Recently, two of the authors have introduced the collapse capacity spectrum methodology, which allows the assessment of the collapse capacity of highly inelastic P delta sensitive regular frame structures without performing non-linear time-history analyses. The main ingredient of this method is the collapse capacity spectrum, which represents the seismic collapse capacity of an inelastic non-deteriorating single-degree-of-freedom system vulnerable to the Pdelta effect as a function of its initial period, negative post-yield stiffness ratio, viscous damping coefficient, and the shape of the hysteretic loop. In the present study, multiple linear regression analyses are applied to provide enhanced analytical expressions of these spectra. The record-torecord uncertainty of the collapse capacity is captured through median, 16th and 84th percentile spectra. For several test systems analytical collapse fragility functions based on these spectra are set in contrast with the corresponding sorted individual collapse capacities. These examples prove the superiority of the proposed analytical expressions compared to its original formulation.

The seismic collapse capacity of a structure is a critical factor in earthquake risk assessment within engineeringpractices. Conventionally, evaluating this capacity involves intricate and time-consuming incremental dynamicanalyses. However, recent progress has brought forth alternative, streamlined methodologies grounded in the use ofstructural behavior curves. This study employs the application of these innovative approaches to comprehensivelyassess the seismic collapse capacity of structures. Embracing advancements, it strives to enhance the efficiency andprecision of seismic risk assessments in engineering practices.In addition to the efficiency of assessment methodologies, it is imperative that the calculation of seismic collapsecapacity aligns with the specific demands of the construction site. This ensures that the seismic risk falls within theestablished allowable limits. This consideration becomes particularly critical for construction sites located in closeproximity to fault zones. In such areas, the presence of directivity pulses heightened attention to seismic collapsecapacity. Recognizing that structural ductility and the pulse period ratio in the near-fault are primary factorsinfluencing seismic collapse capacity, which is demanded in site, this study delves into a detailed numericalinvestigation of these critical elements. Subsequently, the seismic collapse capacity demanded in the near-fault ismeticulously estimated based on these considerations.The extensive investigations undertaken in this study yield insightful revelations. It is evident that heightenedstructural ductility correlates with an augmentation of seismic collapse capacity, both in the near-fault and far-faultscenarios. Conversely, a reduction in seismic collapse capacity in the near-fault is discerned as the pulse period ratioincreases concerning the fundamental period of the structure. To conduct a comprehensive evaluation, the ratio ofseismic collapse capacity in the near-fault to that in the far-fault is calculated, taking into account both ductility andpulse period ratio. This derived parameter, denoted as γ, is then employed to estimate the seismic collapse capacitydemanded in the near-fault. This analysis contributes valuable insights to the understanding of seismic behavior inboth near-fault and far-fault regions.For the assessment of seismic collapse capacity demand at construction sites, the study recommends theutilization of a lower bound of the ratio of near-fault to far-fault seismic collapse capacity. This lower bound,associated with lower ductility and a higher pulse period ratio, is not just conservative but also robust. Importantly,this cautious approach ensures that an increase in this parameter does not significantly escalate the demand at theconstruction site. Such a calculated and conservative estimation of seismic collapse capacity demanded contributesto a more resilient seismic risk assessment for structures situated in near-fault zones.In conclusion, the results indicate that for the assessment of seismic collapse capacity that is demanded atconstruction sites in near-fault zones, utilizing a lower bound of the ratio of near-fault to far-fault seismic collapsecapacity, associated with lower ductility and higher pulse period ratio, is sufficiently conservative. Moreover, anincrease in this parameter does not significantly escalate the demand at the construction site.This approach ensures a cautious estimation of seismic collapse capacity demand, contributing to a more robustseismic risk assessment for structures in near-fault zones.

Selecting and scaling the set of ground-motion records is among the most important challenges in collapse capacity assessment and estimation of structures. In this study, 44 records were considered to estimate the collapse capacity of the structure.Next, the response spectrum matching degree of each ground-motion with the conditional mean spectrum at the inspected hazard level was employed to account for the effect of record selection on the collapse capacity of the structure and appropriate records were selected per each hazard level. Further, the spectral shape factor or epsilon was used to incorporate the effect of spectral shape on collapse capacity. In the first approach, the collapse capacity of the structure was modified for the overall set of selected ground-motions via epsilon according to the level of inspected hazard. In the second approach, the simplified method introduced by Haselton et al. was used to assess the effect of epsilon and modify the collapse capacity of the structure at the inspected hazard level. The results of the modified collapse capacity at the inspected hazard level were collected from three respective methods of record selection. These data were collected using the overall set of ground-motions with the epsilon and the simplified method. The obtained results indicate that the ratio of modified collapse capacity with the effects of record selection and epsilon on collapse capacity, disregarding the record selection and epsilon effects, are 1.216, 1.174, and 1.197 at the hazard level of 2% in 50 years for the three discussed methods, respectively. Ultimately, these three individual methods have rendered approximately equal estimates.

