Inelastic Displacement Ratio Research Articles

Influence of site effects on inelastic displacement ratios for SDOF and MDOF systems

Lateral displacements’ control of structures subjected to earthquake ground motion has now been recognized as a key factor in the assessment of system performance, leading to design approaches that use displacements rather than forces as the starting point for the seismic evaluation of structures. In fact performance-based approaches offer significant advantages in comparison with traditional force-based approaches, since the former are capable of focusing on nonlinear behaviour and consequent damage to the structure, in contrast to the latter. Lateral displacement demand, particularly in structures that exhibit nonlinear behaviour, can be significantly affected by the features of strong ground motion, i.e., amplitude, frequency content and duration. Such characteristics are in turn profoundly influenced by the irregularity and changeability in earthquake ground motions, which should therefore be taken into account appropriately. The great number of strong motion records gathered throughout the last decades in the most widely varying soil-site conditions has made accounting for soil-site effects in the characterization of elastic and inelastic displacement demands feasible. The aim of this paper is to present the results of numerical investigations on the response of both single-degree-of-freedom (SDOF) and multiple-degree-of-freedom (MDOF) systems, through nonlinear time-history analyses performed on the basis of a wide data set of strong motion records. Constant ductility spectra of the ratios of the maximum inelastic displacement to the corresponding maximum elastic demand were derived for this purpose. In particular, the influences of earthquake magnitude, source-to-site distance, local soil-site conditions, ductility and hysteretic behaviour were quantified. Finally, simplified expressions for the ratio of the maximum inelastic to the maximum elastic displacement were established, in order to allow the evaluation of inelastic displacements for new or rehabilitated structures for which the global displacement ductility can be estimated, directly from the knowledge of the corresponding elastic demands.

Effects of hanging wall and forward directivity in the 1999 Chi-Chi earthquake on inelastic displacement response of structures

The characteristics of the inelastic response of structures affected by hanging wall and forward directivity in the 1999 Chi-Chi earthquake are investigated. Inelastic displacement ratios (IDRs) for ground motions impacted by these nearfield effects are evaluated and comprehensively compared to far-field ground motions. In addition, the inelastic displacement responses to hanging wall and footwall ground motions are compared. It is concluded that the inelastic displacement response is significantly affected in the short period range by hanging wall and in the long period range by footwall. Although high peak ground acceleration was observed at hanging wall stations, the IDRs for structures on hanging wall sites are only larger than footwall sites in the very long period range. Forward directivity effects result in larger IDRs for periods longer than about 0.5s. Adopting statistical relationships for IDRs established using far-field ground motions may lead to either overestimation or underestimation in the seismic evaluation of existing structures located in near-field regions, depending on their fundamental vibration periods.

An Empirical Ground-Motion Attenuation Relation for Inelastic Spectral Displacement

This article presents an empirical ground-motion prediction model (attenuation relation) for inelastic (as opposed to elastic) spectral displacement ( Sdi ) for ground motions without forward directivity effects. It is a function of two earthquake parameters, moment magnitude ( M w) and the closest distance to rupture ( R rup), and two bilinear oscillator parameters, an undamped elastic period ( T ) and a yield displacement ( dy ). The dy is introduced via the predicted median strength-reduction factor (![Graphic][1] ), a proxy for the ratio of elastic spectral displacement ( Sde ) to dy , which is identical with the familiar strength-reduction factor ( R ). The proxy ![Graphic][2] recognizes that R can only be estimated indirectly because it implicitly contains the random variable, Sde , which cannot be known a priori ; therefore, the median estimate or predicted median ( Ŝde ) from a conventional (elastic) ground-motion prediction model is used instead to calculate ![Graphic][3] = Ŝde / dy . For enhanced generality, the inelastic spectral displacement prediction model here is based on a ratio concept, that is, the total model is a (any) conventional elastic prediction model coupled with a new inelastic displacement ratio prediction model, with proper statistical correlation between the two. We empirically consider the dependence of this ratio on source and path effects (i.e., M w and R rup), and find that M w is significant, but R rup is not. The resulting prediction model can easily be added to existing probabilistic seismic-hazard analysis (psha) software packages with only one extra structure-specific parameter, dy of the oscillator. In practical engineering applications, this will likely have been estimated from the conventional static pushover analysis of the multi-degree-of-freedom (mdof) structure under consideration. The resulting psha product is a hazard curve for Sdi , the inelastic spectral displacement of a nonlinear oscillator. Such a curve can provide a more direct hazard- based target displacement for nonlinear static procedures (Federal Emergency Management Agency [fema] 356, 2000) and/or a basic input function for new probabilistic seismic-demand analyses that is based on Sdi (as opposed to Sde ) as an efficient and sufficient intensity measure. This new attenuation relationship will be particularly useful in evaluating the performance of existing structures and specified designs with known lateral strength. In particular, unlike most past studies, it does not pre-fix the ductility level. [1]: /embed/inline-graphic-1.gif [2]: /embed/inline-graphic-2.gif [3]: /embed/inline-graphic-3.gif

