Site dependent and spatially varying response spectra

The goal of this study is to provide a stochastic method to investigate the effects of the randomness of soil properties due to their natural spatial variability on the response spectra spatial variation at sites with varying conditions. For this purpose, Monte Carlo Simulations are used to include the variability of both incident ground motion and soil parameters in the response spectra by mean of an appropriate coherency loss function and a site-dependent transfer function, respectively. The approach is built on the assumption of vertical propagation of SH type waves in soil strata with uncertain parameters. The response spectra are obtained by numerical integration of the governing equation of a single-degree-of-freedom(SDOF) system under non-stationary site-dependent and spatially varying ground motion accelerations simulated with non-uniform spectral densities and coherency loss functions. Numerical examples showed that randomness of soil properties significantly affects the amplitudes of the response spectra, indicating that as the heterogeneity induced by the randomness of the parameters of the medium increases, the spectral ordinates attenuate.     

A new model for spectral velocity ordinates at long periods

The estimation of peak linear response via elastic design (response) spectra continues to form the basis of earthquake‐resistant design of structural systems in various codes of practice all over the world. Many response spectrum‐based formulations of peak linear response require an additional input of the spectral velocity (SV) ordinates consistent with the specified seismic hazard. SV ordinates have been conventionally approximated by pseudo spectral velocity (PSV) ordinates, which are close to the SV ordinates only over the intermediate frequency range coinciding with the velocity‐sensitive region. At long periods, PSV ordinates underestimate the SV ordinates, and this study proposes a formulation of a correction factor (>1) that needs to be multiplied by the PSV ordinates in order to close the gap between the two sets of ordinates. A simple model is proposed in the form of a power function in oscillator period to estimate this factor in terms of two governing parameters which are in turn estimated from two single‐parameter scaling equations. The parameters considered for the scaling equations are (1) the period at which the PSV spectrum is maximized and (2) the rate of decay of the pseudo spectral acceleration (PSA) amplitudes at long periods. For a given damping ratio, four regression coefficients are determined for the scaling equations with the help of 205 ground motions recorded in western USA. A numerical study undertaken with the help of several design PSA spectra and ensembles of spectrum‐compatible ground motions illustrates the effectiveness of the proposed correction factor, together with the proposed scaling models, in comparison with the PSV approximation in a variety of design situations. Both the input parameters mentioned above can be easily obtained from the specified design spectrum, and thus the proposed model is convenient to use.     

Effect of site amplification on inelastic seismic response

The available models for eff ective periods of site and structure are reviewed in context of frequency tuning in the inelastic seismic response of soil-structure system. The eff ect of seismic intensity and ductility demand, on the eff ective periods, is investigated, and inelastic site amplifi cation is shown to be strongly correlated to the normalized eff ective period. Two non-dimensional parameters, analogous to the conventional site amplifi cation factors in codes, are defi ned to quantify the inelastic site amplifi cation. It is shown that the inelastic site amplifi cation factor (i.e. ratio of constant ductility spectral ordinates at soil site to those at rock outcrop) is able to represent the site eff ects more clearly, as compared to the inelastic site amplifi cation ratio (i.e. ratio of inelastic spectral ordinates at soil site to the corresponding elastic spectral ordinates at rock outcrop). Further, the peak in the amplifi cation factor corresponding to the eff ective site period diminishes rapidly with increasing ductility demand.     

Prediction of elastic displacement response spectra in Europe and the Middle East

Empirical equations are presented for the prediction of displacement response ordinates for damping ratios of 2, 5, 10, 20 and 30% of critical and for response periods up to 4s, using 532 accelerograms from the strong‐motion databank from Europe and the Middle East. The records were all re‐processed and only employed for regressions at periods within the usable range, defined as a fraction of the filter cut‐off and depending on the instrument type (digital or analogue), earthquake magnitude and site class. The equations can be applied to predict the geometric mean displacement and pseudo‐acceleration spectra for earthquakes with moment magnitudes ( M ) between 5 and 7.6, and for distances up to 100km. The equations also include style‐of‐faulting and site class as explanatory variables. The predictions obtained from these new equations suggest that earlier European equations for spectral displacements underestimate the ordinates at longer periods as a result of severe filtering and the use of the spectral ordinates at periods too close to the filter cut‐off. The results also confirm that the period defining the start of the constant displacement plateau in the Eurocode 8 (EC8) spectrum is excessively short at 2s. The results not only show that the scaling factor defined in EC8 for estimating the spectral ordinates at damping ratios different from 5% of critical are a good general approximation, but also that this scaling varies with magnitude and distance (reflecting the influence of duration) and also displays a mild dependence on response period. Copyright © 2007 John Wiley & Sons, Ltd.     

