Stability of the time‐domain analysis method including a frequency‐dependent soil–foundation system

A number of methods have been proposed that utilize the time‐domain transformations of frequency‐dependent dynamic impedance functions to perform a time‐history analysis. Though these methods have been available in literature for a number of years, the methods exhibit stability issues depending on how the model parameters are calibrated. In this study, a novel method is proposed with which the stability of a numerical integration scheme combined with time‐domain representation of a frequency‐dependent dynamic impedance function can be evaluated. The method is verified with three independent recursive parameter models. The proposed method is expected to be a useful tool in evaluating the potential stability issue of a time‐domain analysis before running a full‐fledged nonlinear time‐domain analysis of a soil–structure system in which the dynamic impedance of a soil–foundation system is represented with a recursive parameter model. Copyright © 2015 John Wiley & Sons, Ltd.     

Response of unbounded soil in scaled boundary finite‐element method

The scaled boundary finite‐element method is a powerful semi‐analytical computational procedure to calculate the dynamic stiffness of the unbounded soil at the structure–soil interface. This permits the analysis of dynamic soil–structure interaction using the substructure method. The response in the neighbouring soil can also be determined analytically. The method is extended to calculate numerically the response throughout the unbounded soil including the far field. The three‐dimensional vector‐wave equation of elasto‐dynamics is addressed. The radiation condition at infinity is satisfied exactly. By solving an eigenvalue problem, the high‐frequency limit of the dynamic stiffness is constructed to be positive definite. However, a direct determination using impedances is also possible. Solving two first‐order ordinary differential equations numerically permits the radiation condition and the boundary condition of the structure–soil interface to be satisfied sequentially, leading to the displacements in the unbounded soil. A generalization to viscoelastic material using the correspondence principle is straightforward. Alternatively, the displacements can also be calculated analytically in the far field. Good agreement of displacements along the free surface and below a prism foundation embedded in a half‐space with the results of the boundary‐element method is observed. Copyright © 2001 John Wiley & Sons, Ltd.     

Time‐domain viscoelastic analysis of earth structures

In this paper two causal models that approximate the nearly frequency‐independent cyclic behaviour of soils are analysed in detail. The study was motivated by the need to conduct time‐domain viscoelastic analysis on soil structures without adopting the ad hoc assumption of Rayleigh damping. First, the causal hysteretic model is introduced in which its imaginary part is frequency independent the same way that is the imaginary part of the popular non‐causal constant hysteretic model. The adoption of an imaginary part that is frequency independent even at the zero‐frequency limit, in conjunction with the condition that the proposed model should be causal, yields a real part that is frequency dependent and singular at zero frequency. The paper shows that the causal hysteretic model, although pathological at the static limit, is the mathematical connection between the non‐causal constant hysteretic model and the physically realizable Biot model. The mathematical structure of the two causal models is examined and it is shown that the causal hysteretic model is precisely the high‐frequency limit of the Biot model. Although both models have a closed‐form time‐domain representation, only the Biot model is suitable for a time‐domain viscoelastic analysis with commercially available computer software. The paper demonstrates that the simplest, causal and physically realizable linear hysteretic model that can approximate the cyclic behaviour of soil is the Biot model. The proposed study elucidates how the dynamic analysis of soil structures can be conducted rigorously in terms of the viscoelastic properties of the soil material and not with the ad hoc Rayleigh damping approach which occasionally has been criticized that tends to overdamp the higher vibration modes. The study concludes that under pulse‐type motions the Rayleigh damping approximation tends to overestimate displacements because of the inappropriate viscous type of dissipation that is imposed. Under longer motions that induce several cycles, the concept of equivalent viscous damping is more appropriate and the Rayleigh damping approximation results to a response that is comparable to the response computed with a rigorous time‐domain viscoelastic finite element analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

Recursive evaluation of interaction forces and property matrices from unit-impulse response functions of unbounded medium based on balancing approximation

Starting from the unit-impulse response matrix of the unbounded medium, a discrete-time formulation permitting the recursive evaluation of the interaction forces and a continuous-time formulation yielding property matrices corresponding to a model with a finite number of degrees of freedom are discussed. This is achieved using the balancing approximation method which is easily automated, guarantees stability and leads to highly accurate results. © 1998 John Wiley & Sons Ltd.     

