Winkler model for dynamic response of composite caisson-piles foundations: Lateral response

As the first part of a sequence focusing on the dynamic response of composite caisson-piles foundations (CCPFs¹), this paper develops a simplified method for the lateral response of these foundations. A Winkler model for the lateral vibration of the CCPF is created by joining the two components, the caisson and the pile group, where the four-spring Winkler model is utilized for the caisson and axial–lateral coupled vibration equations are derived for the pile group. For determining the coefficients of the four-spring Winkler model for the caissons, embedded footing impedance is used and a modification on the rotational embedment factor is made for the sake of the geometrical difference between shallow footings and caissons. Comparisons against results from finite element simulations demonstrate the reliability of this modified four-spring Winkler model for caissons in both homogenous and layered soils. The proposed simplified method for the lateral vibration of CCPFs is verified also by 3D finite element modeling. Finally, through an example, the idea of adding piles beneath the caisson is proved to be of great significance to enhance the resistance of the foundation against lateral dynamic loads.     

Seismic response of bridges with massive foundations

The dynamic response of vehicular overpasses with massive foundations built in highly populated earthquake prone regions is studied to assess the massive foundation potential of being a technically sound mean to reduce the structural response during major earthquakes. The study consists on numerical simulations using 3-D finite element models. Two typical supports of a major 23 km long vehicular overpass recently built in the north east part of Mexico City valley were considered in this research. This zone is characterized by the presence of stiff soils comprised by dense and very dense silty sands and sandy silts, randomly intercalated by stiff clays layers of variable thickness. Initially, a conventional raft foundation structurally connected to four precast closed-end concrete piles was considered. Then, the potential beneficial effect of a massive foundation of variable depth was assessed. Sets of 3-D finite element models were developed and the response of the systems was evaluated for a typical seismic scenario such as that prevailing at the zone, assuming a potential 8.1 M_w seismic event, and for a hypothetical 8.7 M_w extreme event. Important attenuations of about 39% to 48% at the upper deck spectral accelerations and of 21% to 30% in maximum lateral foundation displacements were achieved with the massive soil improvement for the cases analyzed. Thus, the massive foundations seem to be a convenient alternative to reduce the overall structural seismic response.     

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.     

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.     

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.     

Filtering action of embedded massive foundations: New analytical expressions and evidence from 2 instrumented buildings

This paper deals with the effect of the foundation mass on the filtering action exerted by embedded foundations. The system under examination comprises a rigid rectangular foundation embedded in a homogeneous isotropic viscoelastic half‐space under harmonic shear waves propagating vertically. The problem is addressed both theoretically and numerically by means of a hybrid approach, where the foundation mass is explicitly included in the kinematic interaction between the foundation and the surrounding soil, thus referring to a “quasi‐kinematic” interaction problem. Based on the results of an extensive parametric study, it is shown that the filtering problem depends essentially on three dimensionless parameters, i.e.: the dimensionless frequency of the input motion, the foundation width‐to‐embedment depth ratio, and the foundation‐to‐soil mass density ratio. In complements to the translational and rotational kinematic interaction factors that are commonly adopted to quantify the filtering effect of rigid massless foundations on the free‐field motion, an additional kinematic interaction factor is introduced, referring to the horizontal motion at the top of a rigid massive foundation. New analytical expressions for the above kinematic interaction factors are proposed and compared with foundation‐to‐free‐field transfer functions computed from available earthquake recordings on two instrumented buildings in LA (California) and Thessaloniki (Greece). Results indicate that the foundation mass can have a strong beneficial effect on the filtering action with increasing foundation‐to‐soil mass density and foundation width‐to‐embedment depth 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.     

Simplified approach for design of raft foundations against fault rupture. Part Ⅱ： soil-structure interaction

This is the second paper of two, which describe the results of an integrated research effort to develop a four-step simplified approach for design of raft foundations against dip-slip （normal and thrust） fault rupture. The first two steps dealing with fault rupture propagation in the free-field were presented in the companion paper. This paper develops an approximate analytical method to analyze soil-foundation-structure interaction （SFSI）, involving two additional phenomena： （i） fault rupture diversion （Step 3）; and （ii） modification of the vertical displacement profile （Step 4）. For the first phenomenon （Step 3）, an approximate energy-based approach is developed to estimate the diversion of a fault rupture due to presence of a raft foundation. The normalized critical load for complete diversion is shown to be a function of soil strength, coefficient of earth pressure at rest, bedrock depth, and the horizontal position of the foundation relative to the outcropping fault rupture. For the second phenomenon （Step 4）, a heuristic approach is proposed, which ＂scans＂ through possible equilibrium positions to detect the one that best satisfies force and moment equilibrium. Thus, we account for the strong geometric nonlinearities that govern this interaction, such as uplifting and second order （P-△） effects. Comparisons with centrifuge-validated finite element analyses demonstrate the efficacy of the method. Its simplicity makes possible its utilization for preliminary design.     

