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This paper examines various parameters that provide a measure of spectral shape and studies how they relateto the potential of ground motion records to cause the collapse of a given structure. It is shown that whenmeasuring the ground motion intensity by the spectral acceleration at the first-mode period of the structure,Sa(T1), records causing collapse at low ground motion intensities typically have significantly differentspectral shapes than those that do not cause collapse until much higher ground motion intensities. A spectralshape typical of damaging records is identified, and a metric for quantifying the spectral shape of a recordcalled SaRatio is proposed and evaluated. SaRatio is defined as the ratio between Sa ( T1) and the averagespectral value over a period range. The ability of SaRatio to predict the collapse intensity, i.e. the minimumintensity at which a giv en ground motion causes the collapse of a given structure, is compared to otherrecently proposed spectral shape metrics including epsilon (e), eta (η) and Np. The results demonstrate thatSaRatio is typically a much better predictor of collapse intensity than other spectral shape metrics.

This paper investigates the performance of spectral acceleration averaged over a period range (Saavg) as an intensity measure (IM) for estimating the collapse risk of structures subjected to earthquake loading. The performance of Saavg is evaluated using the following criteria: efficiency, sufficiency, the availability or ease of developing probabilistic seismic hazard information in terms of the IM and the variability of collapse risk estimates produced by the IM. Comparisons are also made between Saavg and the more traditional IM: spectral acceleration at the first-mode period of the structure (Sa(T1)). Though most previous studies have evaluated IMs using a relatively limited set of structures, this paper considers nearly 700 moment-resisting frame and shear wall structures of various heights to compare the efficiency and sufficiency of the IMs. The collapse risk estimates produced by Saavg and Sa(T1) are also compared, and the variability of the risk estimates is evaluated when different ground motion sets are used to assess the structural response. The results of this paper suggest that Saavg, when computed using an appropriate period range, is generally more efficient, more likely to be sufficient and provides more stable collapse risk estimates than Sa(T1). © 2015 John Wiley & Sons, Ltd.

SUMMARY A procedure to estimate the seismic motion at the base of a building from measured acceleration response at two or more floors is presented. The proposed method is comprised of two steps. In the first step, the dynamic characteristics of the building are inferred by using an output-only system identification procedure. In the second step, the motion of the base of the building is estimated by using the transfer function of a simplified building model consisting of a shear and flexural continuous beam together with dynamic properties obtained in the first step. The proposed method is validated first with an analytical model subjected to the 1940 El Centro ground motion and then with an instrumented building in California that experienced the 1994 Northridge earthquake, and the ground motions at the base of the building are available. It is shown that the proposed method is capable of providing very good estimates of the motion at the base. The use of the proposed method is finally illustrated on an instrumented building, where the sensor at the base of the building did not function during the 1994 Northridge earthquake. Copyright © 2013 John Wiley & Sons, Ltd.

SUMMARY This study examines the expected cost of repairing earthquake damage in a set of 30 archetype reinforced concrete moment frame buildings. Varying in height from 1 to 20 stories, these archetype buildings are representative of office buildings with special RC moment frames, designed according to modern seismic codes. Economic losses associated with repair are computed using a performance-based earthquake engineering framework, which integrates the site-specific seismic hazard, structural response, damage to building components and contents, and the resulting repair costs. Results are presented in terms of expected losses under the design basis earthquake, expected annual losses, and present value of life cycle losses. Expected annual economic losses for the set of RC frame buildings are approximately 1.0% of building replacement cost for the high seismic California site considered. Repair costs are shown to vary significantly depending on building height and other architectural and structural design parameters. These metrics provide important information about the level of economic losses expected in code-conforming structures and the effect of design decisions on the earthquake losses. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY Assessing the probability of collapse is a computationally demanding component of performance-based earthquake engineering. This paper examines various aspects involved in the computation of the mean annual frequency of collapse (λc) and proposes an efficient method for estimating the sidesway collapse risk of structures in seismic regions. By deaggregating the mean annual frequency of collapse, it is shown that the mean annual frequency of collapse is typically dominated by earthquake ground motion intensities corresponding to the lower half of the collapse fragility curve. Uncertainty in the collapse fragility curve and mean annual frequency of collapse as a function of the number of ground motions used in calculations is also quantified, and it is shown that using a small number of ground motions can lead to unreliable estimates of a structure's collapse risk. The proposed method is shown to significantly reduce the computational effort and uncertainty in the estimate. Copyright © 2012 John Wiley & Sons, Ltd.

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.

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.

Results 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.

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 record-to-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. Copyright © 2006 John Wiley & Sons, Ltd.

