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.     

Discrete models for vertical vibrations of surface and embedded foundations

A five-parameter discrete model that approximates the dynamic force4isplacement relationship for rigid foundations undergoing vertical vibrations on a uniform elastic half-space is presented. The model involves a combination of two springs, two viscous dampers and a mass. Values of the parameters for circular, square and rectangular foundations placed on the surface or embedded in an elastic half-space are listed. The parameters are obtained by minimizing the discrepancy between the force4isplacement relation for the model and that obtained by solution of the mixed boundary-value problem of the rigid foundation on an elastic half-space. The definition of an appropriate input motion to represent wave excitation is also discussed. The input motion to the discrete model differs from the input motion that should be used in a continuum model.     

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.     

The dynamics of a rigid foundation on the surface of an elastic half-space

The response of a rigid rectangular foundation block resting on an elastic half-space has been determined by considering first the displacement functions for any position on the surface of an unloaded half-space due to a harmonic point force. The influence of the foundation has been taken into account by assuming a relaxed condition at the interface, i.e. a uniform displacement under the foundation and that the sum of the point forces must equal the total applied force. The three motions of vertical, horizontal and rocking have been considered and numerical values for the in-phase and the quadrature components of the displacement functions are presented for a Poisson's ratio of 0.25. The effect of the mass and inertia of the foundation can be allowed for by an impedance matching technique. Response curves and non-dimensional resonant frequency curves are given for a square and a rectangular foundation for different mass and inertia ratios and for several values of Poisson's ratio. These curves are for design purposes and are an addition to similar published curves for circular and infinitely long rectangular foundations. Some of the calculated results have been verified by a laboratory experiment.     

Three-dimensional elastic wave scattering and diffraction due to a rigid cylinder embedded in an elastic medium by a point source

The formal solutions of displacement field to the problem of elastic wave scattering and diffraction due to an infinitely long rigid cylinder embedded in an infinite elastic medium by an impulsive point source have been obtained in the integral form. The integrals for the reflected and the diffracted waves both in the shadow zone and in the illuminated zone are evaluated asymptotically for the early time motion by the Reisdue-Cagniard method and the Saddle-point-Cagniard method.Numerical results of the diffractedP, S andPS waves at a fixed circum-distance from the surface of the rigid cylinder show noticeably that (1) the energy partition for the diffractedS wave is small in comparison with that for the diffractedP wave, (2) the wave form of the diffractedS wave is broader and more diffused than that of the diffractedP wave, (3) the direction of the radial motions of the diffractedP andS waves varies as a function of the observational point, and (4) the energy partition for the diffractedP wave is much smaller than that for the direct or the reflectedP waves.This paper has been presented at the 46th Annual International Meeting of Society of Exploration Geophysicists in Houston, Texas, Oct. 28, 1976.     

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.     

Dynamic stiffness of rigid rectangular foundations on the half-space

The dynamic soil–structure interaction of a rigid rectangular foundation with the subsoil represents a mixed-boundary value problem. This problem is formulated in terms of a system of coupled Fredholm integral equations of the first kind. The subsoil is modelled by a homogeneous, linear-elastic and isotropic half-space which is perfectly bonded to the rigid, rectangular foundation. An approximate solution for the resultant loads between the foundation and the half-space due to a unit forced displacement or rotation is obtained using the Bubnov–Galerkin method. Using this method the displacement boundary value conditions are exactly satisfied and the contact stress distributions between the foundation and the half-space are approximated by series expansions of Chebyshev polynomials. This method provides a simple means of studying the soil-structure interaction of rectangular foundations with different inertia properties.     

Effects of the wave passage and the embedment depth for in-plane building-soil interaction

Building foundation-soil interaction is studied in the frequency domain using a two-dimensional analytical model. The building is represented by an infinitely long shear wall resting on a circular foundation, embedded into an elastic homogeneous half-space. Deep and shallow foundations are considered (with depth-to-half-width ratios of 1 and 0·5). Both the dynamic interaction and the wave passage effects are included. The excitation is a plane P- or SV-wave,or a surface Rayleigh wave. The results show that for incident waves which are long relative to the width of the foundation, the foundation driving forces are larger when the embedment is deeper. For shorter incident waves, the input base rotation is larger for shallow foundations and, therefore, the relative building response may then be larger. It is also shown that the input base rotation may contribute significantly to the building excitation and that neglecting it may cause nonconservative estimates for the forces in the building.     

