SUMMARYResults 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 di erent 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 180 m=s are considered. The in uence 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 in uence on constant relative strength inelastic displacement ratios. However, for periods smaller than 1 s earthquake magnitude and soil conditions have a moderate in uence 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 1:0 s 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 1:0 s. Finally, results from non-linear regression analyses are presented that provide a simpliÿed expression to be used to approximate mean inelastic displacement ratios during the evaluation of existing structures built on ÿrm sites.

SUMMARYSix approximate methods to estimate the maximum inelastic displacement demand of single-degree-offreedom systems are evaluated. In all methods, the maximum displacement demand of inelastic systems is estimated from the maximum displacement demand of linear elastic systems. Of the methods evaluated herein, four are based on equivalent linearization in which the maximum deformation is estimated as the maximum deformation of a linear elastic system with lower lateral sti ness and with higher damping coe cient than those of the inelastic system. In the other two methods the maximum inelastic displacement is estimated as a product of the maximum deformation of a linear elastic system with the same lateral sti ness and the same damping coe cient as those of the inelastic system for which the maximum displacement is being estimated, times a modifying factor. Elastoplastic and sti nessdegrading models with periods between 0.05 and 3:0 s are considered when subjected to 264 ground motions recorded on ÿrm sites in California. Mean ratios of approximate to exact maximum displacements corresponding to each method are computed as a function of the period of vibration and as a function of the displacement ductility ratio. Finally, comments on the advantages and disadvantages of each method when applied to practical situations are given.

SUMMARYResults 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-offreedom systems having 6 levels of relative lateral strength when subjected to an ensemble of 240 earthquake ground motions recorded in stations placed on ÿrm sites. The results were statistically organized to evaluate the in uence of the following parameters: period of vibration, level of relative lateral strength, site conditions, earthquake magnitude, and distance to the source. In addition, the in uence of post-yield sti ness ratio in bilinear systems and of the unloading sti ness in sti ness-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-torecord variability and, therefore, this dispersion should be taken into account when estimating residual displacements. A simpliÿed expression is presented to estimate mean residual displacements ratios for elastoplastic systems during the evaluation of existing structures built on ÿrm soil sites.

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SUMMARYThis paper summarizes the results of a comprehensive statistical study aimed at evaluating peak lateral inelastic displacement demands of structures with known lateral strength and sti ness built on soft soil site conditions. For that purpose, empirical information on inelastic displacement ratios which are deÿned 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 singledegree-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 in uence 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 e ects of post-yield sti ness and of sti ness and strength degradation on inelastic displacement ratios are also investigated. It is concluded that magnitude and distance to the source have negligible e ects on constant-strength inelastic displacement ratios. Results also indicate that weak and sti ness-degrading structures in the short spectral region could experience inelastic displacement demands larger than those corresponding to non-degrading structures. Finally, a simpliÿed 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.

This paper summarizes results of a comprehensive analytical study aimed at evaluating the amplitude and heightwise distribution of residual drift demands in multi-storey moment-resisting frames after earthquake excitation. For that purpose, a family of 12 one-bay two-dimensional generic frame models was subjected to an ensemble of 40 ground motions scaled to different intensities. In this investigation, an inelastic ground motion intensity measure was employed to scale each record, which allowed reducing the recordto-record variability in the estimation of residual drift demands. The results were statistically processed in order to evaluate the influence of ground motion intensity, number of stories, period of vibration, frame mechanism, system overstrength, and hysteretic behaviour on central tendency of residual drift demands. In addition, a special emphasis was given to evaluate the uncertainty in the estimation of residual drift demands. Results of incremental dynamic analyses indicate that the amplitude and heightwise distribution of residual drift demands strongly depends on the frame mechanism, the heightwise system structural overstrength and the component hysteretic behaviour. An important conclusion for performance-based assessment is that the evaluation of residual drift demands involves significantly larger levels of uncertainty (i.e. record-to-record variability) than that of maximum drift demands, which suggests that this variability and corresponding uncertainty should be explicitly taken into account when estimating residual drift demands during performance-based seismic assessment of frame buildings.

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. a site [1][2][3][4]. Using the geometry and location with respect to the site of all possible seismic sources, the probability distribution of earthquake magnitudes at each source and attenuation relationships, a conventional PSHA permits the estimations of the mean annual frequency (MAF) of exceedance of a certain peak ground motion parameter (e.g. peak ground acceleration, etc.) or a linear elastic response spectral ordinate (e.g. pseudo-acceleration, S a ) by integration over all possible sources, earthquake magnitudes and distances. However, conventional PSHA only provides probabilistic estimates of demands on linear elastic systems while most structures are likely to undergo significant inelastic deformations in the event of strong, or even moderate, earthquake ground motions. Therefore, it is of utmost importance to develop rational methods to estimate lateral displacement demands on inelastic systems.Although most structures do not behave like single-degree-of-freedom (SDOF) systems, various studies have shown that inelastic SDOF systems may provide the basis for estimating global deformation demands of buildings [5][6][7][8][9][10][11]. Based on this observation, non-linear static procedures have been introduced in recent seismic recommendations for design of new structures as well as for assessment and rehabilitation of existing structures [12][13][14][15][16][17] in which roof displacements are estimated from maximum displacements of SDOF inelastic systems. In these design provisions the estimation of maximum inelastic SDOF displacements is done through simplified procedures by either applying modification factors on maximum SDOF elastic displacement demands or by considering equivalent SDOF systems with elongated fundamental period and increased damping ratio [18]. However, the inherent uncertainty introduced by approximating maximum inelastic displacements from maximum elastic displacements is neglected.The objective of thi...