A practical method to transform frequency dependent impedance to time domain

In order to perform time history earthquake response analyses with consideration to both the dynamic soil–structure interaction and the non‐linear behaviour of the structure, it is important to transform the soil impedance in the frequency domain to the impulse response in the time domain. In this paper, a new transform method with high practicality is proposed. First, the formulation of the proposed transform method is described. Next, the validity of the method is examined using an example problem whose impulse response is analytically obtained. Then, the impedance of the rigid foundation on 2‐layered soil is transformed to the time domain, and the characteristics of the impulse response are investigated. Finally, time history earthquake response analyses of a structure on the soil using the obtained impulse response are carried out. The validity and the efficiency of the proposed method are confirmed through these investigations. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Soil–structure interaction in the time domain using halfspace Green's functions

A time-domain formulation is proposed for the transient response analysis of general, three-dimensional structures resting on a homogeneous, elastic halfspace subjected to either external loads or seismic motions. The formulation consists of two parts: (a) the time domain formulation of the soil behaviour and (b) the coupling of the corresponding soil algorithms to the Finite Element Code ANSYS. As far as the structure is concerned, this coupling opens the way for the analysis of non-linear soil–structure interaction. The approach is based on halfspace Green's functions for displacements elicited by Heaviside time-dependent surface point loads. Hence, the spatial discretisation can be confined to the contact area between the foundation and the soil, i.e. no auxiliary grid beyond the foundation as for conventional boundary element formulations is required. The method is applied to analyse the dynamic response of a railway track due to a moving wheel set by demonstrating the influence of ‘through-the-soil coupling’.     

A novel predictor–corrector algorithm for sub‐structure pseudo‐dynamic testing

A new predictor–corrector (P–C) method for multi‐site sub‐structure pseudo‐dynamic (PSD) test is proposed. This method is a mixed time integration method in which computational components separable from experimental components are solved by implicit time integration method (Newmark β method). The experiments are performed quasi‐statically based on explicit prediction of displacement. The proposed P–C method has an important advantage as it does not require the determination of the initial stiffness values of experimental components and is thus suitable for representing elastic and inelastic systems. A parameter relating to quality of displacement prediction at boundaries nodes is introduced. This parameter is determined such that P–C method can be applicable to many practical problems. Error‐propagation characteristics of P–C method are also presented. A series of examples including linear and non‐linear soil–foundation–structure interaction problem demonstrate the performance of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

Systematic assessment of irregular building–soil interaction using efficient modal analysis

In this paper an efficient methodology applying modal analysis is developed to assess systematically the combined soil–structure interaction and torsional coupling effects on asymmetric buildings. This method is implemented in the frequency domain to accurately incorporate the frequency‐dependent foundation impedance functions. For extensively extracting the soil–structure interaction effects, a diagonal transfer matrix in the modal space is derived. A comprehensive investigation of asymmetric building–soil interaction can then be conveniently conducted by examining various types of response quantities. Results of parametric study show that the increasing height‐to‐base ratio of a structure generally amplifies its translational and torsional responses. Moreover, both the translational and torsional responses are reduced for the case where the two resonant frequencies are well separated and this reduction is enhanced with the decreasing values of the relative soil stiffness and the height‐to‐base ratio. The most noteworthy phenomenon may be the fact that the SSI effects can enlarge the translational response if the structure is slender and the two resonant frequencies are very close. Copyright © 2001 John Wiley & Sons, Ltd.     

Free‐vibration analysis of a three‐dimensional soil–structure system

A procedure which involves a non‐linear eigenvalue problem and is based on the substructure method is proposed for the free‐vibration analysis of a soil–structure system. In this procedure, the structure is modelled by the standard finite element method, while the unbounded soil is modelled by the scaled boundary finite element method. The fundamental frequency, and the corresponding radiation damping ratio as well as the modal shape are obtained by using inverse iteration. The free vibration of a dam–foundation system, a hemispherical cavity and a hemispherical deposit are analysed in detail. The numerical results are compared with available results and are also verified by the Fourier transform of the impulsive response calculated in the time domain by the three‐dimensional soil–structure–wave interaction analysis procedure proposed in our previous paper. The fundamental frequency obtained by the present procedure is very close to that obtained by Touhei and Ohmachi, but the damping ratio and the imaginary part of modal shape are significantly different due to the different definition of damping ratio. This study shows that although the classical mode‐superposition method is not applicable to a soil–structure system due to the frequency dependence of the radiation damping, it is still of interest in earthquake engineering to evaluate the fundamental frequency and the corresponding radiation damping ratio of the soil–structure system. Copyright © 2001 John Wiley & Sons, Ltd.     