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.     

Equivalent fixed-base models for soil-structure interaction systems

To simplify the consideration of the soil-structure interaction (SSI) effects, a single degree-of-freedom (SDOF) replacement oscillator has been successfully utilized to represent an SSI system with SDOF structural model. In the present paper, this approximation is first extended to an equivalent fixed-base model with modified system parameters. Based on this generalization, a methodology is then proposed to determine the equivalent fixed-base models of a general multi degree-of-freedom SSI system using simple system identification techniques in the frequency domain. Various fixed-base models are formulated and their accuracy is compared for a five-story shear building resting on soft soil. It is shown that the actual SSI system can be accurately represented with an appropriate fixed-base model.     

Estimating kinematic interaction of raft foundations from earthquake records and its effects on structural response

A practical method for estimating kinematic interaction from earthquake records is presented. The kinematic interaction is characterized by a two-parameter model and these parameters can be estimated by using a frequency-domain systems identification method. The simple model can be used to model both wave passage effects and the effects of incoherent wave fields. Numerical simulation tests show that kinematic interaction parameters can be estimated to their best accuracy by using building base responses and the free-field excitation and can also be estimated by using building responses, base responses and the free-field excitation. The method was applied to two buildings with raft foundations and it was found that kinematic interaction was significant during earthquakes. Published theoretical models (wave passage effect) for vertically incident SH waves can be used to estimate the transfer functions up to 4–5 Hz and the models for horizontally propagating waves under-predict the estimated transfer functions by a significant amount at frequencies beyond about 1–2 Hz. Theoretical models for a massless rigid foundation under the excitation of an incoherent wave field predict the general trend of the estimated transfer function reasonably well over a large frequency range. The results of numerical examples show that the recorded response spectral attenuation of basement records at high frequencies with respect to the free-field is mainly caused by kinematic interaction, while the changes in storey shear and overturning moment in a structure due to soil flexibility are mainly the results of inertial interaction.     

Efficient 3D nonlinear Winkler model for shallow foundations

This paper presents a new practical modeling approach, based on the beam-on-a-nonlinear Winkler foundation (BNWF) model, to simulate the 3D rocking, vertical and horizontal responses of shallow foundations using structural elements that are readily available in the element library of commercially available structural analysis programs. An assemblage of a moment-rotation hinge, shear hinge connected in series with an elastic frame member attached to the bottom end of ground story columns was proposed to model the response of the footing under combined action of vertical, horizontal and moment loading. To couple the responses of these hinges, two bounding surfaces equations were introduced and derived mathematically: a surface that defines the interaction between the rocking and vertical capacities of the footing along its width and length; and a surface that defines the interaction between the horizontal capacities of the footing along its width and length. Simple calculation steps to evaluate the geometric and mechanical properties of the proposed assemblage of structural elements are provided. The proposed modeling approach was verified using experimental results from large scale model foundations subjected to cyclic loading. Based on this study, it was found that the proposed assemblage can be reliably used in modeling the rocking and horizontal responses of shallow foundations under cyclic loading.     

Winkler model for lateral response of rigid caisson foundations in linear soil

A generalized spring multi-Winkler model is developed for the static and dynamic response of rigid caisson foundations of circular, square, or rectangular plan, embedded in a homogeneous elastic. The model, referred to as a four-spring Winkler model, uses four types of springs to model the interaction between soil and caisson: lateral translational springs distributed along the length of the caisson relating horizontal displacement at a particular depth to lateral soil resistance (resultant of normal and shear tractions on the caisson periphery); similarly distributed rotational springs relating rotation of the caisson to the moment increment developed by the vertical shear tractions on the caisson periphery; and concentrated translational and rotational springs relating, respectively, resultant horizontal shear force with displacement, and overturning moment with rotation, at the base of the caisson. For the dynamic problem each spring is accompanied by an associated dashpot in parallel. Utilising elastodynamic theoretical available in the literature results for rigid embedded foundations, closed-form expressions are derived for the various springs and dashpots of caissons with rectangular and circular plan shape. The response of a caisson to lateral static and dynamic loading at its top, and to kinematically-induced loading arising from vertical seismic shear wave propagation, is then studied parametrically. Comparisons with results from 3D finite element analysis and other available theoretical methods demonstrate the reliability of the model, the need for which arises from its easy extension to multi-layered and nonlinear inelastic soil. Such an extension is presented in the companion papers by the authors [Gerolymos N, Gazetas G. Development of Winkler model for lateral static and dynamic response of caisson foundations with soil and interface nonlinearities. Soil Dyn Earthq Eng. Submitted companion paper; Gerolymos N, Gazetas G. Static and dynamic response of massive caisson foundations with soil and interface nonlinearities—validation and results. Soil Dyn Earthq Eng. Submitted companion paper.].     

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.     

