Vertical vibration of ring foundations

A comprehensive analysis is made of the harmonic response of vertically excited, massless, rigid ring foundations supported at the surface of an homogeneous elastic halfspace. The parameters considered include the thickness to radius ratio for the ring, the frequency of the exciting force and of the resulting steady-state response, and Poisson's ratio for the supporting medium. The response quantities examined include the stiffness and damping coefficients of the system in an equivalent spring–dashpot representation of the supporting medium, the displacements of the foundation and of points on the ground surface away from the foundation, and the normal pressure at the foundation–medium interface. The results in each case are compared with those obtained for a solid disk having the same radius as the outer radius of the ring, and a simple approximate model is used to interpret the results and to provide insight into the action of the system. The reported data are computed by a method of analysis that takes due account of the mixed boundary conditions at the surface of the halfspace, and are believed to be of high accuracy.     

Dynamic response of surface foundations on layered media

The steady-state dynamic responses of both flexible and rigid massless foundations with complex geometries on layered media are studied by the finite layer method. The discussion covers both the horizontal and vertical responses including the rocking mode of vibration. The effect of various parameters, namely, the size of the domain, the number of harmonic terms and the material damping of the soil media, on the convergence and accuracy of the results is investigated. Non-homogeneous soil profiles are treated as a matter of course. The results obtained are compared with solutions by other theoretical methods.     

Rocking response of flexible circular foundations on layered media

The analysis of the response of a flexible circular foundation on layered media due to an arbitrarily distributed vertical loading is presented. The analysis is based on the ‘ring method’ approach, i.e. discretization of the foundation in a set of concentric rings. The arbitrarily distributed loading is expanded in the circumferential direction in a Fourier series. The influence coefficient matrix of soil for each element of the series is evaluated utilizing the stiffness matrix approach. The stiffness matrix of the foundation is obtained from the finite difference energy method approach. Numerical examples illustrate the influence of several soil-foundation parameters on the rocking response of a foundation. Results are presented in terms of displacement and soil reaction distributions and impedance functions point to significantly different responses of flexible and rigid foundations.     

Dynamic interaction between rigid foundations in a layered half-space

A computationally efficient boundary integral equation technique to calculate the dynamic response of a group of rigid surface foundations bonded to a layered viscoelastic half-space and subjected to external forces and seismic waves is presented. The technique relies on an iterative scheme which minimizes in-core memory requirements and takes advantage of any geometrical symmetry of the foundations. Extensive results for the case of two rigid square foundations placed at different separations and bonded to a viscoelastic half-space are presented. It was found that the choice of discretization of the foundations has a marked effect on the calculated impedance functions for extremely small separations. Illustrative results for a case of several closely-spaced foundations bonded to a layered half-space are also presented.     

Parametric study of vertical vibrations of circular flexible foundations on layered media

The paper presents results of a parametric study of vertical oscillations of a flexible circular plate on the surface of an elastic half-space and an elastic layered system. The solution of the problem is based on the ‘ring method’. Vertical oscillations have been analysed to determine the displacement and soil reaction distributions at the soil-plate interface and the impedance functions. Parameters of the study include material and geometrical properties of a soil system and a plate and the load distribution on the plate. The results indicate significantly different behaviour of a flexible plate from that of a rigid one. Based on the observed behaviour, a classification of plates has been suggested.     

Vertical and rocking impedances for rigid rectangular foundations on soils with bounded non-homogeneity

The vertical and rocking response of rigid rectangular foundations resting on a linear-elastic, compressible, non-homogeneous half-space soil model is studied. The non-homogeneity is described by a continuous yet bounded increase of shear modulus with depth. The mixed boundary value problem is solved by means of the semi-analytical method of the subdivision of the foundation/soil contact area whereby the influence functions for the sub-regions are determined by integration of the corresponding surface-to-surface Green's functions for the particular soil model. Impedance functions are given for representative values of the non-homogeneity parameters, the Poisson's ratio and the foundation geometry over a wide range of frequencies. Significant features associated with the soil non-homogeneity are pointed out. Copyright © 1999 John Wiley & Sons Ltd.     

