Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Khanjari, Madihe
Master of Applied Science - MASc
University of British Columbia
Large earthquakes, though infrequent, rank among the most devastating natural disasters, posing significant threats to human life and causing widespread damage to the built environment. The restoration of pre-event conditions after such events requires extended recovery periods, which amplify the long-term impacts of seismic events and result in numerous indirect consequences. As a result, the evaluation of seismic resilience—which measures a structure's ability not only to withstand but also to recover from earthquake events—is critical for ensuring a comprehensive assessment of seismic performance and fostering a more resilient built environment. In this study, we emphasize the importance of accurately determining recovery time, a key factor in improving the accuracy and reliability of the assessed resilience index. To address this, we have developed a comprehensive approach for modeling the repair time process, which unfolds in three sequential steps. The process begins with the allocation of the appropriate number of workers to each repair activity, thereby avoiding the under- or over-estimation of repair durations. The model then integrates these durations using a suitable repair logic network, providing an initial estimation of the project’s completion date. Finally, it accounts for resource congestion effects throughout the project’s duration, ensuring that overly optimistic projections of the building's final restoration date are mitigated. Our proposed repair time model has been incorporated into a probabilistic framework designed to calculate the seismic resilience of buildings. To demonstrate the practical application of this model and the methodology for defining the required input parameters for each component of the resilience framework, we utilized a newly designed 20-story timber structure as a case study. The results of the case study indicate that, at the highest intensity levels of applied ground motions, overlooking the final step of our proposed repair time model—specifically, ignoring resource restrictions—can lead to an underestimation of the overall required repair time by an average of 30 percent. However, despite this significant change in repair time, the resilience index at the highest intensity level scenarios is only marginally reduced by an average of 2.6 percent when resource restrictions are considered. This modest reduction aligns with our findings that the building maintains a substantial level of resilience across all intensity levels.
Development of an Advanced Repair Time Model forAssessing the Seismic Resilience of BuildingsbyMadihe KhanjariA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Civil Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)August 2024© Madihe Khanjari, 2024The following individuals certify that they have read, and recommend to theCollege of Graduate Studies for acceptance, the thesis entitled:Development of an Advanced Repair Time Model for Assessing theSeismic Resilience of BuildingsSubmitted by Madihe Khanjari in partial fulfillment of the requirements for thedegree of Master of Applied ScienceExamining Committee:Solomon Tesfamariam, School of EngineeringSupervisorAbbas Milani, School of EngineeringSupervisory Committee MemberSiddiqua Sumi, School of EngineeringSupervisory Committee MemberiiAbstractLarge earthquakes, though infrequent, rank among the most devastating naturaldisasters, posing significant threats to human life and causing widespread dam-age to the built environment. The restoration of pre-event conditions after suchevents requires extended recovery periods, which amplify the long-term impactsof seismic events and result in numerous indirect consequences. As a result, theevaluation of seismic resilience—which measures a structure’s ability not only towithstand but also to recover from earthquake events—is critical for ensuring acomprehensive assessment of seismic performance and fostering a more resilientbuilt environment.In this study, we emphasize the importance of accurately determining recoverytime, a key factor in improving the accuracy and reliability of the assessed re-silience index. To address this, we have developed a comprehensive approach formodeling the repair time process, which unfolds in three sequential steps. The pro-cess begins with the allocation of the appropriate number of workers to each repairactivity, thereby avoiding the under- or over-estimation of repair durations. Themodel then integrates these durations using a suitable repair logic network, provid-ing an initial estimation of the project’s completion date. Finally, it accounts forresource congestion effects throughout the project’s duration, ensuring that overlyoptimistic projections of the building’s final restoration date are mitigated.Our proposed repair time model has been incorporated into a probabilisticframework designed to calculate the seismic resilience of buildings. To demon-strate the practical application of this model and the methodology for defining therequired input parameters for each component of the resilience framework, we uti-lized a newly designed 20-story timber structure as a case study.iiiThe results of the case study indicate that, at the highest intensity levels ofapplied ground motions, overlooking the final step of our proposed repair timemodel—specifically, ignoring resource restrictions—can lead to an underestima-tion of the overall required repair time by an average of 30 percent. However,despite this significant change in repair time, the resilience index at the highestintensity level scenarios is only marginally reduced by an average of 2.6 percentwhen resource restrictions are considered. This modest reduction aligns with ourfindings that the building maintains a substantial level of resilience across all in-tensity levels.ivLay SummaryThe imperative of constructing buildings with reliable seismic performance hasprompted a deeper exploration into their post-earthquake functionality, emphasiz-ing seismic resilience calculation. Despite existing research in this area, significantstudy is still necessary to fully grasp and effectively address its complexities. Thisthesis endeavors to advance a robust model for precise prediction of building re-covery times, a critical component in assessing seismic resilience. Application ofthis model to a case study building demonstrates its effectiveness, particularly inscenarios of resource congestion. Neglecting such periods can lead to substantialinaccuracies in final repair time calculations, potentially overestimating calculatedresilience.vPrefaceMadihe Khanjari prepared all the contents of this thesis, including the comprehen-sive literature review, model development, analysis, interpretation of results, andthe writing and editing of the manuscript, under the supervision of Dr. SolomonTesfamariam.AI tools were employed exclusively to enhance the readability and clarity ofthe thesis text, ensuring that the content is presented in the most effective andunderstandable manner.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Seismic Performance Assessment . . . . . . . . . . . . . . . . . 72.3 Definitions and formulations of Resilience . . . . . . . . . . . . . 10vii2.4 Resilience Enhancement: Strategies and Outcomes . . . . . . . . 132.5 Probabilistic Framework for Seismic Resilience Assessment . . . 152.5.1 Fragility Analysis . . . . . . . . . . . . . . . . . . . . . . 162.5.2 Consequence Analysis . . . . . . . . . . . . . . . . . . . 262.5.3 Recovery Analysis . . . . . . . . . . . . . . . . . . . . . 292.5.4 Functionality Analysis . . . . . . . . . . . . . . . . . . . 462.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 463 Probabilistic Framework for Assessing Seismic Resilience of Buildings 483.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Probabilistic Framework for Seismic Resilience Assessment . . . 493.3 Earthquake Hazard Model . . . . . . . . . . . . . . . . . . . . . 503.4 Building Structural Model . . . . . . . . . . . . . . . . . . . . . 513.5 Building Characteristics Model . . . . . . . . . . . . . . . . . . . 523.6 Building Fragility Model . . . . . . . . . . . . . . . . . . . . . . 523.6.1 Structural Fragility Curves . . . . . . . . . . . . . . . . . 533.6.2 Non-Structural Fragility Curves . . . . . . . . . . . . . . 543.7 Component Consequence Model . . . . . . . . . . . . . . . . . . 553.8 Building Recovery Model . . . . . . . . . . . . . . . . . . . . . . 563.8.1 Recovery Time Model . . . . . . . . . . . . . . . . . . . 573.8.2 Recovery Path Model . . . . . . . . . . . . . . . . . . . . 753.9 Building Functionality Model . . . . . . . . . . . . . . . . . . . . 763.10 Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 773.11 Seismic Risk Analysis . . . . . . . . . . . . . . . . . . . . . . . 773.12 Seismic Resilience Quantification . . . . . . . . . . . . . . . . . 793.12.1 Time Analysis . . . . . . . . . . . . . . . . . . . . . . . 793.12.2 Functionality Analysis . . . . . . . . . . . . . . . . . . . 803.12.3 Resilience calculation . . . . . . . . . . . . . . . . . . . 804 Application of Developed Resilience Framework on a 20-Story Tim-ber Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2 Building Structural Model . . . . . . . . . . . . . . . . . . . . . 83viii4.3 Earthquake Hazard Model . . . . . . . . . . . . . . . . . . . . . 844.4 Building Characteristics Model . . . . . . . . . . . . . . . . . . . 854.5 Building Fragility Model . . . . . . . . . . . . . . . . . . . . . . 874.5.1 Structural Fragility Curves . . . . . . . . . . . . . . . . . 874.5.2 Non-Structural Fragility Curves . . . . . . . . . . . . . . 884.6 Component Consequence Model . . . . . . . . . . . . . . . . . . 904.7 Building Recovery Model . . . . . . . . . . . . . . . . . . . . . . 914.7.1 Recovery Time Model . . . . . . . . . . . . . . . . . . . 914.7.2 Recovery Path Model . . . . . . . . . . . . . . . . . . . . 964.8 Building Functionality Model . . . . . . . . . . . . . . . . . . . . 964.9 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 964.9.1 Component-Level Monterey Loss Assessment . . . . . . . 974.9.2 Floor-Level Monterey Loss Assessment . . . . . . . . . . 1004.9.3 Intensity-Level Monterey Loss Assessment . . . . . . . . 1034.9.4 Resilience Assessment . . . . . . . . . . . . . . . . . . . 1045 Conclusions and Recommendations for Future Research . . . . . . 1085.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.2 Chapter Summaries . . . . . . . . . . . . . . . . . . . . . . . . . 1095.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.5 Recommendations for future research . . . . . . . . . . . . . . . 111Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112ixList of TablesTable 2.1 The four dimensions of resilience . . . . . . . . . . . . . . . . 14Table 2.2 Recommended classification of components and their respec-tive workforce numbers in REDi™ . . . . . . . . . . . . . . . 33Table 2.3 REDi™-defined maximum number of workers based on build-ing’s height across classified component categories . . . . . . . 33Table 4.1 Considered non-structural elements and their directional quan-tities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Table 4.2 Structural component-level fragility curves . . . . . . . . . . . 88Table 4.3 Non-Structural component-level fragility curves . . . . . . . . 89Table 4.4 Labor allocations for structural components . . . . . . . . . . 93Table 4.5 Labor allocations for non-structural components . . . . . . . . 93Table 4.6 Seismic resilience indices across various intensity levels . . . 107xList of FiguresFigure 2.1 A Schematic illustration of Resilience Curve. Adopted from [49]. 13Figure 2.2 An illustrative diagram highlighting the primary factors be-hind the derivation of fragility functions for structural systems.Adopted from [15] . . . . . . . . . . . . . . . . . . . . . . . 17Figure 2.3 Illustration of: (a) a representative lognormal fragility func-tion; and (b) the assessment of probabilities pertaining to indi-vidual damage states. . . . . . . . . . . . . . . . . . . . . . . 19Figure 2.4 Advantages and Disadvantages of Each Fragility Curve Devel-opment Approach. Adopted from [46] . . . . . . . . . . . . 21Figure 2.5 Understanding Key Advantages and Limitations Across Vari-ous Analytical Approaches in the Literature. Adopted from [15] 24Figure 2.6 Schematic illustration of the conventional method for develop-ing fragility curves: (a) Incremental Dynamic Analysis Curve,(b) Distribution of IMs at a specified damage state, (c) Corre-sponding failure probability for the considered damage state,and (d) Resulting fragility curves. Adopted from [14] . . . . . 25Figure 2.7 Sequence of Repairs Proposed by REDi™ at Each Floor Level,Denoted as ’Typical Repair Sequences.’ Adopted from [2]. . 35Figure 2.8 Representation of the repair time model as proposed by Terzicet al. and Yoo. Adopted from [52]. . . . . . . . . . . . . . . 38Figure 2.9 Flowchart illustrating the resource scheduling method proposedby Terzic et al. and Yoo. Adopted from [52]. . . . . . . . . . 43Figure 2.10 Recovery functions. Adopted from [10]. . . . . . . . . . . . 45xiFigure 3.1 Probabilistic Framework for Assessing Seismic Resilience ofBuildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 3.2 The Input and Output Information of the Building Character-istics Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 3.3 The Input and Output Information of the Building FragilityModel for Structural Components. . . . . . . . . . . . . . . . 54Figure 3.4 The Input and Output Information of the Building FragilityModel for Non-Structural Components. . . . . . . . . . . . . 54Figure 3.5 Typical Consequence Function of FEMA P-58 [19]. . . . . . 56Figure 3.6 The Input and Output Information of the Component Conse-quence Model. . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 3.7 Flowchart of the resource restriction for a single realization ofthe Monte Carlo simulation. . . . . . . . . . . . . . . . . . . 65Figure 3.8 The Input and Output Information of the Response Analysis. 78Figure 3.9 Schematic representation of the resilience curve. . . . . . . . 81Figure 4.1 Three-dimensional view of the prototype model alongside itsstructural components. Adopted from [60]. . . . . . . . . . . 84Figure 4.2 Flowchart of the considered repair logic network for 20-storybuilding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Figure 4.3 Illustration of the distribution of mean repair costs along differ-ent components across six seismic intensity groups: (a) Sa=0.0755g ; (b) Sa=0.151 g; (c) Sa=0.2265 g; (d) Sa=0.3020 g; (e)Sa=0.3775 g; (f) Sa=0.4530 g. . . . . . . . . . . . . . . . . . 98Figure 4.4 Illustration of the distribution of mean repair costs along dif-ferent components. . . . . . . . . . . . . . . . . . . . . . . . 100Figure 4.5 Illustration of the distribution of mean repair costs along eleva-tion across six seismic intensity groups: (a) Sa=0.0755 g ; (b)Sa=0.151 g; (c) Sa=0.2265 g; (d) Sa=0.3020 g; (e) Sa=0.3775g; (f) Sa=0.4530 g. . . . . . . . . . . . . . . . . . . . . . . . 102Figure 4.6 Illustration of the distribution of mean repair costs along ele-vation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103xiiFigure 4.7 Contribution of structural and non-structural elements to totalrepair costs at various spectral acceleration levels. . . . . . . . 104Figure 4.8 Illustration of recovery curves across six seismic intensity groups:(a) Sa=0.0755 g ; (b) Sa=0.151 g; (c) Sa=0.2265 g; (d) Sa=0.3020g; (e) Sa=0.3775 g; (f) Sa=0.4530 g. . . . . . . . . . . . . . . 106xiiiAcknowledgmentsI would like to express my appreciation to my supervisor for the opportunity tocomplete my master’s under his supervision. His invaluable comments, guidance,and support have been critical to my success. I am also thankful to my teammatesfor their unwavering support and collaboration, which have greatly enriched myacademic experience.xivDedicationTo my parents and brother, whose unwavering support and love have been thecornerstone of my journey through this master’s program. Their belief in me andconstant encouragement kept me on the right path and made this accomplishmentpossible.xvChapter 1Introduction1.1 BackgroundThe transition from understanding natural phenomena to comprehending their im-plications for populated areas is crucial in addressing the concept of natural disas-ters. Natural phenomena, driven by dynamic forces and inherent processes, show-case the complexity of the natural world. These phenomena occur independentlyof human presence, though human activities can often amplify their impact. Whensuch phenomena lead to significant hazards and consequences in populated areas,they are classified as natural disasters [1]. This classification emphasizes the po-tential harm to human life, property, and infrastructure, distinguishing natural dis-asters from the broader spectrum of natural phenomena by their direct impact oninhabited regions.As global urbanization accelerates, driven by population growth, the expansionof cities becomes increasingly prominent. This trend raises global concerns abouta corresponding rise in natural disasters, making them a focal point in discussionsof urbanization. The increased focus stems from the potential of natural disastersto induce sudden and profound disruptions in the technical, organizational, so-cial, and economic fabric of communities. Addressing the impact of disasters andreducing the vulnerability of densely populated urban areas requires a proactiveapproach, not only for disaster prevention but also for effective recovery. Withoutsuch measures, communities face the risk of significant casualties and economic1losses.Large earthquakes, though infrequent, are among the most catastrophic naturaldisasters, with severe consequences for both human life and the built environment.These events are triggered by seismic activity, which arises from temporary me-chanical instability within the Earth’s crust, driven by internal physical processes.When this instability occurs, accumulated energy is released as seismic waves, re-sulting in an earthquake. The point of origin beneath the Earth’s surface, knownas the hypocenter, projects upward to the surface as the epicenter. The impact of alarge earthquake can extend far beyond the epicenter, affecting areas hundreds ofkilometers away.When densely populated cities are located near these seismic regions, the po-tential for severe damage and adverse effects on human activities is significantlyheightened. This increased risk threatens buildings, infrastructure, and essentialsystems, often resulting in loss of life, injuries, and extensive property destruc-tion. Additionally, earthquakes can trigger secondary hazards such as landslides,tsunamis, and fires, further compounding the overall damage and escalating therisks faced by the affected population. Regardless of whether these secondary haz-ards occur, the recovery from the direct losses caused by an earthquake can takeyears, as the affected community works to fully or partially address the conse-quences. As a result, the impact of large earthquakes extends beyond immediatedestruction to include indirect losses, the nature and severity of which depend onthe duration required to restore the affected community to its pre-event condition.In recent years, the primary challenge in seismic engineering has shifted to-wards addressing the long-term repercussions of seismic events, with a particularfocus on enhancing the resilience of buildings. Advances in engineering and con-struction techniques have led to the development of structures that perform reliablyduring earthquakes, emphasizing the reduction of immediate damage. While thecapacity to withstand the initial impact is critical for minimizing catastrophic casu-alties and immediate economic losses, it is equally important to consider the build-ing’s performance during the recovery phase. Understanding building performancethroughout this recovery period is crucial for protecting densely populated urbanareas from secondary cascading effects, thereby ensuring their resilience againstnatural disasters.2Various studies have proposed frameworks to model the recovery phase aftera disaster. Some focus exclusively on recovery time, others on recovery paths,and some address both aspects. Among the approaches to calculating the timerequired to complete the repair process of a building, three models are frequentlyreferenced in the literature: those provided by FEMA P-58 [18], the Resilience-based Earthquake Design Initiative (REDi™) Rating System [2], and Terzic et al.[50].The FEMA P-58 model [18]adopts a simplified approach to defining the num-ber of workers and the repair strategy. While this approach offers ease of use, it hascertain limitations, particularly due to its generalized assumptions. The REDi™model [2] attempts to address some of these limitations by incorporating adjust-ments and additional considerations specific to each building’s unique character-istics. However, despite these improvements, the REDi™ model still has short-comings. In 2016, Terzic et al.[50] introduced a repair time model that overcomesmany of the limitations inherent in previous methodologies. Although this modelis more complex than the other two, it still includes some simplified assumptions,particularly in scenarios involving resource congestion periods. These models arediscussed in detail in Chapter 2.1.2 Problem StatementGenerally, Seismic resilience of a structure, community, or system refers to itsability to withstand and recover from the effects of an earthquake. To accuratelycalculate this index for a building, one of the most critical aspects is determiningthe duration required for the damaged building to restore its full functionality. Thisperiod includes the time needed to prepare all necessary resources for performingrepairs and the time allocated for repairing all damaged components. The severityof damage in each component dictates the number of resources required to com-plete all necessary repair activities to restore it to its original condition, therebydetermining its repair time duration.A significant challenge is aggregating these calculated times to determine thetotal repair time for the building. Since the effects of an earthquake in a regioncan be extensive, a shortage of sufficient resources is not uncommon, meaning3that the initially allocated resources may need adjustments. This resource limita-tion and subsequent adjustments can lead to an extension of the final repair time.Therefore, there is a need to define a comprehensive repair time model that canappropriately assign the necessary number of workers to the damaged units andensure that the calculated repair timeline is realistic by accounting for the effectsof resource congestion.1.3 Aims and ObjectivesThis thesis provides a framework for analyzing a building’s seismic resilience. Akey milestone that differentiates this framework from similar works is the devel-opment of a more comprehensive model for calculating repair time. This proposedframework has been rigorously applied to a 20-story case study timber building,demonstrating its practical application and effectiveness in enhancing our under-standing of seismic resilience and recovery.Consequently, the aims and objectives of this thesis are delineated as follows:1. Develop an enhanced repair time model that addresses and overcomes thelimitations found in existing models.2. Design a robust framework for assessing the seismic resilience of buildings,utilizing the proposed repair time model as a foundational component.1.4 Thesis OverviewThis thesis is organized into 5 chapters:Chapter 2 provides a thorough literature review on the concept of resilienceand the proposed methods of its assessment, outlining all the necessary steps tocalculate it precisely.Chapter 3 contains the proposed framework for calculating the building’s re-silience, which includes the developed repair time model as one of its key compo-nents.Chapter 4 contains the application of the proposed framework on a case studybuilding model, aiming to verify the developed framework and clarify the func-tioning of each phase within it.4Chapter 5 contains the conclusion of all chapters, describes the limitations ofthe model, and proposes topics for future work.5Chapter 2Literature reviewIf I have seen farther it is by standing on the shoulders of Giants.