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.     

Influence of long‐period filter cut‐off on elastic spectral displacements

The effect of the long‐period filter cut‐off, T_c, on elastic spectral displacements is investigated using a strong ground‐motion database from Europe and the Middle East. The relation between the filter and oscillator responses is considered to observe the influence of T_c for both analogue and digital records, and the variations with site classification, magnitude, filter order and viscous damping. Robust statistics are derived using the re‐processed European data to generalize the effects of the long‐period filter cut‐off on maximum oscillator deformation demands as a function of these seismological and structural features. Statistics with a 95% confidence interval are derived to suggest usable period ranges for spectral displacement computations as a function of T_c. The results indicate that the maximum period at which spectral displacements can be confidently calculated depend strongly on the site class, magnitude and filter order. The period range where reliable long‐period information can be extracted from digital accelerograms is twice that of analogue records. Copyright © 2006 John Wiley & Sons, Ltd.     

Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records

A new set of empirical equations for prediction of displacement response spectral ordinates from 20 Hz to T = 20 s is illustrated. The coefficients of the equations were obtained by regressing a dataset based on 1,155 tri-axial digital and 9 analog accelerometer records from 60 earthquakes worldwide. Long period disturbances in the accelerograms were evaluated and removed using a very recent method, aimed at preserving the long-period spectral content of the records. Analysis of variance has disclosed only little evidence for regional dependence of ground motions, while a carefully conducted evaluation of site effects resulted in clearly differentiated spectral amplification bands associated to the main ground types B, C, and D of Eurocode 8. Spectral ordinates for vibration periods >5 s were found to scale with magnitude quite consistently with theoretical scaling from Brune’s model. On the other hand, comparison of results with those yielded by recent prediction models in Europe and the United States (NGA), indicated that the latter may not be uniformly reliable at long periods. The proposed empirical equations are easily implemented in computer programs for seismic hazard assessment, being characterized by a simple functional form and a restricted number of predictor variables. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

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.     

On the reliability of long‐period response spectral ordinates from digital accelerograms

Using records from co‐located broadband and digital strong motion (SM) instruments, it is first shown that the displacement waveforms obtained by double integration of the accelerogram need not be free of unrealistic baseline drift to yield reliable spectral ordinates up to at least 10 s. Secondly, to provide objective criteria for selecting reliable digital SM records for ground motion predictions at long periods, a set of synthetic accelerograms contaminated by random long‐period noise has been used, and the difference between the original accelerograms and the spurious ones in terms of response spectra has been quantified, by introducing a noise index that can be easily calculated based on the velocity waveform of the record. The results of this study suggest that high‐pass filtering the digital acceleration record from a cutoff period selected to suppress baseline drifts on the displacement waveform appears to be in most cases too conservative and unduly depletes reliable information on long‐period spectral ordinates. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

A stochastic ground motion model with separable temporal and spectral nonstationarities

A fully nonstationary stochastic model for strong earthquake ground motion is developed. The model employs filtering of a discretized white‐noise process. Nonstationarity is achieved by modulating the intensity and varying the filter properties in time. The formulation has the important advantage of separating the temporal and spectral nonstationary characteristics of the process, thereby allowing flexibility and ease in modeling and parameter estimation. The model is fitted to target ground motions by matching a set of statistical characteristics, including the mean‐square intensity, the cumulative mean number of zero‐level up‐crossings and a measure of the bandwidth, all expressed as functions of time. Post‐processing by a second filter assures zero residual velocity and displacement, and improves the match to response spectral ordinates for long periods. Copyright © 2008 John Wiley & Sons, Ltd.     

Use of the conditional spectrum to incorporate record‐to‐record variability in simplified seismic assessment of RC wall buildings

A procedure for incorporating record‐to‐record variability into the simplified seismic assessment of RC wall buildings is presented. The procedure relies on the use of the conditional spectrum to randomly sample spectral ordinates at relevant periods of vibration. For inelastic response, displacement reduction factors are then used to relate inelastic displacement demand to the spectral displacement at the effective period for single‐degree‐of‐freedom systems. Simple equations are used to convert back and forth between multi‐degree‐of‐freedom RC wall buildings and equivalent single‐degree‐of‐systems so that relevant engineering demand parameters can be obtained. Consideration is also given to higher‐mode effects by adapting existing modal combination rules. The proposed method is applied to several case study buildings, showing promising results in the examination of inter‐storey drift ratio and shear forces. The proposed method captures the variation in the distribution of structural response parameters that occurs with variations in structural configuration, intensity, engineering demand parameter of interest and site characteristics. Discussion is provided on possible ways to improve the accuracy of the procedure and suggestions for additional future work. Copyright © 2015 John Wiley & Sons, Ltd.     