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.

Direct estimation of the seismic demand and capacity of oscillators with multi-linear static pushovers through IDA

SUMMARY SPO2IDA is introduced, a software tool that is capable of recreating the seismic behavior of oscillators with complex quadrilinear backbones. It provides a direct connection between the Static Pushover (SPO) curve and the results of Incremental Dynamic Analysis (IDA), a computer-intensive procedure that offers thorough demand and capacity prediction capability by using a series of nonlinear dynamic analyses under a suitably scaled suite of ground motion records. To achieve this, the seismic behavior of numerous single-degree-of-freedom (SDOF) systems is investigated through IDA. The oscillators have a wide range of periods and feature pinching hysteresis with backbones ranging from simple bilinear to complex quadrilinear with an elastic, a hardening and a negativestiffness segment plus a final residual plateau that terminates with a drop to zero strength. An efficient method is introduced to treat the backbone shape by summarizing the analysis results into the 16%, 50% and 84% fractile IDA curves, reducing them to a few shape parameters and finding simpler backbones that reproduce the IDA curves of complex ones. Thus, vast economies are realized while important intuition is gained on the role of the backbone shape to the seismic performance. The final product is SPO2IDA, an accurate, spreadsheet-level tool for Performance-Based Earthquake Engineering that can rapidly estimate demands and limit-state capacities, strength reduction R-factors and inelastic displacement ratios for any SDOF system with such a quadrilinear SPO curve. Copyright c ∞ 2005 John Wiley & Sons, Ltd.

Residual displacement ratios for assessment of existing structures

AbstractResults of an analytical study aimed at evaluating residual displacement ratios, Cr, 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.

Errata for “Inelastic Displacement Ratios for Design of Structures on Soft Soils Sites” by Jorge Ruiz-García and Eduardo Miranda M.ASCE

Errata for “Inelastic Displacement Ratios for Design of Structures on Soft Soils Sites” by Jorge Ruiz-García and Eduardo Miranda M.ASCE

Inelastic Displacement Ratios for Design of Structures on Soft Soils Sites

This paper summarizes the results of a comprehensive statistical study of inelastic displacement ratios that allow the estimation of maximum lateral inelastic displacement demands from maximum elastic displacement demands for structures built on soft soil sites. These ratios were computed for single-degree-of-freedom systems undergoing six levels of inelastic deformation when subjected to 116 earthquake ground motions recorded on bay-mud sites of the San Francisco Bay Area and on sites in the former lake-bed zone of Mexico City. These soft soil deposits are characterized by low shear wave velocities, high water contents, and high plasticity indices. The influence of period of vibration normalized by the predominant period of the ground motion, the level of inelastic deformation, earthquake magnitude, and epicentral distance are evaluated and discussed. Mean inelastic displacement ratios and their corresponding dispersion are presented. The effect of stiffness degradation on inelastic displacement ratios is also considered. For this purpose, mean ratios of maximum inelastic displacement demands of stiffness degrading systems to maximum inelastic displacement demands of nondegrading systems are presented. Finally, a simplified equation to estimate mean inelastic displacement ratios obtained through nonlinear regression analyses is provided to aid designers estimate inelastic displacement demands of structures built on soft soil sites.