Influence of Boundary Condition Types on Unstable Density‐Dependent Flow

Boundary conditions are required to close the mathematical formulation of unstable density‐dependent flow systems. Proper implementation of boundary conditions, for both flow and transport equations, in numerical simulation are critical. In this paper, numerical simulations using the FEFLOW model are employed to study the influence of the different boundary conditions for unstable density‐dependent flow systems. A similar set up to the Elder problem is studied. It is well known that the numerical simulation results of the standard Elder problem are strongly dependent on spatial discretization. This work shows that for the cases where a solute mass flux boundary condition is employed instead of a specified concentration boundary condition at the solute source, the numerical simulation results do not vary between different convective solution modes (i.e., plume configurations) due to the spatial discretization. Also, the influence of various boundary condition types for nonsource boundaries was studied. It is shown that in addition to other factors such as spatial and temporal discretization, the forms of the solute transport equation such as divergent and convective forms as well as the type of boundary condition employed in the nonsource boundary conditions influence the convective solution mode in coarser meshes. On basis of the numerical experiments performed here, higher sensitivities regarding the numerical solution stability are observed for the Adams‐Bashford/Backward Trapezoidal time integration approach in comparison to the Euler‐Backward/Euler‐Forward time marching approach. The results of this study emphasize the significant consequences of boundary condition choice in the numerical modeling of unstable density‐dependent flow.     

Response spectral ratios

At many sites on soft ground, spectral ratios (ratios of smoothed Fourier amplitude spectral ordinates at the site to those at a station on firm ground) for distant earthquakes are little sensitive to focal mechanism and coordinates and to magnitude. Spectral ratios furnish directly expected Fourier amplitude spectral ordinates at the site of interest. The corresponding response spectra can be estimated through the use of random vibration theory. This step is obviated by resorting directly to ratios of response spectral ordinates. Through comparisons for several sites on the Valley of Mexico we find that these ratios are as stable as those of Fourier amplitude spectral ordinates.     

Distance scaling of vertical and horizontal response spectra of the Loma Prieta earthquake

Attenuations of the vertical and horizontal response spectra of the 17 October 1989 Loma Prieta, California, earthquake are developed through analyses of the ground motion at 53 sites within a 100 km radius of the source. The analyses are performed on the spectral ordinates for 16 incremental periods ranging from 0.05 to 2.0 sec. The response spectra are modelled empirically for two different site conditions characterized by rock and stiff-soil geologies. Data analysis is performed by the application of a non-linear multivariate regression procedure allowing for distance and site factor as independent variables. Variation of the vertical-to-horizontal (V/H) spectral ratios with wave frequency and distance shows the same behaviour as observed previously in the widely separated geographic regions of northeastern Taiwan and east-central Iran. The predicted ratios at sites underlain by stiff soil are generally higher than the commonly used value of 2/3 at high frequencies ( > 5 Hz) in the near-source region (R < 30 km), but reduce to 1/2 or less at longer periods and farther distances. This behaviour is also observed at rock sites; however, it is somewhat less pronounced. With a faster attenuation of spectral ordinates at higher frequencies, the shape of the response spectrum is found to change with distance. As expected, the spectral attenuation with distance is generally higher for the vertical spectrum than for the horizontal spectrum. The difference is particularly significant at the higher-frequency end of spectrum. Site amplification factors for stiff soil with respect to rock geology varies between 1.17 and 1.72 for horizontal spectrum and 1.01 and 1.81 for vertical spectrum. Spectral amplifications at four sites underlain by soft soil and artificial fill, are also evaluated. This is done by a comparison of the observed spectra with those predicted for rock geology at corresponding distances. As expected, the resulting amplification factors at soft-soil sites show significant increase relative to those at sites underlain by rock.     