Response of L-shaped rigid foundations embedded in a uniform half-space to traveling seismic waves

A simplified indirect boundary element method is applied to compute the impedance functions for L-shaped rigid foundations embedded in a homogeneous viscoelastic half-space. In this method, the waves generated by the 3D vibrating foundation are constructed from radiating sources located on the actual boundary of the foundation. The impedance functions together with the free-field displacements and tractions generated along the soil–foundation interface are used to calculate the foundation input motion for incident P, S and Rayleigh waves. This is accomplished by application of Iguchi's averaging method which, in turn, is verified by comparison with results obtained rigorously using the relation between the solutions of the basic radiation (impedance functions) and scattering (input motions) problems. Numerical results are presented for both surface-supported and embedded foundations. It is shown how the seismic response of L-shaped foundations with symmetrical wings differs from that of enveloping square foundations. The effects of inclination and azimuth of the earthquake excitation are examined as well. These results should be of use in analyses of soil–structure interaction to account for the traveling wave effects usually overlooked in practice.     

Out-of-plane (SH) soil-structure interaction: Semi-circular rigid foundation revisited

The model studied in this paper presents an extension of previous work for a shear wall on a semi-circular rigid foundation in an isotropic homogeneous and elastic half-space. The objective is to develop a soil-structure interaction model that can later be applied to the case of a flexible foundation. As shown in the Introduction below, Luco considered the case of a rigid foundation subjected to vertical incident plane SH waves, and Trifunac extended the solution for the same rigid foundation subjected to SH waves but for arbitrary angles of the incidence. In this paper, a new approach and model are presented for the same semi-circular rigid foundation with a tapered-shape (instead of rectangular) superstructure. The analytical expression for the deformation of the semi-circular rigid foundation below this tapered shear wall with soil-structure interaction in an isotropic homogeneous and elastic half-space is thus derived. Results are then compared with those of Trifunac discussed in the section below. This problem formulation can and will later be extended in the case of a flexible foundation that is semi-circular or arbitrarily shaped.     

Displacement solutions for dynamic loads in transversely-isotropic stratified media

Solutions for the displacements caused by dynamic loads in a viscoelastic transversely-isotropic medium are derived. The medium extends horizontally to infinity, but is bounded below by a rigid base. Stratification of the medium presents no difficulties. The medium is discretized in the vertical direction only; discretization in the horizontal direction is obviated by use of analytical solutions to the equations of motion. Application of the displacement solutions to soil-structure interaction is illustrated. A soil flexibility matrix (and hence, a stiffness matrix) for a surface foundation follows directly from the displacement solutions. A simple modification to obtain the soil stiffness for an embedded foundation of arbitrary geometry is described. Stiffnesses of rigid surface and embedded foundations are computed and compared with previously published results. In addition, the dynamic stiffness of a rigid surface foundation on a soil layer with linearly increasing shear modulus is compared to that for a homogeneous soil layer. A reduction in radiation damping is found to result from the inhomogeneity.     

Scattering of shear waves by an elliptical cavity in a radially inhomogeneous isotropic medium

Complex function and general conformal mapping methods are used to investigate the scattering of elastic shear waves by an elliptical cylindrical cavity in a radially inhomogeneous medium. The conformal mappings are introduced to solve scattering by an arbitrary cavity for the Helmholtz equation with variable coefficient through the transformed standard Helmholtz equation with a circular cavity. The medium density depends on the distance from the origin with a power-law variation and the shear elastic modulus is constant. The complex-value displacements and stresses of the inhomogeneous medium are explicitly obtained and the distributions of the dynamic stress for the case of an elliptical cavity are discussed. The accuracy of the present approach is verified by comparing the present solution results with the available published data. Numerical results demonstrate that the wave number, inhomogeneous parameters and different values of aspect ratio have significant influence on the dynamic stress concentration factors around the elliptical cavity.     

