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.     

Footings under seismic loading: Analysis and design issues with emphasis on bridge foundations

The paper provides state-of-the-art information on the following aspects of seismic analysis and design of spread footings supporting bridge piers: (1) obtaining the dynamic stiffness (“springs” and “dashpots”) of the foundation; (2) computing the kinematic response; (3) determining the conditions under which foundation–soil compliance must be incorporated in dynamic structural analysis; (4) assessing the importance of properly modeling the effect of embedment; (5) elucidating the conditions under which the effect of radiation damping is significant; (6) comparing the relative importance between kinematic and inertial response. The paper compiles an extensive set of graphs and tables for stiffness and damping in all modes of vibration (swaying, rocking, torsion), for a variety of soil conditions and foundation geometries. Simplified expressions for computing kinematic response (both in translation and rotation) are provided. Special issues such as presence of rock at shallow depths, the contribution of foundation sidewalls, soil inhomogeneity and inelasticity, are also discussed. The paper concludes with parametric studies on the seismic response of bridge bents on embedded footings in layered soil. Results are presented (in frequency and time domains) for accelerations and displacements of bridge and footing, while potential errors from some frequently employed simplifications are illustrated.     

Seismic response of slender rigid structures with foundation uplifting

The rocking of rigid structures uplifting from their support under strong earthquake shaking is investigated. The structure is resting on the surface of either a rigid base or a linearly elastic continuum. A large-displacement approach is adopted to extract the governing equations of motion allowing for a rigorous calculation of the nonlinear response even under near-overturning conditions. Directivity-affected near-fault ground motions, idealized as Ricker wavelets or trigonometric pulses, are used as excitation. The conditions under which uplifting leads to large angles of rotation and eventually to overturning are investigated. A profoundly nonlinear rocking behavior is revealed for both rigid and elastic soil conditions. This geometrically nonlinear response is further amplified by unfavorable sequences of long-duration pulses in the excitation. Moreover, through the overturning response of a toppled tombstone, it is concluded that the practice of estimating ground accelerations from overturning observations is rather misleading and meaningless.     

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.     

Calibration of dynamic analysis methods from field test data

In view of the heterogeneity of natural soil deposits and approximations made in analysis methods, in situ methods of determining soil parameters are highly desirable. The problem of interest here is the nonlinear dynamic behavior of pile foundations. It is shown in this paper that soil parameters needed for simplified dynamic analysis of a single pile may be back-calculated from the dynamic response of the pile measured in the field. A pile was excited by applying a large horizontal dynamic force at the pile-head level, and the response measured. In this paper, two different (simplified) methods of modeling the dynamic response of the pile are considered. One of the methods is based on the Winkler foundation approach, with the spring constant characterized by the so-called nonlinear p–y springs. The second method is based on the equivalent-linear finite element approach, with the nonlinearity of shear modulus and damping accounted for by employing the so-called degradation relationships. In the latter, the effect of interface nonlinearity is also considered. Starting with best estimates of soil parameters, the experimental data on the response of pile is used to fine-tune the values of the parameters, and thereby, to estimate parameters that are representative of in situ soil conditions.     

Kinematic bending moments in pile foundations

In this paper the kinematic seismic interaction of single piles embedded in soil deposits is evaluated by focusing the attention on the bending moments induced by the transient motion. The analysis is performed by modeling the pile like an Euler–Bernoulli beam embedded in a layered Winkler-type medium. The excitation motion is obtained by means of a one-D propagation analysis. A comprehensive parametric analysis is carried out by varying the main parameters governing the dynamic response of piles like the soil properties, the bedrock location, the diameter and embedment in the bedrock of piles. On the basis of the parametric analysis, a new design formula for predicting the kinematic bending moments for both the cross-sections at the deposit–bedrock interface and at the pile head is proposed.     

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.     

Embedded foundation in layered soil under dynamic excitations

The critical step in the substructure approach for the soil–structure interaction (SSI) problem is to determine the impedance functions (dynamic-stiffness coefficients) of the foundations. In the present study, a computational tool is developed to determine the impedance functions of foundation in layered soil medium. Cone frustums are used to model the foundation soil system. Cone frustums are developed based on wave propagation principles and force-equilibrium approach. The model is validated for its ability to represent the embedded foundation in layered medium by comparing the results with the rigorous analysis results. Various degrees of freedom, such as, horizontal, vertical and rocking are considered for this study.     

Static and dynamic response of massive caisson foundations with soil and interface nonlinearities—validation and results

The static, cyclic, and dynamic response of a massive caisson foundation embedded in nonlinear layered or inhomogeneous soil and loaded at its top is investigated. The caisson is supported against horizontal displacement and rotation by four types of inelastic springs and dashpots, described with the BWGG model that was developed in the preceding companion paper [Gerolymos N, Gazetas G. Development of winkler model for static and dynamic response of caisson foundations with soil and interface nonlinearities. Soil Dyn Earthq Eng, submitted companion paper]. The prediction of the model is satisfactorily compared with results from 3D-finite element analysis. Some experimental corroboration of the method is provided with the help of a 1/3-scale lateral load test that had been conducted in the field by EPRI. An illustrative example of a caisson embedded in linearly-inhomogeneous clay and subjected to static and dynamic loading is analysed. Characteristic results are presented highlighting the role of soil inelasticity and its interplay with the two dominant interface nonlinearities: separation (gapping) of the caisson shaft from the surrounding soil, and uplifting of the base from the underlying soil.     

Seismic rotational displacement of gravity walls by pseudo-dynamic method: Passive case

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.     

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.     

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.     