Limitations associated with deterministic methods to quantify demands and develop rational acceptance criteria have led to the emergence of probabilistic procedures in performance-based seismic engineering. The Pacific Earthquake Engineering Research performance-based methodology is one such approach. In this paper, the impact of certain modelling decisions made at different stages of the evaluation process on the performance assessment of a typical multi-bent viaduct is examined. Modelling, in the context of this paper, covers hazard modelling, structural modelling and loss modelling. The specific application considered in this study is a section of an existing viaduct in California: the 1-880 interstate highway. Several simulation models of the viaduct are developed, a series of nonlinear time-history analyses are carried out to predict demands, measures of damage are evaluated and the probability of closure of the viaduct is estimated using the specified hazard for the site. It is concluded that the methodology offers several advantages over existing deterministic performance-based procedures. Results of the investigation indicate that the assessment methodology is particularly sensitive to the reliability of decisions made by bridge inspectors following a seismic event, and to the dispersion in the demand estimation, which in turn is influenced by several factors including soil-structure interaction effects and ground motion scaling procedures.

SUMMARY The damping reduction factors are used in a few building codes in order to estimate the elastic response spectrum with high damping ratios from its 5%-damped counterpart. At present, much research regarding this factor has been published. The purpose of this paper is to evaluate the accuracy ofve dierent models of damping reduction factors. In addition, the damping reduction factors adopted in the recent U.S.A. seismic codes are also evaluated. Statistical studies are carried out according to a total of 216 earthquake ground motions recorded onrm sites in California. Considered here are elastic single- degree-of-freedom (SDOF) systems with natural vibrating periods between 0.1 and 6:0 s. Mean ratios and dispersion of approximate to exact maximum elastic displacement demands corresponding to various damping reduction factors are computed as functions of the natural vibrating period and the viscous damping ratio. Copyright ? 2005 John Wiley & Sons, Ltd.

SUMMARY 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 dierent 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 inuence 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 ra- tios 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 inuence on constant relative strength inelastic displacement ratios. However, for periods smaller than 1 s earth- quake magnitude and soil conditions have a moderate inuence on these ratios. Strain hardening de- creases 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 dis- placement 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 pre- sented that provide a simplied expression to be used to approximate mean inelastic displacement ratios during the evaluation of existing structures built onrm sites. Copyright ? 2003 John Wiley & Sons, Ltd.

Six approximate methods to estimate the maximum inelastic displacement demand of single-degree-of-freedom 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 stiffness and with higher damping coefficient 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 stiffness and the same damping coefficient as those of the inelastic system for which the maximum displacement is being estimated, times a modifying factor. Elastoplastic and stiffness-degrading models with periods between 0.05 and 3.0 s are considered when subjected to 264 ground motions recorded on firm 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. Copyright © 2001 John Wiley & Sons, Ltd.

Results from analytical studies conducted on an instrumented ten-storey reinforced concrete building which experienced ground accelerations in excess of 0.6g during the 1987 Whittier-Narrows California earthquake and suffered only minimal damage are presented. Using the dynamic characteristics inferred from accelerations recorded in the building during the earthquake, a mathematical model was calibrated to study the response of the building and to explain its good behaviour despite the apparent severity of the motions recorded in the basement of the building. Very good correlation was obtained between the computed and recorded response of the building. Non-linear analyses were conducted to evaluate the strength and deformation capacity of the building and to estimate its response in the event of more severe earthquake ground motions. Special emphasis is given to the evaluation of the overstrength of the building. Lateral overstrengths larger than 4.2 and larger than 5.7 were computed for the longitudinal and transverse directions of the building, respectively. It is concluded that these high levels overstrength in the building played an important role in limiting the damage during the Whittier-Narrow earthquake. Since the estimation of inelastic deformations during severe earthquake ground motions depends on the actual strength of the building, it is recommended to consider explicitly probable values of this overstrength in the strength reduction factors.

This paper presents a probabilistic approach to the estimation of lateral strengths required to provide an adequate control of inelastic deformations in structures during severe earthquake ground motions. In contrast to a deterministic approach, the approach presented herein accounts explicitly for the variability of the response of non-linear systems due to the inherent uncertainties in the intensity and characteristics of the input excitation by considering the probability distribution of maximum inelastic strength demands. This study is based on the computation of non-linear strength demands of single-degree-of-freedom (SDOF) systems experiencing different levels of inelastic deformation when subjected to 124 recorded earthquake ground motions. Using empirical cumulative distribution functions site-dependent probabilistic non-linear spectra were computed for six probabilities of exceedance of different levels of inelastic deformation. It is concluded that the lateral strength required to control displacement ductility demands is significantly affected by the maximum tolerable inelastic deformation, the system's period of vibration, the local site conditions and the level of risk in exceeding the maximum tolerable deformations.