Rocking vibrations of a rigid circular foundation on poroelastic half-space to elastic waves

A study is carefully conducted for the rocking response of a rigid circular foundation resting on a poroelastic half-space when subjected to seismic waves under the framework of Biot’s theory. The free-field waves, rigid-body scattering field waves and radiation scattering field waves are introduced to consider the complex behavior of the soil owing to the scattering phenomena caused by the existence of the foundation. The contact surface between the soil and the foundation is supposed to be perfectly bonded and fully permeable. Combining with the divided wave fields, two sets of dual integral equations elaborating the mixed boundary-value conditions are established, and then reduced to Fredholm integral equations. Therefore, with a semi-analytical method, the expressions of the rocking displacements are obtained. The numerical results of the rocking vibration of the foundation for incident P, SV and Rayleigh waves are presented. The influences of certain parameters, such as the permeability of the soil, the incident angle, Poisson’s ratio and the mass of the foundation, on the rocking vibration of the foundation are explored and studied. Different reactions are found when the foundation is excited by different waves.     

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.     

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.     

基础柔性对土-结构相互作用系统响应影响的一个解析解

采用波函数展开法,通过SH波入射均匀半空间中二维埋置半圆形刚柔复合基础-单质点模型,推导土-刚柔复合基础-上部结构动力相互作用的解析解,并验证解的正确性。研究表明:基础柔性对于系统响应峰值与系统频率有较大影响。考虑基础柔性后,上部结构相对响应峰值相比全刚性基础结果均有一定减小,且系统频率也会产生向低频偏移的现象。     

Response of a circular foundation on a uniform half-space to elastic waves

An alternative technique to obtain the dynamic response of a massless rigid circular foundation resting on a uniform elastic half-space when subjected to harmonic plane waves is presented. The technique relies on the use of an integral representation involving the free-field ground motion and the contact tractions obtained in the course of calculating the dynamic force–displacement relationship of the foundation for external forces. Tables listing the translational and rotational components of the response of the foundation for non-vertically incident SH, P, SV and Rayleigh waves are presented.     

Vibrations of square and circular foundations with concentric openings on elastic half space

A study on the dynamic characteristics of rigid foundations with special geometries such as square or circular with concentric internal holes, is presented. The foundations are resting on a homogeneous, linear elastic halfspace and are subjected to external forces or seismic wave excitation. Both ‘relaxed’ and ‘non-relaxed’ boundary conditions at the interface between the foundation and the halfspace are considered, and several parametric studies are conducted to assess the influence of either type of boundary conditions upon each of the possible modes of vibration. Results for massive and massless foundations are presented in time and frequency domains for impulsive and harmonic excitations, respectively. A time domain boundary element method (BEM) developed by the authors for the solution of a class of 3-D soil-structure interaction (SSI) problems is used for all the analyses reported in this work. The accuracy and efficiency of the method and the BEM models developed in this work are assessed on the basis of comparison studies with published results.     

Soil-structure interaction at higher frequency factors

Approximate solutions to the forced vibrations of a rigid circular plate attached to the surface of an elastic halfspace are presented for large values of the frequency factor. These results are important when solving soil-structure interaction problems when such problems involve high-frequency factors. This situation arises when high-frequency components of earthquakes are associated with a relatively rigid foundation of a large base area and located on a soft terrain. Similar situations occur in cases of blast loadings and impact and in the foundations of large high-speed machinery. These solutions are used to solve the problem of the motion of a rigid mass on an elastic halfspace subjected to steady state and transient horizontal accelerations. From these results, it is deduced that a large rigid mat foundation located on soft terrain significantly attenuates input accelerations and consequently may be useful as the foundation of large massive rigid structures such as a nuclear power station.     

Dynamic stiffness matrices for two circular foundations

A systematic procedure is presented for generating dynamic stiffness matrices for two independent circular foundations on an elastic half-space medium. With the technique reported in References 1–3, the analytic solution of three-dimensional (3D) wave equations satisfying the prescribed traction due to the vibration of one circular foundation can be found. Since there are two analytic solutions for two prescribed tractions due to the vibrations of two circular foundations, the principle of superposition must be used to obtain the total solution. The interaction stresses (prescribed tractions) are assumed to be piecewise linear in the r-directions of both cylindrical co-ordinates for the two circular foundations. Then, the variational principle and the reciprocal theorem are employed to generate the dynamic stiffness matrices for the two foundations. In the process of employing the variational principle, a co-ordinate transformation matrix between two cylindrical co-ordinate systems is introduced. Some numerical results of dynamic stiffness matrices for the interaction of two identical rigid circular foundations are presented in order to show the effectiveness and efficiency of the present method, and some elaborations for its future extensions are also discussed.