Effects of pulse period of near‐field ground motions on the seismic demands of soil–MDOF structure systems using mathematical pulse models

In this paper, the effects of pulse period associated with near‐field ground motions on the seismic demands of soil–MDOF structure systems are investigated by using mathematical pulse models. Three non‐dimensional parameters are employed as the crucial parameters, which govern the responses of soil–structure systems: (1) non‐dimensional frequency as the structure‐to‐soil stiffness ratio; (2) aspect ratio of the superstructure; and (3) structural target ductility ratio. The soil beneath the superstructure is simulated on the basis of the Cone model concept. The superstructure is modeled as a nonlinear shear building. Interstory drift ratio is selected as the main engineering demand parameter for soil–structure systems. It is demonstrated that the contribution of higher modes to the response of soil–structure system depends on the pulse‐to‐interacting system period ratio instead of pulse‐to‐fixed‐base structure period ratio. Furthermore, results of the MDOF superstructures demonstrate that increasing structural target ductility ratio results in the first‐mode domination for both fixed‐base structure and soil–structure system. Additionally, increasing non‐dimensional frequency and aspect ratio of the superstructure respectively decrease and increase the structural responses. Moreover, comparison of the equivalent soil–SDOF structure system and the soil–MDOF structure system elucidates that higher‐mode effects are more significant, when soil–structure interaction is taken into account. In general, the effects of fling step and forward directivity pulses on activating higher modes of the superstructure are more sever in soil–structure systems, and in addition, the influences of forward directivity pulses are more considerable than fling step ones. Copyright © 2013 John Wiley & Sons, Ltd.     

A hybrid time frequency domain formulation for non-linear soil–structure interaction

Methods that combine frequency and time domain techniques offer an attractive alternative for solving Soil–Structure-interaction problems where the structure exhibits non-linear behaviour. In the hybrid-frequency-time-domain procedure a reference linear system is solved in the frequency domain and the difference between the actual restoring forces and those in the linear model are treated as pseudo-forces. In the solution scheme explored in this paper, designated as the hybrid-time-frequency-domain (HTFD) procedure, the equations of motion are solved in the time domain with due consideration for non-linearities and with the unbounded medium represented by frequency-independent springs and dampers. The frequency dependency of the impedance coefficients is introduced by means of pseudo-forces evaluated in the frequency domain at the end of each iteration. A criterion of stability for the HTFD approach is derived analytically and its validity is sustained numerically. As is often the case, the criterion takes the form of a limit of unity on the spectral radius of an appropriately defined matrix. Inspection of the terms in this matrix shows that convergence can be guaranteed by suitable selection of the reference impedance. The CPU times required to obtain converged solutions with the HTFD are found, in a number of numerical simulations, to be up to one order of magnitude less than those required by the alternative hybrid-frequency-time-domain approach. © 1998 John Wiley & Sons, Ltd.     

Response‐only modal identification of structures using strong motion data

Dynamic characteristics of structures — viz. natural frequencies, damping ratios, and mode shapes — are central to earthquake‐resistant design. These values identified from field measurements are useful for model validation and health‐monitoring. Most system identification methods require input excitations motions to be measured and the structural response; however, the true input motions are seldom recordable. For example, when soil–structure interaction effects are non‐negligible, neither the free‐field motions nor the recorded responses of the foundations may be assumed as ‘input’. Even in the absence of soil–structure interaction, in many instances, the foundation responses are not recorded (or are recorded with a low signal‐to‐noise ratio). Unfortunately, existing output‐only methods are limited to free vibration data, or weak stationary ambient excitations. However, it is well‐known that the dynamic characteristics of most civil structures are amplitude‐dependent; thus, parameters identified from low‐amplitude responses do not match well with those from strong excitations, which arguably are more pertinent to seismic design. In this study, we present a new identification method through which a structure's dynamic characteristics can be extracted using only seismic response (output) signals. In this method, first, the response signals’ spatial time‐frequency distributions are used for blindly identifying the classical mode shapes and the modal coordinate signals. Second, cross‐relations among the modal coordinates are employed to determine the system's natural frequencies and damping ratios on the premise of linear behavior for the system. We use simulated (but realistic) data to verify the method, and also apply it to a real‐life data set to demonstrate its utility. Copyright © 2012 John Wiley & Sons, Ltd.     