Flow in unsaturated random porous media,nonlinear numerical analysis and comparison to analytical stochastic models

This work presents a rigorous numerical validation of analytical stochastic models of steady state unsaturated flow in heterogeneous porous media. It also provides a crucial link between stochastic theory based on simplifying assumptions and empirical field and simulation evidence of variably saturated flow in actual or realistic hypothetical heterogeneous porous media. Statistical properties of unsaturated hydraulic conductivity, soil water tension, and soil water flux in heterogeneous soils are investigated through high resolution Monte Carlo simulations of a wide range of steady state flow problems in a quasi-unbounded domain. In agreement with assumptions in analytical stochastic models of unsaturated flow, hydraulic conductivity and soil water tension are found to be lognormally and normally distributed, respectively. In contrast, simulations indicate that in moderate to strong variable conductivity fields, longitudinal flux is highly skewed. Transverse flux distributions are leptokurtic. the moments of the probability distributions obtained from Monte Carlo simulations are compared to modified first-order analytical models. Under moderate to strong heterogeneous soil flux conditions (σ²_y≥1), analytical solutions overestimate variability in soil water tension by up to 40% as soil heterogeneity increases, and underestimate variability of both flux components by up to a factor 5. Theoretically predicted model (cross-)covariance agree well with the numerical sample (cross-)covarianaces. Statistical moments are shown to be consistent with observed physical characteristics of unsaturated flow in heterogeneous soils.©1998 Elsevier Science Limited. All rights reserved     

Influence of irregular topography and random soil properties on coherency loss of spatial seismic ground motions

Coherency functions are used to describe the spatial variation of seismic ground motions at multiple supports of long span structures. Many coherency function models have been proposed based on theoretical derivation or measured spatial ground motion time histories at dense seismographic arrays. Most of them are suitable for modelling spatial ground motions on flat‐lying alluvial sites. It has been found that these coherency functions are not appropriate for modelling spatial variations of ground motions at sites with irregular topography (Struct. Saf. 1991; 10 (1):1–13). This paper investigates the influence of layered irregular sites and random soil properties on coherency functions of spatial ground motions on ground surface. Ground motion time histories at different locations on ground surface of the irregular site are generated based on the combined spectral representation method and one‐dimensional wave propagation theory. Random soil properties, including shear modulus, density and damping ratio of each layer, are assumed to follow normal distributions, and are modelled by the independent one‐dimensional random fields in the vertical direction. Monte‐Carlo simulations are employed to model the effect of random variations of soil properties on the simulated surface ground motion time histories. The coherency function is estimated from the simulated ground motion time histories. Numerical examples are presented to illustrate the proposed method. Numerical results show that coherency function directly relates to the spectral ratio of two local sites, and the influence of randomly varying soil properties at a canyon site on coherency functions of spatial surface ground motions cannot be neglected. Copyright © 2010 John Wiley & Sons, Ltd.     

Non‐stationary seismic response of tanks with soil interaction by wavelets

A wavelet‐based random vibration theory has been developed for the non‐stationary seismic response of liquid storage tanks including soil interaction. The ground motion process has been characterized via estimates of statistical functionals of wavelet coefficients obtained from a single time history of ground accelerations. The tank–liquid–soil system has been modelled as a two‐degree‐of‐freedom (2‐DOF) system. The wavelet domain equations have been formulated and the wavelet coefficients of the required response state are obtained by solving two linear simultaneous algebraic equations. The explicit expression for the instantaneous power spectral density function (PSDF) in terms of the functionals of the input wavelet coefficients has been obtained. The moments of this PSDF are used to estimate the expected pseudo‐spectral acceleration (PSA) response of the tank. Parametric variations are carried out to study the effects of tank height, foundation natural frequency, shear wave velocity of soil and ratio of the mass of tank (including liquid) to the mass of foundation on the PSA responses of tanks. Copyright © 2001 John Wiley & Sons, Ltd.     

成层地基一维土层对地震的随机反应分析

首先基于改进的一维剪切梁模型，对成层土层推导了确定自振频率、振型函数、参与系数及稳态动力响应的封闭型解析表达式，首次证明了成层土层振型函数的正交性，然后在此基础上，利用随机振动理论，研究了成层土层对地震的随机动力反应问题，关于基岩输入地震加速度的功率谱密度函数，考虑了两种形式：白噪声谱和过滤白噪声谱。数值计算结果表明：对这两种谱，土层的最大期望反应是不相同的；平稳输入与输出过高估计了土层的随机反应。     

The high‐frequency limit of usable response spectral ordinates from filtered analogue and digital strong‐motion accelerograms

The range of response frequencies for which spectral ordinates obtained from accelerograms may be considered reliable is limited by several factors, primary among them being the effects of filters that are routinely applied to remove noise from the records. Considerable attention has been focused on the low‐frequency limit of the usable spectral ordinates because of various engineering applications requiring long‐period spectral accelerations or displacements but only recently have rational approaches to selecting the high‐frequency limit been proposed. Since there are applications for which the high‐frequency spectral ordinates are important, the approaches to this issue presented in the recent studies are reviewed and their application to the ground‐motion database from Europe and the Middle East is explored. On the basis of the results of these analyses, it is concluded that a large proportion of this dataset can be used to provide reliable estimates of response spectral ordinates at much shorter periods than may have previously been considered feasible. Copyright © 2011 John Wiley & Sons, Ltd.     