Numbers of scaled and matched accelerograms required for inelastic dynamic analyses

Selecting, scaling and matching accelerograms are critically important to engineering design and assessment, enabling structural response to be determined with greater confidence and through fewer analyses than if unscaled accelerograms are employed. This paper considers the response of an 8‐storey multiple‐degree‐of‐freedom reinforced concrete structure to accelerograms selected, linearly scaled or spectrally matched using five different techniques. The first method consists of selecting real records on the basis of seismological characteristics, while the remaining methods make an initial selection on the basis of magnitude and spectral shape before (1) scaling to the target spectral acceleration at the initial period; (2) scaling to the target spectrum over a range of periods; (3) using wavelet adjustments to match the target spectrum and (4) using wavelet adjustments to match multiple target spectra for multiple damping ratios. The analyses indicate that the number of records required to obtain a stable estimate of the response decreases drastically as one moves through these methods. The exact number varies among damage measures and is related to the predictability of the damage measure. For measures such as peak roof and inter‐storey drift, member end rotation and the Park and Ang damage index, as few as one or two records are required to estimate the response to within ±5% (for a 64% confidence level) if matching to multiple damping ratios is conducted. Bias checks are made using predictive equations of the expected response derived from the results of 1656 nonlinear time‐domain analyses of the structure under the action of unscaled accelerograms. Copyright © 2008 John Wiley & Sons, Ltd.     

Site-dependent design spectra and strength modification factors, based on records from Greece

Elastic and inelastic spectra are derived, based on a representative sample of acceleration records from Greece, carefully selected based on magnitude, distance and peak ground acceleration criteria, and grouped into three ground condition categories according to the 2004 Eurocode 8 (EC8) provisions. Using software developed in-house, elastic (pseudoacceleration, pseudovelocity and displacement), as well as inelastic (strength and displacement) spectra are computed for various critical damping ratios and ductility levels. After appropriate scaling, mean spectra are computed both irrespective of, as well as for each different, ground condition, and comparisons with EC8 provisions are made. As a further evaluation of the code spectra, three additional earthquake scenarios are considered representing ground-motion characteristics not reflected in the compiled dataset of records. Subsequently, modification factors for strength (q_μ) are derived from statistical analysis of constant ductility spectra, and corresponding empirical relationships, suitable for design purposes, are proposed.     

PREDICTION OF VERTICAL RESPONSE SPECTRA IN EUROPE

Equations for the prediction of vertical peak and absolute acceleration spectral ordinates in terms of magnitude, source-distance and site geology are presented. Comparison to similarly derived horizontal equations shows vertical spectral values to be 1/2–1/4 of the horizontal. The influence of site geology on vertical ground motion is reduced with respect to the horizontal. Ratios of peak vertical to peak horizontal ground acceleration in the near-field of thrust faults are magnitude and distance dependent, reaching values in excess of one very near the fault of large magnitude events. For strike-slip faults the ratio exceeds one for moderate events, decreasing for larger events, and is distance independent. Spectral acceleration ratios exceed one at short periods but are less than one at intermediate and long periods, irrespective of the source mechanism.     

Inelastic displacement demands in steel structures and their relationship with earthquake frequency content parameters