Inelastic spectra for infilled reinforced concrete frames

AbstractIn two companion papers a simplified non‐linear analysis procedure for infilled reinforced concrete frames is introduced. In this paper a simple relation between strength reduction factor, ductility and period (R–µ–T relation) is presented. It is intended to be used for the determination of inelastic displacement ratios and of inelastic spectra in conjunction with idealized elastic spectra. The R–µ–T relation was developed from results of an extensive parametric study employing a SDOF mathematical model composed of structural elements representing the frame and infill. The structural parameters, used in the proposed R–µ–T relation, in addition to the parameters used in a usual (e.g. elasto‐plastic) system, are ductility at the beginning of strength degradation, and the reduction of strength after the failure of the infills. Formulae depend also on the corner periods of the elastic spectrum. The proposed equations were validated by comparing results in terms of the reduction factors, inelastic displacement ratios, and inelastic spectra in the acceleration–displacement format, with those obtained by non‐linear dynamic analyses for three sets of recorded and semi‐artificial ground motions. A new approach was used for generating semi‐artificial ground motions compatible with the target spectrum. This approach preserves the basic characteristics of individual ground motions, whereas the mean spectrum of the whole ground motion set fits the target spectrum excellently. In the parametric study, the R–µ–T relation was determined by assuming a constant reduction factor, while the corresponding ductility was calculated for different ground motions. The mean values proved to be noticeably different from the mean values determined based on a constant ductility approach, while the median values determined by the different procedures were between the two means. The approach employed in the study yields a R–µ–T relation which is conservative both for design and performance assessment (compared with a relation based on median values). Copyright © 2004 John Wiley & Sons, Ltd.

Characterization of displacement demand for elastic and inelastic SDOF systems

The results of a research concerning the characterization of elastic and inelastic displacement spectral demand as a function of magnitude, source-to-site distance, and soil type are presented. The displacement spectra were computed for single degree of freedom systems subjected to a large set of strong ground motion records. In the elastic case, design displacement spectra, modeled in a simplified way with a bilinear shape in the period range 0–4 s, are then proposed for the estimation of the displacement demand to structures located on different local soil condition, at different distance from the causative fault, and for different levels of magnitude. In order to evaluate the reliability of the proposed design displacement spectra, probabilistic displacement spectra corresponding to different levels of probability of non-exceedance were also carried out. The inelastic displacement demand to elasto-plastic systems was analyzed through the ratio between inelastic and elastic spectral displacements. Simplified relationships of the inelastic displacement ratio are then proposed as a function of displacement ductility, soil condition and period of vibration. Finally, as a comparison, the inelastic displacement ratios were also estimated considering other constitutive models.

Inelastic displacement ratios for evaluation of existing structures

AbstractResults 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.

Estimation of Inelastic Deformation Demands of SDOF Systems

Two recently proposed and closely related approximate methods to estimate the maximum lateral inelastic deformations demands of single-degree-of-freedom (SDOF) systems subjected to earthquake ground motions are investigated. In both methods, maximum inelastic displacements are estimated from maximum elastic displacements through the use of parameters that relate the response of elastic and inelastic SDOF systems. In the first method, the maximum inelastic displacement is estimated from the maximum elastic displacement through mean strength reduction factors derived from statistical studies. In the second method, the maximum inelastic displacement is estimated from the maximum elastic displacement through inelastic displacement ratios also derived from statistical studies. It is shown that the first method is actually a first-order approximation of the second method and that only in the absence of variability the two methods provide the same result. Furthermore, it is shown that the systematic error introduced by the first method leads to underestimations of maximum inelastic displacement demands. The error is evaluated and examples using results from statistical results of the seismic response of SDOF systems are presented.

Inelastic Displacement Ratios for Structures on Firm Sites

The results of a comprehensive statistical study of inelastic displacement ratios that permit the estimation of maximum lateral inelastic displacement demands on a structure from maximum lateral elastic displacement demands are presented. These ratios were computed for single-degree-of-freedom systems undergoing different levels of inelastic deformation when subjected to a relatively large number of recorded earthquake ground motions. The study is based on 264 acceleration time histories recorded on firm sites during various earthquakes in California. Three types of soil conditions with shear-wave velocities higher than 180 m/s are considered. The influences of period of vibration, level of ductility demand, site conditions, earthquake magnitude, and epicentral distance are carefully evaluated and discussed. Inelastic displacement ratios associated with mean values are presented. Special emphasis is given to the dispersion of the results. It is concluded that for sites with average shear-wave velocities higher than 180 m/s the influence of soil conditions is relatively small and can be neglected for design purposes. Finally, results from nonlinear regression analyses are presented that provide a simplified expression to be used in the design to approximate mean inelastic displacements ratios for structures on firm sites.