— Sir Isaac Newton2.1 IntroductionOver decades and millennia, substantial earthquakes such as the 2004 Sumatra off-shore earthquake, the 2008 Wenchuan earthquake in China, the 1995 Kobe earth-quake, the 2011 Tohoku earthquake with a magnitude of 9.0, the 2005 NorthernPakistan earthquake (resulting in over 75,000 casualties), and the 2010 Haiti earth-quake have provided valuable lessons through their devastating impacts. Despitethese lessons, mysteries and uncertainties persist about the timing of earthquakeoccurrences, the resulting ground shaking, and the performance of structures andinfrastructure subjected to shaking.Numerous contributors to earthquake analysis, including domain experts inearthquake physics and engineering, seek to deepen their understanding of thesephenomena. Research endeavors range from predicting earthquakes to utilizinghazard analysis methods for characterizing natural phenomena resulting from earth-quakes, and risk analysis to understand the consequences of such hazards, includ-ing structural failures, fatalities, and economic costs. The pursuit of knowledge in”Seismic Hazard and Risk Analysis” remains critical in enhancing our understand-ing and preparedness for seismic events.62.2 Seismic Performance AssessmentThe importance of disaster management becomes evident when we consider themultifaceted challenge posed by seismic threats, including preserving lives, pro-tecting critical infrastructure, and ensuring the long-term stability of communitiesin the aftermath of such calamities. An integral part of addressing this challenge isgaining a clear understanding of the risks associated with seismic activities. Thisunderstanding plays a pivotal role in guiding resource allocation for actions to betaken before, during, and after a seismic event. While engineers can provide expertinsights into the technical aspects of earthquake probability and impact, decision-makers often prioritize clear, concise, and actionable information. This encom-passes quantified economic losses, projected repair timelines, and the expectednumber of casualties, aligning with their decision-making process.In the endeavor to minimize potential losses, reduce the likelihood of disruptiveevents, and avert failures resulting from seismic events, traditional seismic researchhas harnessed risk management strategies and dedicated substantial resources tothe development of mitigation options[26]. These options take the form of bothprevention and protection mechanisms, aimed at creating systems renowned fortheir exceptional reliability. While the pursuit of heightened system reliability re-mains a crucial proactive approach, Yodo and Wang’s [58] research underscoresthat a singular emphasis on high reliability is no longer sufficient, and this is forseveral compelling reasons. Foremost, the achievement of high system reliabil-ity necessitates a significant financial investment. Additionally, in considerationof probability theory, the likelihood of attaining a flawless record of zero systemfailures is exceedingly low due to the infinite range of potential values. As a result,certain failures are, inevitably, beyond our control, particularly those triggered bynatural events.The aforementioned challenges underscore the inevitability of natural eventsand their associated consequences, reinforcing the imperative for precision in eval-uating the critical determinants that gauge the seismic performance of diverse sys-tems. These determinants include economic loss, casualty rates, and system re-covery time, and they are intrinsically linked with catastrophic events throughoutthe entire lifespan of any given system, whether it pertains to individual structures,7infrastructural components, or entire communities, in the aftermath of seismic in-cidents. Substantial scholarly investigation within the field of seismic loss has beenconducted to comprehensively compute and apprehend these vital variables.In the context of structural assessment, specialized tools, such as the FEMA P-58 [17, 19] Seismic Performance Assessment of Buildings, alongside its associatedsoftware application, the Performance Assessment Calculation Tool (PACT) [18],developed under the auspices of the Pacific Earthquake Engineering Research Cen-ter, assume a pivotal role in the meticulous quantification of repair expenses, theestimation of restoration timelines, and the assessment of casualties arising fromseismic events.Given the complexity and multifaceted nature of earthquake impacts, no singlevariable can provide a complete understanding of the system’s condition after adisaster. Therefore, the concept of measuring the seismic performance of a systemby combining these variables into a unique decision variable (DV), referred to asa resilience index according to the terminology of the Multidisciplinary Center ofEarthquake Engineering to Extreme Event (MCEER) [9], has become crucial. Theresilience index offers a comprehensive and standardized approach that takes intoaccount various decision variables, resulting in a more comprehensive assessmentof the system’s capacity to endure and recover from an earthquake. It effectivelyaddresses the limitations of reporting each variable individually. Yodo and Wang[58] have aptly characterized this concept as a transformative development in re-cent research endeavors, owing to its capacity to address the constraints of ensuringa highly reliable system, one that can endure failures and facilitate post-disaster re-covery. In essence, the resilience concept furnishes a holistic framework for theassessment and enhancement of a system’s earthquake performance.In 2003, Bruneau et al. [7] made a significant contribution by laying the foun-dation for a seminal framework that comprehensively defined the concept of seis-mic resilience. This framework was built around the concept of the recovery curvefor communities and encompassed a wide array of strategies aimed at reducinglosses associated with seismic hazards. Following the publication of this pio-neering paper, numerous subsequent studies have significantly advanced the ex-ploration of seismic resilience assessment [8, 10, 12, 58]. As a result, seismicresilience has evolved into a pivotal and extensively investigated subject within the8field of disaster risk reduction and management.Over time, the assessment of the resilience of complex and extensive civil en-gineering systems has evolved, encompassing various levels of study and system-atically categorizing resilience quantification into four essential components, asdelineated by Lu et al. [34]. Initially, the focus lies on evaluating the resilienceof city and community systems, considering their multifaceted engineering, so-cial, and economic functions. Secondly, the assessment extends to urban subsys-tems, delving into the analysis of various infrastructures and distributed networks,encompassing transportation, power, production, multitier supply chains, generalinfrastructure, and healthcare systems. A pivotal emphasis is placed on examin-ing building-level resiliency as the third aspect. Finally, the fourth part involvesconducting resilience limit-state analysis.Community resilience gains particular significance in the wake of severe nat-ural disasters, such as earthquakes. While immediate lifesaving efforts are vital,there’s an equally essential need to ensure the well-being and overall quality of lifewithin the affected community. Overlooking this aspect could lead to long-lastingand irreversible consequences that are challenging to address adequately. To bol-ster community resilience, it’s imperative that all subsystems, including infrastruc-ture, networks, and individual buildings, are designed to minimize vulnerabilityand maximize resilience. Although individual infrastructures, networks, and spe-cialized buildings hold greater weight in terms of community resilience comparedto single residential or commercial buildings, it’s crucial to emphasize that thesheer number of residential and commercial structures underscores the importanceof assessing and enhancing the resilience of each one.Significant research efforts have been dedicated to evaluating the seismic per-formance of structures, with a focus on surpassing the seismic performance ofbuildings designed according to conventional codes. Ongoing research exploresadvanced techniques and materials to enhance a building’s capacity to resist lateralforces, spanning both initial design considerations and retrofitting strategies for ex-isting structures. In addition to these advancements, the current building evaluationmethods have concentrated on quantifying seismic losses resulting from immedi-ate, direct damages after earthquakes. While this approach is effective in address-ing economic assessments, life safety, and collapse prevention, it often overlooks9the vital aspect of a building’s potential for re-occupancy following earthquakedamage.The resilience assessment method, which examines a structure’s ability to with-stand and recover from external hazards, plays a pivotal role in addressing the chal-lenge posed by conventional seismic assessment of buildings. This comprehensiveassessment considers the lasting impact of earthquakes during a building’s recoveryphase. Notably, it includes downtime calculations for structures unable to swiftlyreturn to their pre-earthquake state. Embracing both immediate damage and recov-ery, this holistic approach to building performance encourages the development ofmore resilient and robust structures, better equipped to endure the unpredictableforces of seismic activity.Simultaneously, the growing importance of evaluating seismic resilience andincorporating it into building design has given rise to various relevant applicationsand reports. Notable examples include the Resilience-based Earthquake DesignInitiative (REDi™) Rating System [2] and similar resources, highlighting the in-creasing emphasis on fortifying structures to withstand seismic challenges effec-tively.2.3 Definitions and formulations of ResilienceUnderstanding the concept of resilience is paramount when it comes to quantifyingthe resilience of a system effectively. When it comes to civil systems, such as com-munities or structures, this comprehension is of utmost importance as it sets thestage for optimizing the effectiveness of resilience assessment methods. As high-lighted by Bruneau et al.[7] in their research, achieving a meaningful analysis andassessment requires a clear and precise definition of resilience, the identification ofits multiple dimensions, and the development of measurement and quantificationtechniques. These elements are indispensable in the process of achieving meaning-ful analysis and assessment of resilience in all systems. In essence, knowing theresilience concept is essential because it provides the necessary groundwork forevaluating and enhancing the resilience of communities and infrastructure.Resilience, stemming from the Latin word ’resilio,’ which originally meant ”tojump back,” has evolved to embrace a contemporary understanding. According to10the dictionary, resilience is defined as ”the ability to recover from or adapt easilyto misfortune or change.” This multifaceted concept finds applications in diversefields, from psychology to engineering. Consequently, numerous conceptual def-initions of resilience have been proposed and are currently under discussion. It’sworth noting that many of these definitions share similarities, reflecting the sharedunderstanding of resilience across various disciplines.In the field of civil engineering, the concept of resilience holds a pivotal po-sition and has garnered significant attention in academic literature, prompting ex-tensive efforts to establish a precise definition for this term. One particularly com-prehensive and widely recognized definition comes from Bruneau et al.[7], whodefined seismic resilience for communities as ”the ability of social units (e.g., or-ganizations, communities) to mitigate hazards, contain the effects of disasters whenthey occur, and carry out recovery activities in ways that minimize social disrup-tion and mitigate the effects of future earthquakes.” Building upon this foundation,Cimellaro et al.[8, 10] further refined the concept by providing a detailed definition,stating that ”Resilience is defined as a normalized function indicating the capabil-ity to sustain a level of functionality or performance for a given building, bridge,lifeline, networks, or community over a specified period of time (TLC, life cycle,life span, etc.), including the recovery period after damage in an extreme event.”The quest to define the concept of seismic resilience remains dynamic [6, 58],with researchers adjusting their definitions to align with specific study objectives,while some pursue universally applicable definitions. Despite the variety of pro-posed definitions, they all converge on the core idea of seismic resilience as a met-ric evaluating an engineering system’s capacity to maintain its functionality after amajor earthquake and efficiently recover from any losses to return to its standardcondition.In the literature [8, 10, 11, 58] ,the authors utilized a graphical representationto explicate the concept of disaster resilience, illustrating the performance of anengineering system across a specified timeframe. The x-axis precisely marks time,detailing the control time within the relevant period, while the y-axis graphicallydelineates the engineered system’s performance, measured in dimensionless per-centages. It spans from 0% denoting a complete absence of available services to100% representing full functionality.11When an engineered system faces uncertain, unpredictable, and potentially se-vere conditions, like an earthquake at time instant t0, it may incur substantial dam-age, resulting in an abrupt decline in performance. In such scenarios, two outcomescan emerge.Firstly, a nonresilient-engineered system, as defined by Yodo and Wang [58],may be unable to withstand the disruption, leading to a gradual or immediate de-cline in performance until it reaches a significantly low level, ultimately resultingin complete failure or collapse.Conversely, a resilient-engineered system possesses the capacity to allocate re-sources for restoring performance post-earthquake disruption. This system mayrecover to a functionality level greater, equal to, or lower than the pre-earthquakestate. At the recovery’s end, resolution of pre-existing issues could elevate the sys-tem’s performance beyond the original level. However, irretrievable losses mightlead to a performance level lower than the baseline before the disaster.Building on the illustrated performance of the system within the specified timespan, termed the ”recovery period,” the resilience index is graphically defined asthe normalized shaded area beneath the system’s functionality curve.As previously emphasized, literature has delved into diverse perspectives todelineate seismic resilience, leading to the development of numerous methods forquantifying this crucial aspect. Cimellaro et al.[10] stand out with a notable con-tribution, introducing an equation acknowledged as a fundamental approach forassessing engineering resilience, particularly in the context of individual incidents.Employing a graphical definition of resilience, the authors have meticulously for-mulated the following Eq. 2.1:R =∫ tOE+TLCtOEQ(t)TLCdt (2.1)Figure 2.1 visually represents Eq. 2.1. In this illustration, the resilience in-dex (R) is denoted by the area beneath the recovery curve, while concurrently, theresilience loss (RL) is depicted by the area above the recovery curve in this illustra-tion. To quantify the resilience loss experienced by the system, Eq. 2.2 is appliedas follows:12Figure 2.1: A Schematic illustration of Resilience Curve. Adopted from [49].RL =∫ trtOE[100−Q(t)]dt (2.2)In these equations, Q(t) symbolizes the system’s recovery function at time t,with tOE pinpointing the moment of an extreme event occurrence, and tr indicatingthe moment at which the system regains full functionality. The term TLC designatesthe time horizon, representing the system’s control time. Furthermore, R ∈ [0,1]serves as the seismic resilience index, with higher values suggesting greater systemresilience and, consequently, a reduced value of resilience loss RL.2.4 Resilience Enhancement: Strategies and OutcomesPreviously underscored, seismic resilience signifies a system’s prompt recoveryfrom a significant earthquake. Figure 2.1 visually embodies the essence of re-silience, and when considered alongside the provided equation, both convey thatenhancing the system’s resilience after a severe event, such as an earthquake, isachievable by reducing the probabilities of failure, minimizing repercussions, and13shortening the recovery time. Building upon this understanding, Bruneau et al.[7]have identified four dimensions crucial for enhancing resilience, defined in termsof the 4 R’s (robustness, redundancy, resourcefulness, and rapidity). The defini-tions for these dimensions, as provided by Bruneau et al.[7] and Cimellaro et al.[10, 11], can be found in Table 2.1.Table 2.1: The four dimensions of resilienceBruneau et al. [7] Cimellaro et al. [10, 11]Robustness Strength, or the abilityof elements, systems, andother units of analysis towithstand a given levelof stress or demand with-out suffering degradationor loss of functionThe residual functionalitythat persists in the after-math of an extreme eventRapidity The capacity to meet prior-ities and achieve goals in atimely manner in order tocontain losses and avoid fu-ture disruptionThe slope of the functional-ity curve during the recov-ery timeRedundancy The extent to which ele-ments, systems, or otherunits of analysis exist thatare substitutable, i.e., capa-ble of satisfying functionalrequirements in the eventof disruption, degradation,or loss of functionalityThe availability of alterna-tive resources in the loss orrecovery processResourcefulness The capacity to iden-tify problems, establishpriorities, and mobilizeresources when conditionsexist that threaten to dis-rupt some element, system,or other unit of analysisConsisting of the ability toapply material (i.e. mon-etary, physical, technologi-cal and informational) andhuman resources in theprocess of recovery to meetestablished priorities andachieve goals.Recognizing the pivotal dimensions essential for fortifying a system’s resilience14against challenges is a strategic imperative. Through the careful integration ofthese fundamental elements, the journey toward establishing a highly resilient sys-tem not only becomes plausible but also holds significant promise. Bocchini etal. [6] intricately delineate specific outcomes associated with a resilient system:Enhanced Reliability, Swift Functional Recovery, and Minimized SocioeconomicConsequences. They shed light on the rationale behind each outcome. Firstly, byaugmenting the system’s robustness, the likelihood of reaching higher limit statesdiminishes, resulting in less reduction of functionality and rendering the systeminherently more reliable. Moreover, the speed at which a system can regain nor-mal functioning stands as a crucial indicator of its resilience, making the rapidrestoration of functionality after a disaster a hallmark of high-resilience systems.The societal impact of extreme events is ultimately minimized when a reliable sys-tem swiftly restores its pre-event functionality. Considered at an individual build-ing level, greater seismic resilience translates into less immediate damage post-earthquake and rapid recovery, collectively reducing downtime and repair costs,thereby minimizing socioeconomic consequences. Additionally, Bruneau et al.[7]emphasize the broader benefits of enhancing seismic resilience for communities,including the reduction of loss of life, injuries, and economic losses, contributingto a lesser decline in people’s quality of life in the aftermath of earthquakes.2.5 Probabilistic Framework for Seismic ResilienceAssessmentTo ascertain the resilience index of a given system, the construction of its func-tionality curve is indispensable, delineating the level of functionality throughout aspecified time frame and its capability to return to full operation following a haz-ardous event. The development of the functionality curve is intricately tied to thequantification and characterization of both the functionality and recovery functionsof the system under scrutiny.In the literature, various approaches to assessing the recovery process of dam-aged buildings have been discussed, reflecting a broad array of probabilistic frame-works [3, 11, 32, 48]. Although the specific steps and organizational structuresof these algorithms differ, they commonly incorporate four principal analytical15phases, excluding seismic hazard analysis as foundational. These phases are namedFragility Analysis, Consequence Analysis, Recovery Analysis, and FunctionalityAnalysis. The first phase models the severity of incurred damage, while the secondphase provides information on the consequences of this damage. The third phasefocuses on recovery strategies, and the fourth phase determines the functionalitylevel from the aftermath of earthquake occurrence until the completion of the re-covery process. Detailed discussions of all these phases are provided in subsequentsections.2.5.1 Fragility AnalysisThe assessment of seismic risk and resilience in buildings relies heavily on theestimation and application of fragility curves. These curves are integral to vari-ous seismic assessment procedures, particularly in seismic probabilistic risk andresilience evaluations. The significance of Fragility Analysis (FA) stems from itspivotal role in comprehending how a building’s structural and nonstructural sys-tems respond to damages corresponding to specific earthquake intensity levels.Sarno and Elnashai [15] conducted a thorough investigation into the fundamen-tal elements involved in deriving fragility functions for structural systems. Theysummarized the key factors essential for deriving these seismic fragility functionsin Figure 2.2.2.5.1.1 Fragility Curve Fundamentals: Definition and GeneralFormulationIn the context of earthquake engineering, fragility curves are defined as statisticaldistributions employed to express the conditional probability of failure or surpass-ing a predefined damage level for crucial structural and nonstructural elements, ormore broadly, vulnerable systems, as function of a specified seismic intensity mea-sure (IM) (e.g., Peak Ground Acceleration (PGA), Peak Ground Velocity (PGV ),Spectral Acceleration (SA)), or a predictive Engineering Demand Parameter (EDP)(e.g., story drift and floor acceleration, etc.).The lognormal cumulative distribution function (CDF) is a common analyticaltool for defining seismic fragility functions. However, it’s essential to recognize16Figure 2.2: An illustrative diagram highlighting the primary factors behindthe derivation of fragility functions for structural systems. Adoptedfrom [15]that this selection may not be universally deemed the most optimal or consistentlyappropriate, as the distribution might not always provide the best fit. The lognormalCDF is characterized by its median value and logarithmic standard deviation.Porteret al. [43] have introduced the mathematical expression for this CDF fragilityfunction, outlined in Eq. 2.3.Fd(x) = P [D≥ d | X = x] d ∈ {1,2, ...,ND}=Φ(ln(x/θd)βd)(2.3)Here, Fd(x) represents the fragility function for a given component, indicatingthe likelihood of the component being in damage state d or a more severe state asa function of the level x. The notation P[A | B] denotes the probability of event17A occurring given that event B has already occurred, expressed as a conditionalprobability. The variable D signifies the uncertain damage state of a specific com-ponent, with D belonging to the set {1,2, ...,nD}. Notably, D = 0 signifies theundamaged state, and D = 1 denotes the 1st damage state, etc. The parameter drepresents a specific value of D without any uncertainty, while nD stands for thetotal number of potential damage states nD ∈ {1,2, ...