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.     

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 response of axisymmetric embedded foundations

The boundary element method is used to obtain dynamic stiffness functions of rigid cylindrical foundations embedded in a uniform or layered viscoelastic half-space. Dynamic stiffness functions of hemispherical foundations embedded in a uniform half-space are also computed. The direct integral equation formulation is used in combination with the complete space point load fundamental solution that is integrated numerically along the azimuthal coordinate. The approach is easy to implement because of the simplicity of the fundamental solution. The numerical results obtained by this method for cylindrical and hemispherical foundations are very close to corresponding published results obtained by different procedures. A parametric study shows the important effects of the Poisson's ratio on the dynamic stiffness functions of cylindrical foundations embedded in a uniform viscoelastic half-space. The effect of the bedrock compliance on the stiffness functions is also shown in the case of cylindrical foundations embedded in a soil layer that rests on a bedrock.     

A response-based simplified model for vertical vibrations of embedded foundations

A simplified damped oscillator model is proposed to simulate unbounded soil for the vertical vibration analysis of rigid embedded foundations. Based on the dynamic responses of a foundation–soil system, an optimal equivalent model is determined as the best simplified model. Magnification responses of a foundation–soil system simulated by the optimal equivalent model are well consistent with those obtained by the half-space theory and by a widely used computer program even as embedment depth or vibrating mass increases. The optimal equivalent model utilizing only three parameters can result in responses as accurate as the existing models, which use more parameters. This proposed method uses much simpler procedure than optimization techniques used by most existing discrete models. This proposed method may also be easily and accurately applied to practical soil–structure interaction analysis.     

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.     

Dynamic behaviour of structures with embedded foundations

A study of the dynamics of building-soil interaction is presented that includes embedding of the foundation and material damping. By considering buildings on rigid footings embedded into linear elastic soil with hysteretic damping, it is shown that the earthquake response of the building-foundation model may be found from the response to modified excitation of equivalent one-degree-of-freedom linear, viscously damped oscillators resting on rigid ground. For a single-storey building approximate formulas are developed for the modified natural frequency and damping ratio. Results show that the natural frequency and damping in the system increase with embedding. Effective damping also increases with internal friction in the soil. Ignoring these two factors may underestimate considerably the effective natural frequency and damping in the system. In spite of additional sources of energy dissipation provided by the soil, damping in the equivalent oscillator may be greater or smaller than that corresponding to the superstructure alone, depending upon the system parameters. For lightly damped superstructures, the peak amplitude of the steady-state overturning moment at the base of a building supported on flexible soil is significantly smaller than that corresponding to rigid ground. This result has practical implications for earthquake design.     

Dynamic interaction between adjacent foundations

The results of a parametric study on the dynamic interaction between adjacent foundations are presented. Addressed in this study are the effects of distance, direction of alignment, embedment and structural inertia. Several conclusions are drawn regarding the significance of each of these factors.     

Impedance matrices for circular foundation embedded in layered medium

A numerical scheme is developed in the paper for calculating torsional, vertical, horizontal, coupling and rocking impedances in frequency domain for axial-symmetric foundations embedded in layered media. In the scheme, the whole soil domain is divided into interior and exterior domains. For the exterior domain, the analytic solutions with unknown coefficients are obtained by solving three-dimensional (3D) wave equations in cylindrical coordinates satisfying homogeneous boundary conditions. For the interior domain, the analytical solutions are also obtained by solving the same 3D wave equations satisfying the homogeneous boundary conditions and the prescribed boundary conditions. The prescribed conditions are the interaction tractions at the interfaces between embedded foundation and surrounding soil. The interaction tractions are assumed to be piecewise linear. The piecewise linear tractions at the bottom surface of foundation will be decomposed into a series of Bessel functions which can be easily fitted into the general solutions of wave equations in cylindrical coordinates. After all the analytic solutions with unknown coefficients for both interior and exterior domains are found, the variational principle is employed using the continuity conditions (both displacements and stresses) at the interfaces between interior and exterior domains, interior domain and foundation, and exterior domain and foundation to find impedance functions.     

A discrete model for response estimation of soil-structure systems with embedded foundations

The need for simplifi ed physical models representing frequency dependent soil impedances has been the motivation behind many researches throughout history. Generally, such models are generated to capture impedance functions in a wide range of excitation frequencies, which leads to relatively complex models. That is while there is just a limited range of frequencies that really in? uence the response of the structure. Here, a new methodology based on the response-matching concept is proposed, which can lead to the development of simpler discrete models. The idea is then used to upgrade an existing simple model of surface foundations to the case of embedded foundations. The applicability of the model in both frequency domain and time domain analyses of soil-structure systems with embedded foundations is discussed. Moreover, the accuracy of the results is compared with another existing discrete model for embedded foundations.     

Dynamic interaction between flexible strip foundations

This paper is concerned with the dynamic response of a system of flexible strip foundations resting in smooth contact with a homogeneous isotropic viscoelastic half space. An arbitrary number of foundations with different flexibilities and geometries subjected to time-harmonic distributed loadings are considered in the formulation. The response of each strip foundation is governed by the classical plate theory and its transverse deflection is represented by an admissible function containing a set of generalized co-ordinates. A coupled variational-Green's function scheme is employed to establish the equations of motion of the strip foundation system. The numerical stability and convergence of the solution scheme are established. The influence of the foundation flexibility, distance between adjacent foundations and frequency of motion on the response of the foundation system is investigated in the numerical study.     

