident SH waves

A solution for the two-dimensional scattering and diffraction of plane SH waves by canyons of arbitrary shape in an elastic half space is presented. The wave field for arbitrary geometry in this paper is computed numerically by the method of weighted residues (moment method). The wave displacement field computed by the present residual method for the case of a semi-circular canyon was shown to agree analytically and numerically with that computed by the exact closed form series solution. The same observations about ground amplifications, their dependence on frequencies and orientations of the incident waves, can be stated here for canyons of arbitrary shape as previously made for circular canyons.     

In-plane foundation-soil interaction for embedded circular foundations

Foundation soil interaction is studied using an analytical two-dimensional model, for circular foundations embedded in a homogeneous elastic half-space and for incident plane P- and SV- and for surface Rayleigh waves. The scattered waves are expanded in complete series of cyclindrical wave functions. A detailed analysis is presented of the foundation response to unit amplitude incident waves as a function of the type of incident waves and angle of incidence, the depth of the embedment and the foundation mass per unit length.It is shown that free-field translations and point rotation approximate well the foundation input motion only for very long incident waves. For shorter incident waves, those in general overestimate the foundation input motion. Neglecting the rotation of the foundation input motion (which is usually done in practice) may eliminate a major contribution to the base excitation of buildings and may cause nonconservative estimates of the forces in these buildings. Incident waves appear as ‘longer’ to a shallow foundation than to a deeper foundation. Therefore, deeper foundations are more effective in reflecting and scattering the short incident waves.     

Soil-structure interaction on shallow rigid circular foundation: plane SH waves from far-field earthquakes

A closed-form wave function analytic solution of two-dimensional scattering and diffraction of incident plane SH-waves by a fl exible wall on a rigid shallow circular foundation embedded in an elastic half-space is presented. This research generalizes the previous solution by Trifunac in 1972, which tackled only the semi-circular foundation, to arbitrary shallow circular-arc foundation cases, and is thus comparatively more realistic. Ground surface displacement spectra at higher frequencies are also obtained. As an analytical series solution, the accuracy and error analysis of the numerical results are also discussed. It was observed from the results that the rise-to-span ratio of the foundation profi le, frequency of incident waves, and mass ratios of different media(foundation-structure-soil) are the three primary factors that may affect the surface ground motion amplitudes near the structure.     

Diffraction of plane SV waves by a shallow circular-arc canyon in a saturated poroelastic half-space

An analytical solution for the scattering and diffraction of incident plane SV waves by a shallow circular-arc canyon in a saturated poroelastic half-space is derived by the wave function expansion method. The solution is utilized to analyze the dependence of the computed surface motions on the incident frequencies, incident angles, porosity, boundary drainage and Poisson's ratio. It is shown that, depending on the incident angles, the surface displacement amplitudes around a canyon in a dry poroelastic half-space and saturated poroelastic half-space can be very different. The surface displacement amplitudes of an undrained saturated poroelastic half-space are close to those of a drained saturated poroelastic half-space. For low porosity, the surface displacement amplitudes of a saturated poroelastic half-space are almost identical to those of a dry poroelastic half-space, and drainage condition has little influence on the surface displacement amplitudes. But for high porosity, the effect of drainage condition becomes significant, and for the same porosity, the displacement amplitudes of an undrained saturated half-space will be larger than those of a drained saturated half-space. Poisson's ratio is also an important factor affecting the surface displacement amplitudes around the canyon, both in drained and undrained conditions, but leads to larger effects for an undrained saturated half-space than for a drained saturated half-space. Large pore pressures are found around the canyon and their amplitudes depend on the incident angles and frequencies. Below the surface, the amplitudes of pore pressures are less than they are at the surface, especially for high 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.     

Transient scattering of elastic waves by dipping layers of arbitrary shape. Part 2: Plane strain model

Scattering of elastic waves by dipping layers of arbitrary shape embedded within an elastic half-space is investigated for a plane strain model by using a boundary method. Unknown scattered waves are expressed in the frequency domain in terms of wave functions which satisfy the equations of motion and appropriate radiation conditions at infinity. The steady state displacement field is evaluated throughout the elastic medium for different incident waves so that the continuity conditions along the interfaces between the layers and the traction-free conditions along the surface of the half-space are satisfied in the least-squares sense. Transient response is constructed from the steady state one through the Fourier synthesis. The results presented show that scattering of waves by dipping layers may cause locally very large amplification of surface ground motion. This amplification depends upon the type and frequency of the incident wave, impedance contrast between the layers, component of displacement which is being observed, location of the observation station and the geometry of the subsurface irregularity. These results are in agreement with recent experimental observations.