层状饱和场地中圆柱形基础的阻抗函数

本文利用间接边界元法,建立层状饱和场地中三维土-基础动力相互作用模型,研究了孔隙中流体的存在、场地动力特性和圆柱形基础的高度对基础阻抗函数的影响。数值结果表明,孔隙中流体的存在对基础阻抗函数有明显的影响,尤其是水平和竖向分量;场地动力特性也对基础阻抗函数有明显的影响,层状饱和场地中阻抗函数围绕相应均匀半空间结果振荡,且基岩与土层刚度比越大或土层厚度越小,振荡的幅度越大,土层厚度越小,振荡的周期越小;另外,基础高度也对阻抗函数有明显的影响,随基础高度增大,阻抗函数逐渐增大。     

Vertical vibration of an embedded rigid foundation in a poroelastic soil

This paper considers time-harmonic vertical vibration of an axisymmetric rigid foundation embedded in a homogeneous poroelastic soil. The soil domain is represented by a homogeneous poroelastic half space that is governed by Biot's theory of poroelastodynamics. The foundation is subjected to a time-harmonic vertical load and is perfectly bonded to the surrounding half space. The contact surface can be either fully permeable or impermeable. The dynamic interaction problem is solved by employing an indirect boundary integral equation method. The kernel functions of the integral equation are the influence functions corresponding to vertical and radial ring loads, and a ring fluid source applied in the interior of a homogeneous poroelastic half space. Analytical techniques are used to derive the solution for influence functions. The indirect boundary integral equation is solved by using numerical quadrature. Selected numerical results for vertical impedance of rigid foundations are presented to demonstrate the influence of poroelastic effect, foundation geometry, hydraulic boundary condition along the contact surface and frequency of excitation.     

Stochastic response of rigid foundations

While the study of kinematic interaction (i.e. the dynamic response of massless foundations to seismic loads) calls, in general, for advanced analytical and numerical techniques, an excellent approximation was proposed recently by Iguchi.^1,2 This approximation was used by the authors to analyse embedded foundations subjected to spatially random SH-wave fields, i.e. motions that exhibit some degree of incoherence. The wave fields considered ranged from perfectly coherent motions (resulting from seismic waves arriving from a single direction) to chaotic motions, resulting from waves arriving simultaneously from all directions. Additional parameters considered were the shape of the foundation (cylindrical or rectangular) and the degree of embedment. It was found that kinematic interaction usually reduces the severity of the motions transmitted to the structure, and that incoherent motions do not exhibit the frequency selectivity (i.e. narrow valleys in the foundation response spectra) that coherent motions do.     

Impedances of embedded rigid strip foundations

This paper is concerned with the dynamic response of rigid strip foundations of arbitrary geometry embedded in a homogeneous elastic half-space. The embedded rigid foundation is modelled by an equivalent domain in a uniform half-space which is subjected to an appropriate body force field. The components of the impedance matrix are determined through the solution of a linear simultaneous equation system which is established by invoking rigid body displacements of discrete locations within the equivalent domain and appropriate equilibrium consideration. It is found that high numerical efficiency and flexibility can be achieved using the body force model when compared to boundary integral formulations through the selection of appropriate displacement influence functions and a ‘parent domain’ in the analysis. Numerical results are presented to illustrate the influence of the embedment ratio, frequency of excitation, foundation geometry and Poisson's ratio on the vertical, horizontal, rocking and coupled impedances of a single embedded foundation. The effect on the impedance due to the presence of an adjacent embedment is investigated for various distances between foundations and embedment ratios.     

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.     

