Dynamic analysis of 3-D flexible embedded foundations by a frequency domain BEM-FEM

A study on the dynamic response of three-dimensional flexible foundations of arbitrary shape, embedded in a homogenous, isotropic and linear elastic half-space is presented. Both massive and massless foundations are considered. The soil-foundation system is subjected to externally applied forces, and/or to obliquely incident seismic waves. The numerical method employed is a combination of the frequency domain Boundary Element Method, which is used to simulate the elastic soil medium, and the Finite Element Method, on the basis of which the stiffness matrix of the foundation is obtained. The foundation and soil media are combined by enforcing compatibility and equilibrium conditions at their common interface. Both relaxed and completely bonded boundary conditions are considered. The accuracy of the proposed methodology is partially verified through comparison studies with results reported in the literature for rigid embedded foundations.     

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.     

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.     

Antiplane SH-deformations near a surface rigid foundation above a subsurface rigid circular tunnel

The problem on the dynamic response of a rigid embedded foundation in the presence of an underground rigid tunnel and subjected to excitation of incident anti-plane SH waves is analyzed. By using the exact analytical solution for the two-dimensional SH-wave propagation in and around both the surface rigid foundation and subsurface rigid tunnel, those aspects of the resulting ground motions that are of special interest and importance for seismic resistant design in earthquake analyses have been examined. The computed amplitudes of the resulting periodic ground motions display a very complicated wave-interference between the surface foundation and underground tunnel that lead to observed standing wave patterns, together with abrupt changes in the wave amplitudes and large amplification of the incident motions. Supported by: National Science Foundation grant CMS 97-14859     

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.     

不规则场地上基础的动阻抗及其在入射波作用下的输入运动

本文根据边界元方法建立了位不规则场上刚体的动阻抗和在入射平面波作用下的有效输入运动的分析模型，分析模型考虑了不规则场地和基础对入射波的散射作用以及土与基础的相互作用，通过验证确认了本方法的正确性，文中计算了凹陷，高地和盆地三种不规则场地土不同条件基础的动阻和有效输入的运动，并与半空间地基上相应基础的情况作了对比，计算表明，当基础尺寸与不规则场地范围可比时有必要用本文模型分析不规则场地的影响和土一结     

Dynamic response of flexible strip-foundations by boundary and finite elements

A time domain Boundary Element-Finite method is employed to determine the dynamic response of flexible surface two-dimensional foundations under conditions of plane strain placed on an elastic soil medium and subjected either to transient external forces or to obliquely incident seismic waves. The elastic, isotropic, and homogeneous soil medium is treated by the time domain Direct Boundary Element Method, while the flexible foundation is treated by the Finite Element Method. The two methods are appropriately combined through equilibrium and compatibility considerations at the soil-foundation interface. Parametric studies examining the effect of the relative stiffness between the foundation and the soil and the spatial distribution of the dynamic disturbances on the foundation response are presented.     

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.     

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.     

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.     

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 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.     

Response of pile foundations to rayleigh waves and obliquely incident body waves

The paper presents results of a study on the harmonic response of piles and pile groups embedded in a halfspace to various forms of seismic waves. These include the Rayleigh wave as well as obliquely incident P, SV and SH waves. The pertinent mixed boundary value problems of pile-soil-pile interaction are solved by a numerical model of the boundary integral nature. All modes of foundation vibrations, i.e. translational, rocking and torsional, are included in the model. The results presented are used to highlight the salient features of the seismic response of piles. In addition, the influence of certain pile-soil parameters, such as pile rigidity and pile spacing, on the seismic behaviour of pile foundations is investigated.     

Response of seismically isolated structures to rocking‐type excitations

In this paper, the origin of rocking‐type excitations and their effects on the response of base isolated structures are studied. In particular, the role of kinematic interaction in the determination of the rocking excitation is highlighted. The cases of surface foundations subjected to horizontally propagating waves, as well as of embedded foundations under vertically incident shear waves are examined. The validity of the kinematic interaction based on the rigid base mat assumption is discussed. It is shown that, in the case of classical horizontal isolation, rocking input may amplify significantly the response of the lower non‐isolated modes. The examination of full three‐dimensional isolation and active and semi‐active control methods demonstrates the efficacy of these methods to improve the performance of seismically isolated structures subjected to rocking excitations. Copyright © 2009 John Wiley & Sons, Ltd.     

Kinematic soil–structure interaction for long-span cable-supported bridges

A general procedure is presented to study the dynamic soil–structure interaction effects on the response of long-span suspension and cable-stayed bridges subjected to spatially varying ground motion at the supporting foundations. The foundation system is represented by multiple embedded cassion foundations and the frequency-dependent impedance matrix for the multiple foundations system takes into account also the cross-interaction among adjacent foundations through the soil. To illustrate the potential implementation of the analysis, a numerical example is presented in which the dynamic response of the Vincent–Thomas suspension bridge (Los Angeles, CA) subjected to the 1987 Whittier earthquake is investigated. Although both kinematic and inertial effects are included in the general procedure, only the kinematic effects of the soil–structure interaction are considered in the analysis of the test case. The results show the importance of the kinematic soil–foundation interaction on the structural response. These effects are related to the type, i.e. SH-, SV-, P- or Rayleigh waves and to the inclination of the seismic wave excitation. Moreover, rocking components of the foundation motion are emphasized by the embedment of the foundation system and greatly alter the structural response.     

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.     

On the relation between radiation and scattering problems for foundations embedded in an elastic half-space

The relation between the solutions of the boundary-value problems for the dynamic response of embedded foundations to external forces and seismic excitation is examined. It is shown that the contact tractions obtained in the course of calculating the response to external forces can also be used to determine the response of the foundation to an arbitrary seismic excitation.     

An approximate method for soil-structure interaction for sh-waves—the born approximation

An approximate method is proposed for the scattering of SH-waves by foundations of irregular shape and the resulting soil-structure interaction problems. The scattering of elastic waves by the rigid foundation embedded in half-space is solved approximately by using integral representation of the wave equation. The procedure is the Born approximation which has been widely used in quantum mechanics for collision and scattering theory though not well-known in elastodynamics. This paper extends the previous work of the authors on the scattering of waves to account for soil-structure interaction. The motion of the foundation is evaluated by the balance of momentum under stresses due to the incident waves as well as the waves generated by its own motion and the forces coming from the superstructure. The model investigated consists of an infinitely long elastic shear wall of height H and thickness h erected on a rigid infinitely long foundation. Results are presented for the cases with circular, elliptical and rectangular foundations. For a circular foundation, excellent agreement is found with the exact solutions for the foundation displacement and the relative displacement between the top and bottom of the structure for the entire range of wave numbers. For an elliptical foundation, accuracy decreases with increasing wave numbers. Foundation displacements are compared for foundation shapes that are shallow elliptical, deep elliptical, rectangular and circular. It is observed that foundation displacements are dependent on the angle of incidence except for a semi-circle. The results on the details of the scattered field are, however, not as accurate.     

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.     

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.