Time-domain representation of frequency-dependent foundation impedance functions

Foundation impedance functions provide a simple means to account for soil–structure interaction (SSI) when studying seismic response of structures. Impedance functions represent the dynamic stiffness of the soil media surrounding the foundation. The fact that impedance functions are frequency dependent makes it difficult to incorporate SSI in standard time-history analysis software. This paper introduces a simple method to convert frequency-dependent impedance functions into time-domain filters. The method is based on the least-squares approximation of impedance functions by ratios of two complex polynomials. Such ratios are equivalent, in the time-domain, to discrete-time recursive filters, which are simple finite-difference equations giving the relationship between foundation forces and displacements. These filters can easily be incorporated into standard time-history analysis programs. Three examples are presented to show the applications of the method.     

任意刚性基础竖向动阻抗的简化计算

对于均质弹性半空间上的任意形状的刚性明置和埋置基础,其动刚度和阻尼系数的确定,已有很多这方面的研究。通常基础的任意形状用其外包的规则几何形状代替原有的不规则基础形状,以达到确定动刚度和阻尼系数的目的,而且这两个参数的确定仅仅是对单独刚性基础的,无法考虑相邻基础对其产生的影响。针对上述两方面不完善之处作了进一步探讨,引入相邻基础动力相互作用因子的概念,并利用地基为平面应变假定以求之。推荐的方法经验证,非常准确。     

A practical method for estimating dynamic soil stiffness on surface of multi‐layered soil

It is important to estimate the influence of layered soil in soil–structure interaction analyses. Although a great number of investigations have been carried out on this subject, there are very few practical methods that do not require complex calculations. In this paper, a simple and practical method for estimating the horizontal dynamic stiffness of a rigid foundation on the surface of multi‐layered soil is proposed. In this method, waves propagating in the soil are traced using the conception of the cone model, and the impulse response function can be calculated directly and easily in the time domain with a good degree of accuracy. The characteristics of the impedance, that is the transformed value to the frequency domain of the obtained impulse response, are studied using two‐ to four‐layered soil models. The cause of the fluctuation of impedance is expressed clearly from its relation to reflected waves from the lower layer boundary in the model. Copyright © 2005 John Wiley & Sons, Ltd.     

Simplified model for torsional foundation vibrations

A simplified model is presented to simulate unbounded soil for torsional foundation vibration problems. Based on the criterion of equivalent displacement response, a group of equivalent models are developed for a foundation-soil system. An optimal equivalent model is then determined to represent the best simplified model. The parameters of the optimal equivalent model may be obtained by a much easier and more efficient method than lengthy optimization techniques used by most existing lumped-parameter models. The dynamic torsional responses of the foundation-soil system using the optimal equivalent models are very consistent with those obtained by the half-space theory and by the existing models. With fewer parameters, the optimal equivalent model is also found to be as accurate as most existing models. This proposed method may be effectively applied to practical torsionally vibrating problems involving soil–structure interaction.     

Experimental study of the dynamic interaction between the foundation of the NEES/UCSD Shake Table and the surrounding soil: Reaction block response

An extensive experimental study of the dynamic interaction between the foundation block for the NEES/UCSD Large High Performance Outdoor Shake Table and the surrounding soil was conducted in 2003. The vibrations induced by the two NEES@UCLA large eccentric mass shakers were recorded at multiple stations within the reinforced concrete foundation block and on the surface of the surrounding soil up to distances of 270 m from the block. The present paper focuses on analysis of the data recorded within the reaction block including the average rigid body motion of the foundation and its dependence on frequency, and the deformation of the block for longitudinal (EW), transverse (NS), and torsional excitation. Comparison of the reaction block response during shaker induced vibrations with that for the much stronger actuator forces shows that linearity holds for the range of forces involved. Comparisons with analytical results for a simplified model of the foundation show good agreement between experimental and theoretical results.     

Analytical moment–rotation curves for rigid foundations based on a Winkler model

Analytical equations for the moment–rotation response of a rigid foundation on a Winkler soil model are presented. An equation is derived for the uplift-yield condition and is combined with equations for uplift- and yield-only conditions to enable the definition of the entire static moment–rotation response. The results obtained from the developed model show that the inverse of the factor of safety, χ, has a significant effect on the moment–rotation curve. The value of χ=0.5 not only determines whether uplift or yield occurs first but also defines the condition of the maximum moment–rotation response of the footing. A Winkler model is developed based on the derived equations and is used to analyze the TRISEE experiments. The computed moment–rotation response agrees well with the experimental results when the subgrade modulus is estimated using the unload–reload stiffness from static plate load–deformation tests. A comparison with the recommended NEHRP guidelines based on the FEMA 273/274 documents shows that the choice of value of the effective shear modulus significantly affected the comparison.     

Analytical evaluation of the dynamic distress of rigid fixed-base retaining systems

The current study proposes an analytical closed-form solution for the dynamic distress of rigid fixed-base retaining systems aiming at evaluating the main assumptions and limitations of the pertinent available elasticity-based methods. The new solution is actually an extension of the well-known model of Wood and is capable of evaluating the dynamic distress of either a single or a pair of rigid fixed-base walls interacting with each other, in the case of harmonic base loading. Wall distress is mainly evaluated in terms of dynamic earth pressures, shear forces and bending moments, while the original concept of a “distress spectrum” is introduced as a potential new tool for the seismic design of retaining structures. Distress and wall deformation are interrelated in a number of three-dimensional graphs, where dynamic interaction phenomena are evident. Finally, given the rigorous nature of the new solution, its results verify qualitatively and quantitatively the negligible amplitude of the computational errors of the approximate elasticity-based solutions proposed in the literature.