Scattering of plane SH waves by a circular-arc hill with a circular tunnel

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Surface motion of a semi-elliptical hill for incident plane SH waves

A closed-form analytical solution of surface motion of a semi-elliptical cylindrical hill for incident plane SH waves is presented. Although some previous analytical work had already dealt with hill topography of semi-circular and shallow circular, our work aims at calculating surface motion of very prolate hill for high incident frequency, and explaining the special vibrating is checked by boundary conditions, numerical results for and some conclusions are obtained. properties of very prolate hill. Accuracy of the solution surface motion of oblate and prolate hills are calculated,

Dynamic soil-tunnel interaction in layered half-space for incident plane SH waves

The dynamic soil-tunnel interaction is studied by indirect boundary element method （IBEM）, using the model of a rigid tunnel in layered half-space, which is simplified to a single soil layer on elastic bedrock, subjected to incident plane SH waves. The accuracy of the results is verified through comparison with the analytical solution. It is shown that soil-tunnel interaction in layered half-space is larger than that in homogeneous half-space and this interaction mechanism is essentially different from that of soil-foundation-superstructure interaction.

2.5D scattering of incident plane SH waves by a canyon in layered half-space

This paper presents 2.5D scattering of incident plane SH waves by a canyon in layered half-space by the indirect boundary element method (IBEM). The free field response is carried out to give the displacements and stresses on the line which forms boundary of the canyon. The fictitious uniform moving loads are applied to the same line to calculate the Green's functions for the displacements and stresses. The amplitudes of the loads are determined by the boundary conditions. The displacements due to the free field and from the fictitious uniform moving loads have to be added to obtain the whole motion. The numerical results are carried out for the cases of a canyon in homogenous and in one layer over bedrock. The results show that the 2.5D wave scattering problem is essentially different from the 2D case, and there exist distinct differences between the wave amplification by a canyon in layered half-space and that in homogeneous half-space. The reasons for the distinct difference are explored, and the effects of the thickness and stiffness of the layer on the amplification are discussed.

Three-dimensional Scattering of Obliquely Incident Plane SH Waves by an Alluvial Valley in a Layered Half-space

The indirect boundary element method （IBEM） is used to study three-dimensional scattering of obliquely incident plane SH waves by an alluvial valley embedded in a layered half-space. The free-field response of the layered half-space is calculated by the direct stiffness method, and dynamic Green＇s functions of moving distributed loads acting on inclined lines in a layered half-space are calculated to simulate the scattering wave field. The presented method yields very accurate results since the three-dimensional dynamic stiffness matrix is exact and the moving distributed loads can act directly on the valley boundary without singularity. Numerical results and analyses are performed for amplification of obliquely incident plane SH waves around an alluvial valley in a uniform half-space and in single layer over half-space. The results show that the three-dimensional responses are distinctly different from the two-dimensional responses, and the displacement amplitudes around alluvial valleys in a uniform haft-space are obviously different from those in a layered half-space.

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.

Anti-plane (SH) waves diffraction by an underground semi-circular cavity:analytical solution

Diffraction of a two-dimensional （2D） semi-circular cavity in a half-space under incident SH-waves is studied using the classic wave function expansion method with a new de-coupling technique. This so-called ＂improved cosine half- range expansion＂ algorithm exhibits an excellent performance in reducing displacement residual errors at two rim points of concern. The governing equations are developed in a manner that minimizes the residues of the boundary conditions. Detailed derivation and analysis procedures as well as truncation of infinite linear governing equations are presented. The semi-circular cavity model presented in this paper, due to its simple profile, is expected to be used in seismic wave propagation studies as a benchmark for examining the accuracies of various analytical or numerical methods for mixed-boundary wave propagation problems.

3D Diffraction of obliquely incident SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space

Abstract This paper studies three-dimensional diffraction of obliquely incident plane SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space using indirect boundary element method. The approach is validated by comparison with the literature, and the effects of cavity interval, incident frequency, and boundary drainage condition on the diffraction are studied through numerical examples. It is shown that, the interaction between two cavities is significant and surface displacement peaks become large when two cavities are close, and the surface displacement may be significantly amplified by twin cavities, and the influence range with large amplification can be as wide as 40 times of the cavity radius. Surface displacements in dry poroelastic case and saturated poroelastic cases with drained and undrained boundaries are evidently different under certain circumstances, and the differences may be much larger than those in the free-field response.