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。     

Dynamic stiffness of foundation embedded in layered halfspace based on wave propagation in cones

An Erratum has been published for this article in Earthquake Engineering & Structural Dynamics 33(6) 2004, 793. The dynamic stiffness of a foundation embedded in a multiple‐layered halfspace is calculated postulating one‐dimensional wave propagation in cone segments. In this strength‐of‐materials approach the sectional property of the cone segment increases in the direction of wave propagation. Reflections and refractions with waves propagating in corresponding cone segments occur at layer interfaces. Compared to rigorous procedures the novel method based on cone segments is easy to apply, provides conceptual clarity and physical insight in the wave propagation mechanisms. This method postulating one‐dimensional wave propagation in cone segments with reflections and refractions at layer interfaces is evaluated, calculating the dynamic stiffness of a foundation embedded in a multiple‐layered halfspace. For sites resting on a flexible halfspace and fixed at the base, engineering accuracy (deviation of ±20%) is achieved for all degrees of freedom with a vast parameter variation. The behaviour below the cut‐off frequency in an undamped site fixed at its base is also reliably predicted. The accuracy is, in general, better than for the method based on cone frustums, which can lead to negative damping. Copyright © 2003 John Wiley & Sons, Ltd.     

A generalized hysteresis model for biaxial response of pile foundations in sands

We present the development and calibration of a macroelement model that captures the response of piles in cohesionless soils subjected to biaxial lateral loading. The model is founded on actual physical mechanism of soil resistance and provides the framework for extending a uniaxial model to biaxial case by means of a single cross-stiffness parameter. Both upper and lower bounds for the cross-stiffness parameter are also presented. The model is calibrated and verified using three-dimensional finite element (FE) simulations of soil-pile interaction for uniformly prescribed displacement along the pile length. Comparison of predictions from uniaxial and biaxial models with the FE results for transient loading indicates that the response assuming no coupling between the two horizontal directions for biaxial loading can differ significantly from the ‘true’ response for some cases. Accounting for coupling in the lateral direction, the proposed model captures the transverse pile response with very good accuracy while retaining the simplicity and computational efficiency of macroelement formulations compared to 3D FE analyses.     

Development of Winkler model for static and dynamic response of caisson foundations with soil and interface nonlinearities

As an extension of the elastic multi-spring model developed by the authors in a companion paper [Gerolymos N, Gazetas G. Winkler model for lateral response of rigid caisson foundations in linear soil. Soil Dyn Earthq Eng; 2005 (submitted companion paper).], this paper develops a nonlinear Winkler-spring method for the static, cyclic, and dynamic response of caisson foundations. The nonlinear soil reactions along the circumference and on the base of the caisson are modeled realistically by using suitable couple translational and rotational nonlinear interaction springs and dashpots, which can realistically (even if approximately) model such effects as separation and slippage at the caisson–soil interface, uplift of the caisson base, radiation damping, stiffness and strength degradation with large number of cycles. The method is implemented in a new finite difference time-domain code, NL-CAISSON. An efficient numerical methodology is also developed for calibrating the model parameters using a variety of experimental and analytical data. The necessity for the proposed model arises from the difficulty to predict the large-amplitude dynamic response of caissons up to failure, statically or dynamically. In a subsequent companion paper [Gerolymos N, Gazetas G. Static and dynamic response of massive caisson foundations with soil and interface nonlinearities—validation and results. Soil Dyn Earthq Eng; 2005 (submitted companion paper).], the model is validated against in situ medium-scale static load tests and results of 3D finite element analysis. It is then used to analyse the dynamic response of a laterally loaded caisson considering soil and interface nonlinearities.     

Simple formulations of ground impedance functions for rigid surface foundations

A differential equation is formulated for the dynamic response of ground medium by using a simplified ground model. Applying Galerkin's procedure for weighted residual, this equation leads to a governing equation only at the ground surface. The equation indicates that the ground surface behavior can be computed even further by a simplified model. By solving the governing equation for the boundary conditions along the surface, expressions in simple closed forms are developed for the dynamic response analysis of a massless rigid foundation that rests on the ground surface. Despite their significant simplicity, the developed expressions compute the values very close to those computed by far more complex rigorous solutions. They are found to be capable of capturing the important characteristics of the dynamic ground behavior well.     

A damage model for structures with degrading response

The paper presents a hysteretic damage model for the response simulation of structural components with strength and stiffness deterioration under cyclic loading. The model is based on 1D continuum damage mechanics and relates any 2 work‐conjugate response variables such as force‐displacement, moment‐rotation, or stress‐strain. The strength and stiffness deterioration is described by a continuous damage variable. The formulation uses a criterion based on the hysteretic energy and the maximum or minimum deformation for damage initiation with a cumulative probability distribution function for the damage evolution. A series of structural component response simulations showcase the ability of the model to describe different types of hysteretic behavior. The relation of the model's damage variable to the Park‐Ang damage index is also discussed. Because of its consistent and numerically robust formulation, the model is suitable for the large‐scale seismic response simulation of structural systems with strength and stiffness deterioration.