The collapse risk role has increasingly drawn engineers’ attention in the performance-based design field and engineers tend to design the structures in a way to be qualified enough to resist earthquakes, especially at near-fault sites. Due to specific characteristics of near-fault records, structures in near-fault required more collapse capacity in comparison with far-fault sites. Furthermore, it is necessary to determine the collapse capacity that structures should be designed to comply with standards and to meet the collapse risk limit in the given site. In this research, the ratio method is presented to determine the design collapse capacity of structures based on the risk value of 1% in 50 years as well as the site hazard stemming from the integration scenario for near-fault. In this method, the structure behavior and fundamental period are incorporated, and effect of pulse period is considered as well. This method utilizes the ratio of the collapse capacity of the structure in near-fault to that of far-fault named γ. Consequently, efficient procedures based on nonlinear static pushover are used for obtaining the collapse capacity in far-fault and near-fault. Then, the ratio method is employed on a mid-rise RC frame and the design collapse capacities are acquired for two amounts of Tp/T. The result shows ratio method can be used for any Tp/T values especially those corresponding to the governing Tp at site. Moreover, the least value of γ can be used conservatively since the design collapse capacity of the structure in near-fault is raised by reducing γ.

The aim of the present contribution is quantify the effect of material deterioration on the seismic collapse capacity of single-degree-of-freedom systems vulnerable to the destabilizing effect of gravity loads (P-delta effect). The outcomes are presented in terms of collapse capacity spectra, where for characteristic structural parameters median and dispersion of record-dependent collapse capacities is plotted against the initial structural period. The ultimate goal of this study is to isolate sets of material deterioration parameters, which do not affect substantially the collapse capacity of P-delta sensitive structures. These threshold values depend primarily on the post-yield stiffness ratio, the assigned hysteretic law, and the initial structural period.

One of the situations that distorts the safety of concrete moment frame structures is the rare seismic events. Bam earthquake in 2003 was one of the rare seismic events that caused damages to many newly built structures. In this paper, the capacity of structures was evaluated according to the standard record sets of FEMA P695 and maximum considered earthquake (MCE). A comprehensive method should be used to express the seismic behavior of structures for assessing the collapse capacity. Incremental dynamic analyses proposed by Vamvatsikos and Cornell in 2002. This method is used for assessing the collapse capacity of structures in this study. The proposed methodology is used for collapse assessing of an individual as well as a group of buildings with due attention to rare seismic events and incremental dynamic analyses method. Illustrative results show that, if structures provide minimum acceptable requirements of FEMA P695, they would have been secured against rare seismic events.
Development of nonlinear models for collapse stimulation is the first step of collapse assessing methodology. All of the structures have been designed according to ASCE 7-05 code, and for expressing of nonlinear behavior of materials, Mander and Menegotto-Pinto model has been considered. Selection of ground motion record sets for collapse assessment of building structures is very important. Both far-field and near-field records have been considered in FEMA P695, but in this paper, the far-field records were used. Three analyses have been considered in assessing the collapse capacity. Eigenvalue analyses, incremental dynamic analyses and static pushover analyses are required for assessing the collapse capacity. Incremental dynamic analyses is one the suitable methods for expressing of seismic behavior of structures. The basic idea of this analysis was described by Bertero in 1997. In 2002, this method was accompanied with big progress by Vamvatsikos and Cornell. Illustrative results show where the incremental dynamic analyses curve slope is equal to 20% of the elastic while the point also belongs to softening branch defined as collapse point. Additionally, another candidate point is displacement ratio of 10%. Illustrative results show that where the incremental dynamic analyses curve lining to infinity is being defined as collapse point. The incremental dynamic analyses curves show record to record variability, thus it is essential to summarize such data. The fragility fitting approach has been used widely for defining the median collapse acceleration. Adjusted collapse margin ratio is the most important parameter for assessing the collapse capacity of structures. According to FEMA P695, the acceptable value of the adjusted collapse margin ratio for each individual model within a performance group should exceed ACMR (20%). Additionally, the average value of adjusted collapse margin ratio for each performance group should exceed ACMR (10%).
Finally, collapse capacity of 5 and 10 story concrete moment frame structures are defined. Both structures have acceptable adjusted collapse margin ratio and both of them have acceptable safety according to rare seismic events. Structures that could not satisfy the FEMA’s conditions must increase their lateral strength and re-evaluate.

One of the important steps in seismic collapse assessment of structures by using nonlinear dynamic analyses is the appropriate selection of ground motion records. Epsilon (εSa), eta (η), and gamma (γ) for long-period structures, are proxies that have been recently proposed for Ground Motion Record Selection (GMRS). In this study, two parameters, named γs, are proposed, which have considerable correlation with the collapse capacity of short-period structures having fundamental period less than 1 s. One of these parameters is a linear combination of εSa, epsilon of Pseudo Spectral Acceleration (PSA) at 1.5 times of the fundamental period of the structure (εSa(1.5T1)) and εPGV. The other one is a linear combination of εSa, εSa(1.5T1) and epsilon of spectrum intensity, εSI. To obtain and optimize γs, the Particle Swarm Optimization (PSO) algorithm is applied. Since the parameters proposed as γs have significant correlation with the collapse capacity of short-period structures, they can be used as efficient proxies for GMRS in seismic collapse assessment of short-period structures. The results show that GMRS using γs leads to reduction in the dispersion of structural collapse capacity in comparison with GMRS based on εSa or η.

