Antiplane response of isosceles triangular hill to incident SH waves

In this paper, antiplane response of an isosceles triangular hill to incident SH waves is studied based on the method of complex function and by using moving coordinate system. The standing wave function, which can satisfy the governing equation and boundary condition, is provided. Furthermore, numerical examples are presented; the influences of wave number and angle of the incident waves and the angle of the hill‘s peak on ground motion are discussed.     

SH波入射时柔性基础上等腰三角形坝体结构的出平面反应

本文利用复变函数和坐标移动方法研究了SH波入射对柔性基础上等腰三角形坝体结构的影响。首先建立问题的数学模型并根据分区和辅助函数法将模型分割为2部分,其1为等腰三角形和半圆形组成的区域Ⅰ,其余为区域Ⅱ;其2在区域Ⅰ内构造1个满足等腰三角形两边应力自由的驻波解,在区域Ⅱ内构造满足水平边界应力自由的散射波;通过移动坐标在区域Ⅰ、Ⅱ的公共边界实现位移和应力的连续,建立起求解该问题的无穷代数方程组;最后,本文给出了例题和数值结果并对其进行了讨论,并通过算例强调了与文献[9]的本质区别。     

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。     

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.     

SH波作用下浅埋裂纹附近半圆形凸起的地震动

利用Green函数、复变函数及多极坐标法研究了浅埋裂纹附近的半圆形凸起地形对SH波的散射问题。首先求解出适合本问题的Green函数,即含有半圆形凸起的弹性半空间内任意一点承受时间谐和的出平面线源荷载作用时位移场的解答。然后,从半圆形凸起地形对SH波的散射问题出发,利用本问题的Green函数在基体内构造裂纹。最后给出了浅埋裂纹附近的半圆形凸起地形对SH波散射的位移场。通过算例讨论了裂纹对地表位移幅值的影响,结果表明裂纹在一定情况下具有减震作用。     

SH波入射时多个半圆形沉积谷地附近浅埋圆形孔洞的动力分析

本文采用辅助函数的思想,利用复变函数和多级坐标的方法给出了SH波入射条件下多个半圆形沉积谷地附近浅埋圆形孔洞动力分析问题的解答。将整个求解区域分割成两部分来处理,区域I为多个半圆形沉积谷地,区域II为浅埋圆形孔洞附近带半圆形凹陷的半无限弹性空间。在区域I和II中分别构造位移解,并在二个区域的“公共边界”上实施位移应力的连续条件,建立求解该问题的无穷代数方程组。最后,本文给出了算例和数值结果,并对其进行了讨论。     

SH波入射时半圆形凸起与凹陷地形的地震动

研究了弹性半空间中半圆形凸起与凹陷相连地形对SH波的散射问题。将整个求解区域分割为2部分,在其中分别构造满足边界条件的位移解,通过移动坐标使之满足“公共边界”以及半圆形凹陷表面上的边界条件,从而建立起求解该问题的无穷代数方程组。最后,给出了地表位移幅值的数值结果以及凸起地形顶点和凹陷地形最低点处位移幅值的反应谱并进行了讨论。     

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.     

双等腰三角形凸起地形在SH波入射时的地表位移函数

利用"分区、辅助函数"思想,将求解的区域分为三部分,区域Ⅰ为一个带有半圆形弧线的等腰三角形区域,区域Ⅱ也为一个带有半圆形弧线的等腰三角形区域,区域Ⅲ为两个带有半圆形凹陷的弹性半空间,并利用波函数展开法分别在区域Ⅰ、Ⅱ和区域Ⅲ中构造满足等腰三角形在其斜面上应力自由和在弹性半空间中满足水平面上应力自由的驻波函数,以此为基础,通过区域Ⅰ、Ⅱ和区域Ⅲ的"公共边界"位移应力连续条件,建立对该问题进行求解的无穷代数方程组,并采用截断有限项的方法对其进行求解。     

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。     

Antiplane response of a dike with flexible soil-structure interface to incident SH waves

