Scattering of plane P waves by a semi-cylindrical hill： analytical solution

An analytical solution for scattering of plane P waves by a semi-cylindrical hill was derived by using the wave function expansion method, and convergence of the solution and accuracy of truncation were verified. The effect of incident frequency and incident angle on the surface motion of the hill was discussed, and it was shown that a hill greatly amplifies incident plane P waves, and maximum horizontal displacement amplitudes appear mostly at the inclined incidence of waves, which are located at the half-space; and maximum vertical displacement amplitudes emerge mostly at the vertical incidence of waves, which are situated at the hill.     

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 properties of very prolate hill. Accuracy of the solution is checked by boundary conditions, numerical results for surface motion of oblate and prolate hills are calculated, and some conclusions are obtained.     

Diffraction of anti-plane SH waves by a semi-circular cylindrical hill with an inside concentric semi-circular tunnel

A closed-form analytic solution of two-dimensional scattering and diffraction of plane SH waves by a semicylindrical hill with a semi-cylindrical concentric tunnel inside an elastic half-space is presented using the cylindrical wave functions expansion method. The solution is reduced to solving a set of infinite linear algebraic equations. Fourier expansion theorem with the form of complex exponential function and cosine function is used. Numerical solutions are obtained by truncation of the infinite equations. The accuracy of the presented numerical results is carefully verified.     

任意圆弧形凸起地形中隧洞对入射平面SH波的影响

借助辅助函数的思想， 采用波函数展开法给出了圆弧形凸起地形中隧洞对入射平面SH波影响问题的一个解析解．数值结果表明，凸起地形中隧洞的存在以及隧洞大小，对凸起地形表面运动，以及凸起地形中的隧洞的动应力集中均具有显著影响．      

Surface motion of a semi-cylindrical hill for incident plane SV waves： Analytical solution

Introduction Scattering of incident waves by local topographies is one of the most attractive topics in the field of engineering seismology. They can be resolved by either a numerical method or an analyti-cal method. The numerical methods include finite difference method, finite element method, boundary element method, etc. The analytical method is the wave function expansion method. Al-though numerical methods can be used for arbitrary-shaped topographies, the analytical solutions are still …     

Surface motion of a cylindrical hill of circular-arc cross-section for incident plane SH waves

A closed-form solution of two-dimensional scattering of plane SH waves by a cylindrical hill of circular-arc cross-section in a half-space is presented using the wave functions expansion method. The solution is reduced to solving a set of infinite linear algebraic equations, using the auxiliary functions and the exterior region form of the Graf's addition theorem. Numerical solutions are obtained by truncation of the infinite equations and their accuracies are demonstrated by convergence of the numerical results and by the extent to which the numerical results fit the exact boundary conditions with increasing the truncation order. The numerical results for some typical cases are then presented for checking accuracies of various numerical methods. The effects of the height-to-width ratio of the hill on surface ground motion are finally illustrated.     

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。     

Scattering of SH waves by a scalene triangular hill

<正>The influence of local landforms on ground motion is an important problem.The antiplane response of a scalene triangular hill to incident SH waves is studied in this paper by using a complex function,moving coordinates and auxiliary functions.First,the model is divided into two domains:a scalene triangular hill with a semi-circular bottom;and a half space with a semi-circular canyon.Wave functions that satisfy the zero-stress condition at the triangular wedges and at the horizontal surface are constructed in both domains.Then,considering the displacement continuity and stress equilibrium, algebraic equations are established.Finally,numerical examples are provided to illustrate the influence of the geometry of the hill and the characteristics of the incident waves on the ground motions.     

Scattering of plane P waves by circular-arc layered alluvial valleys: An analytical solution

IntroductionThe site effect of local inhomogeneity and irregularity on seismic wave propagation is one of the most attractive topics in seismology. The problem may be resolved by either analytical method or numerical method. Here, the analytical method is the wave function expansion method; numerical methods include the finite difference, finite element, boundary integration equation, and discrete wave number method, etc. Although these numerical methods can be applied for sites of arbitrary s…     

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.     

含多个圆弧状沉积层河谷场地对平面SH波散射问题的解析解

利用波函数展开法给出了含多个圆弧状沉积层河谷(盆地)场地对平面SH波二维散射问题的一个解析解,并以具有三个沉积层的河谷为例,分析了河谷中沉积层层数与排列次序对SH波散射的影响。     

Rayleigh波在浅圆凹陷地形附近的散射:高频解答

利用波函数展开法给出了Rayleigh波在浅圆凹陷地形附近散射的一个高频解析解，并分析了入射频率、凹陷地形宽度和深度等因素对波散射的影响。数值结果表明，由于Rayleigh波幅值随深度而衰减，凹陷地形表面位移幅值整体上较小，且随着凹陷地形深度的增加而减小；由于Rayleigh波幅值还随频率而衰减，随着入射频率的升高，凹陷地形表面位移幅值逐渐减小；由于凹陷地形的屏障作用，在人射波的近端，地表位移分布变得相对复杂，地表位移峰值出现在左角点附近，而在入射波远端，地表位移分布相对简单，地表位移峰值出现在距凹陷地形较远的地方。     