Accurate determination of the collapse moment of structures by nonlinear analyses is one of the major challenges for engineers in the seismic design of buildings. Although, structural damage can be assessed at various levels, the collapse of buildings is one of the worst events in the construction industry where casualties reach their maximum. In this research, 3D steel moment-resisting frame structures with 4, 8 and 12 story with special ductility have been subjected to nonlinear analysis including nonlinear static analysis, incremental nonlinear dynamic analysis and finally to investigate their collapse capacity, the fragility curves were used and earthquakes were considered according to FEMA P695 instruction including a pair of 22 far fault records, 14 near fault records with pulse and 14 near fault records without pulse. The models are 3D structures designed in ETABS 2016 software. The design of the structures and their seismic criteria control are based on fully validated according to standard 2800 Fourth Edition. Nonlinear structural models are also created in 3D state in OpenSees2.5.0 software. The effect of stiffness and strength deterioration is considered based on the results of the experimental models and the collapse capacity of the three-dimensional structures of the special steel moment-resisting frame is investigated probabilistically. The results show that the collapse capacity of 4, 8 and 12-story structures is the highest under far fault earthquakes and the lowest under near fault earthquakes without pulse and among the low-rise structures, The 4-story has less collapse capacity. For example, in the 4-story structure, the structural collapse capacity at statistical level 84% under near fault with and without pulse and far fault ground motions is 3.21 g, 3.61 g and 4.14 g, respectively.

One of the concerns of assessing structural collapse performance is the appropriate selection of ground motions for use in the nonlinear dynamic collapse simulation. For this purpose, the selected earthquakes must be stronger than the earthquake which used in the initial design of structure. For modern buildings which have the ductile behavior against earthquake, ground motions that cause collapse are expected to be rare high-intensity motions associated with a large magnitude earthquake. The researches have shown that rare high-intensity ground motions have a peaked spectral shape that should be considered in groundmotion selection and scaling. They have shown that spectral shape, in addition to ground-motion intensity, is a key characteristic of ground motions affecting structural response. One method to account for this spectral shape effect is through the selection of a set of ground motions that is specific to the building’s fundamental period and the site hazard characteristics. The using of this method for each building faces challenges as need to selecting specific sets of ground motions. The selection of a specific ground motion set for each building is not practical; nor is it desirable because the goal is to generalize the collapse assessment results across seismic design categories. Thus, another method was presented in which uses a general ground-motion set, selected without regard to ε values, and then corrects the calculated structural response distribution to account for the target epsilon expected for the specific site and hazard level. To provide a more practical method for adjusting the collapse capacity, a simplified version of the second method presented that can be used to determine the appropriate adjustment factors for the collapse capacity distribution without requiring the computation of the epsilon values for each record and then performing a regression analysis. The development of simplified method causes that reduced computation. The collapse capacity results of three different methods were compared by several examples and the results of past research. The results show that the first two methods produce nearly identical results, with the predictions of the mean collapse capacity differing by only. Comparison of the Simplified Method and Method 2 indicates that the results are very close to each other, so that the full regression analysis results yield 0.257, which agrees very well with the simplified value of 0.254. The corresponding mean collapse capacity from Method 2 is 2.63 g as compared to the simplified value of 2.83 g. This difference of about 8% is reasonable for most applications, particularly in contrast to the alternative of neglecting the spectral shape effects. The calculated dispersion from Method 2 is 0.35, which is about 10% lower than the slightly conservative value of 0.40 used in the simplified method. This paper indicates all methods that modified the resulting collapse capacity by the spectral shape effects. Also, these methods have been compared and introduced the best method by the examples provided.

Collapse capacity is one of the fundamental factors for evaluating of collapse risk in performance-based design engineering field. Calculation of this parameter has been time consuming during past decade. This issue has prevented engineers from determining this parameter in a prevalent and practical way. Furthermore, defining of this value has been found more challenging in a near-source region due to special characteristics of its pulse-like records which make the collapse capacity more dependent on period ratio, T/Tp. In this study, amethod is proposed to obtain collapse capacity of reinforced concrete (RC) structures considering two main variables effecting columns behavior: axial load ratio and confinement ratio. The mentioned methodeschews the intensive computational challenges of incremental dynamic analyses to find collapse probability. By the proposed approach, the pulse period impact is incorporated into collapse risk using probabilistic equations. After the role of axial load ratio was illustrated,the resulted collapse probability distributions and the corresponding risk values are obtained for a near-fault site. The resultsexplain that asthe confinement ratio descends, the collapse capacity with near-fault pulse effect is decreased and the risk values are raised consequently. In addition, the results are found in compliance with ASCE acceptable risk value.