This paper deals with the estimation of peak inelastic displacements of SDOF systems, representative of typical steel structures, under constant relative strength scenarios. Mean inelastic deformation demands on bilinear systems (simulating moment resisting frames) are considered as the basis for comparative purposes. Additional SDOF models representing partially‐restrained and concentrically‐braced (CB) frames are introduced and employed to assess the influence of different force‐displacement relationships on peak inelastic displacement ratios. The studies presented in this paper illustrate that the ratio between the overall yield strength and the strength during pinching intervals is the main factor governing the inelastic deformations of partially‐restrained models and leading to significant differences when compared with predictions based on bilinear structures, especially in the short‐period range. It is also shown that the response of CB systems can differ significantly from other pinching models when subjected to low or moderate levels of seismic demand, highlighting the necessity of employing dedicated models for studying the response of CB structures. Particular attention is also given to the influence of a number of scalar parameters that characterise the frequency content of the ground motion on the estimated peak displacement ratios. The relative merits of using the average spectral period T_aver, mean period T_m, predominant period T_g, characteristic period T_c and smoothed spectral predominant period T_o of the earthquake ground motion, are assessed. This paper demonstrates that the predominant period, defined as the period at which the input energy is maximum throughout the period range, is the most suitable frequency content scalar parameter for reducing the variability in displacement estimations. Finally, noniterative equivalent linearisation expressions based on the secant period and equivalent damping ratios are presented and verified for the prediction of peak deformation demands in steel structures. Copyright © 2011 John Wiley & Sons, Ltd.     

Inelastic displacement ratios for evaluation of existing structures

Results of a detailed statistical study of constant relative strength inelastic displacement ratios to estimate maximum lateral inelastic displacement demands on existing structures from maximum lateral elastic displacement demands are presented. These ratios were computed for single‐degree‐of‐freedom systems with different levels of lateral strength normalized to the strength required to remain elastic when subjected to a relatively large ensemble of recorded earthquake ground motions. Three groups of soil conditions with shear wave velocities higher than 180m/s are considered. The influence of period of vibration, level of lateral yielding strength, site conditions, earthquake magnitude, distance to the source, and strain‐hardening ratio are evaluated and discussed. Mean inelastic displacement ratios and those associated with various percentiles are presented. A special emphasis is given to the dispersion of these ratios. It is concluded that distance to the source has a negligible influence on constant relative strength inelastic displacement ratios. However, for periods smaller than 1s earthquake magnitude and soil conditions have a moderate influence on these ratios. Strain hardening decreases maximum inelastic displacement at a fairly constant rate depending on the level of relative strength for periods of vibration longer than about 1.0s while it decreases maximum inelastic displacement non‐linearly as the period of vibration shortens and as the relative‐strength ratio increases for periods of vibration shorter than 1.0s. Finally, results from non‐linear regression analyses are presented that provide a simplified expression to be used to approximate mean inelastic displacement ratios during the evaluation of existing structures built on firm sites. Copyright © 2003 John Wiley & Sons, Ltd.     

Response spectra for differential motion of columns paper II: Out-of-plane response

It is shown that the common response spectrum method for synchronous ground motion can be extended to make it applicable for earthquake response analyses of extended structures experiencing differential out-of-plane ground motion. A relative displacement spectrum for design of first-story columns SDC (T, T_T, ζ, ζ_T, τ, δ) is defined. In addition to the natural period of the out-of-plane response, T, and the corresponding fraction of critical damping, ζ, this spectrum also depends on the fundamental period of torsional vibrations, T_T, and the corresponding fraction of critical damping, ζ_T, on the “travel time,” τ (of the waves in the soil over a distance of about one-half the length of the structure), and on a dimensionless factor δ, describing the relative response of the first floor. The new spectrum, SDC, can be estimated by using the empirical scaling equations for relative displacement spectra, SD, and for peak ground velocity, v_max. For recorded strong-motion acceleration, and for symmetric buildings, the new spectrum can be computed from Duhamel's integrals of two uncoupled equations for dynamics equilibrium describing translation and rotation of a two-degree-of-freedom system. This representation is accurate when the energy of the strong-motion is carried by waves in the ground the wavelengths of which are one order of magnitude or more longer than the characteristic length of the structure.     

Probabilistic estimation of maximum inelastic displacement demands for performance‐based design

A probabilistic approach to estimate maximum inelastic displacement demands of single‐degree‐of‐freedom (SDOF) systems is presented. By making use of the probability of exceedance of maximum inelastic displacement demands for given maximum elastic spectral displacement and the mean annual frequency of exceedance of elastic spectral ordinates, a simplified procedure is proposed to estimate mean annual frequencies of exceedance of maximum inelastic displacement demands. Simplifying assumptions are thoroughly examined and discussed. Using readily available elastic seismic hazard curves the procedure can be used to compute maximum inelastic displacement seismic hazard curves and uniform hazard spectra of maximum inelastic displacement demands. The resulting maximum inelastic displacement demand spectra provide a more rational way of establishing seismic demands for new and existing structures when performance‐based approaches are used. The proposed procedure is illustrated for elastoplastic SDOF systems having known‐lateral strength located in a region of high seismicity in California. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

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.     