Three-dimensional dynamic soil–structure interaction analysis in the time domain

A new numerical procedure is proposed for the analysis of three-dimensional dynamic soil–structure interaction in the time domain. In this study, the soil is modelled as a linear elastic solid, however, the methods developed can be adapted to include the effects of soil non-linearities and hysteretic damping in the soil. A substructure method, in which the unbounded soil is modelled by the scaled boundary finite-element method, is used and the structure is modelled by 8–21 variable-number-node three-dimensional isoparametric or subparametric hexahedral curvilinear elements. Approximations in both time and space, which lead to efficient schemes for calculation of the acceleration unit-impulse response matrix, are proposed for the scaled boundary finite-element method resulting in significant reduction in computational effort with little loss of accuracy. The approximations also lead to a very efficient scheme for evaluation of convolution integrals in the calculation of soil–structure interaction forces. The approximations proposed in this paper are also applicable to the boundary element method. These approximations result in an improvement over current methods. A three-dimensional Dynamic Soil–Structure Interaction Analysis program (DSSIA-3D) is developed, and seismic excitations (S-waves, P-waves, and surface waves) and externally applied transient loadings can be considered in analysis. The computer program developed can be used in the analysis of three-dimensional dynamic soil–structure interaction as well as in the analysis of wave scattering and diffraction by three-dimensional surface irregularities. The scattering and diffraction of seismic waves (P-, S-, and Rayleigh waves) by various three-dimensional surface irregularities are studied in detail, and the numerical results obtained are in good agreement with those given by other authors. Numerical studies show that the new procedure is suitable and very efficient for problems which involve low frequencies of interest for earthquake engineering. Copyright © 1999 John Wiley & Sons Ltd     

On the performance of lumped parameter models with gyro‐mass elements for the impedance function of a pile‐group supporting a single‐degree‐of‐freedom system

Lumped parameter models with a so called “gyro‐mass” element (GLPMs) have been proposed recently in response to a strong demand for efficiently and accurately representing frequency‐dependent impedance functions of soil–foundation systems. Although GLPMs are considered to be powerful tools for practical applications in earthquake engineering, some problems remain. For instance, although GLPMs show fairly close agreement with the target impedance functions, the accuracy of the transfer functions and the time‐histories of dynamic responses in structural systems comprising GLPMs have never been verified. Furthermore, no assessment has been performed on how much difference appears in the accuracy of dynamic responses obtained from GLPMs and those from conventional Kelvin–Voigt models comprising a spring and a dashpot arranged in parallel with various frequency‐independent constants. Therefore, in this paper, these problems are examined using an example of 2×4 pile groups embedded in a layered soil medium, supporting a single‐degree‐of‐freedom system subjected to ground motions. The results suggest that GLPMs are a new option for highly accurate computations in evaluating the dynamic response of structural systems comprising typical pile groups, rather than conventional Kelvin–Voigt models. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

PROPERTY MATRICES IDENTIFICATION OF UNBOUNDED MEDIUM FROM UNIT-IMPULSE RESPONSE FUNCTIONS USING LEGENDRE POLYNOMIALS: FORMULATION

A systematic procedure to construct the (symmetric) static-stiffness, damping and mass matrices representing the unbounded medium is presented addressing the unit-impulse response matrix corresponding to the degrees of freedom on the structure–medium interface. The unit-impulse response matrix is first diagonalized which then permits each term to be modelled independently from the others using expansions in a series of Legendre polynomials in the time domain. This leads to a rational approximation in the frequency domain of the dynamic-stiffness coefficient. Using a lumped-parameter model which provides physical insight the property matrices are constructed.     

A discontinuous Galerkin method for seismic soil–structure interaction analysis in the time domain

A technique for modeling transient wave propagation in unbounded media is extended and applied to seismic soil–structure interaction analysis in the time domain. The technique, based on the discontinuous Galerkin method, requires lower computational cost and less storage than the boundary element method, and the time‐stepping scheme resulting from Newmark's method in conjunction with the technique is unconditionally stable, allowing for efficient and robust time‐domain computations. To extend the technique to cases characterized by seismic excitation, the free‐field motion is used to compute effective forces, which are introduced on the boundary of the computational domain containing the structure and the soil in the vicinity of the structure. A numerical example on a dam–foundation system subjected to seismic excitation demonstrates the performance of the method. Copyright © 2003 John Wiley & Sons, Ltd.     

Effects of foundation embedment during building–soil interaction

A numerical solution for evaluating the effects of foundation embedment on the effective period and damping and the response of soil–structure systems is presented. A simple system similar to that used in practice to account for inertial interaction effects is investigated, with the inclusion of kinematic interaction effects for the important special case of vertically incident shear waves. The effective period and damping are obtained by establishing an equivalence between the interacting system excited by the foundation input motion and a replacement oscillator excited by the free-field ground motion. In this way, the use of standard free-field response spectra applicable to the effective period and damping of the system is permitted. Also, an approximate solution for total soil–structure interaction is presented, which indicates that the system period is insensitive to kinematic interaction and the system damping may be expressed as that for inertial interaction but modified by a factor due to kinematic interaction. Results involving both kinematic and inertial effects are compared with those obtained for no soil–structure interaction and inertial interaction only. The more important parameters involved are identified and their influences are examined over practical ranges of interest. © 1998 John Wiley & Sons, Ltd.