Seismic response analysis of pipes by a probabilistic approach

Seismic response of buried pipes in longitudinal direction is studied. The effect of the variation of geotechnical properties of the surrounding soil on the stiffness, mass and damping of the soil is considered. The soil–structure interaction depends on pipe stiffness, joint stiffness, the variation of the soil stiffness and the soil mass and damping. Variations of the properties of the surrounding soil along the pipe are described by the random field theory. A numerical model is developed in order to simulate the effects of the variation of the soil on displacements, bending moments in the pipe and also to carry out a statistical analysis. The influence of different parameters regarding design and safety level of the pipe is conducted.     

The effects of torsion and motion coupling in site response estimation

Soil amplification characteristics are investigated using data from the Chibaken‐Toho‐Oki earthquake and its aftershocks recorded at Chiba dense array in Japan. The frequency‐dependent amplification function of soil is calculated using uphole‐to‐downhole spectral ratio analysis, considering the horizontal components of shear wave. The identified spectral ratios consistently demonstrate the splitting of peaks in their resonance frequencies and low amplification values in comparison with a 1D model. The torsional behaviour and horizontal ground motion coupling are clarified as the reasons for these phenomena at the site. To prove the hypothesis, the torsional motion is directly evaluated using the data of the horizontal dense array in different depths at the site. The comparison between Fourier spectra of torsional motion and identified transfer functions reveals the peaks at the same frequencies. The wave equation including torsion and horizontal motion coupling is introduced and solved for the layered media by applying wave propagation theory. Using the developed model, the effects of torsional motion with horizontal motion coupling on soil transfer function are numerically examined. Splitting and low amplification at resonance frequencies are confirmed by the results of numerical analysis. Furthermore, the ground motion in two horizontal directions at the site is simulated using site geotechnical specification and optimizing the model parameters. The simulated and recorded motions demonstrate good agreement that is used to validate the hypothesis. In addition, the spectral density of torsional ground motions are compared with the calculated one and found to be well predicted by the model. Finally, the results are used to explain the overestimation of damping in back‐calculation of dynamic soil properties using vertical array data in small strain level. Copyright © 2003 John Wiley & Sons, Ltd.     

Inelastic displacement ratio of near‐source pulse‐like ground motions

Near‐source pulse‐like records resulting from rupture's directivity have been found to depart from so‐called ordinary ground motions in terms of both elastic and inelastic structural seismic demands. In fact, response spectra may be strong if compared with what is expected from common ground motion prediction equations. Moreover, because not all spectral ordinates are affected uniformly, a peculiar spectral shape, with an especially amplified region depending on the pulse period, may follow. Consequently, inelastic seismic demand may show trends different to records not identified as pulse‐like (i.e., ordinary). This latter aspect is addressed in the study reported in this short communication, where a relatively large dataset of identified impulsive near‐source records is used to derive an analytical‐form relationship for the inelastic displacement ratio. It is found that, similar to what was proposed in literature for soft soil sites, a double‐opposite‐bumps form is required to match the empirical data as a function of the structural period over the pulse period ratio. The relationship builds consistently on previous studies on the topic, yet displays different shape with respect to the most common equations for static structural assessment procedures. Copyright © 2012 John Wiley & Sons, Ltd.     