Inelastic displacement ratios for evaluation of structures built on soft soil sites

This paper summarizes the results of a comprehensive statistical study aimed at evaluating peak lateral inelastic displacement demands of structures with known lateral strength and stiffness built on soft soil site conditions. For that purpose, empirical information on inelastic displacement ratios which are defined as the ratio of peak lateral inelastic displacement demands to peak elastic displacement demands are investigated. Inelastic displacement ratios were computed from the response of single‐degree‐of‐freedom systems having 6 levels of relative lateral strength when subjected to 118 earthquake ground motions recorded on bay‐mud sites of the San Francisco Bay Area and on soft soil sites located in the former lake‐bed zone of Mexico City. Mean inelastic displacement ratios and their corresponding scatter are presented for both ground motion ensembles. The influence of period of vibration normalized by the predominant period of the ground motion, the level of lateral strength, earthquake magnitude, and distance to the source are evaluated and discussed. In addition, the effects of post‐yield stiffness and of stiffness and strength degradation on inelastic displacement ratios are also investigated. It is concluded that magnitude and distance to the source have negligible effects on constant‐strength inelastic displacement ratios. Results also indicate that weak and stiffness‐degrading structures in the short spectral region could experience inelastic displacement demands larger than those corresponding to non‐degrading structures. Finally, a simplified equation obtained using regression analyses aimed at estimating mean inelastic displacement ratios is proposed for assisting structural engineers in performance‐based assessment of structures built on soft soil sites. Copyright © 2006 John Wiley & Sons, Ltd.     

地震动特性对隔震结构弹塑性位移反应谱的影响研究

根据特定震源机制、震级、断层距和场地条件选取69条地震动记录并进行分组,利用Nspectra软件计算隔震结构的弹塑性位移反应谱,分析断层距、场地条件、震级、阻尼比对弹塑性位移谱的影响,探讨隔震层的力学参数对地震能量耗散的影响。研究结果表明:相较于远场,处于近场的隔震结构最为不利,隔震层位移谱值受场地条件、地震加速度和速度大小影响较大;随着断层距的增大,位移谱值衰减较快,且在软土场地中隔震层的位移谱值衰减幅度大于硬土场地;地震震级大小对位移谱形状的影响不明显,但能够使隔震层的位移谱值产生整体缩放效应;阻尼比在小于0.4的范围内,隔震层在不同地震动特性作用下位移谱值差别较大,但在大于0.4以后,位移谱值及谱形基本趋于一致;屈服力较小(恢复力/重力小于等于1)的隔震层随自振周期增大其耗能性能更加突出。     

Residual displacement ratios for assessment of existing structures

Results of an analytical study aimed at evaluating residual displacement ratios, C_r, which allow the estimation of residual displacement demands from maximum elastic displacement demands is presented. Residual displacement ratios were computed using response time‐history analyses of single‐degree‐of‐freedom systems having 6 levels of relative lateral strength when subjected to an ensemble of 240 earthquake ground motions recorded in stations placed on firm sites. The results were statistically organized to evaluate the influence of the following parameters: period of vibration, level of relative lateral strength, site conditions, earthquake magnitude, and distance to the source. In addition, the influence of post‐yield stiffness ratio in bilinear systems and of the unloading stiffness in stiffness‐degrading systems was also investigated. A special emphasis is given to the uncertainty of these ratios. From this study, it is concluded that mean residual displacement ratios are more sensitive to changes in local site conditions, earthquake magnitude, distance to the source range and hysteretic behaviour than mean inelastic displacement ratios. In particular, residual displacement ratios exhibit large levels of record‐to‐record variability and, therefore, this dispersion should be taken into account when estimating residual displacements. A simplified expression is presented to estimate mean residual displacements ratios for elastoplastic systems during the evaluation of existing structures built on firm soil sites. Copyright © 2005 John Wiley & Sons, Ltd.