}.Additionally, the variable X is associated with an uncertain seismic intensitymeasure (IM) or an EDP. The term Φ(s) denotes cumulative distribution functionCDF of the standard normal (Gaussian) distribution, calculated at the value s. Fur-thermore, θd represents the median value of the probability distribution, measuredin units equivalent to X , and βd denotes the total logarithmic standard deviation.Considering the assumption that damage states unfold consecutively, the con-ditional probability of the considered component being in a specific damage condi-tion, while taking into account the component’s exposure to intensity level X , canbe expressed as follows [17]:P [D | x] = Fd+1(x)−Fd(x) (2.4)Where the conditional probability denoted by Fd+1(x) represents the likelihoodthat the component will incur damage up to state d+1 or a more severe condition,while Fd(x) adheres to the previously established definition. Figure 2.3(a) depictsa conventional lognormal fragility function, portrayed as a cumulative distribu-tion function. Meanwhile, Figure 2.3(b) outlines the procedure for calculating theprobability of a component being in a specified damage state, denoted as ”d,” at aparticular level of IM or EDP.2.5.1.2 Integrated Approaches in the Development of FragilityCurvesTo achieve the precise calibration of Eq. 2.3 for a specific system, a pivotal processinvolves conducting a thorough analysis to derive accurate estimates for the equa-tion’s parameters—explicitly denoted as the values of θ and β . Over an extensiveperiod, these estimates have undergone meticulous refinement through rigorous18(a) (b)Figure 2.3: Illustration of: (a) a representative lognormal fragility function;and (b) the assessment of probabilities pertaining to individual damagestates.computations in numerous research studies. The overarching goal of these endeav-ors is to craft robust fragility curves tailored to a diverse spectrum of commonbuilding types, ranging from steel frame and reinforced concrete to masonry andtimber, along with their associated structural and nonstructural elements.In order to facilitate the accessibility and dissemination of this valuable infor-mation, several publicly available fragility databases have been established. Note-worthy among them are prominent databases such as Hazus [16] and FEMA P-58 guidelines [17–19], developed by the U.S. Federal Emergency ManagementAgency (FEMA). These databases function as comprehensive libraries of fragili-ties, offering profound insights into the vulnerability of both structural and non-structural components.The methodologies employed in these research studies rely on various generalapproaches to create fragility functions, conventionally classified into four distinctcategories: empirical, analytical, expert elicitation, and hybrid methodologies. Theselection of the most suitable method depends on the conditions of data availability,whether real or experimental, or the feasibility of modeling the system.An empirical fragility function is established through the utilization of obser-vational data, which is derived from a significant volume of laboratory test dataor gathered through post-earthquake surveys and on-site observations of tangibledamage in real-world scenarios.Conversely, an analytical fragility function is formulated in circumstances where19data availability is limited, yet it remains feasible to construct mathematical mod-els capable of analyzing structural behavior and estimating the threshold demandlevel at which a specific damage state is anticipated to occur for a given asset class.In situations where observational data is inaccessible and conducting an an-alytical analysis is not feasible, a judgment-based fragility function is crafted asa final recourse. One or more seasoned individuals with expertise in the relevantasset class provide insights based on their experience and engineering judgment.This fragility function relies on expert opinions to estimate the level of demand atwhich damage is likely to occur, serving as a valuable approach in the absence ofempirical data or analytical feasibility.Lastly, a hybrid fragility function is crafted by integrating various techniques,such as empirical, analytical, or judgment-based methods, to address and compen-sate for the individual limitations of each approach. This hybridization aims tomitigate drawbacks such as modeling complexities, variations in data source avail-ability, and uncertainties related to both modeling and seismic factors.Figure 2.4 presents the table provided by Schultz et al. [46], offering a com-prehensive summary that delineates the inherent advantages and disadvantages as-sociated with the various approaches employed in the creation of fragility curves.Empirical, judgmental, and hybrid methodologies, which represent a fusion ofboth empirical and judgmental approaches, have played a pivotal role in shapingfragility functions predominantly generated in databases. The judgmental methodemerges as a pragmatic and realistic alternative in scenarios where empirical datais scarce or testing proves economically burdensome and time-intensive.In thedatabases, to define fragility function parameters, a systematic grouping of fragilitycurves has been employed. These groupings, established through meticulous clas-sification of structural, nonstructural elements, and content, take into account com-parable characteristics among sub-components. Key attributes influencing the clas-sification encompass construction features, potential damage modes, and the con-sequential outcomes of damage. This methodology streamlines the data collectionprocess for nonstructural components and content, as the resulting fragility curvesexhibit a notable degree of independence from specific geological and geotechnicalconditions, owing to the shared characteristics and standardized design criteria ofmost nonstructural elements or contents.20Figure 2.4: Advantages and Disadvantages of Each Fragility Curve Develop-ment Approach. Adopted from [46]However, challenges may arise for structural elements when employing em-pirical or judgmental methods, especially when the available data is generic ornot precisely aligned with the specific characteristics of the building in question.Structural damages, intricately linked to a spectrum of properties such as geometry,material characteristics, morphological features, age, seismic design level, equip-ment anchorage, local seismicity, soil types, environmental factors, and foundationdetails, pose significant hurdles to the universal applicability of available fragilityfunctions across diverse regions.Therefore, in the pursuit of a more nuanced understanding of seismic vulnera-bility, the potential application of analytical fragility functions emerges as a com-pelling avenue. This analytical approach not only holds the promise of deriving21more accurate fragility functions for structural elements but also allows for metic-ulous control over collected data. By encompassing effective properties withinstructural models, this approach minimizes bias related to structural configurations,built environments, and the complex interplay of geotechnical and seismotectoniccharacteristics within a seismogenic area.2.5.1.3 Analytical Fragility DerivationIn the context of analytical approaches for estimating fragility functions, a diverseset of methodologies is employed for conducting nonlinear dynamic structuralanalyses. These include Incremental Dynamic Analysis (IDA), Multiple-StripeAnalysis (MSA) and the Cloud Method . Each method brings unique insights intothe seismic performance of structures, contributing to a comprehensive data col-lection process for fragility function estimation.◦ Incremental Dynamic Analysis (IDA)The IDA method assumes prominence as the principal analytical approachfor fragility assessment, distinguished by its straightforward implementa-tion. Through a meticulous evaluation of the seismic performance of struc-tures, this methodology facilitates accurate predictions of structural responses.The methodology entails a series of nonlinear dynamic analyses conductedunder a comprehensive set of ground motion records, incrementally ampli-fied across the entire spectrum from elastic behavior to the point of identify-ing the intensity measure level at which each ground motion triggers globaldynamic instability, ultimately leading to structural collapse [56]. The culmi-nation of this process involves the establishment of an Incremental DynamicAnalysis curve (IDA curve), which delineates the relationship between aconsidered EDP as a measure of damage and the IM derived from the ap-plied scaled accelerograms.◦ Multiple-Stripe Analysis (MSA)The MSA stands out as a second prevalent non-linear dynamic structuralanalysis method, denoting a series of ’stripe’ analyses conducted at distinctIM levels. Each of these levels encompasses a distinct set of ground motions22[28]. Notably, this approach eliminates the necessity of conducting the anal-ysis up to the IM level where all ground motions lead to collapse. Moreover,by employing diverse ground motions at each IM level, the analyst may notobserve a strict correlation between increasing fractions of collapse and as-cending IM values.◦ Cloud MethodThe method of cloud analysis is widely embraced in the field of structuralengineering due to its low computational cost, making it a popular choicefor conducting time-history analyses to generate fragility curves. In this ap-proach, the analysis is primarily executed using a suite of unscaled records,exactly as recorded, encompassing various IMs [29, 36]. While this methodshares similarities with both Incremental Dynamic Analysis (IDA) and Mul-tiple Stripes Analysis (MSA), where the resulting response values are plottedagainst the IM, it differentiates itself by presenting these points as a cloudrather than connecting them with splines or straight lines, as IDA does, ororganizing them into stripes of response values, as seen in MSA.Each of these analysis methods comes with its own set of limitations, andthe choice between them depends on several factors, including record availabil-ity, the sophistication of the developed structural model, computational costs, andthe availability of resources, among other considerations. Figure 2.5, showcasingthe table proposed by Sarno and Elnashai [15], is a valuable resource offering acomprehensive list of the primary advantages and shortcomings of each method,aiding practitioners in making informed choices and selecting the most suitablemethod based on their specific project requirements and constraints.Within the domain of defined dynamic analysis methods aimed at construct-ing analytical fragility curves, this research specifically concentrates on exploringfragility functions through IDA. The forthcoming emphasis will delve into the in-tricacies of this specialized approach, aiming to illuminate its unique methodologyand underscore its substantial contributions to the broader field of seismic vulner-ability assessment.23Figure 2.5: Understanding Key Advantages and Limitations Across VariousAnalytical Approaches in the Literature. Adopted from [15]2.5.1.4 Determining Analytical Fragility Functions through IDAThe IDA curve illustrates the relationship between the selected IM and EDP. Toderive the parameters of a fragility function, specifically the mean θ and standarddeviation β , from the IDA curve data, the conventional method involves severalsteps, as depicted in Figure 2.6 by Dabiri et al. [14].Building upon the illustrated steps in Figure 2.6, following the establishmentof IDA curves and acknowledging the variability between records where the sameIM from different ground motions can yield varying values of the considered EDPfor a given structure, the subsequent step involves identifying the IM values cor-responding to each ground motion that lead to a specific level of damage. Thisidentification is accomplished by plotting a vertical line from the associated EDPvalue, as shown in Figure 2.6(b). Subsequently, the logarithms of these IM valuesare computed to derive the mean (lnθ ) and standard deviation (β ) for each damagelevel using Eq. 2.5 and Eq. 2.6 [5].24Figure 2.6: Schematic illustration of the conventional method for developingfragility curves: (a) Incremental Dynamic Analysis Curve, (b) Distri-bution of IMs at a specified damage state, (c) Corresponding failureprobability for the considered damage state, and (d) Resulting fragilitycurves. Adopted from [14]ln θ̂ =1nn∑i=1ln IMi (2.5)β̂ =√1n−1n∑i=1(ln(IMiθ̂))2(2.6)The parameter n denotes the total number of earthquake ground motions usedin conducting the IDA, while the term IMi represents the IM value of the ith groundmotion associated with the specific damage condition under investigation.Two widely used statistical approaches for parameter estimation, such as es-timating fragility function parameters from data, are the method of moments andthe maximum likelihood method. The method employed here is referred to asthe ”method of moments estimator.” This method is named for its use of sample25distribution statistics to estimate population parameters. The method of momentsensures that the resulting distribution has equivalent moments (e.g., mean and stan-dard deviation) to the sample moments of the observed data.2.5.2 Consequence AnalysisIn the aftermath of massive earthquakes, the extensive damage typically resultsin a scarcity of available resources. It becomes imperative to efficiently prioritizebuildings that have not collapsed and are deemed repairable to ensure the effec-tive allocation of these limited resources. This prioritization facilitates swift andtargeted repair efforts that address the most critical needs first. Central to this pro-cess is the use of clear and meaningful decision variables, which are crucial forperforming a comprehensive assessment of the condition of different structures.In addition, another crucial aspect arises during the design phase of buildings,emphasizing the importance of clear and meaningful parameters to evaluate theseverity of seismic damage. Selecting the most appropriate earthquake-resistantsystem for a building requires a comprehensive evaluation of all proposed systemsacross different levels of earthquake intensity. Assessing the seismic performanceof these systems involves using parameters that distinctly highlight performancevariations among designs. It is vital that these parameters offer stakeholders trans-parent insights into the operational effectiveness of each system during seismicevents, thereby supporting informed decision-making and enhancing the overallsafety of the structures.While EDPs and probabilities of damage states derived from fragility analysisprovide valuable insights into structural responses, they alone are inadequate forthorough comparisons of damage conditions between buildings or for evaluatingvarious seismic resistance systems. These metrics do not offer a holistic view ofoverall structural integrity. Consequently, a more robust approach is required—onethat integrates this data into decision variables that consider both the individualresponses of components and their collective impact on the building’s overall in-tegrity and functionality. To bridge this gap effectively, it is crucial to quantify theconsequences of earthquakes in terms of potential losses.Cimellaro et al. [10] identify two distinct categories of potential losses resulting26from earthquakes, each further subdivided into specific subcategories. The first cat-egory, Direct Losses (LD), comprises the immediate damage inflicted at the time ofthe seismic event. The second category, Indirect Losses (LI), encompasses the con-sequential impacts that unfold over time, primarily due to the complete or partialloss of functionality. Furthermore, within these primary categories, the losses aresubdivided into Economic losses (LE), which include the financial ramifications,and Casualty losses (LC), which account for injuries and fatalities.Based on the established categorization, direct and indirect losses might ini-tially seem to be independent; however, a significant interrelationship clearly ex-ists between them. Substantial direct losses, resulting from severe initial damage,invariably lead to extended repair periods. This prolongation significantly exac-erbates time-dependent losses, illustrating that the magnitude of indirect losses isdirectly linked to direct losses. This correlation underscores the complex dynam-ics of earthquake impacts, emphasizing the necessity for a precise assessment ofthe initial damage to accurately evaluate both direct and indirect losses. Notably,while direct losses are relatively easier to measure, indirect losses—such as busi-ness interruption, relocation expenses, and rental income losses—present consid-erable challenges in quantification [8]. Accurately quantifying these indirect lossesdepends crucially on the initial correct calculation of direct losses, which entailsa precise determination of repair timelines and associated costs in the aftermathof an earthquake, a process that inherently relies on comparisons to pre-disasterconditions.The inherent uncertainties in loss estimation necessitate a precise and com-prehensive approach to documenting all processes essential for restoring damagedcomponents to their original state. A reliable estimate must encompass a detailedrepair plan that address both the actual damage and the collateral works vital foraccessing the affected areas. To streamline this complex task and mitigate un-certainties, FEMA P-58 [17] offers an extensively detailed and practical suite ofrepair descriptions for all damage states identified within its established fragilitycurves. This robust guidance results from collaborative efforts with engineeringconsultants and cost estimators, each bringing profound expertise in various repairtechniques and a deep understanding of the associated collateral works and costs.The FEMA P-58 framework [17] prescribes repair actions adhering to the prin-27ciple of ’in-kind’ repair and replacement. This method ensures that restorationefforts precisely replicate original components without integrating any new en-hancements or modifications. Building upon this principle, the framework detailsall necessary repair materials and labor requirements for each specified damagestate. These comprehensive specifications serve as the foundational elements inthe development of FEMA P-58’s consequence functions [18], which are advancedanalytical tools that predict distributions of probable damage outcomes. By trans-lating earthquake impacts into quantifiable metrics—including repair and replace-ment costs, repair timelines, and other vital performance indicators—these func-tions significantly enhance the accuracy and efficiency of disaster loss assessments.Utilizing these consequence functions allows for the precise calculation of neces-sary repair costs and time at the component level. This detailed analysis allows forthe aggregation of individual component losses, which are then used to calculatethe total earthquake losses for the building.2.5.2.1 Component-Level Repair Cost and Time AnalysisThe median repair cost and time for various damage states of a component can beaccurately determined using the repair consequence functions outlined in FEMAP-58 [18]. However, because structural simulations do not pinpoint the exact dam-age condition of a component, relying on the time and cost required to repair asingle damage state to represent the total repair cost and time of the component isimpractical. Instead, a comprehensive evaluation of the component’s overall repaircost and time necessitates integrating values calculated for each damage state, ad-justed by their respective weights. This weighted integration method ensures thatthe repair consequences for each damage state reflect its likelihood of occurrence,thereby allowing for a nuanced approach to estimating cost and time. This ap-proach aligns with established academic research [8, 33, 48], where the aggregaterepair cost and repair time of a component are derived using a function that inte-grates the probabilities of its damage states with their associated repair costs andtimes, respectively. The relationship between these variables is clearly expressedin the following formula:28L =N∑i=1Li ·PDSi|IM (2.7)Equation 2.7 calculates the total repair cost or time L for a damaged component.It sums the product of each damage state’s repair consequence function Li and itsprobability PDSi|IM given an earthquake intensity IM. Here, N represents the totalnumber of damage states for the component.2.5.3 Recovery AnalysisRegardless of the type of functionality defined to showcase the performance of theconsidered system, seismic hazards can result in a proportional loss of the sys-tem’s defined functionality. The potential for restoring the system to regain itsfunctioning, based on various considerations, prompts the need to undergo a com-prehensive recovery phase. For an accurate assessment of the system’s resilience, aprecise prediction of the recovery phase is crucial, encompassing both the recoverytime and the recovery path.In addressing the recovery phase, Cimellaro et al.[8, 10] have defined the re-covery time (TRE) as the required period to bring back a structure or infrastructuresystem’s functionality to a desired level—below, equal to, or better than its pre-event functionality. The authors highlight multiple factors contributing to the def-inition and calculation of recovery time, including the level of damage (dependenton earthquake intensities), the system’s location, the chosen repair path, and theavailability of resources such as materials and labor. Each of these variables in-troduces uncertainty into the recovery time, transforming it into a random variablewith significant uncertainty, thereby making accurate prediction challenging.2.5.3.1 Recovery TimeIn the aftermath of an earthquake event, commencing the repair process proveschallenging due to the necessity of completing essential pre-construction activi-ties. These activities encompass critical steps such as inspection, strategic repairplanning, and obtaining mandatory permits. The completion of this series of tasksintroduces inevitable delays in initiating repair procedures. Consequently, the total29recovery time extends beyond the repair duration to also include this mobilizationperiod.Estimating this recovery time is complex, given the myriad variables and un-certainties inherent in both mobilization and repair phases. Unforeseen challengesfurther complicate the accuracy of these predictions [2]. Despite these difficulties,creating a robust functionality curve, which is fundamental for system resilience,demands an accurate assessment of the total recovery time. In the subsequent para-graphs, detailed exploration of the models developed to capture mobilization andrepair phases will be provided.Mobilization TimeThe mobilization phase involves essential activities that are generally predictableregardless of project location or scale. Therefore, it is crucial to identify and com-pile these activities based on information gathered from reliable sources such asdata collected post-real earthquakes or insights from experts in relevant fields.REDi™[2] categorizes all potential activities necessary to initiate the repair pro-cess for a damaged building into six groups, referred to as impeding factors. Ac-cording to REDi™ approach, determining the time required for each of these fac-tors involves using lognormal cumulative distribution functions. These functionsare characterized by their median θ and standard deviation β , details of which areelaborated in the REDi™ documentation.Repair TimeOnce the mobilization phase to address impeding factors is complete, the repairprocess can commence. To determine the total repair duration, the initial step in-volves establishing the duration required for each repair activity, addressing everydamaged component on every floor, measured in ’days’. Following this, it is vitalto develop network logic that accurately describes the sequence of performing re-pair activities at each floor level and throughout the building, reflecting real-worldscenarios as closely as possible.The repair times provided by FEMA P-58’s consequence function [18] are interms of ’worker-days’. However, in practice, multiple workers may contribute to30repairing a component, leading to shorter repair times than initially estimated [2].Therefore, by multiplying the number of workers involved in each repair activity,the FEMA P-58 [18] repair time results can be converted into repair duration mea-sured in ’days’. Thus, the key parameter for calculating the length of repair time foreach repair activity, which impacts the overall repair schedule and, consequently,the final length of the building’s repair time, is the assumed number of workers foreach repair activity. While allocating more labor can expedite the repair processfor each component and the building as a whole, factors such as resource availabil-ity, workspace constraints, and site access may limit the maximum workforce [2].Therefore, to prevent underestimation or overestimation of repair duration, certainassumptions should be made regarding the number of workers used, ensuring thefinal results are applicable.Building upon the insights into repair duration and workforce dynamics, theFEMA P-58 [19] methodology introduces a system for allocating workers basedon a ”maximum worker per square foot” principle. This allocation is linked tothe building’s occupancy status, prescribing worker densities of 0.1% for occu-pied and 0.2% for unoccupied conditions during repair operations, regardless ofthe specific nature of the repair or the building’s condition. Consequently, FEMAP-58 ensures an equitable distribution of labor for all repair tasks on each floor,based on the specific area of that floor and assumes repairs on each floor are under-taken sequentially. It then delineates two diametrically opposed repair strategies.The first, considered the lower limit, operates under the assumption that repairsacross all floors are conducted concurrently, with the repair duration of the build-ing equated to the longest repair period observed on any single floor. Conversely,the upper limit strategy posits that repairs are executed sequentially, floor by floor,commencing from the bottom.The developed framework by FEMA P-58 for estimating repair time of dam-aged buildings faces critical shortcomings due to its simplified approach in re-source allocation and repair strategy. This framework attempts to provide a repairtime range but significantly underestimates the impact of crucial factors such asbuilding height and the extent of damage, leading to a wide discrepancy in time pre-dictions. Although FEMA P-58 uses two extreme repair sequence logic as bound-aries for probable repair times, these often do not reflect realistic repair schedules,31particularly for taller buildings, where the range can become excessively broad[2]. Contrary to FEMA P-58’s repair model, which suggests that repairs on a sin-gle floor proceed sequentially—regardless of whether the building’s overall