Dynamic response of flexible rectangular foundations on an elastic half-space

An approximate method for the analysis of the dynamic interaction between a flexible rectangular foundation and the soil with consideration of the out-of-plane deformation of the foundation is presented. The procedure is based on an extension of the subdivision method developed by Wong and Luco for rigid foundations. Numerical results describing the influence of the flexibility of the foundation on the vertical and rocking impedance functions and on the contact stresses between the foundation and the soil are presented. The possibility of representing a flexible foundation by an equivalent rigid foundation having the same force-displacement relationships is also discussed. The results obtained indicate that at low frequencies, the dynamic stiffness coefficients for flexible foundations are lower than those for a rigid foundation of the same area. At higher frequencies the opposite behaviour is observed. The radiation damping coefficients for flexible foundations are significantly lower than those for a rigid foundation of the same area.     

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.     

表面矩形基础阻抗函数的集中参数模型

本文在系统地研究了Ｗｏｌｆ二阶集中参数模型的基础上,完整地给出了表面刚性矩形基础阻抗函数的集中参数模型,同以往的工作相比,本文给出的集中参数模型能在更宽的频段上反映精确解的变化。     

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.     

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 stiffness of deep foundations with inclined piles

The influence of inclined piles on the dynamic response of deep foundations and superstructures is still not well understood and needs further research. For this reason, impedance functions of deep foundations with inclined piles, obtained numerically from a boundary element–finite element coupling model, are provided in this paper. More precisely, vertical, horizontal, rocking and horizontal–rocking crossed dynamic stiffness and damping functions of single inclined piles and 2 × 2 and 3 × 3 pile groups with battered elements are presented in a set of plots. The soil is assumed to be a homogeneous viscoelastic isotropic half‐space and the piles are modeled as elastic compressible Euler–Bernoulli beams. The results for different pile group configurations, pile–soil stiffness ratios and rake angles are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Dynamic stiffness of foundation on layered soil half-space using cone frustums

Approximate dynamic-stiffness coefficients of a disk on the surface of a single layer on a half-space may be calculated using cone models. This concept is generalized to the case of a horizontally stratified site consisting of many layers on a homogeneous half-space. After constructing the so-called ‘backbone cone’ determining the radii of the disks at all interfaces, the dynamic-stiffness matrices of the layers (modelled as cone frustums) and the dynamic-stiffness coefficient of the underlying half-space (modelled as a cone) are assembled to that of the site. The dynamic-stiffness matrix of a layer is a complex-valued function of frequency because radiation of energy in the horizontal direction is considered. In this model of the layered half-space the properties of the cone reproduce themselves (cloning). The advantages of using cone models are also present for the layered half-space; in particular, no transformation to the wave-number domain is performed.     

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.     

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.     

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.     

Dynamic response of rectangular foundations to obliquely incident seismic waves

A study is made of the harmonic response of a rigid massless rectangular foundation bonded to an elastic half-space and subjected to the action of both external forces and obliquely incident plane seismic waves. The associated mixed boundary value problem is discretized and solved numerically. The results obtained indicate that the angle of incidence of the seismic wave has a marked effect on the nature and magnitude of the foundation response.     

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.     

Nonlinear rocking stiffness of foundations

The response of surface foundations to large overturning moments is studied under undrained conditions. Rigid circular, strip, and rectangular footings of various aspect ratios are considered, with the soil modeled as an inelastic homogeneous deposit, characterized by an elastic (small-strain) shear modulus G_o, an undrained shear strength S_u, and a G/G_o versus γ curve appropriate for medium-plasticity clays. Three stages of foundation performance, ranging from the initial elastic fully-bonded response, to the nearly-elastic but nonlinear response with the foundation partially detaching and uplifting from the soil, and finally to the ultimate stage where full mobilization of soil bearing failure mechanisms develop. Simple to use formulas or charts are developed for all stages of response in terms of dimensionless parameters, prominent among which is the static factor of safety against bearing-capacity failure under purely-vertical loading.     

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.     

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 interaction between 3-D rigid surface foundations and comparison with the ATC-3 provisions

The dynamic behaviour of a system of three-dimensional, massless, rigid, surface foundations of arbitrary shape perfectly bonded to the elastic half-space is numerically studied with the frequency domain boundary element method. This method employs the dynamic Green's function for the surface of the half-space and this results in a discretization of only the soil-foundation interfaces. In addition, use of isoparametric quadratic quadrilateral boundary elements increases the accuracy of the method, which is confirmed by comparison with other known numerical solutions. Externally applied loads, harmonically varying with time, are considered. The through the soil coupling effect between the foundations as a function of distance and frequency is assessed through extensive parametric studies involving two and four rigid foundations being isolated or interconnected. It is found that the assertion of ATC-3 regulations that omission of coupling effects leads to conservative results is not always correct for all frequencies.     

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.     

Dynamic response of 3-D rigid surface foundations by time domain boundary element method

The dynamic response of three-dimensional rigid surface foundations of arbitrary shape is numerically obtained. The foundations are placed on a linear elastic, isotropic and homogeneous half-space representing the soil medium and are subjected to either external dynamic forces or seismic waves of various kinds and directions, with a general transient time variation. The problem is formulated in the time domain by the boundary element method and the response is obtained by a time step-by-step integration. Two examples dealing with three-dimensional rectangular foundations are presented in detail, together with comparisons with other methods, in order to document the accuracy of the method. The main advantages of the proposed method are that, unlike frequency domain techniques, it provides directly the transient response and forms the basis for extension to the case of non-linear behaviour.