CALCULATING THE DYNAMIC STIFFNESS MATRIX OF 2-D FOUNDATIONS BY DISCRETE WAVE NUMBER INDIRECT BOUNDARY ELEMENT METHODS

Apart from some special cases, calculating the dynamic stiffness matrix of foundations on a layered half-space, especially for embedded foundations, is computationally expensive. An efficient method for two-dimensional foundations in a horizontally layered soil media is presented in this paper. This method is based on indirect boundary element methods and uses discrete wave number solution methods for calculating Green's functions for displacements and analytical methods for the integrations over the boundary. For surface foundations, the present method applies at all frequencies. For embedded foundations or for constructing energy transmitting boundaries, because the free-field part is modelled by boundary elements and the excavated part is modelled by finite elements, the present method applies only at low frequencies for the spring coefficients (the real parts of the dynamic stiffness matrix) but applies at all frequencies for the damping coefficients (the imaginary part of the dynamic stiffness matrix) for undamped sites. The novelty of the method can be used for three-dimensional foundations. © 1997 by John Wiley & Sons, Ltd.     

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.     

A coupled fluid layer–rigid disk–poroelastic half-space vibration problem

This paper proposes a coupled fluid layer–foundation–poroelastic half-space vibration model to study how still water affects foundations operating underwater. As an example, we consider the problem of the vertical vibration of a rigid disk on a poroelastic half-space covered by a fluid layer having a finite depth. The solution of the disk vibration problem is obtained using the boundary conditions at the free surface of the fluid layer and the boundary conditions at the fluid layer–poroelastic medium interface. The solution is expressed in terms of dual integral equations that are converted into Fredholm integral equations of the second kind and solved numerically. Selected numerical results for the vertical dynamic impedance coefficient are examined based on different water depths, poroelastic materials, disk permeabilities and frequencies of excitation. Based on the numerical results, it is proposed that the hydrodynamic pressure caused by the foundation vibration is the intrinsic reason that the existence of a fluid layer has such a great effect on the dynamic characteristics of the foundation. In many cases, the hydrodynamic pressure caused by the foundation vibration cannot be ignored when designing dynamic underwater foundations. These results are helpful in understanding the dynamic response of foundations under still water without water waves, such as foundations in pools, lakes and reservoirs.     

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.     

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.     

Application of a second-order paraxial boundary condition to problems of dynamics of circular foundations on a porous layered half-space

A half-space finite element and a consistent transmitting boundary in a cylindrical coordinate system are developed for analysis of rigid circular (or cylindrical) foundations in a water-saturated porous layered half-space. By means of second-order paraxial approximations of the exact dynamic stiffness for a half-space in plane-strain and antiplane-shear conditions, the corresponding approximation for general three-dimensional wave motion in a Cartesian coordinate system is obtained and transformed in terms of cylindrical coordinates. Using the paraxial approximations, the half-space finite element and consistent transmitting boundary are formulated in a cylindrical coordinate system. The development is verified by comparison of dynamic compliances of rigid circular foundations with available published results. Examination of the advantage of the paraxial condition vis-á-vis the fixed condition shows that the former achieves substantial gain in computational effort. The developed half-space finite element and transmitting boundary can be employed for accurate and effective analysis of foundation dynamics and soil–structure interaction in a porous layered half-space.     

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.     

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.     

地基半空间等效集总体系的比拟法与实测分析

先略述莱斯默比拟法的形成;再由半空间理论等效为质弹体系,得出辐射阻尼比、刚度及参振土质量,并论述两体系的结合;最后经实测、分析和使用,考虑土体非匀质性折减阻尼比以作修正,使其更为实用。这有助于消除在我国长期认为阻尼比大而不安全、不便使用的疑虑,以便推动半空间理论在我国的实用化。     

Vertical vibration of three-dimensional rigid foundations on layered media

A numerical method of analysis is presented for the determination of the steady-state vertical vibration of rigid foundations with arbitrary three-dimensional geometries resting on the surface of a layered soil medium. The method utilizes the flexibility concept applied to steady-state periodic problems and it is solved in the frequency domain. The accuracy of the method is verified by comparison with several published solutions for massless, smooth rigid rectangular foundations on a homogeneous, isotropic elastic half-space. Parametric solutions are presented to study the dynamic behaviour of massless, smooth rigid rectangular foundations on a homogeneous, elastic stratum.     

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.     

