Antiplane response of a dike on flexible embedded foundation to incident SH-waves

The response of an elastic circular wedge on a flexible foundation embedded into a half-space is investigated in the frequency domain for incident pane SH-waves. The problem is solved by expansion of the motion in all three media (wedge, foundation and half-space) in cylindrical wave functions (Fourier-Bessel series). The structural model is simple, but accounts for both differential motions of the base and for the effects of soil-structure interaction. Usually, structural models in earthquake engineering consider either differential ground motion, but ignore soil-structure interaction, or consider soil-structure interaction, but for a rigid foundation, thus ignoring differential ground motion. The purpose of the study is to find how stiff the foundation should be relative to the soil so that the rigid foundation assumption in soil-structure interaction models is valid. The shortest wavelength of the incident waves considered in this study is one equal to the width of the base of the wedge. It is concluded that, for this model, a foundation with same mass density as the soil but 50 times larger shear modulus behaves as ‘rigid’. For ratio of shear moduli less than 16, the rigid foundation assumption is not valid. Considering differential motions is important because of additional stresses in structures that are not predicted by fixed-base and rigid foundation models.     

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。     

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 solutions available. The soil-structure interaction problem is one of the most classic problems connecting the two disciplines 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 superstructure. This interaction effect is important across many structure, foundation, and subgrade types but is most pronounced when a rigid superstructure is founded on a relatively 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 superstructure. 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 a superstructure from its substructure resting on a shaking half-space.     

Dynamic response of a group of flexible foundations to incident seismic waves

The dynamic response of a finite number of flexible surface foundations subjected to harmonic incident Rayleigh or SH waves is presented. The foundations are assumed to be resting on an elastic half-space. The results show that the foundation stiffness has a marked effect on the vertical response, while there is only a minor effect on the horizontal displacements. In general, the dynamic response decreases with increasing foundation stiffness. In cases of Rayleigh wave incidence, the existence of an adjacent foundation generates a certain amount of horizontal response in the direction perpendicular to the incident wave and subsequently causes the system to undergo a torsional motion; while in cases of horizontally incident SH waves, a vertical response has been observed and its magnitude is comparable to the response in the direction of the incident wave.     

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.     

Antiplane response of isoceles 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.     

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.     

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.     

Effects of wave passage on the relevant dynamic properties of structures with flexible foundation

An evaluation of the wave passage effects on the relevant dynamic properties of structures with flexible foundation is presented. A simple soil–structure system similar to that used in practice to take into account the inertial interaction effects by the soil flexibility is studied. The kinematic interaction effects due to non‐vertically incident P, SV and Rayleigh waves are accounted for in this model. The effective period and damping of the system are obtained by establishing an equivalence between the interacting system excited by the foundation input motion and a replacement oscillator excited by the free‐field ground motion. In this way, the maximum structural response could be estimated from standard free‐field response spectra using the period and damping of the building modified by both the soil flexibility and the travelling wave effects. Also, an approximate solution for the travelling wave problem is examined over wide ranges of the main parameters involved. Numerical results are computed for a number of soil–structure systems to identify under which conditions the effects of wave passage are important. It comes out that these effects are generally negligible for the system period, but they may significantly change the system damping since the energy dissipation within the soil depends on both the wave radiation and the diffraction and scattering of the incident waves by the foundation. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Scattering of SH waves from a partially debonded rigid elliptic cylinder

Scattering of SH waves by an embedded rigid elliptic cylinder of finite length, which is partially debonded from elastic soil is studied. The debonding regions are modeled as multiple elliptic arc-shaped interface cracks with non-contacting faces. The scattered wave field is expressed as a Mathieu function expansion with unknown coefficients. The mixed boundary conditions of the problem lead to a set of singular integral equations of the first type in terms of the dislocation density functions of the cracks. A quadrature method is used to solve these integral equations numerically. The results for dynamic stress intensity factors, far-field pattern of the displacement and scattering cross sections are presented. We particularly discuss the effects of the ratio of the short radius to long radius of the cylinder.     

Torsional response of a rigid circular foundation on a saturated half-space to SH waves

An analytical approach is used to study the torsional vibrations of a rigid circular foundation resting on saturated soil to obliquely incident SH waves. Biot’s poroelastic dynamic theory is considered to characterize the saturated soil below the foundation, which is solved by Hankel transform later. In order to consider the scattering phenomena caused by the existence of the foundation, the total wave field in soil is classified into free-field, rigid-body scattering field and radiation scattering field. According to the classification of wave field and the mixed boundary-value conditions between the soil and the foundation, torsional vibrations of the foundation are formulated in two sets of dual integral equations. Then, the dual integral equations are reduced to Fredholm integral equation of the second kind to be solved. Combining with the dynamic equilibrium equations of the foundation, the expressions for the torsional vibrations of the foundation are obtained. Numerical results are presented to demonstrate the influence of excitation frequency, incident angle, the torsional inertia moment of the foundation and permeability of the saturated half-space on the torsional vibrations of the foundation.     

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。     

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.     

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 …     

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。     

Kinematic response of single piles to vertically incident P‐waves

Dynamic response of single piles to seismic waves is fundamentally different from the free‐field motion because of the interaction between the pile and the surrounding soil. Considering soil–pile interaction, this paper presents a new displacement model for the steady‐state kinematic response of single piles to vertically incident P‐waves on the basis of a continuum model. The governing equations and boundary conditions of the two undetermined functions in the model are obtained to be coupled by using Hamilton's principle. Then, the two unknown functions are decoupled and solved by an iterative algorithm numerically. A parametric study is performed to investigate the effects of the properties of the soil–pile system on the kinematic response of single piles. It is shown that the effects of the pile–soil modulus ratio, the slenderness ratio of the pile, and the frequency of the incident excitations are very significant. By contrast, the influence of soil damping on the kinematics of the system is slight and can be neglected. Copyright © 2013 John Wiley & Sons, Ltd.     

Influence of water saturation on horizontal and vertical motion at a porous soil interface induced by incident P wave

This paper describes a study on the influence of water saturation on horizontal and vertical motion at the interface between porous soil and rock formation due to an incident P wave. The soil is modeled as a partially water-saturated porous material with a small amount of air inclusions, while the underlying rock is approximately regarded as an ordinary one-phase solid. Theoretical formulation is developed for the computation of motion amplitudes in both horizontal and vertical components, which are considered as functions of the degree of saturation, the angle of incidence as well as the frequency. Numerical results are provided to illustrate the effect of saturation on displacement amplitudes in both components and their ratios. The results show that even a slight decrease of complete saturation may lead to a substantial change in the motion amplitudes in both horizontal and vertical components. In general, partial saturation may cause lower horizontal-to-vertical ratios over the entire range of incident angles except the normal incidence, implying the potential importance of saturation effects in the interpretation of field observations.