repairstrategy is sequential or parallel across floors—real-world practices often see mul-tiple trades working simultaneously on the same floor to enhance cost-efficiency[51]. This divergence results in FEMA P-58’s framework overestimating repairdurations. Additionally, FEMA P-58’s labor allocation model is unrealistic, inac-curately predicting repair times for minor damages because it overlooks the intri-cacies of damage severity and resource requirements. To address these limitationsand enhance decision-making, the Resilience-based Earthquake Design Initiative(REDi™) Rating System [2] proposes an alternative approach, incorporating ad-justments and additional considerations specific to each building’s unique charac-teristics, thereby offering a more realistic and practical framework for repair timeestimation.REDi™ revolutionizes FEMA P-58’s approach by merging RS Means data(Reed Construction Data Incorporated 2013) with expert insights, significantly im-proving repair time estimates. This advancement is primarily due to a compre-hensive overhaul of labor allocation and sequencing strategies, which entails cate-gorizing all structural and non-structural components into seven principal groups.Depending on the group, labor is meticulously assigned based on square footageor the count of damaged units, as detailed in Table 2.2.In line with FEMA P-58 guidelines, REDi™ implements a 0.2% labor densityfor unoccupied buildings, setting a precedent of one worker per 500 ft2 for indi-vidual trades managing structural repairs. This ratio is modified to one worker per1000 ft2 for simultaneous repairs in Sequences A and B. For sequences C throughF, REDi™ calculates the number of workers based on the average number of dam-aged units across all components within a type.Building on this foundation, REDi™ introduces a further refinement by imple-menting restrictions on the total number of workers allowed across all levels forthe repair of specific component types. This measure addresses the challenges as-sociated with securing a workforce with the requisite skills. As delineated Table 7from REDi™—reproduced in this document as Table 2.3, the resource allocationfor distinct repair sequences is proportionally adjusted to the height of the struc-32Table 2.2: Recommended classification of components and their respectiveworkforce numbers in REDi™Repair Sequences Component Type Number of Workers(per ft2 or damaged units)Structural Structure 1 worker/500 sfSequence APipes/Sprinklers1 worker/1000 sfHVAC distributionPartitionsCeilingsSequence BExterior Partitions1 worker/1000 sfCladding/GlazingSequence C Mechanical Equipment 3 workers/damaged unitSequence D Electrical Systems 3 workers/damaged unitSequence E Elevators 2 workers/damaged unitSequence F Stairs 2 workers/damaged unitture. This approach ensures that resource distribution is commensurate with thescale and complexity of larger projects.Table 2.3: REDi™-defined maximum number of workers based on building’sheight across classified component categoriesRepair Low-rise Medium-rise High-riseSequences (Less than 5 stories) (Between 6 and 20 stories) (Greater than 20 stories)A 15 30 45B 15 30 45C 9 18 27D 9 18 27E 6 12 18E 6 12 18Acknowledging the critical importance of balancing expedited repair processeswith the overarching goals of safety and efficiency in the post-disaster recoveryphase, it becomes imperative to carefully manage the expansion of the workforce.33This equilibrium necessitates the implementation of precise limitations on the per-missible size of the workforce, not just on individual floors but across the entiresite, while considering an array of essential factors. In response to this neces-sity, REDi™ has established a constraint predicated upon the building’s gross area,which serves to limit the total number of workers operating concurrently within thebuilding. The formulation of this constraint is captured in Eq. 2.8, which delineatesthe calculation for the upper limit of workers present on site, represented as Nmaxin the equation,in relation to the total floor area of the building, symbolized as Atotand measured in square feet.Nmax = 2.5×10−4Atot +10 20≤ Nmax ≤ 260 (2.8)Upon determining the requisite workforce for repairing structural elements andeach type of non-structural component, categorized into seven distinct groups, andsubsequently verifying the number of workers against predefined limitations, it be-comes possible to calculate the necessary repair time for the building. This calcu-lation hinges upon meticulously organized repair sequences, delineating the orderof operations aligned with practical sequences carried out by contractors.In line with the fundamental principle of ensuring the safety of both workersand occupants, the REDi™ methodology emphasizes the paramount importanceof securing the building’s structural stability before initiating any subsequent re-pair activities. This crucial phase is attained after the thorough completion of re-pairs to any structural damage identified within the building. According to thisprinciple, the initiation of repairs to non-structural components is strictly contin-gent upon achieving the complete restoration of the building’s structural elementsacross all floor levels. Furthermore, the strategy for addressing both structural andnon-structural repairs adopts a methodical approach, scheduling the repair work toprogress systematically, one floor at a time, beginning from the lowest level andmoving upwards.In accordance with the REDi™ framework, following the completion of struc-tural repairs throughout the building, the subsequent repair of non-structural el-ements on each floor follows a predefined path known as ”Typical Repair Se-quences.” As previously highlighted, REDi™ categorizes all non-structural ele-34ments into seven sequences. These sequences serve not only for determining therequired workforce but also for defining the repair sequences of non-structural el-ements at each floor level. The proposed repair sequence, based on these sevensequences, is meticulously detailed in Figure 2.7.The REDi™ repair strategy is founded on the premise that it is feasible to un-dertake concurrent repairs across all sequences at a particular floor level, assumingthere is no interference between the activities within these repairs. However, it isessential to note that within an individual sequence, the repairs must be executedsequentially. For instance, in Sequence A, the repair of partitions cannot com-mence until the repair of HVAC distribution systems is completed, as the integrityof the latter depends on the intact condition of pipes. In accordance with this repairstrategy, the duration necessary to complete repairs on a given floor is determinedby the sequence with the longest repair time. This methodology is consistentlyapplied across all floor levels.Figure 2.7: Sequence of Repairs Proposed by REDi™ at Each Floor Level,Denoted as ’Typical Repair Sequences.’ Adopted from [2].Within the context of this framework [2], it is critical to understand that therepair processes for elevators and stairs, identified as sequences E and F respec-tively, are contingent upon the prior completion of repairs to structural elements.To support the continuous advancement of structural repairs, the deployment of35temporary elevators for the efficient transportation of materials and equipment isplanned at the construction site. This strategic provision is aimed at ensuring thatthe repair operations proceed without interruption.According to the REDi™ repair framework, once the repair of a particularcomponent on a given floor is completed, repair operations for that componenton the floor above can commence, provided that the prerequisite elements of itssequence have been repaired on that floor. This enables repairs to proceed con-currently across different floor levels. However, it is imperative to ensure that thenumber of workers on-site aligns with the criterion outlined by Eq. 2.8. As long asthis condition is met, there is no need to postpone the repair of any component. Fi-nally, based on all these assumptions and processes, the repair time of the buildingcan be determined.While the REDi™ framework has made strides in addressing some of the sim-plifications inherent in the FEMA P-58 guidelines for estimating repair times, itstill harbors certain assumptions regarding workforce allocations and the sequenc-ing of repairs that can lead to inaccuracies in project timelines.The REDi™ model proposes a rigid repair schedule that assigns fixed se-quences between predefined groups, lacking the flexibility needed to adapt to vary-ing project demands. In reality, the repair process for buildings is subject to abroad array of factors, making each project distinct. Even among buildings of sim-ilar types, the need for adjustments in the repair process can arise due to differentrecovery speed requirements, deadlines for restoring full functionality, or variancesin resource availability. This rigid adherence to a predetermined repair sequencecould be impractical or ineffective in meeting the diverse and dynamic needs ofreal-world repair projects.Furthermore, the allocation of labor is a critical component in precisely pre-dicting the total repair time required for buildings, reflecting the intricacies of real-world projects. REDi™ advances beyond the FEMA P-58 guidelines by rejectingthe notion of a static workforce for all repair activities, instead introducing a modelthat varies labor allocation based on the classification of components into specificgroups. This approach recognizes the dynamic nature of repair work but is notwithout its limitations.A significant limitation within the REDi™ framework is its simplistic method36of labor assignment, which fails to account for the severity of damage to compo-nents. The framework assumes that the number of workers allocated is based solelyon the count of damaged units within a group, ignoring the critical aspect of dam-age severity. Such a uniform strategy for labor distribution overlooks the variedcomplexity and skill requirements needed for repairs, depending on the extent ofdamage. This oversight results in a disconnect between the theoretical labor allo-cation model proposed by REDi™ and the realities observed during actual repairoperations.Moreover, REDi™’s guidelines on the composition of repair crews—whichdirectly influences the number of workers deployed—are predicated on the totalnumber of damaged units. For instance, the model suggests that the repair of threeunits within an electrical system, identified as sequence D, would require the for-mation of three crews, each with three workers, totaling nine individuals. However,practical experiences frequently demonstrate that fewer crews are often adequatefor such repairs [51], underscoring a disparity between REDi™’s theoretical frame-work and the practical necessities encountered on the ground.In 2016, Terzic et al. [50] and Yoo [59] introduced a groundbreaking repair timemodel through a collaborative effort with general contractors, marking a significantadvancement in the field. This innovative model addresses and overcomes the limi-tations inherent in existing methodologies by introducing a highly adaptable frame-work equipped with enhanced functionalities. It skillfully manages the dichotomybetween the straightforward nature of new construction projects in stable condi-tions and the complex challenges associated with repairing earthquake-damagedbuildings amidst the chaotic aftermath of a disaster. The model highlights howthe increased demand for construction materials and labor, compounded by spatialrestrictions at the site due to damaged components, significantly hampers resourceallocation efficiency. This situation leads to a notable slowdown in project pro-gression, illustrating a clear contrast to the resource scheduling observed in newconstruction projects.Unlike new construction projects, which generally employ consistent materialsand components across floors with similar occupancy types, the post-earthquakelandscape introduces a layer of unpredictability. Components may experiencevastly different conditions depending on their specific locations within a build-37ing, thereby complicating repair efforts and necessitating varied repair tasks andresource allocation for each floor.The innovative model introduced by Terzic et al. [50] and Yoo [59] represents asignificant advancement beyond traditional resource allocation strategies, specifi-cally tailored to meet the distinct challenges posed by post-disaster scenarios. Thismodel exhibits remarkable precision in adjusting worker distribution, ensuring thatthe allocation of resources is meticulously calibrated to address the specific needsof each component. By considering both the severity of damage and the quantity ofaffected units, it facilitates a strategic and realistic approach to labor deployment.Moreover, the model showcases exceptional adaptability by diverging fromconventional fixed repair schedules. It adopts a flexible strategy that treats eachrepair task as a distinct operation, rather than aggregating tasks into predefinedgroups. This refined methodology takes into account the complex interdepen-dence’s between components, acknowledging these relationships both within in-dividual floors and across the building’s entire vertical structure. Such a sophisti-cated approach ensures the model’s comprehensive applicability to a wide range ofbuildings, transcending limitations related to size, occupancy, or the diverse levelsof damage induced by earthquakes of varying intensities.Presented in Figure 2.8 is a schematic depiction of the innovative repair timemodel developed by Terzic et al. [50] and Yoo [59]. This figure highlights themodel’s architecture, emphasizing its division into two fundamental components.The foundational element of the repair time model, designated as the repair schedul-ing method, is distinguished by its exceptional adaptability.Figure 2.8: Representation of the repair time model as proposed by Terzic etal. and Yoo. Adopted from [52].Their proposed repair scheduling method commences with the implementation38of a specialized logic network, depicted in Figure 2.8, a meticulously structuredsystem that integrates a strategic repair sequence. This sequence is specificallycrafted to navigate the complex web of interrelationships among building compo-nents, methodically establishing transparent predecessor and successor links forevery repair task. Furthermore, it precisely delineates the scope for concurrentrepairs across multiple floors for each designated activity.Upon the construction of the logic network, the repair scheduling methodologydeveloped by Terzic et al. [50] and Yoo [59] moves forward to accurately estimatethe duration required for the repair of all damaged components, integrating prac-tical labor allocations. Central to this phase is their pioneering approach, whichskillfully calculates the total workforce needed for the restoration of each compo-nent type across different floor levels. This is achieved by determining the numberof crews and the specific makeup of each crew, tailored for the repair of each com-ponent on every floor.Terzic et al. [50] and Yoo [59] have highlighted the imperative of strategicallydetermining optimal crew sizes for repair tasks, underlining the direct correlationbetween such allocations and the extent of damage that components endure. Theypostulate that severely damaged components require the formation of larger, morecomplex teams composed of individuals with specialized expertise, specifically se-lected to address the intricate challenges of extensive repair tasks. At each floorlevel the variability of damage across similar components, often due to directionalfactor, poses a significant challenge in accurately determining optimal crew sizes.In practice, this challenge, can be effectively addressed through detailed inspec-tions that enable the precise assessment of each unit’s condition within a compo-nent type. Such evaluations facilitate the tailoring of crew assemblies to the uniquedemands of repair tasks, thereby ensuring both efficient and effective restorationprocesses.To effectively address the complexities inherent in theoretical modeling of crewsize determination for repair tasks, Terzic et al. [50] and Yoo [59] have introduceda refined model that streamlines this intricate process. Their methodological in-novation lies in the utilization of the average condition of damaged units within acomponent on a given floor as a standardized metric,as delineated in Eq. 2.9, to as-certain the requisite crew size. This innovative strategy presents a practical solution39to the intricacies involved in crew allocation, effectively bridging the gap betweentheoretical considerations and their practical application in the repair process.ADS =∑ki=1 niDMin(2.9)Where ADS denotes the average damage state across a specific component ona single floor level, encompassing a total of n units. Here, k represents the totalnumber of distinct damage states identified for that component, while ni indicatesthe number of units within the ith damage state.Within the domain of repair operations, it is a recognized standard that spe-cialized number of crews are allocated to handle a defined range of damaged unitswithin specific component types. Incrementally increasing the number of crewsin direct response to each rise in the NDU is impractical and uneconomical. Toenhance operational efficiency, it is essential to determine the optimal quantity ofunits that a crew can restore effectively, which neither necessitates additional hours(overtime) nor leads to periods of inactivity (idle time). This calibrated approachensures that repairs are completed efficiently and economically, maintaining high-quality outcomes and the well-being of the crew.In light of this perspective, Terzic et al.[50] and Yoo[59] have developed amodel that is structured around the establishment of predefined ranges that linkthe number of damaged units to the requisite number of repair crews for eachtype of building’s component. This model requires the setting of upper thresh-olds for damaged units, above which the engagement of additional crew membersbecomes essential. Consequently, this framework facilitates the accurate computa-tion of necessary crews for any component at a given floor by comparing the actualnumber of damaged units per component against established ranges. This method-ical approach significantly mitigates uncertainties associated with assigning a stan-dard number of crews across all projects by enabling the definition of these rangesthrough the utilization of data from local contractors. This data-driven strategy,augmented by engineering judgment tailored to each specific building, ensures thatcrew allocations are precisely adjusted to meet the distinct requirements of eachproject.As highlighted in Figure 2.8, upon establishing the total workforce allocated to40each building component across different floors, the repair scheduling method de-veloped by Terzic et al.[50] and Yoo[59] progresses to estimate the duration neededfor all repair activities. Following this phase, the critical path method (CPM) isemployed within their model to ascertain the overall project duration. This calcu-lation begins with the meticulous scheduling of start and finish dates for all repairactivities, which is systematically orchestrated based on pre-defined predecessorrelationships within the adopted repair logic network. For each component on anygiven floor, the initiation date is determined by the latest completion date amongits predecessors, with the addition of the required repair duration determining thefinish date.In the phase dedicated to scheduling repairs, the failure to anticipate potentialdelays triggered by essential adjustments in resource allocation, aimed at adheringto daily resource ceilings, can result in the creation of an optimistic repair timeline.The imperative for such adjustments is acutely felt in scenarios characterized byan escalated demand for construction resources or a surge in labor congestion sub-sequent to a disaster. This discrepancy underscores the critical need for additionalanalysis to align the repair scheduling with realistic post-disaster repair timelines.As illustrated in Figure 2.8, Terzic et al.[50] and Yoo[59] address this issue in thesecond component of their repair time model through the introduction of a resourcescheduling method. This innovative component is designed to efficiently managethe workforce during periods of labor congestion, thereby enhancing the accuracyof the estimated total repair duration.In their repair scheduling model, Terzic et al.[50] and Yoo[59] introduce foun-dational assumptions critical to the model’s effectiveness. Central to these is thepremise that the resource allocation for each repair task is designed to remain con-stant for the entirety of its duration. This assumption holds unless there is a com-pelling need to adjust resources to comply with daily limitations.Specifically, themodels anticipate scenarios where daily resource limits may mandate a recalibra-tion of resource allocations, leading to a corresponding adjustment in the scheduleddurations of repair tasks to align with the revised workforce distributions. Fur-thermore, alterations to the project’s completion date is considered a measure oflast resort. Such changes are permissible solely in instances where adherence tothe initially established timeline proves to be untenable within the confines of the41specified resource constraints.Building upon these foundational assumptions, Terzic et al.[50] have incor-porated an iterative process into the fabric of their repair scheduling framework,designed to ensure strict compliance with the prescribed daily resource thresh-olds. This methodology is detailed methodically in a comprehensive series of steps,which is effectively depicted in the flowchart shown in Figure 2.9.The algorithm commences with the critical task of establishing daily resourceceilings, which serve to guide the distribution of resources throughout each floorand across the building. As underscored by Terzic et al.[50], a core aspect of theseceilings’ development is the imperative that they must never fall below the min-imum crew size designated for any specific repair task. Essentially, the smallestoperational team sets the minimum threshold for these limits, though the actuallimits may exceed this minimum. This provision is indispensable for ensuring op-erational efficiency and adherence to safety protocols, guaranteeing that regardlessof any variations in resource availability, all tasks are allocated a sufficient work-force.Subsequent to establishing resource ceilings, the proposed model by Terzicet al.