A note on the surface motion of a semi-cylindrical canyon for incident cylindrical SH waves radiated by a finite fault

The plane-wave assumption for incident SH waves can be a good approximation for cylindrical and spherical waves radiated from finite sources, even when the source is as close as twice the size of the inhomogeneity, and when the source and the inhomogeneity are described within the same coordinate system. However, in a more general setting, and when the fault＇s radiation pattern must be considered, the plane-wave approximation may not yield satisfactory answers for arbitrary orientation of the fault. Jalali et al. （2015） demonstrated this for a semi-cylindrical, sedimentary valley, and in this study we extend their results to a case in which the semi-circular, sedimentary valley is replaced by a canyon. We describe the effects of incident cylindrical waves on the amplitudes of surface motion in and near the semi-cylindrical canyon when the causative faults are at different distances and have different curvatures and orientations.

Out-of-plane (SH) soil-structure interaction： a shear wall with rigid and flexible ring foundation

Soil-structure interaction （SSI） of a building and shear wall above a foundation in an elastic half-space has long been an important research subject for earthquake engineers and strong-motion seismologists. Numerous papers have been published since the early 1970s; however, very few of these papers have analytic closed-form solu- tions available. The soil-structure interaction problem is one of the most classic problems connecting the two dis- ciplines of earthquake engineering and civil engineering. The interaction effect represents the mechanism of energy transfer and dissipation among the elements of the dynamic system, namely the soil subgrade, foundation, and super- structure. This interaction effect is important across many structure, foundation, and subgrade types but is most pro- nounced when a rigid superstructure is founded on a rela- tively soft lower foundation and subgrade. This effect may only be ignored when the subgrade is much harder than a flexible superstructure： for instance a flexible moment frame superstructure founded on a thin compacted soil layer on top of very stiff bedrock below. This paper will study the interaction effect of the subgrade and the super- structure. The analytical solution of the interaction of a shear wall, flexible-rigid foundation, and an elastic half- space is derived for incident SH waves with various angles of incidence. It found that the flexible ring （soft layer） cannot be used as an isolation mechanism to decouple asuperstructure from its substructure resting on a shaking half-space.

Time-history responses on the surface by regularly distributed enormous embedded cavities:Incident SH-waves

The time-history responses of the surface were obtained for a linear elastic half-plane including regularly distributed enormous embedded circular cavities subjected to propagating obliquely incident plane SH-waves. An advanced numerical approach named half-plane time-domain boundary element method（BEM）, which only located the meshes around the cavities, was used to create the model. By establishing the modified boundary integral equation（BIE）independently for each cavity and forming the matrices, the final coupled equation was solved step-by-step in the timedomain to obtain the boundary values. The responses were developed for a half-plane with 512 cavities. The amplification patterns were also obtained to illustrate the frequencydomain responses for some cases. According to the results,the presence of enormous cavities affects the scattering and diffraction of the waves arrived to the surface. The introduced method can be recommended for geotechnical/mechanical engineers to model structures in the fields of earthquake engineering and composite materials.

Effects of interlayer on SH wave propagations

Considering properties of materials can not abruptly change, an interface layer model with the linear continuous transitive properties is proposed for the double half planes to investigate the problems of SH waves propagation in the double half planes with the interface layer taken into account, the reflective and refractive waves would exhibit phase shifts. The amplitudes of the phase shifts of the reflective and refractive waves vary greatly with the thickness of the interface layer, properties of media and incident angle. It suggests that there is probably considerable errors in understanding the problems of wave propagation under conditions of neglecting the interface layer structures.