Cone model for two surface foundations on layered soil

In this paper, the cone model is applied to the vibration analysis of two foundations on a layered soil half space. In the analysis, the total stress field in the subsoil is divided into the free-field and the scattering field. Seed＇s simplified method is adopted for the free-field analysis, while the cone model is proposed for analyzing the dynamic scattering stress wave field. The shear stress field and the compressive stress field in the layered stratum with two scattering sources are calculated by shear cone and compressive cone, respectively. Furthermore, the stress fields in the subsoil with two foundations are divided into six zones, and the P wave and S wave are analyzed in each zone. Numerical results are provided to illustrate features of the added stress field for two surface foundations under vertical and horizontal sinusoidal force excitation. The proposed cone model may be useful in handling some of the complex problems associated with multi-scattering sources.     

On rigid foundations subjected to seismic waves

In the analysis of structural foundations for seismic loads, it is customary to distinguish two types of soil-structure interaction effect: kinematic interaction (or wave passage), and inertial interaction. The former refers to the phenomenon of wave scattering, which occurs because the foundation is much stiffer than the surrounding soil and cannot accommodate to its distortions. Inertial interaction, on the other hand, is caused by feedback of kinetic energy of the structure into the soil. This paper is concerned only with the first phenomenon. The rigorous analysis of rigid, embedded foundations subjected to seismic disturbances requires, in general, substantial computational effort. Indeed, a typical analysis would normally require models with finite elements and/or boundary elements. Although such methods may be used to find an accurate solution to the problem of kinematic interaction, their use is not always warranted, given the many uncertainties involved and the multitude of assumptions that must be considered. Hence, approximate solutions are attractive for this problem. One such approximate method is the remarkably simple algorithm proposed by Iguchi.³ This paper presents first an appraisal of this method by way of a comparison with accurate numerical solutions for cylindrical foundations; next the algorithm is applied to rectangular (prismatic) foundations. It is found that Iguchi's method gives results that are adequate for engineering purposes, even if not entirely accurate.     

A note on the dynamic response of rigid embedded foundations

The problem of the dynamic response of rigid embedded foundations subjected to the action of external forces and seismic excitation is analysed. It is shown that to calculate the response of rigid embedded foundations, or the response of flat rigid foundations subjected to non-vertically incident seismic waves, it is necessary to obtain not only the impedance matrix for the foundation, but also the forces induced by the incident seismic waves. Under these general conditions, rocking and torsional motion of the foundation is generated in addition to translation. The case of a two-dimensional rigid foundation of semi-elliptical cross-section is used as an example to illustrate the effects of the embedment depth and angle of incidence of the seismic waves on the response of the foundation.     

Dynamic response of rigid foundations of arbitrary shape

An approximate numerical procedure for calculation of the harmonic force-displacement relationships for a rigid foundation of arbitrary shape placed on an elastic half-space is presented. This procedure is used to evaluate the vertical, rocking and horizontal compliance functions for rigid rectangular foundations and the vertical compliance for a rigid square foundation with an internal hole. Several comparisons between the results obtained by the proposed approach and other methods are also presented.     

Dynamic behaviour of rigid foundations of arbitrary shape on a halfspace

An approximate method for computation of the compliance functions of rigid plates resting on an elastic or visco-elastic halfspace excited by forces and moments in all degrees of freedon is presented. The method is based on a Green's function approach. These functions are given for all degrees of freedom in form of well-behaved integrals. The numerical procedure is described and is used to evaluate the vertical, horizontal, rocking and torsion compliance functions of rectangular plates with side ratios 1 ≤ b/a ≤ 10 and non-dimensional frequency 0≤a₀≤10. It is shown how this method can be extended to problems concerning a linear visco-elastic halfspace and a halfspace with variable stiffness.     

Seismic response of foundations embedded in a layered half-space

An integral equation technique to determine the response of foundations embedded in a layered viscoelastic half-space when subjected to various types of seismic waves is presented. The technique is validated by comparison with previous results for rigid hemispherical and cylindrical foundations embedded in a uniform half-space. Illustrative results for rigid cylindrical foundations embedded in layered media are also presented.