[50] advances with the deployment of a comprehensive aggregation strategy.This strategy is systematically designed for both individual floors and the buildingas a whole, aimed at meticulously recording the total number of workers presenton a daily basis, irrespective of the variety of tasks to which they are assigned. Asdelineated in their detailed flowchart, if the daily allocation of resources consis-tently remains within the predefined limits throughout the project’s duration, theresource scheduling method concludes successfully, without necessitating any ad-justments to resource distribution. In such instances, adherence to the project’sscheduled completion date is assured, as determined by the Critical Path Methodduring the repair scheduling phase. Conversely, in the event of exceeding estab-lished resource thresholds, the algorithm identifies specific intervals experiencinglabor congestion. Subsequently, it meticulously enumerates all the repair tasks tobe undertaken within each interval where resource limits are exceeded.The model then systematically ranks tasks identified in each interval accordingto their repair duration, applying criteria that prioritize tasks with either the short-est duration or the longest FF. This prioritization forms the basis for their resource42scheduling strategy, aimed at reducing crew numbers during times of labor con-gestion by refining the list to exclude tasks that can only be managed by a singlecrew.The strategy progresses with a calculated reduction in the workforce assignedto the task of highest priority by one crew, an approach meticulously crafted to min-imize any adverse effects on the project’s comprehensive timeline that could resultfrom elongating the task’s duration due to decreased labor. Subsequently, this stepis followed by a comprehensive aggregation of the current workforce for compar-ison against the predetermined daily ceilings. If this recalibrated total workforcestill surpasses the ceilings, necessitating further adjustments. A new list is thengenerated, featuring tasks that require more than one crew and are within to thespecific interval in question, organized by repair duration from shortest to longest.This iterative process of crew reduction is repeated for the given interval until theworkforce aligns with the resource ceilings.Figure 2.9: Flowchart illustrating the resource scheduling method proposedby Terzic et al. and Yoo. Adopted from [52].This methodical and recursive application across all periods where labor allo-cation surpasses established daily ceilings requires an extension of repair durationsfor activities undergoing resource adjustments. As a result, with repair durationsre-calibrated and integrated into the project’s timeline, the precise calculation of43the building’s ultimate repair duration becomes feasible through the application ofthe CPM. By effectively preventing resource congestion at any construction phase,this enhanced scheduling technique significantly narrows the gap between the pro-jected and actual completion dates, ensuring a more accurate final project timeline.Integrating both repair and resource scheduling methodologies within the com-prehensive repair time model, as proposed by Terzic et al. [50], and employingMonte Carlo simulations, enables the creation of a repair time fragility function.This function adeptly captures the range of potential outcomes across varied sce-narios. Furthermore, for each scenario meticulously selected, this sophisticatedapproach facilitates the generation of a Gantt chart, in tandem with strategies fordaily resource allocation. This output not only provides a visual depiction ofproject timelines but also establishes a strategic operational framework for effi-cient project management, particularly in post-hazard situations where rapid andjudicious decision-making is imperative. Building upon this foundation, in 2021,Terzic et al. [52] expanded their research to introduce a comprehensive probabilis-tic framework aimed at modeling the functionality recovery process of buildingsystems. Within this framework, their earlier proposed repair time model [50]emerges as a fundamental element, reinforcing the synergy between theoreticaldevelopment and practical application in disaster recovery planning.2.5.3.2 Recovery PathThe recovery path meticulously outlines the outcomes from targeted strategies andactions implemented to restore a system to its operational baseline following anatural disaster. This path’s configuration is typically shaped by several criticalfactors: the nature of the disaster, the extent of the damage, and the availability ofvital resources, which include financial, human, and technological assets, as well asthe community’s preparedness. To effectively represent the recovery trajectory asa time-varying function that captures the dynamic interplay between the disaster’simpact and the system’s recovery capacity, various modeling approaches can beutilized. These may range from simplified recovery functions to more complexmodels that integrate both organizational and socio-political dynamics [61].In resilience literature [8, 10, 11, 61], three simplified recovery functions are44frequently discussed: linear, exponential, and trigonometric recovery functions, asdepicted in Figure 2.10. Each function offers a distinct perspective on the recoverytrajectory, highlighting the critical need to assess the level of societal preparednessin which the system operates. The linear recovery function is typically preferredwhen data on societal response and resource availability are scarce. The exponen-tial recovery model is generally applied in scenarios characterized by rapid initialrecovery, spurred by an immediate influx of resources, which gradually decreasesover time. In contrast, the trigonometric function is ideal for situations where re-covery begins slowly due to organizational or resource limitations but acceleratesas external support intensifies or as planning becomes more effective.Figure 2.10: Recovery functions. Adopted from [10].In the formulation of the linear, exponential, and trigonometric recovery func-tions as specified in Eq. 2.10, the term t0E designates the precise moment of thehazard event’s occurrence, which signals the commencement of system disruption.In contrast, TRE quantifies the duration required for the compromised system tofully recover its operational capabilities. Additionally, the constants a and b serveas adjustable parameters that are specific to each given scenario, determined bycurve fitting to the particular context of the disruption.linear: frec(t) = a(t− t0ETRE)+b;exponential: frec(t) = aexp[−b(t− t0ETRE)];trigonometric: frec(t) =a2[1+ cos(πb(t− t0E)TRE)];(2.10)452.5.4 Functionality AnalysisSystem functionality is expressed as a dimensionless percentile metric, represent-ing the normalized current functionality in comparison to the available functional-ity before a significant seismic event. This metric operates on a scale from zero toone. Prior to any risk occurrence in the functionality curve, the system’s functional-ity is equivalent to a state of complete functioning, denoted as one. In a post-hazardstate, if the system experiences a total loss of function, its functionality drops tozero. Conversely, if no loss of function is observed, the functionality remains atone, mirroring the pre-hazard condition.Various methodologies can be employed to characterize and quantify the func-tionality of a system, depending on the specific issue at hand. The accurate def-inition of functionality for resilience measurement presents a challenging processowing to its crucial importance. Consequently, multiple function indexes have beendeveloped to accommodate diverse systems and research objectives, taking into ac-count specific problem-solving approaches and system complexity. For example,Cimellaro et al.[11] introduced two metrics to assess the performance of health-care systems. In the context of a hospital, the normalized waiting time serves as ameasure of system functionality. The authors evaluate the proportion of the healthypopulation in a hospital network during an extreme event, normalized to the totalhealthy population before that event, as an indicator of functioning. In a separatestudy by Shang et al.[48], the resilience of hospital systems was examined concern-ing the lack of data on functionality-related indicators such as waiting time. Theygauged seismic resilience by measuring economic loss resulting from the damageor failure of components, as these incidents directly impact functionality and canbe quantified through the ensuing economic loss.2.6 Chapter SummaryUntil this juncture, we have diligently traced the trajectory of quantifying a sys-tem’s resilience. Consequently, we have discussed the essential steps common inmost proposed frameworks for assessing the seismic resilience of buildings. Thecomprehensive literature review of each of these steps reveals gaps that have notyet been addressed, despite their direct impact on the accuracy of resilience assess-46ment. By identifying and bridging these gaps, future frameworks can better capturethe complex dynamics of building resilience, ensuring more robust and informeddecision-making processes in seismic risk mitigation.47Chapter 3Probabilistic Framework forAssessing Seismic Resilience ofBuildingsIf you can’t explain it simply, you don’t understand it well enough.— Sir Albert Einstein3.1 IntroductionReporting the seismic resilience of a building offers a comprehensive view of itsperformance after experiencing earthquake events, as it encompasses the building’srecovery process. One crucial aspect in accurately modeling the recovery processis providing a detailed model that can precisely estimate the recovery time of thebuilding based on the required time to repair all damaged components. Simplifiedrecovery time models can result in optimistic or pessimistic values of resilience,which may lead to incorrect decisions and require significant time and cost to rec-tify.Although recent advancements have introduced more sophisticated models toaccurately estimate recovery times following earthquakes [50, 52], simplificationsstill persist. In this study, we have endeavored to augment existing models by ad-48dressing their overlooked aspects, thereby taking a strategic step toward achievinga more realistic seismic resilience index. In line with this goal, we proposed aresilience easement framework to exert more control over the process of provid-ing the required input data for our developed repair time model. The next sectionprovides an overview of our proposed methodology at a macro level, followed bydetailed discussions in subsequent sections.3.2 Probabilistic Framework for Seismic ResilienceAssessmentIn general, every probabilistic framework comprises sequential steps that mustbe meticulously followed in their prescribed order to achieve the desired objec-tive. Typically, data integrated to achieve the desired result can be categorized intotwo types: raw information, requiring no post-processing and usable as collected;and information needing modifications to fit the input format of this combination.The latter category may undergo single-step calibration or multiple post-processingsteps to become usable. Additionally, adjusting specific types of input data ofteninvolves integrating them with other inputs, which may include raw data initiallycalled for this step or outputs from other post-processing steps on the same or dif-ferent types of data.By considering one post-processing step or group of them as a single unit basedon their input and output information, all the necessary components for creating aflowchart to achieve the desired objective can be assembled. To organize these unitseffectively, foundational units that do not depend on outputs from others are placedat the initial level. Subsequently, units whose inputs depend on outputs from thesefoundational steps are sorted into the second level, and this process continues untilreaching the final level where their outputs are combined for the last time to achievethe framework’s objective. This systematic approach facilitates the development ofsophisticated algorithms and the construction of a new framework by identifyingnecessary steps and prioritizing them appropriately.Based on this structured approach to identifying and then organizing sequentialsteps in a probabilistic framework, the development of our resilience frameworkwas systematically guided by the integration of distinct post-processing units into49a cohesive flowchart, as illustrated in Figure 3.1.Figure 3.1: Probabilistic Framework for Assessing Seismic Resilience ofBuildingsThe required inputs and outputs of each unit in our developed flowchart, alongwith their correlations with inputs and outputs of other units, which led to theirarrangement in this structure, have been meticulously discussed in the followingsection.3.3 Earthquake Hazard ModelPerforming a dynamic analysis, especially time history analysis, is essential foraccurately evaluating the seismic performance of a building model. This processcannot proceed without subjecting the model to earthquake loads that realisticallysimulate the scenarios it may experience during its operational lifetime at its spe-cific location. Therefore, a fundamental step toward achieving precise seismic re-silience assessment is selecting a suitable set of ground motion records that accu-50rately reflect the seismic hazard of the region. This approach minimizes uncertaintyin the obtained structural response and ensures more reliable final results.Selecting appropriate earthquake records, whether artificial accelerograms orrecorded data from past earthquakes, must align with the specific geological con-ditions of the site [17, 45]. The variety of models and data sources available tomeet this requirement, that prioritizing and opting one over the others dependsheavily on the specific context of the problem. The chosen model serves as thecore of our “Earthquake Hazard Model,” generating pairs of ground motions as itsoutput.3.4 Building Structural ModelPerforming nonlinear time history analysis necessitates employing nonlinear mod-eling techniques for the building under consideration. Extensive guidelines existto outline the procedures for conducting accurate simulations of buildings in non-linear conditions. Various software tools have been developed to facilitate thisprocess, such as OpenSees [35].Regardless of the chosen nonlinear modeling approach and software, a criti-cal decision revolves around whether to develop a two-dimensional (2D) buildingmodel or a three-dimensional (3D) model. Evaluating the resilience of a build-ing depends significantly on accurately assessing the damaged condition of all itscomponents. Whether the building plan is symmetrical or not, variations in thecharacteristics of applied seismic forces across different horizontal directions re-sult in differing responses of the building components at the same floor level but indifferent directions. This variability leads to varying levels of damage among thesecomponents.To capture these differences in damage conditions across directions, utilizing3D building models enhances the reliability of seismic resilience analysis. Ourframework recommends prioritizing 3D models over 2D models for this reason.However, it’s crucial to acknowledge that analyzing 3D models, particularly for tallbuildings, can incur significant computational costs. In cases where a 2D model ischosen, our suggestion is to assuming that the modeled direction yields maximumstructural responses, and positioning all vulnerable components accordingly in that51direction.3.5 Building Characteristics ModelTo calculate a building’s functionality and account for the impact of non-structuralelements, it is essential to select the appropriate types and quantities of these ele-ments. This selection process should consider the building’s occupancy type, floorplan, and the interior design of each floor. Such meticulous consideration ensuresthat the non-structural elements are perfectly aligned with the specific needs andlayout of the building, thereby avoiding any overestimation or underestimation thatcould compromise the building’s overall functionality assessment.In this study’s framework, the selection of non-structural elements for eachfloor is based on 25 groups and their subcategories, as outlined in FEMA P-58 [19].These selections are informed by the building’s occupancy types and its architec-tural features, both interior and exterior. The Normative Quantity Estimation Toolfrom FEMA P-58 [18] is then employed to evaluate the quantities of these ele-ments, taking into account each floor’s occupancy type and gross square footage(GSF).The resulting normative quantities of non-structural elements are expressed inunits consistent with those used in FEMA P-58 [19] for developing fragility curvesand calculating repair costs and durations. It’s crucial to note that these quantitiesshould be adjusted based on the building’s layout and engineering judgment for3D models, ensuring the total count of each component is appropriately divided bydirectional orientation.The output of this step, known as the “Building Characteristics Model”, pro-vides a detailed, floor-by-floor and directional list of non-structural elements andtheir quantities,as illustrated in Figure 3.2. This serves as an input for both theFragility Model and the Component Consequence Model.3.6 Building Fragility ModelAfter measuring structural responses in terms of Engineering Demand Parameters(EDPs), fragility curves become indispensable tools for quantifying the extent ofdamage to building components. Each element’s fragility curve is defined by a52Figure 3.2: The Input and Output Information of the Building CharacteristicsModel.specific EDP that most accurately corresponds to potential damage triggers for thatelement. These curves translate EDPs into probabilities of various damage levels,reflecting uncertainties in the damage condition of each component at specific floorlevels and directions.In this study, a crucial component is the “Building Fragility Model.” This stepinvolves deriving essential fragility curves for both structural and non-structuralelements, predominantly using FEMA P-58’s fragility functions [18].3.6.1 Structural Fragility CurvesTo evaluate the structural condition of a building following an earthquake, it isessential to derive appropriate fragility curves for vulnerable structural elementswithin the building model. This process begins with identifying a list of vulner-able structural elements. These elements are then matched with component-levelstructural fragility curves provided by FEMA P-58 [18] to select the most suitablefragility curves.If the building’s structural characteristics do not align with the classified fragilitygroups prescribed by FEMA P-58 [18], or if enhanced precision is required forstructural damage assessment, alternative analytical methods should be employed.These methods, discussed in detail in Chapter 2, allow for the establishment offragility curves tailored to the specific characteristics of the building, consideringall stories and directions.As shown in Figure 3.3, the “Building Fragility Model” for structural parts uti-lizes outputs from the ”Building Structural Model.” This integration can result in53component-level structural fragility curves extracted from FEMA P-58, story-leveldirectional fragility curves derived from analytical methods, or a combination ofboth, to provide a comprehensive assessment of the building’s structural vulnera-bility.Figure 3.3: The Input and Output Information of the Building FragilityModel for Structural Components.3.6.2 Non-Structural Fragility CurvesFor the non-structural components selected from FEMA P-58 [18], their corre-sponding fragility curves are also derived from this guideline. As illustrated in Fig-ure 3.4, the ”Building Fragility Model” step for non-structural elements results incomponent-level fragility curves based on the list of vulnerable non-structural com-ponents within the building, as identified by the ”Building Characteristics Model.”Figure 3.4: The Input and Output Information of the Building FragilityModel for Non-Structural Components.543.7 Component Consequence ModelTo quantify earthquake-induced damage in terms of potential losses, this study uti-lizes the repair cost and repair time consequence functions from FEMA P-58 [19].These functions estimate the necessary cost and time to repair each unit of a com-ponent at its various damage states.To incorporate uncertainty into the calculations, FEMA P-58’s consequencefunctions [19] recommend probability distributions for estimating the unit cost andtime values for each damage level. These distributions can be either normal or log-normal, depending on which distribution best fits the data. For all damage statesof a component, the distribution type and its dispersion parameter are provided,but the median value must be inferred from the component’s volume on each floor.This calculated median value reflects economies of scale and efficiencies in con-struction operations associated with different extents of any component at the samelevel of damage. The FEMA P-58’s approach acknowledges the practical realitythat certain tasks are essential for performing repairs at any level of damage, ir-respective of the number of units affected. Consequently, for each component, asthe the extent of its damaged units increases, the need to repeat certain preparatorytasks diminishes. This leads to a reduction in the calculated median cost and timeper unit of the component, as associated expenses and durations for similar typesof work are distributed over a larger quantity.Figure 3.5 presents the typical demonstration of FEMA P-58’s consequencefunctions [19], employed for estimating the unit cost and unit time correspondingto each damage level of a component. The model elucidates that the cost and timeof each unit reach their maximum value when the quantity of damaged units fallsbelow a certain threshold, termed the Minimum quantity in the figure, indicatinga lack of economies of scale. Subsequently, as the number of units surpasses thisthreshold, the model demonstrates a gradual decrease in the cost or time per unitof the component, leading to a phase where economies of scale or operational effi-ciencies are once again unattainable, as denoted by the point of Maximum quantity.Each point along this figure yields the median of the fitted distribution function cor-responding to that level of damage.The “Component Consequence Model” relies on the previously provided list55Figure 3.5: Typical Consequence Function of FEMA P-58 [19].of floor-level quantities of components, encompassing both structural and non-structural elements, as its input data.In this phase, the corresponding FEMA P-58’sconsequence functions [19] are identified for all types of considered componentsthroughout the building. The unit cost and time of any damage state of a com-ponent are calculated by leveraging the provided data from these consequencefunctions, including the quantity threshold, unit value, dispersion, and fitted dis-tribution. Considering the suggested dispersion for generating random unit values,and the potential variation in median values corresponding to the units of a com-ponent at each floor level, different amounts of unit cost and time may result forthe same damage state of a component at different levels of the building. The dataoutput from this step provides the associated repair cost, measured in dollars, andthe repair time, measured in worker-days, for each unit of a component across allits damage states and floors. Figure 3.6 illustrates the inputs and outputs of theComponent Consequence Model.3.8 Building Recovery ModelThe seismic resilience loss factor of a building is significantly influenced by twokey aspects: the time required to restore the building’s functionality and the specific56Figure 3.6: The Input and Output Information of the Component Conse-quence Model.path taken to achieve this restoration.3.8.1 Recovery Time ModelAfter completing the recovery period, a damaged building can reestablish its pre-disruption state. This recovery process consists of two sequential phases. Thepreparatory phase involves establishing critical groundwork without which com-mencing the restoration process is impractical, such as securing resources, obtain-ing permits, and conducting safety assessments. The next phase then proceeds withthe actual repair activities, ensuring that all structural and functional aspects of thebuilding are restored efficiently and effectively. Together, these phases contributeto a comprehensive and structured recovery process, with the total recovery timebeing the aggregate of the time required to complete each phase.In our developed framework, the proposed “Recovery Time Model” consists oftwo sub-models representing the two recovery process phases, and their combinedoutputs determine the overall downtime for a building. The time required to com-plete the first phase is addressed through the “Mobilization Time Model,” whichis then added to the time value obtained for completing the second phase from the“Repair Time Model.” These sub-models are discussed in detail in their respectivesections.3.8.1.1 Mobilization Time ModelThe delay time associated with the preparatory phase carries inherent uncertainty,influenced by factors such as project scale, community preparedness to provide57resources, and the severity of building damage. Relevant literature on resiliencecalculation or repair time estimation addresses these uncertainties through proba-bilistic models. Notable examples include the downtime calculation model intro-duced by REDi™ [2], and the repair time model proposed Terzic et al.[50] andYoo[59].In our study framework, the “Recovery Time Model” focuses primarily on pro-viding a realistic estimate of repair durations for damaged components. Conse-quently, we accept a degree of uncertainty by adopting a fixed value for mandateddelay time, rather than incorporating detailed modeling of the preparatory phase.This simplification will be addressed in future work by developing a probabilisticmodel for the preparatory phase. Our resilience calculation framework includesa specific segment for this enhancement, designated as the “Mobilization TimeModel.” While the duration of the preparatory phase in this research is determinedthrough engineering judgment, it serves as the output of the “Mobilization TimeModel.”3.8.1.2 Repair Time ModelThe total repair time of a building is determined by assessing the time requiredto repair each damaged element and incorporating these times into a repair logicnetwork that defines the sequence and dependencies of repairs. To ensure the es-timated overall repair timeline is realistic, resource restrictions must also be inte-grated into this process. In the framework developed in this study, three models arecombined to create the “Repair Time Model,” each of which is discussed in detailin the following sections.Crew Schedule ModelTo determine the repair time for all activities within the framework of thisstudy, the repair scheduling method proposed by Terzic et al.[50] and Yoo[59]has been utilized. This method outlines the total number of workers assigned toeach repair activity, combining the required number of crews and their sizes, asdetailed in Chapter2. By allocating an appropriate number of workers to each typeof damaged component at each floor level, the raw repair times, initially calculated58in worker-days, can be converted into calendar days.According to Terzic et al.