Analytic wave series solution of out-of-plane(SH) waves diffraction by an almost semi-circular shallow cylindrical hill

A closed-form wave equation analytic solution of two-dimensional scattering and diffraction of outof-plane(SH) waves by an almost semi-circular shallow cylindrical hill on a flat, elastic and homogeneous half space is proposed by applying the discrete Fourier series expansions of sine and cosine functions. The semi-circular hill problem is discussed as a special case for the new formulated equation.Compared with the previous semi-circular cases solutions, the present method can give surface displacement amplitudes which agrees well with previous results. Although the proposed equation can only solve the problem of SH-waves diffracted by almost semi-circular shallow hills, the stress and displacement residual amplitudes are numerical insignificantly everywhere. Moreover, the influences of the depth-towidth ratio(a parameter defined in this paper to evaluate the shallowness of the topography of hills) on ground motions are presented and summarized. The limitations and errors of truncation from Graf's addition theorem and Fourier series equations in the present paper are also discussed.     

Scattering of Rayleigh waves by a shallow circular alluvial valley： High-frequency solution

Introduction The effect of local site conditions on wave propagation is one of the most attractive topics in engineering seismology. It may be resolved by either a numerical method or an analytical method. Numerical methods include the finite difference, finite element, boundary element method, etc. The analytical method is the wave function expansion method. The advantage of these numerical methods is that they can be applied to local inhomogeneity of arbitrary shapes, but analytical method …     

Antiplane SH-deformations near a surface rigid foundation above a subsurface rigid circular tunnel

The problem on the dynamic response of a rigid embedded foundation in the presence of an underground rigid tunnel and subjected to excitation of incident anti-plane SH waves is analyzed. By using the exact analytical solution for the two-dimensional SH-wave propagation in and around both the surface rigid foundation and subsurface rigid tunnel, those aspects of the resulting ground motions that are of special interest and importance for seismic resistant design in earthquake analyses have been examined. The computed amplitudes of the resulting periodic ground motions display a very complicated wave-interference between the surface foundation and underground tunnel that lead to observed standing wave patterns, together with abrupt changes in the wave amplitudes and large amplification of the incident motions. Supported by: National Science Foundation grant CMS 97-14859     

Scattering of plane P waves by canyons containing multiple circular-arc-shaped layers

By expressing the wave functions in the form of Fourier-Bessel series,the analytical solution for the two-dimension scattering problem of plane P waves by the cylindrical canyon topography that contains arbitrary number of circular-arc-shaped layers is presented firstly.And then,the convergence of the proposed series solution with the truncation number of terms is discussed,which demonstrates that the analytical solution can converge even for very high frequencies of the incident P wave.Finally,using this solution,the influences that are imposed on the stationary ground motion by the number and the sequence of alluvial layers,as well as the stiffness of soft interlayer contained in the canyon,are studied.     

斜入射平面SH波在层状半空间中沉积谷地周围的三维散射

采用直接刚度法计算自由场动力响应,以层状半空间中移动斜线均布荷载动力格林函数模拟散射波场,采用间接边界元方法求解了层状半空间中沉积谷地对斜入射平面SH波的三维散射问题.由于文中采用的层状场地三维动力刚度矩阵是精确的,且用于模拟散射波场的均布移动斜线荷载可以直接施加在沉积交界面处而不存在奇异性,所以本文方法具有很高精度.文中以均匀半空间和基岩上单一土层中沉积谷地对入射SH波的散射为例进行了数值计算,研究表明,沉积谷地对地震波的三维散射与二维散射之间存在本质差别;层状半空间中沉积谷地与均匀半空间中沉积谷地附近地表位移存在显著差异.     

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.     

浅埋圆柱形弹性夹杂附近等腰三角形凸起地形引起的SH波的散射

文章利用“契合”思想,给出了地下弹性夹杂与地面上的等腰三角形凸起地形引起的SH波散射问题的解析解答。利用复平面下坐标移动,通过区域Ⅰ和区域Ⅱ以及区域Ⅱ和区域Ⅲ的“公共边界”位移应力连续条件,建立起求解该问题的无穷代数方程组并截断有限项进行求解,最后通过具体算例及结果分析得出相应结论。     

Effects of a covering layer in a circular-arc canyon on incident plane SV waves

Introduction The effect of local topographies on seismic wave propagation is one of the most attractive topics in seismology. It may be resolved by either an analytical method or a numerical method. Here, the analytical method is the wave function expansion method; numerical methods include the finite difference, finite element, boundary integration equation, and discrete wave number method, etc. The advantage of these numerical methods is that they can be applied to local inhomogeneity and ir…