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.     

Study of vibrating foundations considering soil-pile-structure interaction for practical applications

An investigation of soil-pile-structure interaction is carried out, based on a large reciprocating compressor installed on an elevated concrete foundation （table top structure）. A practical method is described for the dynamic analysis, and compared with a 3D finite element （FE） model. Two commercial software packages are used for dynamic analysis considering the soilpile-structure interaction （SPSI）. Stiffness and damping of the pile foundation are generated from a computer program, and then input into the FE model. To examine the SPSI thoroughly, three cases for the soil, piles and superstructure are considered and compared. In the first case, the interaction is fully taken into account, that is, both the superstructure and soil-pile system are flexible. In the second case, the superstructure is flexible but fixed to a rigid base, with no deformation in the base （no SSI）. In the third case, the dynamic soil-pile interaction is taken into account, but the table top structure is assumed to be rigid. From the comparison beteen the results of these three cases some conclusions are made, which could be helpful for engineering practice.     

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.     

Modal analysis of circular flexible foundations under vertical vibration

A methodology using modal analysis is presented to evaluate dynamic displacements of a circular flexible foundation on soil media subjected to vertical vibration. The interaction effects between the foundation and the underlying soil are represented using modal soil impedance functions determined by an efficient procedure developed. The displacements of the foundation can then be easily solved by modal superposition. Comparing with existing solutions, the presented method is found to provide accurate results with less computational effort using only a few vibration modes. In addition, parametric studies for modal responses of the flexible foundation indicate that the response of the foundation are significantly influenced by relative stiffness between the foundation and the soil medium, load distributions, vibration frequency range, and the foundation mass. Besides, justification for flexible foundations to be considered as rigid are investigated.     

Boundary differential equation method: simplified dynamic soil stiffnesses for embedded rigid foundations

With a simplified model and Galerkin's weighted residual procedure, two simple differential equations of dynamic behavior of a bounded rectangular medium are established along the boundaries in the x- and y-direction in the medium. Solutions of these equations yield closed form expressions of soil stiffnesses for various cases of a partially embedded rigid foundation, including the stiffnesses per depth of foundation with rectangular base area and the stifnesses of strip foundation. The developed procedure provides the definition of the weight functions, which are used in Galerkin's method for weighted residual. In addition to these weight functions, their conjugators are also suitable for weight functions. When the soil depth is finite, the original weight functions fail to produce physically meaningful results in some frequency range but the conjugators do not fail at any frequencies. The developed equations to compute soil stiffnesses for embedded foundations are simple yet capable of calculating the responses close to those computed by the much more elaborated finite element method.     

Dynamics of rigid strip foundations embedded in orthotropic elastic soils

This study is concerned with the dynamic response of an arbitrary shaped rigid strip foundation embedded in an orthotropic elastic soil. The foundation is subjected to time-harmonic vertical, horizontal and moment loadings. The boundary-value problem related to an embedded foundation is analysed by using the indirect boundary integral equation method. The kernel functions of the integral equations are displacement and traction Green's functions of an anisotropic elastic half plane. Exact analytical solutions are used for the Green's functions. The boundary integral equation is solved by using numerical techniques. Selected numerical results are presented for the impedances of rectangular and semi-circular rigid strip foundations embedded in four types of anisotropic soils. A discussion on the influence of soil anisotropy and frequency of excitation on the impedances is presented. The versatility of the analysis is demonstrated by considering the through soil interaction between two semi-circular strip foundations.     

Torsional vibrations of elastic foundations embedded in an elastic half-space

This paper examines the axisymmetric torsional vibrations of an elastic pile and a hemispherical foundation embedded in a homogeneous elastic half-space. The embedded foundation–half-space system is decomposed into an extended half-space and a fictitious foundation. The deformations of the fictitious system are specified by an admissible function containing a set of generalized coordinates. The Lagrangian equations of motion are used to determine these coordinates associated with the assumed displacement function. Numerical results are presented for torsional impedance of an elastic pile and a hemisphere to illustrate the effects of relative flexibility and geometry. By employing certain simplifications on the pile–half-space system an approximate closed form solution is presented for the torsional impedance of an elastic pile.     

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.     

Plain strain soil–structure interaction model for a building supported by a circular foundation embedded in a poroelastic half-space

A simple theoretical model for soil–structure interaction in water saturated poroelastic soils is presented, developed to explore if the apparent building–foundation–soil system frequency changes due to water saturation. The model consists of a shear wall supported by a rigid circular foundation embedded in a homogenous, isotropic poroelastic half-space, fully saturated by a compressible and inviscid fluid, and excited by in-plane wave motion. The motion in the soil is governed by Biot's theory of wave propagation in fluid saturated porous media. Helmholtz decomposition and wave function expansion of the two P-wave and the S-wave potentials is used to represent the motion in the soil. The boundary conditions along the contact surface between the soil and the foundation are perfect bond (i.e. welded contact) for the skeleton, and either drained or undrained hydraulic condition for the fluid (i.e. pervious or impervious foundation). For the purpose of this exploratory analysis, the zero stress condition at the free surface is relaxed in the derivation of the foundation stiffness matrix, which enables a closed form solution. The implications of this assumption are discussed, based on published comparisons for the elastic case. Also, a closed form representation is derived for the foundation driving forces for incident plane (fast) P-wave or SV wave. Numerical results and comparison with the full-scale measurements are presented in the companion paper, published in this issue.