[50] and Yoo[59], the required number of crews foreach type of component at each floor level corresponds to the number of damagedunits of that element. To apply their framework, different ranges should be definedfor each structural and non-structural component based on available data sources orengineering judgment. Each range specifies the number of damaged units that canbe effectively repaired by a specific number of crews, ensuring optimal efficiencywithout the need for additional crews.Furthermore, their repair scheduling method aligns the number of skilled work-ers in each crew with the severity of damage for each component. This processbegins with accurately determining the crew size for each damage state of all con-sidered vulnerable components within the building, based on detailed damage statedescriptions. Subsequently, to assign the most appropriate make-up of a crew fora component at a specific level, Terzic et al.[50] and Yoo[59] defined Eq. 3.1. Thisequation calculates the average damage level for each component at each floor, ac-counting for the possibility of varying damage levels among the damaged units ofa component, particularly those located in different orientations.In this equation, ADS denotes the average damage state of a specific componenton a single floor level. k represents the total number of distinct damage statesidentified for that component, DMi represents the specific damage state of a unit,expressed numerically, and ni indicates the number of units within the damage stateDMi . The total number of damaged elements on the considered floor is denotedby n.ADS =∑ki=1 niDMin(3.1)By applying the extracted fragility curves, the structural responses in terms ofEDPs are translated into probabilities of various damage states for all componentsthroughout the building. Consequently, the damage condition of a component isreported as probabilities of different damage states, which are consistent for all itsunits located in the same direction at a specific floor level. However, to use Eq. 3.1,it is necessary to specify the number of damaged units of a component in eachexplicit damage state, rather than using these calculated probabilities. This involves59employing the method proposed in [17], where the threshold between 1 to 100 isdivided into ranges based on the obtained probabilities of the component’s damagestates at each specific floor and direction. Each range represents the probabilityof not incurring and exceeding a specific damage state. Then for total units ofthe component at the considered level and direction, a random number should begenerated and compared with the limits of these ranges to assign a specific damagestate to each component’s unit.Generating random numbers for the total number of components throughoutthe building increases both the computational cost and the uncertainty of the simu-lations. While all units of a component in the same direction on a single story havean equal probability of experiencing different degrees of damage, some of theseunits can be assigned an intact damage condition based on their corresponding ran-dom numbers, while others are designated as being in a complete damage state.This randomness can contribute to inaccuracies in crew size assessment.To address these challenges, this study proposes an updated version of Eq.3.1,presented as Eq.3.2, for calculating the ADS for the total number of a componentwithin one story. This refined approach streamlines the process of calculating crewsizes by eliminating the need to assign an explicit damage state to each unit, relyinginstead on the calculated damage states probabilities. By utilizing these probabili-ties, the method significantly reduces computational complexity and enhances theaccuracy of damage state assignments, ultimately leading to more reliable simula-tion outcomes.ADS =∑dj=1 n j∑ki=1 piDMin(3.2)In this equation, d represents both the latitudinal and transversal directions ofthe building. n j defines the number of units of a component on the same floorand in the same direction that share similar damage state probabilities, with eachdamage state probability represented by pi.Overall, the “Crew Schedule Model,” a subsection of the “Repair Time Model,”converts calculated raw repair times into calendar days by dynamically allocatingresources to all structural and non-structural components at different floor levelsbased on the number of damaged units and their average damage condition. As60illustrated in Figure 3.6, the input parameters for this step are derived from twopreceding steps. Firstly, to define the appropriate ranges for the optimal number ofcrews required to manage the repair tasks for specific units of a component, it is es-sential to determine the considered elements and their maximum number across allfloors, as obtained from the “Component Consequence Model.” Secondly, definingthe crew sizes for various potential damage conditions necessitates using the cor-responding fragility curve properties, including the number of damage states and adetailed description of the repair work needed to restore the component to its intactcondition, as provided by the “Building Fragility Model.”Repair Schedule ModelIn the developed framework of this study, instead of providing a predefined re-pair schedule with fixed sequences and steps, the strategy proposed by Terzic et al.[50] and Yoo [59] has been utilized. Recognizing that rigid repair paths can lead tounrealistic and impractical scenarios, resulting in over- or under-estimation of theproject’s ultimate repair duration, their recommendation emphasizes a flexible re-pair schedule specialized to the specific case study building model. This approachensures that all relevant factors in defining the repair logic network to calculate thefinal repair duration of a building are considered.To implement this flexible approach effectively, a repair logic network must bedefined to organize the repair process for all structural and non-structural elements.This network comprises multiple repair paths, each containing sequential steps.Within each step, components are assigned the same repair order, meaning theirrepair initiation is mutually exclusive. Consequently, the approach to defining astep involves grouping components such that completing the repair process for allof them is necessary to start repairs for the components of next step in the path,while their repair conditions do not affect one another within the group.Beyond defining an appropriate repair schedule for the case study building, itis imperative to establish two additional critical parameters to accurately determinethe start date for each repair activity.Firstly, after categorizing a component into its respective group, it is crucial toidentify the stories in which each of its predecessor components must be intact. Aspreviously discussed, a component is assigned to a specific group within a repair61path based on the dependency of starting its repair process at each floor level on theintact condition of all components in the previous step on the same floor. However,certain conditions, such as safety requirements, may impose additional restrictions,necessitating the completion of repair activities for some predecessors across allstories.Secondly, defining the number of stories that can be repaired simultaneouslyfor each component is essential. This can range from a floor-by-floor approach,where only one floor is repaired at a time, to repairing all floors concurrently. Acrucial factor in determining this parameter is the condition of the component interms of damage on the previous story or stories. If repairing a component on aspecific floor requires the intact condition of the component on the preceding floor,then a floor-by-floor repair scenario should be selected. Otherwise, depending onsite conditions and available resources, it may be feasible to repair multiple floorssimultaneously, potentially up to all floors at once.Based on the given parameters and defined predecessor-successor relation-ships, the repair start date for each activity at each floor level can be determined.For each component, the start date is set as the latest repair completion date amongall its predecessors. When a component is repaired one floor at a time, this maxi-mum completion date is influenced by both the finishing dates of its predecessorsand the completion date of the same component on the previous floor. By addingthe calculated repair duration, the finish date of the component can also be estab-lished.By determining the completion dates for all repair activities, the latest date istaken as the overall project duration. The sequence of activities from the start tothe finish of the project that dictates the shortest possible project duration is knownas the critical path. Any delay in these activities will directly impact the project’scompletion date. As noted by Ammar (2003), Total Float (TF) and Free Float (FF)can be calculated for each activity based on the identified Critical Path.Total Float (TF) indicates the amount of time an activity can be delayed withoutimpacting the project’s end date; for activities on the critical path, TF is zero. SinceTF is shared among sequential activities on a path, utilizing a portion of TF for oneactivity reduces the remaining TF for its successors. Conversely, Free Float (FF)represents the exact period an activity can be postponed without affecting either62the final project duration or subsequent activities.Within the designed framework of this study, as also suggested by Terzic etal. (2016) and Yoo (2016) in their respective model, the amount of FF, along withthe the start and finish dates, are calculated for each repair activity. Equation 3.3demonstrates the method for calculating the FF for a component denoted by N.FFN = min(Start Time of SuccessorsN)−Finish TimeN (3.3)By utilizing the output from the “Component Consequence Model” that listsall considered structural and nonstructural elements within the building, the keyparameters of the “Repair Schedule Model” can be effectively defined.Resource Restriction ModelProviding a construction timeline that effectively addresses the technical andexternal challenges arising in the aftermath of an earthquake, which impact therepair process of a damaged building, requires meticulous planning and strategicresource allocation. Major earthquakes often result in extensive damage across theaffected region, leading to simultaneous damage to numerous buildings and a sig-nificant shortage of skilled workers. Beyond human resource constraints, shortagesof construction materials, equipment, and machinery can also impede repair efforts.Additionally, safety concerns may limit the number of personnel allowed on-site.Failure to account for these factors can result in underestimating the duration ofeach repair activity, leading to overly optimistic projections for the building’s finalrestoration date.Terzic et al. [50] and Yoo [59] have introduced the “Resource SchedulingMethod” in their proposed repair time model to address these complexities. Thismodel imposes constraints on the number of laborers allowed on-site and on eachfloor level to accommodate various scenarios that may extend the building’s to-tal restoration period. To adhere to these resource limitations, they developed adetailed algorithm, which is thoroughly discussed in Chapter 2.In the developed framework of this study, potential challenges during the project’srepair timeline are also addressed by imposing constraints on worker density perday. The “Crew Schedule Model” considers the severity of damage at each unit of a63component on a specified floor level as the sole factor for assigning the appropriatenumber of workers to determine the duration of its repair activity in days. Thesecalculated durations are then integrated with the outputs of the “Repair ScheduleModel” to establish the building’s final repair date. According to the developedrepair logic network, the only criterion for delaying repair start dates is the con-dition of each component’s predecessor on the predefined floors, which must becompletely intact. Therefore, there is a need for a model that accounts for re-source restrictions in calculating both all repair activity durations and their startdates. Consequently, the “Resource Restriction Model,” the third sub-model of the“Repair Time Model,” has been developed.Figure 3.7 illustrates the algorithm developed for this study, ensuring that dailyworker density limits are met at each time span and floor level within a single re-alization of the Monte Carlo simulation. This new algorithm improves upon themodel proposed by Terzic et al. [50] and Yoo [59] through eliminating severalsimplifications present in their approach, resulting in a more accurate estimationof the building restoration completion date. The following steps outline the imple-mentation of the algorithm, with each step compared to the earlier methodology.This comparative analysis highlights the differences and improvements made inour approach.• Step 1: Determine the initial conditions by establishing the total number ofworkers that can simultaneously operate on each floor and site.To adhere to daily resource limits, this study follows the methodology pro-posed by Terzic et al. [50] and Yoo [59], adopting a resource reduction strat-egy that adjusts the number of crews rather than their sizes. The minimumcrew size necessary for any repair activity is set as the baseline for resourcelimits. Therefore, it is crucial to ensure that the worker limits on each floordo not go below this minimum threshold.• Step 2: Throughout the project’s duration, the daily number of workerspresent on each floor is aggregated, irrespective of the type of repair ac-tivities they are performing.• Step 3: The algorithm progresses by comparing the results obtained in Step 264Start Set the daily resource limits (per floor and building).Aggregate the total number of daily resources during the lifetime of a project.Find the periods of time when the maximum daily resource limits are exceeded.Is the resource limit exceeded?Identify all activities in the first period that are exceeding daily resource limit.Divide the identified activities into two categories: Single-Crew Activities and Multi-Crew Activities.Is the Multi-Crew Activities list empty?Prioritize the Single-Crew Activities list by sorting based on the higher rank of the EIF, latest starting date, shortest repair duration, maximum FF, largest crew size, and minimal parallel repairs between stories.Update allocated resources by reverting the crew count of the first activity in the list to its original number. Move the first activity from the list to the Postponed Activities list.Prioritize the Multi-Crew Activities list by sorting based on the higher rank of the EIF, shortest repair duration, maximum FF, largest number of crews, largest crew size, and minimal parallel repairs between stories.Update allocated resources by decrementing the workforce of the first activity in the list by one crew.Adjust repair times and FFs according to the updated resource allocations.Calculate the total daily resources.Is the resource limit still exceeded?Calculate resource capacity as the difference between total daily resources and the daily limit.Is resource capacity available?Generate the Crew Expansion list by selecting among modified crew activities that are not postponed, with a crew size meeting or exceeding the remaining resource capacity. Prioritize by higher EIF rank, longest repair duration, smallest crew size, and maximum parallel repairs between stories.NOYESYESYESFinishNoNOIs the Crew Expansion list empty?Calculate the earliest start date for postponed activities as the minimum of the finish dates of remaining activities, based on their original start dates and updated repair durations.YESNOYESUpdate allocated resources by incrementing the workforce of the first activity on the list by one crew.Adjust repair times according to the updated resource allocations.NOStretch the repair durations based on the reduced number of workers and estimate the final repair time utilizing CPM.Calculate the total daily resources.Figure 3.7: Flowchart of the resource restriction for a single realization of theMonte Carlo simulation.with the constraints established in Step 1. The algorithm will be terminated,if throughout the entire repair period, there is no instance where the numberof workers exceeds the specified limits. Nonetheless, it proceeds to the nextstep.• Step 4: In this step, the data from step 3 are analyzed to identify specifictime intervals at each floor level when the number of workers exceeds thespecified limits. For intervals lasting more than one day, the start and finishdates do not require justification if the type of repair activities remains con-stant throughout. If an activity ends or a new one begins during a period,the initial period must be subdivided into smaller intervals. These smallerintervals should encompass the entire duration of the original period, withtheir start and finish dates aligned with the times when the type of activitieschanges.• Step 5: To effectively address intervals where resource reduction is neces-sary to meet imposed conditions, each interval should be handled individ-ually. This means that resource adjustments for one interval must be com-65pleted before starting adjustments on another. Each interval contains at leastone repair activity, so the iterative process of reducing crew numbers willimpact at least one element per interval, thereby increasing its repair du-ration. Depending on the number of repair activities within each interval,this process can extend the repair duration of multiple activities within thatinterval.Changes in the repair durations of activities within one interval can affect notonly the overall project completion time but also the start and finish dates ofsubsequent activities. These changes necessitate re-evaluating all subsequentintervals, which may result in no impact, a complete overhaul, or somethingin between, potentially including new intervals or excluding previous ones.To determine which interval should undergo resource adjustment amongthose identified in step 4, the intervals should be sorted to identify the highestpriority one. Prioritization can be based on factors such as the type and num-ber of elements to be fixed first for safety reasons, the defined repair logicnetwork, and the number of stories. If no criteria exist for ranking intervals,the highest priority should be assigned to the first interval of the first levelfrom the bottom that exceeds the defined worker limits for that story. Sub-sequently, all repair activities concurrently performed in the high-priorityinterval should be determined.• Step 6: After identifying the repair activities within an interval, their numberof crews serves as a criterion to categorize all activities into two groups:those being repaired by a single crew and those assigned to two or morecrews.The identified repair activities within an interval, where the total number ofassigned workers exceeds the predefined resource limits, may involve one ormore crews per activity, with each crew size potentially differing across ac-tivities. Separating repair activities based on the allocated number of crews atthis step causes the subsequent steps of the developed algorithm in this studyto significantly differ from the approaches proposed by Terzic et al. [50]and Yoo [59]. Their methodology assumes that during each exceeding pe-riod, crew compositions remain unchanged, meaning the number of workers66within one crew cannot be reduced. Consequently, resource reduction onlyapplies to repair activities involving more than one crew. If the defined lim-its on the number of workers are not met for that period, the crew reductioncontinues until all activities within the interval are performed by a singlecrew.However, following this approach may still result in periods where the to-tal number of workers for all repair activities within an interval exceeds thepredefined limits, especially when crew sizes are large or the number of ac-tivities is considerable. Therefore, it is necessary to modify this simplifiedassumption to ensure that no day in the project timeline is overcrowded com-pared to the allowable number of workers.• Step 7: If at least one repair activity involves more than one crew, the groupof Multi-Crew Activities from step 6 is not empty, allowing the resource re-duction algorithm to proceed with decreasing the number of crews accordingto steps 8 and 9. Conversely, if no activities involve more than one crew, allidentified activities are listed in the category of Single-Crew Activities fromstep 6, and the process should continue with steps 10 and 11.• Step 8: The algorithm proceeds to this phase only if the list of Multi-CrewActivities contains one or more repair activities, each involving two or morecrews. When the list comprises multiple entries, specific criteria must bedefined to prioritize them. This prioritization will determine which repairprocess will continue with one fewer crew, the assigned crews for the otheractivities remain unchanged.Although Terzic et al. [50] and Yoo [59] only took into account the longestFF and the least repair time duration as the variables to rank multi crew activ-ities, our developed algorithm incorporates six different factors to determinethe top-ranked activity. The top-ranked activity is the one selected for crewreduction by one.The first considered criterion is the Element Importance Factor (EIF), whichreflects the significance of an undamaged state for a particular kind of com-ponent relative to others. Components with higher EIF rankings are those67that do not need urgent repair, while components with lower EIF rankingsmust be repaired as soon as possible. Thus, as much as possible, crew reduc-tion should not apply to the latter.To determine the importance factor for each component in prioritizing itsrepair activity, engineering judgment should be applied based on the spe-cific parameters of the case study building model. Various considerationsregarding the role of each component within the building can be beneficialin assigning this importance factor. Aspects such as the component’s con-tribution to overall building stability, occupancy safety, and its influence onthe functionality of essential systems, or a combination of these aspects, canbe used in this evaluation. If the importance factors have the same values forall components within the interval, it indicates no difference in prioritizingthe completion of their repair activities, and crew reduction can be appliedto any of them. In such cases, the second criterion should be used to guidethe prioritization process.Based on the second considered ranking parameter, the repair activities aresorted from shortest to longest repair activity duration. If the shortest repairduration cannot distinguish one activity from others within the specified pe-riod, the activities with the shortest repair time are compared in terms of freefloat. The activities with the longest free float receive the first rankings underthis third criterion.The subsequent criteria entails ranking activities according to the numberof crews allocated to them, from largest to smallest, to prioritize those withmore crews. Eliminating one crew from an activity with three or more crewshas a lesser impact on increasing its repair duration compared to removingone crew from a two-crew activity, which would double its initial repair time.This approach aims to mitigate significant extensions of the repair period byprioritizing activities assigned more crews.The size of the crews is considered as the fifth factor for sorting multi crewsrepair activities, prioritizing from the largest to the smallest. For componentswith identical EIF, repair duration, FF, and number of crews, reducing onecrew results in a similar increase in repair duration. Therefore, preference68is given to the component with the largest crew size for resource reduction.This approach accelerates reaching the defined resource limit by removingmore workers from a single crew, thereby reducing the need to repeat thisiterative process and minimizing the number of repair activities experiencingresource reduction and duration increase.Finally, if all previous factors fail to distinguish one activity from another forcrew reduction, the last criterion considered is the number of stories under-going concurrent repair. Activities are ranked from those repairing floor-by-floor to those repairing all floors simultaneously. This criterion focuses onthe scope of the repair work, aiming to maximize efficiency and minimizedisruption for large-scale repair activities.By applying these criteria, the algorithm ensures that resource justificationhas been addreseed to meet the defined resource limits while minimizing theimpact on the overall repair schedule.• Step 9: By applying all six criteria mentioned in Step 8, all componentsin the list of Multi-Crew Activities are sorted. In this step, the assignedresources for the top-ranked activity are then updated by decrementing itscrew size by one.