圆弧形层状凹陷地形对平面SH波的散射

利用 Fourier- Bessel级数展开法给出了圆弧形层状凹陷地形对平面 SH波散射二维问题的一个解析解 ,分析了层状凹陷地形沉积排列顺序、软夹层刚度和厚度等因素对 SH波散射的影响。结果表明 ,层状凹陷地形对平面 SH波的散射和单一凹陷地形差别很大 ,凹陷地形表面松软沉积的存在 ,即使厚度很薄 ,也会大大降低凹陷地形本身对入射波的屏障作用 ;层状凹陷地形中沉积排列顺序、软夹层刚度和厚度的变化对散射也具有重要影响。

圆弧形层状沉积河谷场地在平面SH波入射下动力响应分析

利用Fourier Bessel级数展开法在频域内给出了圆弧形层状沉积河谷场地对平面SH波放大作用的一个解析解 。

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(Ⅰ)——方法及验证

利用间接边界元法,求解了弹性层状半空间中无限长洞室对斜入射平面SH波的三维散射问题.通过与文献结果进行对比,验证了本方法的正确性.与工程中常用的二维模型比较表明,工程中将三维散射问题简单地分解为平面内问题和平面外问题的做法存在较大误差.文中并研究了斜入射角度对散射的影响,表明斜入射角度对地表位移幅值有着重要影响.

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.

弹性层状半空间中沉积谷地对入射平面SH波的放大作用

对Wolf理论进行拓展,使之可解决沉积谷地对波的散射问题,进而利用间接边界元法,求解了弹性层状半空间中沉积谷地对入射平面SH波的放大作用问题。通过自由场反应分析,求得假想边界上各点的位移和各单元的应力反应。施加虚拟均布荷载,求得位移和应力的格林函数。根据应力和位移连续边界条件确定虚拟分布荷载,将自由场反应和虚拟分布荷载产生的反应叠加起来,即得到问题的解答。最后,以基岩上单一土层中沉积谷地对入射平面SH波的放大作用为例进行了数值计算分析,结果表明,层状半空间中沉积谷地和均匀半空间中沉积谷地对波的放大作用存在显著差别。文中分析了造成差别的原因,并讨论了覆盖层厚度和刚度对放大作用的影响。

弹性介质中平面SH波通过弹性夹层时的传播特性

运用波的传播理论及Snell定律研究了平面SH波通过弹性层状介质时的传播特性,求得波动方程的解析解,在此基础上,利用数值计算绘出了反射﹑透射系数与介质参数及入射角的关系曲线。结果表明,透射系数不仅与介质参数有关,而且与夹层厚度和入射角有关,严格控制夹层厚度和入射角可获得满意的隔振效果,为屏障隔振理论在工程中的应用提供了理论基础。

多层沉积凹陷地形对平面SH波散射问题的解析解

利用波函数展开法给出了含多个沉积层凹陷地形对平面SH波散射二维问题的一个解析解,并以一具有3个沉积层的凹陷地形为例,分析了凹陷地形中沉积层排列次序对SH波散射的影响。结果表明,凹陷地形中沉积层排列次序将显著改变凹陷地形的卓越频率,沉积层的某个(些)排列次序有可能对入射波产生巨大的放大作用;此外,沉积层的刚度、厚度以及凹陷地形的深宽比等均对凹陷地形的放大效应也有一定的影响。

弹性半空间夹杂群对平面SH波散射的快速多极多域边界元法模拟

结合快速多极子展开技术,发展一种高精度快速多域间接边界元方法,用于求解弹性半空间大规模夹杂群对平面SH波的二维散射。数值检验表明,该方法能够高效精确地求解大规模多域散射问题,同时可大幅度降低计算存储量,据此在普通电脑上可实现数百万自由度二维多域出平面散射问题快速求解,进而分别针对半空间中均匀分布和随机分布的夹杂群,探讨了夹杂群刚度、形状对其周围平面SH波散射的影响规律。研究结果表明:由于弹性波的多次散射相干,使得夹杂群对平面SH波的散射与单夹杂体存在显著差异,总波场空间分布特征和位移频谱特征更为复杂;平面SH波散射特性主要取决于夹杂体的材料软硬程度、几何形状特征和入射波的角度、频率。另外,研究方法和分析结论可为半空间复杂不均质体对SH波散射正、反演分析提供新的技术手段和理论依据。