Studies of the effects of differential ground motions on structural response generally do not consider the effects of the soil-structure interaction. On the other end, studies of soil-structure interaction commonly assume that the foundation of the structure (surface or embedded) is rigid. The former ignore the scattering of waves from the foundation and radiation of energy from the structure back to the soil, while the latter ignore quasi-static forces in the foundations and lower part of the structure deforming due to the wave passage. This paper studies a simple model of a dike but considers both the soil-structure interaction and the flexibility of the foundation. The structure is represented by a wedge resting on a half-space and excited by incident plane SH-waves. The structural ‘foundation’ is a flexible surface that can deform during the passage of seismic waves. The wave function expansion method is used to solve for the motions in the half-pace and in the structure. The displacements and stresses in the structure are compared with those for a fixed-base model shaken by the free-field motion. The results show large displacements near the base of the structure due to the differential motion of the base caused by the wave passage.     

Antiplane response of two scalene triangular hills and a semi-cylindrical canyon by incident SH-waves

Antiplane response of two scalene triangular hills and a semi-cylindrical canyon by SH-waves is studied using wave function expansion and complex function method. Firstly, the analytical model is divided into three parts, and the displacement solutions of wave fields are constructed based on boundary conditions in the three regions. Three domains are then conjoined to satisfy the "conjunction" condition at shared boundary. In addition, combined with the zero-stress condition of semi-cylindrical canyon, a series of infinite algebraic equations for the problem are derived. Finally, numerical examples are provided and the influence of different parameters on ground motion is discussed.     

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.     

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

IntroductionEarthquakedamageinvestigationsandtheoreticalanalysesshowthatahillamplifiesincidentwavestremendouslyduetomulti-reflectionofthewaveswithinthehill;notabledynamicstressconcentrationisobservedonwallofatunnel(Pao,Mow,1973;Lee,Trifunac,1979).Therefore,itmaybeassumedthattheexistenceofatunnelinahillmighthavegreateramplificationeffectonthegroundmotionnearby,andthedynamicstressconcentrationofthetunnelinthehillmightbemorenotable.Itiscommonthattherailwaysorhighwaysarethroughahilloramountainwi…     

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 …     

SH波对浅埋圆柱形弹性夹杂附近多个半圆形凸起的散射

利用复变函数及移动坐标法建立了多个半圆形凸起及附近浅埋弹性夹杂对SH波散射问题的解析方法.求解时将整个求解模型进行分区,区域I为包括多个半圆形凸起在内的多个圆形区域;区域II包括多个半圆形凹陷和1个圆柱形弹性夹杂.然后在两个区域内分别构造满足边界条件的位移解.最后,根据连接面的契合条件,利用移动坐标法将两个区域在公共边界上装配起来,同时考虑弹性夹杂的边界条件,建立起求解该问题的无穷代数方程组.通过算例,讨论了凸起间距,夹杂埋深等参数对地表位移幅值的影响.     

Scattering of SH waves induced by a symmetrical V-shaped canyon: a unified analytical solution

This paper reports a series solution of wave functions for two-dimensional scattering and diffraction of plane SH waves induced by a symmetrical V-shaped canyon with different shape ratios. A half-space with a symmetrical V-shaped canyon is divided into two sub-regions by using a circular-arc auxiliary boundary. The two sub-regions are represented by global and local cylindrical coordinate systems, respectively. In each coordinate system, the wave field satisfying the Helmholtz equation is represented by the separation of variables method, in terms of the series of both Bessel functions and Hankel functions with unknown complex coefficients. Then, the two wave fields are described in the local coordinate system using the Graf addition theorem. Finally, the unknown coefficients are sought by satisfying the continuity conditions of the auxiliary boundary. To consider the phase characteristics of the wave scattering, a parametric analysis is carried out in the time domain by assuming an incident signal of the Ricker type. Surface and subsurface transient responses demonstrate the characteristics and mechanisms of wave propagating and scattering.     

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.     

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.     

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.