Scaling of spectral displacement ordinates with damping ratios

The next generation of seismic design codes, especially those adopting the framework of performance‐based design, will include the option of design based on displacements rather than forces. For direct displacement‐based design using the substitute structure approach, the spectral ordinates of displacement need to be specified for a wide range of response periods and for several levels of damping. The code displacement spectra for damping values higher than the nominal value of 5% of critical will generally be obtained, as is the case in Eurocode 8 and other design codes, by applying scaling factors to the 5% damped ordinates. These scaling factors are defined as functions of the damping ratio and, in some cases, the response period, but are independent of the nature of the expected ground shaking. Using both predictive equations for spectral ordinates at several damping levels and stochastic simulations, it is shown that the scaling factors for different damping levels vary with magnitude and distance, reflecting a dependence of the scaling on the duration of shaking that increases with the damping ratio. The options for incorporating the influence of this factor into design code specifications of displacement response spectra are discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

3-D dynamic foundation-soil-foundation interaction on layered soil

In this work the interaction between adjacent rigid, surface foundations resting on a viscoelastic layered soil medium is studied. A 3-D frequency domain BEM formulation in conjunction with infinite space fundamental solutions and the so called `successive stiffness method', initially developed for elastostatics and adapted here for the solution of elastodynamic problems, are used for the simulation of a layered soil medium. As a result, a discretization of the soil-foundation interface and the surrounding free surface as well as the soil layers' interfaces is necessary. However, it is shown in this work that reasonably accurate results can be obtained by using a substantially reduced discretization scheme involving only a small portion of the free surface surrounding the foundation and the corresponding interfaces of the soil layers. The presented numerical results demonstrate the importance of the dynamic foundation-soil-foundation interaction phenomenon which becomes even more pronounced where the supporting soil medium is made up of relatively shallow layers close to its free surface.     

Delineation of hydrologically similar units in a watershed based on fuzzy classification of soil hydraulic properties

Evaluation of flow and transport processes in a watershed‐scale requires that the watershed be divided into homogenous spatial units referred to as hydrologically similar units (HSUs). Although a few discretization schemes are already in use, a universally acceptable method of obtaining HSUs is yet to emerge. In this study, we developed a fuzzy inference system (FIS) to classify the saturated hydraulic conductivity (K_s) and two water‐retention parameters α and n into fuzzy logic‐based soil hydrologic classes (FSHCs). Analysis of these classes showed that soil properties within an FSHC have less variability and those between two FSHCs have large variability. This result suggested that soils belonging to a specific FSHC may be more similar than those across different FSHCs and may be grouped together to represent an HSU. Soils within a specific hydrologic class were aggregated to delineate HSUs within the watershed. For the Dengei Pahad micro‐watershed (DPW), this approach showed five distinct regions representing a discretized zone having similar soil hydraulic properties. Application of this approach on a larger international database of soil hydraulic properties revealed that the developed hydrologic classes are quite comparable across different databases. The delineated HSUs based on these FSHCs were also better than the soil series map of the watershed in maintaining the soil heterogeneity of the watershed. Moreover, this new discretization scheme using the SWAT modelling environment showed better performance than the soil series‐based discretization approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Extending ground-motion prediction equations for spectral accelerations to higher response frequencies

Ground-motion prediction equations (GMPEs) for spectral accelerations have traditionally focused on the range of response periods most closely associated with the dynamic characteristics of buildings. Providing predictions only in this period range (from 0.1 to 2 or 3 s) has also accommodated the assumed limitations on the usable period range resulting from the processing of accelerograms. There are, however, engineering applications for which estimates of spectral ordinates are required at shorter response periods. Recent work has demonstrated that high-frequency spectral ordinates are relatively insensitive to record processing, contrary to previous assumptions. In the light of this finding, additional regressions are performed to extend a recent pan-European GMPE to higher response frequencies. This model and others that also include coefficients for spectral ordinates at several high response frequencies are used to explore options for interpolating coefficients for equations that do not provide good coverage in this range. The challenges and uncertainties associated with such interpolations are discussed. The paper concludes that a set of standard response frequencies could be usefully established for future GMPEs.     

Optimal design of friction pendulum system properties for isolated structures considering different soil conditions

This study aims at evaluating the optimal properties of friction pendulum bearings to be employed for the seismic protection of elastic isolated structural systems under earthquake excitations with different characteristics in terms of frequency content. A two-degree-of-freedom model is considered to describe the isolated system behavior while accounting for the superstructure flexibility and a non-dimensional formulation of the governing equations of motion is employed to relate the characteristic parameters describing the isolator and structure properties to the response parameters of interest for the performance assessment. Seismic excitations are modeled as time-modulated filtered Gaussian white noise random processes of different intensity within the power spectral density method. The filter parameters control the frequency content of the random excitations and are calibrated to describe stiff, medium and soft soil conditions, respectively. Finally, multi-variate regression expressions are obtained for the optimum values of the friction coefficient that minimize the superstructure displacements relative to the base mass as a function of the structural system properties, of the seismic input intensity and of the soil condition.