• Step 10: When the Multi-Crew Activities list is empty and the total numberof workers from all single-crew activities over a given period still exceedsresource constraints, our algorithm proceeds to this step. The provided solu-tion here is to delay the start of some activities within the list until the sum-mation of workers on the remaining single-crew activities meets the definedresource limits. This strategy has been selected since it makes it possible tomaintain the key premise of Terzic et al. [50] and Yoo [59], that resourcereduction only pertains to crew numbers, not their sizes.To determine which component from the list of single-crew activities shouldbe postponed, all activities are prioritized using six criteria. The first crite-rion is the EIF, which is addressed comprehensively in Step 8, and compo-nents with a greater EIF are favored for postponement over others. If rankingbased on EIF does not yield a unique component, the repair operations are69then compared based on their start dates, with those that began later receiv-ing higher priority. The subsequent parameters for ranking repair operationsinclude the shortest repair period, the longest FF, the largest crew size, andthe fewest concurrent repairs across stories.• Step 11: Step 10 identifies the specific activity that should be delayed andcannot be repaired concurrently with others during the under-process period.This activity is added to the list of “Postponed Activities,” and if its numberof crews had been modified in the resource reduction loop and it was notinitially a single-crew activity, its allotted crews are restored to the originalnumber.• Step 12: The algorithm reaches this step regardless of whether the resourceadjustment loop involves Steps 8 and 9 or Steps 10 and 11. Each set ofsteps results in resource updates for the top-ranked activity chosen throughtheir prioritization procedure. This resource change may involve reducingthe crew size by one if the activity is selected from the Multi-Crew Activi-ties list (outcome of Steps 8 and 9), or restoring the original crew size if theactivity is selected from the Single-Crew Activities list and must be post-poned (outcome of Steps 10 and 11). Consequently, it is necessary to adjustthe repair duration based on the revised number of crews for this prioritizedactivity, irrespective of the resource update process.Although the Free Float (FF) parameter is considered a factor for prioritizingrepair tasks in both the Single- and Multi-Crew Activities lists, accuratelyupdating it during the adjustment process before ensuring the set resourcelimit has been met is impractical. According to the established methodology,the impact of completing the resource adjustment for an interval with anexcess number of workers extends beyond that interval. The estimated startand completion dates for all components throughout the building, initiallyused in modifying the considered interval, will no longer be valid and mustbe recomputed using the revised repair durations. These updated timelinesare then utilized to identify the next time span requiring resource reductionand to compute the new FF for each repair operation based on the new startdates of its successors and its updated end date, using Equation 3.3.70In this study, to minimize errors caused by importing incorrect FF values af-ter applying the resource reduction loop once for each interval, we updatedthe FF only for components with nonzero FF, assuming no change in thestart dates of their subsequent activities. Based on the suggested formulationfor FF 3.3, if the start dates of successive activities for one component re-main constant but the completion date of the activity increases, the FF willdecrease by the same amount as the increase in the completion date.Thus, ifthe allocated crews for an activity decrease by one, its repair duration willincrease, and the FF for this activity in the next execution of the adjustmentloop will be reduced in proportion to its increased duration. The procedureof updating the FF is not applied to delayed activities, as they are not stillinvolved in this period if further postponement is required.• Step 13: By reducing the number of crews for one repair activity withinthe considered interval or postponing one of its initially identified repair ac-tivities, the number of workers now operating concurrently during that timespan has been reduced. Consequently, the total number of workers for theinterval should be recalculated to determine if further adjustments to the startdates of remaining repair activities or their crew numbers are still required.• Step 14: If the total number of workers for all repair activities within thespecified interval is equal to or less than the considered resource limit, thealgorithm proceeds to the next step. Otherwise, it returns to Step 6 andrepeats the loop until the condition on the allowable number of workers ismet.• Step 15: When the resource adjustment loop is repeatedly applied to a multi-repair activity interval to meet the specified resource limit, various repairactivities within the interval may experience crew reductions or postpone-ments. This iterative approach can sometimes lead to an excessive reductionin resources. This over-reduction occurs because, in the final implementationof the loop, the crew size of the selected repair activity—based on definedprioritization criteria—may be larger than the necessary number of workersthat has to be reduced.71Although in the proposed approach by Terzic et al. [50] and Yoo [59], thereduced resources after the last iteration of the resource reduction loop areconsidered final for each repair activity, our developed algorithm goes furtherby calculating a critical metric termed “Resource Capacity.” This parametercaptures the difference between the total number of workers allocated to allremaining activities within the interval and the allowable resource limit. Thevalue of the Resource Capacity parameter determines whether the currentallocation of workers to all repair activities is optimal, ensuring alignmentwith the permissible workforce for that period.• Step 16: If the obtained value of the Resource Capacity parameter fromthe previous step is greater than zero, it indicates an opportunity to increasethe number of workers to achieve optimal utilization within the specifiedconstraints. Consequently, the algorithm advances to the next step to makethe necessary adjustments, enhancing overall efficiency and effectiveness bymanaging the allocated resources for all repair activities. Conversely, if theResource Capacity parameter is zero, it means there is no available capacity,indicating that the maximum allowable workforce has been utilized withinthe interval without any over-reduction resulting from the resource adjust-ment loop. In this case, the algorithm directly proceeds to Step 22.• Step 17: Advancing to this phase, our approach addresses the limitationsof the initial resource reduction loop by ensuring the allocated number ofworkers within an interval fully utilizes the allowable resource capacity perday. This optimal use of workers is achieved through strategic reallocationof resources for repair activities within the specified exceeding period whileadhering to established resource limits. The primary assumption of our pro-posed increment process aligns with that of the resource reduction approach,where changes apply only to the number of crews, not their composition.Thus, while crew sizes for the remaining repair activities within the intervalremain constant, the available worker capacity may be utilized to augmentthe number of crews.To select among non-postponed activities for crew increases, we employ fourranking criteria. However, this ranking process is applied only to repair ac-72tivities eligible for inclusion in the “Crew Expansion” list by meeting twospecific conditions. First, only repair activities that have experienced crewreductions in previous steps are qualified for the list, ensuring that the num-ber of damaged units remains the primary factor in determining the numberof crews needed for each activity. Second, the crew size for these activi-ties must be equal to or less than the available resource capacity to ensurethat adding to their crew numbers does not exceed the defined daily resourcelimit. Once these repair activities are added to the “Crew Expansion” list,they are being sorted using the following criteria.Similar to the first criterion employed for identifying the top-ranked repairactivity in both the crew reduction and postponement approaches, the initialprioritizing factor here is the Element Importance Factor (EIF). Componentswith a lower EIF, as previously mentioned, are those whose repair processesshould be completed as soon as possible. Therefore, within an exceeding in-terval, components with higher EIFs are given priority when reducing crewsor postponing their repair activities. Following this logic, components withlower EIFs are prioritized for an increase in their crew numbers by one. Ifthe EIF cannot distinguish the first repair activity on the “Crew Expansion”list, repair activities are ranked based on their repair durations, from longestto shortest.The third criterion compares repair activities with equivalent EIF and repairduration in terms of their crew sizes. Activities with smaller crews are pri-oritized to avoid utilizing all available capacity for a single repair activity,thereby enhancing the possibility of repeating the resource increment loop.Finally, components undergoing simultaneous repairs across all floors areprioritized for resource expansion over those whose repair process is con-ducted one floor at a time.• Step 18: Although the obtained “Resource Capacity” measurement may be anatural number, this available capacity can be unusable due to all the definedconditions in Step 17. This scenario occurs when the “Crew Expansion” listis empty, making an increase in the number of workers within the intervalimpossible. As a result, the algorithm directly proceeds to Step 22.73Conversely, if the “Crew Expansion” list includes only one repair activity,it indicates that the allocated resources are not in optimal condition. In thiscase, the number of workers is increased in proportion to the crew size ofthe highest-priority repair activity within the list, allowing the algorithm toadvance to the next step.• Step 19: In this step, the resources for the top-ranked component from the“Crew Expansion” list are updated by allocating an additional crew to itsexisting crews.• Step 20: Increasing the number of resources for a repair activity reduces therequired duration to complete its repair process. Consequently, it is neces-sary to adjust the repair duration based on the revised number of allocatedworkers for the activity that underwent crew augmentation in the previousstep.• Step 21: After one execution of the resource increment loop, the numberof workers within the interval increases. To determine whether the initialavailable capacity has been fully utilized or if there is still room to allocatemore workers, the total number of workers assigned to all repair activitiesduring this period must be aggregated. The algorithm then returns to Step 15to repeat the loop if it is still possible to allocate additional workers.• Step 22: The algorithm has reached this step due to either the “ResourceCapacity” being determined to have a value of zero or the “Crew Expansion”list containing no elements. In both scenarios, the resources assigned to eachrepair activity for the specified period are finalized and will not undergofurther changes.Given the stability in the assigned resources for all remaining activities withinthe interval, their completion dates are estimated in this step by adding theirupdated repair durations to their original start dates. The earliest completiontime from these component is then recorded and used as a new condition forthe repair start dates of delayed activities, establishing their earliest possiblestart date.74• Step 23: After modifying an exceeding interval using the developed strategy,it ensures that the established resource limit is satisfied while eliminatingover-reduction through optimal use of available capacity. This adjustmentmay result in the extension of certain repair periods, delayed start dates forother activities, or a combination of both, that their impacts extending be-yond this period. To evaluate the consequences of these modifications on theoverall project duration, the start and completion dates, as well as the FFsfor all repair activities, should be recalculated.Using these updated dates, the algorithm returns to Step 2 to aggregate thenewly assigned daily resources across all floors, determining whether furtherperiod are required to address for meeting the set resource limit by repeatingthe loop. Finally, when all exceeding intervals throughout the building havebeen corrected, the total duration of the repair operation can be calculated.The proposed “Resource Restriction Model” in this study is designed to ac-count for resource congestion while ensuring that addressing these congestion pe-riods aligns with minimizing delays and optimizing resource allocation. This ap-proach ultimately streamlines the achievement of a realistic overall repair duration.3.8.2 Recovery Path ModelThe recovery path outlines the trajectory taken toward full restoration of a build-ing’s functionality. A building achieves its intended functionality when all its mainsystems operate perfectly. It is important to note that the contributions of these sys-tems and their components to the building’s overall functionality are not uniform.Consequently, the rate of progress in restoring the building’s full performance dur-ing repair time is not constant. This results in a recovery path resembling a stepfunction, where each step corresponds to the restoration of partial or full function-ality in one of the main systems. Achieving this precise recovery path necessitatesa comprehensive approach to calculating building functionality. The model devel-oped by Terzic et al. [51] exemplifies such a framework.In cases where the system’s functionality is defined by a simplified function,tracing the exact functionality at each step of the recovery time is not beneficial.Instead, a mathematical function should be selected to represent the recovery path.75In the framework of this study, fitting an appropriate function capable of predict-ing the recovery path is designated as the “Recovery Path Model.” As discussed indetail in Chapter 2, different critical factors affect this selection. This frameworksuggests prioritizing the linear recovery function over exponential and trigonomet-ric functions, as it eliminates the need to gather information about community pre-paredness at the building location. This also reduces the effect of external factorson the calculated resilience factor, focusing more on the exact building damagecondition.3.9 Building Functionality ModelSelecting the appropriate level of detail and sophistication in defining a functional-ity function depends on the specific project or study requirements. While advancedcomputational techniques for defining functionality [51] typically yield precise out-comes, simplified forms of functionality functions [11, 48] stand out for their prac-tical implementation.In situations where quick and reliable outcomes with minimal parameter def-initions are prioritized over high accuracy, our study proposes defining the func-tionality function in terms of monetary loss. This approach simplifies assessmentby directly linking the building’s operational state to financial metrics, which arenot only quantifiable outcomes of consequence functions for each damaged com-ponent but also easy to understand. Equation 3.4 encapsulates the relationshipbetween the building’s performance status and its monetary value.Functionality(Q(t)) =Replacement cost ($)−Repair cost ($)Replacement cost ($)(3.4)According to this functionality function, the building operates at its full func-tionality when all components within it are intact, signifying there are no repaircosts. Conversely, any incurred repair costs indicate damage and result in a reduc-tion in functionality. When the repair cost equals the replacement cost, indicatingirreparability, the building’s functionality is reduced to zero. FEMA P-58 [19]highlights that building owners often prefer replacement rather than repair whenrepair costs exceed 40% of the replacement cost. Therefore, in this study, if re-pair costs surpass 40% of the building’s replacement cost, it is considered to have76experienced a total loss of functionality, regardless of the formula’s output.3.10 Response AnalysisWithin the framework outlined by this study, the initial phase of evaluating a build-ing’s seismic performance is termed “Response Analysis.” This critical step con-centrates on quantifying potential damage to both structural and non-structural el-ements, forming the cornerstone for assessing the building’s seismic risk and re-silience.The outputs from preceding steps, including the corresponding fragility curvesof the case study building alongside their associated demand parameters, serveas indispensable inputs throughout this analytical phase. The step of “FragilityModel” yields a comprehensive array of fragility curves for all components con-sidered within the building. Meanwhile, the demand parameters are sourced fromother stages within the developed framework. Depending on whether the consid-ered fragility curves express the probability of each damage level as a functionof an EDP or a seismic intensity measure (IM), demand parameters may be ex-tracted from the response history analysis conducted within the “Building Struc-tural Model,” or the “Earthquake Hazard Model” respectively.For each structural and non-structural element at every floor level, the asso-ciated demand parameter is identified based on its corresponding fragility curve.This process isn’t solely reliant on the type of the demand parameter specified indeveloping the fragility curve; it also involves considering directional effects andensuring that the floor level aligns with the definition of the demand parameterin the fragility curve. The outcome of this analytical step entails determining theprobability of all damage states for each component across all floors and direc-tions within the building. The inputs and outputs of the Response Analysis step aredepicted in Figure 3.8.3.11 Seismic Risk AnalysisIn our framework for evaluating the seismic performance of buildings, “SeismicRisk Analysis” constitutes the second fundamental analytical process. This stepinvolves combining the calculated probabilities of all damage states with their cor-77Figure 3.8: The Input and Output Information of the Response Analysis.responding consequences. The name reflects its consistency with risk analysis asdescribed by Murphy et al. [38] as “the process of quantifying the probabilities ofpotential consequences in various hazardous scenarios.”Firstly, utilizing the outputs from the steps of the “Component ConsequenceModel” and “Response Analysis,” the unit-level repair costs and times for all com-ponents within the building, across all floors and directions, are calculated usingthe Equation 3.5, as detailed in Chapter 2.L =N∑i=1Li ·PDSi|IM (3.5)The second output of this step involves calculating repair costs and times forall units of a component located in the same direction on a specific floor, giventheir identical damage condition. This is achieved by integrating the number ofdamaged units obtained from the “Building Characteristics Model” by the previ-ously calculated unit-level repair costs and times. Finally, by aggregating thesedirectional repair costs and times, the main output of this step—the total requiredcost and time for completing the repair activity of each component—is calculated.It should be noted that the calculated repair times are reported in units ofworker-days. Therefore, these raw repair time outputs need to be processed us-ing the methods described in subsequent steps to convert them into standard days.783.12 Seismic Resilience QuantificationThis section outlines the methodological approach to assessing structural resilience.Each preceding part is meticulously developed to produce the necessary models ordata, without which resilience quantification would not be practical or accurate. Toproperly integrate their outcomes, two separate analytical sections have been dedi-cated. The “Time Analysis” section incorporates relevant models and informationto assess time-dependent aspects. The “Functionality Analysis” section processesdata to address system functionality. These subsections correspond to the X and Yaxes of the resilience curve, respectively.3.12.1 Time AnalysisThe calculated raw repair time from “Seismic Risk Analysis,” expressed in worker-days for each repair activity, needs to be converted into calendar days. This con-version requires determining the appropriate number of workers using the devel-oped “Crew Schedule Model.” According to the relevant section of the “CrewSchedule Model,” the total number of workers assigned to each activity resultsfrom the interaction of the specified number of crews and their respective sizes.To determine these factors for a specific component, both the number of dam-aged units and the probabilities of all damage states in each direction of a floorlevel—outputs from the “Building Characteristics Model” and “Response Analy-sis,” respectively—must be extracted.Subsequently, the repair start and finish dates, along with the Free Float (FF)values for each repair activity, are determined by integrating these initial estimatesof repair durations into the repair logic network developed in the “Repair ScheduleModel.” To refine these timelines accounting for resource congestion effects, the“Resource Restriction Model” is applied. This results in finalized repair durations,start dates, and finish dates for all repair activities.Aggregating the ultimate project repair duration with the output of the “Mo-bilization Time Model” leads to the calculation of the total recovery time for thebuilding. This final output is the most important result of the comprehensive ana-lytical “Time Analysis” section.793.12.2 Functionality AnalysisObtaining the total required repair cost is indispensable when calculating buildingfunctionality using the proposed function in the “Building Functionality Model”.This critical variable is derived by aggregating the computed repair costs for alldamaged components across the building, as determined in the “Seismic RiskAnalysis”. If an alternative functionality function is employed, its required inputparameters should be sourced from preceding analytical steps or relevant modelsto ensure a comprehensive and accurate assessment.Regardless of the employed functionality function, the operational capacity ofa system immediately following an earthquake is a definite output of the analyticalstep of “Functionality Analysis”.3.12.3 Resilience calculationTo calculate the seismic resilience of a structure, the following steps should beundertaken to plot its corresponding resilience curve, which illustrates the system’sfunctionality from the moment the external hazard occurs through the completionof the recovery process:1. Initial Drop in Functionality: Plot the initial reduction in functionality att = 0 using data derived from the “Functionality Analysis.”2. Mobilization Time: Maintain the system’s functionality constant at the levelof the initial drop throughout the calculated mobilization period. This willresult in extending the initial drop horizontally during the mobilization pe-riod.3. Repair Time: Starting from the end of the mobilization period, plot therecovery trajectory until the system achieves full functionality. If a simpli-fied model defines the functionality function, ensure an appropriate recoveryfunction is specified to plot the path of recovery during repair time. Alterna-tively, if using a comprehensive functionality model, calculate the system’sfunctionality at each stage of the repair process based on this detailed func-tion.80Based on the resilience curve derived from these steps, as illustrated in theschematic shown in Figure 3.9, the resilience index of the system can be com-puted using the equation proposed by Cimellaro et al.