Axisymmetric soil–structure interaction by global–local finite elements

A version of the global–local finite element method is presented for studying dynamic steady-state soil–structure interaction wherein the soil medium extends to infinity. Herein, only axisymmetric behaviour is considered. In this approach, conventional finite elements are used to model the structure and some portion of the surrounding soil medium considered to be homogeneous and isotropic. A complete set of outgoing waves in the form of spherical harmonics for the entire space is used to represent the behaviour in the half-space beyond the finite element mesh and these are termed the global functions. Full traction and displacement continuity is enforced at the finite element mesh interface with the outer region. On the free surface of the half-space in the outer field, traction-free surface conditions are enforced by demanding that a sequence of integrals of the weighted-average tractions must vanish. Numerical examples are presented for the response of different shaped foundations, resting on the free surface or at various submerged levels, due to a normal seismic plane compressional wave. Plots of differential scattering cross-sections show the angular distribution of the energy (its directional nature) of the scattered field.     

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.     

Two- and three-dimensional modelling of half-space and train-track embankment under dynamic loading

Two modelling approaches for the analyses of half-space and train-track embankment on half-space subjected to dynamic loads are presented and compared. A three-dimensional (3D) modelling approach is performed by a coupled Boundary Element–Boundary Element method (BE–BE) and a two-dimensional (2D) one by a coupled Boundary Element–Finite Element method (BE–FE). Both approaches employ time domain algorithms. The comparison between the results of the presented approaches points out whether a problem can be treated as a 2D or as a 3D case. As an application, a parametric study of the wave propagation problem in a train-track embankment with an underlying half-space is presented.     

Effects of the dynamic cross-interaction in the seismic analysis of multiple embedded foundations

A boundary element formulation of the substructure deletion method is presented for the seismic analysis of the dynamic cross-interaction between multiple embedded foundations. This approach is particularly suitable for three-dimensional foundations of any arbitrary geometrical shape and spatial location, since it requires only the discretization of the foundations’ surfaces. The surrounding soil is represented by a homogeneous viscoelastic half-space while the foundations are assumed to be rigid and subjected to incoming SH-, P-, and SV-waves arbitrarily inclined in both the horizontal and vertical planes. The proposed methodology is tested for the case of two identical embedded square foundations for different values of the foundations’ embedment and distance. The effects of the cross-interaction are outlined in the components of the impedance matrix and of the foundation input motion. © 1997 John Wiley & Sons, Ltd.     

埋置基础扭转振动的实用化计算

根据弹性半空间理论及基础振动试验和数值计算的最新成果,采用方程对等法推演出埋置块体基础扭转振动的实用化计算公式。采用该实用化计算公式和弹性半空间理论分别对复杂形状的埋置块体基础在谐和扰力矩作用下的扭转振动动力响应进行计算,并将计算结果进行了比较,两者结果吻合,振幅误差为9.13%。该实用化计算公式具有概念明确,计算简单的优点,对于任意形状、任意埋置状况的动力设备块体基础都实用,对于结构工程的地震响应计算具有参考意义。     

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.     

Discrete model for dynamic through-the-soil coupling of 3-D foundations and structures

An efficient discrete model for predicting the dynamic through-the-soil interaction between adjacent rigid, surface foundations supported by a homogeneous, isotropic and linear elastic half-space is presented. The model utilizes frequency-independent springs and dashpots, and the foundation mass, for the consideration of soil–foundation interaction. The through-the-soil coupling of the foundations is attained by frequency-independent stiffness and damping functions, developed in this work, that interconnect the degrees of freedom of the entire system of foundations. The dynamic analysis of the resulting coupled system is performed in the time domain and includes the time lagging effects of coupled dynamic input due to wave propagation using an appropriate modification of the Wilson-θ method. The basic foundation interaction model is also extended to the evaluation of coupled building-foundation systems. © 1998 John Wiley & Sons, Ltd.     

Transient response of a group of rigid strip surface foundations

A numerical procedure is proposed to investigate the transient response of a group of rigid strip foundations resting on an elastic, homogeneous half-space subjected to either external forces or seismic motions. A fundamental solution is presented for uniform strip loadings with Heaviside function time-dependence applied on the half-space. In the procedure, each of the foundations is discretized into subelements. The tractions between the half-space and the subelements are assumed constant at every time step. The through-soil coupling effects between the foundations are studied numerically.     

DYNAMIC CROSS-INTERACTION BETWEEN FLEXIBLE SURFACE FOOTINGS BY COMBINED BEM AND FEM

A combined boundary and finite element method is developed and applied to study the dynamic behaviour of a system of flexible surface footings of arbitrary shape bearing on an elastic half-space. The proposed method employs the frequency domain Green's function for the surface of the elastic half-space while a layered plate model is used for the flexible footing. Both the footing and the surface of the half-space are discretized by 8-noded quadratical isoparametric elements, and the meshes are identical. Thus, the compatibility of displacements and equilibrium of forces between the footing and the half-space are fully satisfied. This model provides a better approximation of the stress concentration at edges of relatively rigid footings. Numerical examples demonstrating the effects due to the excitation frequency, the relative rigidity and the distance between footings on the interaction between two square footings are presented. The external forces can be either harmonic or transient.