[10], as discussed in detail inChapter 2.Figure 3.9: Schematic representation of the resilience curve.81Chapter 4Application of DevelopedResilience Framework on a20-Story Timber Building4.1 IntroductionIn response to the imperative of sustainable urban development, timber has emergedas a compelling alternative to conventional building materials such as concrete andsteel [25, 37]. This shift is underpinned by significant advancements in EngineeredWood Products (EWPs), which have significantly enhanced timber’s structural in-tegrity, sustainability profile, and seismic performance capabilities [27].In recent years, the construction of timber structures has significantly increased,even in seismic zones. However, their performance in the aftermath of severe earth-quakes, particularly their potential for rapid recovery and minimal loss of function-ality, remains less studied compared to buildings constructed from conventionalmaterials. Noteworthy research investigating timber’s seismic performance beyondimmediate stability following earthquakes is exemplified in the work by Guan etal. [23]. Their study evaluated a 6-story self-centering and welded-moment re-sisting frame in terms of seismic-incurred monetary losses using the methodologyproposed in FEMA P-58 [19]. Similarly, Wilson et al. [57] applied the FEMA P-5882approach to assess economic losses for prototype post-tensioned CLT buildings ofvarying heights (5- and 12-stories). Pei et al. [40–42] have extended their focusbeyond loss assessment, emphasizing the critical aspect of seismic design philos-ophy to develop resilient tall-wood buildings. Furley et al. [20] investigated therequired recovery time for a two-story mass timber building using a combinationof approaches from REDi™ [2] and FEMA P-58 [19]. Furthermore, You et al. [60]have explored deeper by assessing the resilience of tall timber buildings, estimatingseismic losses and resilience of a 20-story CLT-CW system using the FEMA P-58procedure.In this study, we applied the framework developed in Chapter 3 to a buildingmodel similar to that utilized by You et al. [60], originally designed by Tesfamariamas documented in [53, 54]. This chapter mirrors the structure of the precedingone, beginning with determining of the required input variables based on the se-lected case study building model. Subsequently, we present the results obtainedfrom applying the developed framework, including the resilience assessment of thebuilding.4.2 Building Structural ModelThe case study building model used in this research is the result of a recentlycompleted project funded by British Columbia Forestry Innovation InvestmentLtd. Comprehensive building information, design details, modeling, and selectedground motion records for this building are documented in the works of Tesfamariamet al. [53], and Teweldebrhan et al. [54, 55]. Therefore, we provide only a briefoverview of this building to avoid duplicating information readily available in thereferenced documents.The structural system (CLT-CW) for this building consists of two distinct load-carrying systems. Glulam beams and columns are designed exclusively for gravityloading purposes, while CLT balloon shear-walls, steel coupling beams (SCBs)with replaceable shear-links, and buckling-restrained brace (BRB) hold-downs aredesigned to withstand seismic forces. The 3D diagram of this structural system isillustrated in Figure 4.1.To determine the design loads for both structural systems of the building, it83Figure 4.1: Three-dimensional view of the prototype model alongside itsstructural components. Adopted from [60].was assumed that the building is located in Vancouver. The associated snow andseismic loads were determined based on the provisions of the National BuildingCode of Canada (NBC 2020), taking into account a class C soil condition.Analyses were conducted using the ETABS program, and all timber elementswere designed in accordance with the CSA 086-19 standard [13]. Comprehensivedetails of the design procedure, including a numerical example, can be found inTesfamariam et al. [53], Teweldebrhan and Tesfamariam [54] and Teweldebrhan etal. [55].4.3 Earthquake Hazard ModelTo ensure that the suite of selected ground motion records for seismic assessmentof a building accurately reflects the seismic hazard characteristics at its location, astandard approach involves comparing the response spectra of these records witha target response spectrum at the site of interest, determining their alignment. Toachieve this, Tesfamariam et al. ([53])developed an in-house probabilistic seismichazard analysis tool based on Monte Carlo simulations [22] by implementing all84major components of the national seismic hazard model [24]. Subsequently, con-sidering the first fundamental period of the building model and seismic sources(crustal, interface, and inslab) relevant to Vancouver with a return period (TR) of2475 years, three sets of 30 bi-directional ground motion records were selectedusing the multiple-conditional mean spectrum (CMS; Baker 2011)- based recordselection method [21]. For further details on the building and the methodologyused in this section, readers are referred to the main sources [53–55]).In our study, we benefited from access to the results of Incremental DynamicAnalysis (IDA) conducted on this structural building model. These provided resultscomprise a valuable dataset of building responses across various scaled intensitiesof selected ground motion records. This access eliminated the need to repeat theprocess of structural modeling and ground motion selection, enabling us to ded-icate saved time towards focusing extensively on developing a detailed resilienceassessment framework. The accuracy of our repair time assessment represents asignificant advancement compared to the work conducted by You et al. [60] instudying the resilience of this building.4.4 Building Characteristics ModelTable 4.1 presents the list of non-structural elements considered in this study, cate-gorized by their major groups as shown in the first column. Detailed descriptions ofall selected non-structural elements are provided by FEMA P-58 [18], so only thecorresponding FEMA P-58 IDs for these elements are listed in the second column.The quantities for each element at each floor level are provided as total valuesand should be appropriately divided between the building’s orientations based onthe floor plan and engineering judgment. In this case study, only for elementswhose damage depends on the seismic load direction, the total quantity has beendivided between the longitudinal (L) and transverse (T) directions. Due to thesymmetrical building plan on all floors, it is assumed that the quantities are equallydistributed between these two directions. For elements whose damage conditionsdepend on the maximum EDP on its level, regardless of direction, the quantity isreported under the non-directional (ND) column. All reported quantities are in themeasurement units used for each component to develop its corresponding fragility85Table 4.1: Considered non-structural elements and their directional quantitiesNonstructural Component FEMA P-58 Units Location Quantity per DirectionFragility Specifications Classification Number (Floor) L T NDExterior Window Systems B2022.002 30 SF All floors 29.305 29.305Partitions C1011.011a 100 LF All floors 7.033 7.033Stairs C2011.001b 1 EA All floors 1 1Wall FinishesC3011.001a 100 LF All floors 7.571 7.571C3011.001b 100 LF All floors 1.893 1.893Ceilings and Ceiling LightingC3032.003a 250 SF All floors 42.199C3032.003b 600 SF All floors 0.977C3032.003c 1800 SF All floors 0.195C3032.003d 2500 SF All floors 0.094C3034.002 1 EA All floors 48Elevators and Lifts D1014.011 1 EA First floor 8Domestic Water DistributionD2021.013a 1000 LF All floors 1.243D2021.013b 1000 LF All floors 0.373D2021.023a 1000 LF All floors 0.176D2021.023b 1000 LF All floors 0.053D2022.013a 1000 LF All floors 2.485D2022.013b 1000 LF All floors 0.746D2022.023a 1000 LF All floors 0.352D2022.023b 1000 LF All floors 0.105Sanitary Waste Piping SystemD2031.023a 1000 LF All floors 1.418D2031.023b 1000 LF All floors 0.426Chilled Water PipingD2051.013a 1000 LF All floors 0.029D2051.013b 1000 LF All floors 0.009D2051.023a 1000 LF All floors 0.029D2051.023b 1000 LF All floors 0.009Chillers, Cooling Towers, and CompressorsD3031.013e 250 TN Roof 1D3031.023e 250 TN Roof 1Distribution SystemsD3041.011c 1000 LF All floors 0.586D3041.012c 1000 LF All floors 0.088D3041.031b 10 EA All floors 10D3041.041b 10 EA All floors 1Packaged Air Handling Units D3052.013h 20000 CFM Roof 4Fire ProtectionD4011.024a 1000 LF All floors 2.579D4011.053a 100 EA All floors 1.2D4011.074a 100 EA All floors 0.22Electrical Service and DistributionD5012.013c 1 EA First floor 2D5012.023b 225 AP All floors 0.042D5012.033b 1 EA First floor 4D5011.013b 1 EA First floor 12All floors = floors 1-20; EA = each; LF = linear foot; SF = square feet; CFM = cubic feet per minute;HVAC = heating, ventilation, and air conditioning; AP = ampere; L = longitudinal; T = transversal;ND = non-directional.curve, which are listed in the third column.864.5 Building Fragility ModelBased on the developed framework in Chapter 3, for analyzing the damaged con-dition of all vulnerable structural and non-structural elements within the case studybuilding model, their corresponding fragility curves have been identified.4.5.1 Structural Fragility CurvesIn the resilience study conducted by You et al. [60] on a building model similar tothe one used in this research, the vulnerable seismic structural elements have beencategorized into three groups.BRB hold-downs, situated exclusively on the first floor of the case study build-ing model, are identified as one of the building’s vulnerable structural elements.Their corresponding fragility curve has been extracted from FEMA P-58 [18]. Thedemand parameter used to define the rate of damage for this component, based onits fragility curve, is the maximum directional axial strain. While FEMA P-58 con-siders these elements as diagonal braces, they are located vertically in the buildingmodel under investigation. To ensure the fragility curve accurately reflects thisconfiguration, You et al. [60] adjusted the median of the damage level by applyinga scale factor of 0.5.In this building model, CLT shear walls and replaceable steel coupling beams,located across all floors, are also recognized as vulnerable structural elements. AsFEMA P-58 [18] does not provide fragility curves for these components, Youet al.[60] extracted the fragility curves for CLT shear walls from Shahnewaz etal.[47] and for replaceable steel coupling beams from Ji et al. [30, 31].Based onthese fragility curves, three possible damage levels have been considered for bothcomponents. The Maximum Interstory Drift Ratio (MISDR) is the EDP used todescribe potential damage levels for CLT shear walls, while the rotation angle be-tween the ends of the beams is the EDP for replaceable steel coupling beams.We select component-level fragility curves for analyzing vulnerable structuralelements, similar to the three groups of components identified by You et al.[60].The parameters for these fragility curves are detailed in Table4.2.87Table 4.2: Structural component-level fragility curvesComponent Name EDP Damage States Mean (%) DispersionBRB hold-down Axial strain DS1 1.0 0.4Balloon-type CLT walls Story drift ratioDS1 0.3 0.44DS2 0.5 0.3DS3 2.7 0.16Steel Coupling Beams Link rotation angleDS1 5 0.3DS2 9 0.19DS3 11 0.15BRB = buckling restrained brace;EDP= engineering demand parameter; DS= damage state;CLT=cross-laminated timber.4.5.2 Non-Structural Fragility CurvesTo determine the extent of damage that each non-structural element in the casestudy building model may experience under various earthquake scenarios, theirrelevant fragility curves have been derived from FEMA P-58 [18] using the classi-fication numbers provided in Table 4.1.Detailed fragility information for these components is provided in FEMA P-58 [18]. Therefore, Table 4.1 includes only a summary of fragility parametersfor each component, covering the number of damage states, the types of EDPsused to express the components’ damage conditions (such as story drift ratio orfloor acceleration), and the demand locations, indicating whether the EDP for eachelement should be sourced from its respective level or the floor above.88Table 4.3: Non-Structural component-level fragility curvesClassification Number Number of DSs Type of EDP EDP Location(FEMA P-58 IDs) (floor above?)B2022.002 2 SDR NoC1011.011a 3 SDR NoC2011.001b 3 SDR NoC3011.001a 1 SDR NoC3011.001b 1 SDR NoC3032.003a 3 PFA YesC3032.003b 3 PFA YesC3032.003c 3 PFA YesC3032.003d 3 PFA YesC3034.002 1 PFA YesD1014.011 4 PFA NoD2021.013a 2 PFA YesD2021.013b 1 PFA YesD2021.023a 2 PFA YesD2021.023b 2 PFA YesD2022.013a 2 PFA YesD2022.013b 1 PFA YesD2022.023a 2 PFA YesD2022.023b 2 PFA YesD2031.023a 1 PFA YesD2031.023b 1 PFA YesD2051.013a 2 PFA YesD2051.013b 1 PFA YesD2051.023a 2 PFA YesD2051.023b 2 PFA YesD3031.013e 1 PFA NoD3031.023e 1 PFA NoD3041.011c 2 PFA YesD3041.012c 2 PFA YesD3041.031b 1 PFA YesD3041.041b 1 PFA YesD3052.013h 2 PFA NoD4011.024a 2 PFA YesD4011.053a 2 PFA YesD4011.074a 2 PFA YesD5012.013c 1 PFA NoD5012.023b 1 PFA NoD5012.033b 1 PFA NoD5011.013b 1 PFA NoEDP= engineering demand parameter; DS= damage state; SDR= story drift ratio; PFA= peak flooracceleration.894.6 Component Consequence ModelCorresponding consequence functions for non-structural components have beenextracted from FEMA P-58 [19]. While FEMA P-58 offers a wide spectrum ofconsequence functions for assessing potential losses incurred by vulnerable struc-tural elements, for this case study building model, only the BRB hold-downs haverelevant functions available. The other two types of considered structural elementsare not covered by consequence functions in FEMA P-58.To address this data gap, there are two options: one is to use the provided ex-planations for each level of damage for both CLT shear walls and replaceable steelcoupling beams, estimating the necessary time and cost for all required activitiesto restore them to their intact condition. The other option is to identify structuralcomponents in FEMA P-58 that closely resemble these two components in termsof the number of damage states and required repair works.Although the first approach yields more precise repair cost consequence func-tions, it necessitates additional adjustments. It is crucial to account for the fluctu-ating value of money over time when recalculating expenses across different peri-ods. The repair costs for components with corresponding consequence functionssourced from FEMA P-58 are based on the currency value as of the latest update in2019. For components not addressed by FEMA P-58, the estimated costs reflect theeconomic conditions at the time of their calculation. To mitigate currency discrep-ancies between estimated repair costs and those reported by FEMA P-58, currencydevaluation or inflation should be recognized across different time periods. Thisensures that all reported costs align with the same economic conditions.However, acknowledging these potential inaccuracies involves significant com-putational complexity, particularly when converting the monetary values of allselected FEMA P-58 consequence functions to the current time. Therefore, thechosen approach in this study for components lacking consequence functions inFEMA P-58 is to select comparable components from this source, similar to themethodology employed by You et al. [60]. The elements identified from FEMAP-58 for this study, due to their comparable damage conditions to CLT shear wallsand replaceable steel coupling beams, are recognized by the IDs “B1071.011” and“B1035.071,” respectively.904.7 Building Recovery ModelThe precision of resilience assessment outcomes for a building is significantly re-lies on the assumptions and models defining the recovery process. As outlined inChapter 3, the suggested “Building Recovery Model” in this study is structuredinto two main parts, each containing specific subsection models. To ensure thesemodels are effectively applied to the designated case study building, their input pa-rameters must be meticulously calibrated to align with the building’s unique char-acteristics. Subsequent sections discuss the selected input parameters for each sub-section model within the recovery framework and provide the underlying rationalefor their selection.4.7.1 Recovery Time ModelThe “Mobilization Time Model” and the “Repair Time Model” collectively consti-tute the overall “Recovery Time Model”. Thus, there is no need to define additionalinputs beyond those required for these models.4.7.1.1 Mobilization Time ModelInitiating the repair process hinges on completing various preparatory tasks, whichREDi™[2] categorizes into six distinct groups known as impeding factors. Thecalculation of these factors is customized to the specific damage condition of thebuilding. According to REDi™, these factors interact in a combined manner ratherthan strictly sequentially or in parallel. For example, while completing the inspec-tion process is necessary before starting financing and contractor mobilization, thelatter two can progress concurrently.In this study, instead of adopting REDi™’s probabilistic approach, we haverelied on engineering judgment to determine a fixed mobilization time. Utilizingthe median impeding factors data provided by REDi™ for structures similar to our20-story case study building, we have set a mobilization time of 30 days acrossall scenarios, regardless of the building’s damage condition. While this approachoverlooks uncertainties in this parameter, it simplifies the modeling of mobilizationtimes and reduces computational costs. Considering that the variability in imped-ing factors across different damage conditions, as provided by REDi™, does not91significantly affect the duration in terms of days, a constant mobilization time isnot a questionable assumption.4.7.1.2 Repair Time ModelThe key milestone in the resilience calculation framework of this study, setting itapart from previous similar works, is its “Repair Time Model”. Detailed devel-opment and implementation methods of its subsection models have already beencovered in the preceding chapter. Here, our focus shifts to identifying the essentialconstant variables of each subsection model specific to the building case study.Crew Schedule ModelThe effective allocation of workers to each repair task is governed by the inte-gration of crew numbers and their sizes, as outlined in the “Crew Schedule Model”section of Chapter 3.The severity and complexity of damage sustained by a component dictate therequisite workforce within each crew for optimal restoration. Thus crew size deter-mination hinges upon detailed descriptions of each damage state which defines thespecific tasks necessary to return the component to its original operational state.Typically, these tasks necessitate specialized workers with specific skills and train-ing. However, certain aspects may overlap across different specialized works, al-lowing general laborers with diverse skills to contribute effectively. Therefore, thecomposition of each crew at a specific damage level is determined by balancingthe deployment of specialized and general workers according to the specific re-quirements of its repair tasks. For the case study building model, determining crewsizes across all levels of damage for both structural and non-structural elements isachieved by utilizing their respective damage state definitions.Establishing the maximum number of damaged units that one crew can ef-ficiently manage during the repair process relies on various factors, such as thebuilding’s location, number of stories, and worker efficiency per day. This assess-ment results in defining optimal ranges for crew deployment across all components,ensuring that crews operate at maximum efficiency without being overburdened orunderutilized. Based on engineering judgment and considering the maximum units92Table 4.4: Labor allocations for structural componentsComponent Name Workers per Crew Based on ADS Maximum Number of Damaged Units in CrewsADS: 0-1 ADS: 1-2 ADS: 2-3 1 Crew 2 Crews 3 Crews 4 Crews 5 Crews 6 CrewsBRB Hold-down 6 N/A N/A 4 6 12 16 20 24Balloon-type CLT Walls 1 2 4 2 3 6 8 10 12Steel Coupling Beams 3 6 6 2 4 6 8 10 12ADS= average damage state ; BRB = buckling restrained brace; CLT=cross-laminated timber.Table 4.5: Labor allocations for non-structural componentsClassification Number Workers per Crew Based on ADS Maximum Number of Damaged Units in Crews(FEMA P-58 IDs) units ADS: 0-1 ADS: 1-2 ADS: 2-3 ADS: 3-4 1 Crew 2 Crews 3 Crews 4 Crews 5 CrewsB2022.002 30 SF 2 3 3 N/A 20 30 45 60 80C1011.011a 100 LF 1 2 3 N/A 4 6 12 16 20C2011.001b 1 EA 1 3 4 N/A 1 2 3 4 5C3011.001a 100 LF 2 N/A N/A N/A 5 8 12 16 20C3011.001b 100 LF 2 N/A N/A N/A 5 8 12 16 20C3032.003a 250 SF 2 2 3 N/A 20 36 54 68 80C3032.003b 600 SF 3 3 4 N/A 1 1.6 2.4 3.2 4C3032.003c 1800 SF 3 3 4 N/A 0.2 0.36 0.48 0.6 1C3032.003d 2500 SF 3 3 4 N/A 0.05 0.1 0.12 0.16 0.2C3034.002 1 EA 2 N/A N/A N/A 15 28 42 60 70D1014.011 1 EA 2 3 4 5 2 4 6 8 10D2021.013a 1000 LF 2 3 N/A N/A 0.35 0.6 0.9 1.4 1.7D2021.013b 1000 LF 2 N/A N/A N/A 0.35 0.6 0.9 1.4 1.7D2021.023a 1000 LF 2 3 N/A N/A 0.15 0.26 0.36 0.48 0.6D2021.023b 1000 LF 2 2 N/A N/A 0.15 0.26 0.36 0.48 0.6D2022.013a 1000 LF 2 3 N/A N/A 1 1.6 2.7 4 4.5D2022.013b 1000 LF 2 N/A N/A N/A 1 1.6 2.7 4 4.5D2022.023a 1000 LF 2 3 N/A N/A 0.4 0.7 0.9 1.2 1.5D2022.023b 1000 LF 2 2 N/A N/A 0.4 0.7 0.9 1.2 1.5D2031.023a 1000 LF 2 N/A N/A N/A 0.4 0.7 1.2 1.5 2D2031.023b 1000 LF 2 N/A N/A N/A 0.4 0.7 1.2 1.5 2D2051.013a 1000 LF 2 3 N/A N/A 1 1.6 2.7 4 4.5D2051.013b 1000 LF 2 N/A N/A N/A 1 1.6 2.7 4 4.5D2051.023a 1000 LF 2 3 N/A N/A 0.4 0.7 0.9 1.2 1.5D2051.023b 1000 LF 2 2 N/A N/A 0.4 0.7 0.9 1.2 1.5D3031.013e 250 TN 3 N/A N/A N/A 2 4 6 8 10D3031.023e 250 TN 3 N/A N/A N/A 2 4 6 8 10D3041.011c 1000 LF 2 3 N/A N/A 0.4 0.7 0.9 1.2 1.5D3041.012c 1000 LF 2 3 N/A N/A 0.2 0.3 0.6 0.8 1D3041.031b 10 EA 2 N/A N/A N/A 4 6 12 16 20D3041.041b 10 EA 2 N/A N/A N/A 2 4 6 8 10D3052.013h 20000 CFM 1 2 N/A N/A 2 4 6 8 10D4011.024a 1000 LF 2 3 N/A N/A 1.8 3 4.5 6 7.5D4011.053a 100 EA 2 3 N/A N/A 0.4 0.8 1.2 1.6 2D4011.074a 100 EA 2 3 N/A N/A 0.15 0.28 0.45 0.6 0.7D5012.013c 1 EA 2 N/A N/A N/A 2 4 6 8 10D5012.023b 225 AP 1 N/A N/A N/A 0.03 0.048 0.09 0.12 0.15D5012.033b 1 EA 2 N/A N/A N/A 2 4 6 8 10D5011.013b 1 EA 2 N/A N/A N/A 5 8 12 20 25ADS= average damage state .of a component available on-site, the corresponding crew ranges have been estab-93lished for all considered components across the building model.Finally, to validate all determinations, comparative analyses have been con-ducted against similar components in relevant research studies [44, 51]. The de-tails of the selected number of crews and their sizes for both structural and non-structural components are reported in Table 4.4 and Table 4.5 , respectively.Repair Schedule ModelFigure 4.2 illustrates the repair logic network adopted for the 20-story casestudy building model. In alignment with the model proposed by REDi™ [2] non-structural elements are categorized into six repair sequences, allowing for parallelrepairs. However, our approach diverges concerning the initiation of repairs withinthese sequences. Similar to the sequential repair models recommended by Terzicet al. [50] and Yoo [59], we have not restricted the initiation of elevator and stairrepairs until the completion of structural element repairs. In contrast, the repair ofthe other four non-structural sequences is contingent upon the intact condition ofall structural elements within the building.Ballon-Type CLT Walls Stairs ElevatorCold WaterService, Cold orHot PotableHot Water Service, Heating Hot WaterChilled WaterPipingSanitary WastePiping Exterior WindowHVACDuctingHVACdrops/diffusers insuspendedceilingsFire Sprinkler WaterPipingFire Sprinkler Drop PartitionsSuspended CeilingLow VoltageSwitchgear IndependentPendant LightingHVAC FanIn Line FanVariable Air Volume (VAV)Box ChillerCoolingTowerPackaged AirHandling UnitWall paperDistributionPanelStart Of RepairEnd Of RepairStructural Repair Finishes at Each FloorABCDEFGBRB Hold-DownSteel Coupling BeamsMotor ControlCenterTransformerFigure 4.2: Flowchart of the considered repair logic network for 20-storybuilding.Another critical parameter influencing repair start and finish dates is the possi-bility of parallel repairs across different stories for each component. In our study,we assume that parallel repair is feasible only for structural elements across all 20stories. Although, repairs for non-structural elements proceed sequentially, floor-94by-floor. This sequential approach means that repairs on lower floors must be com-pleted before initiating repairs on upper floors.Resource Restriction ModelThe total number of workers able to work simultaneously on one floor is theprimary factor contributing to extending the project finish date. In this study, we de-termined this parameter using the guidelines proposed by REDi™ [2], extensivelydiscussed in Chapter 2. According to REDi™, the maximum number of workerson-site cannot exceed 260 people. Additionally, for mid-rise buildings like ourcase study model, the total number of daily workers for different non-structural se-quences varies between 12 to 20. Based on these criteria, we assumed a consistentrestriction of 13 workers per day per floor throughout the project’s duration.As discussed in Chapter 3, applying these defined limits of allowable workersto the calculated repair timeline results in identifying periods where these limitsare exceeded. According to our algorithm, adjustments to allocated resources dur-ing these periods are carried out one at a time, requiring prioritization of whichintervals to address first. For the case study model, we have devised a method forranking exceeding intervals that aligns with the established repair logic network,where the repair of all non-structural elements (except stairs and elevators) is con-tingent upon the completion of structural repairs across the building.We use the last day of completing the repair works for all structural elementsacross the building as the reference point to categorize intervals on each floor intotwo parts. The first part comprises intervals occurring before this defined date, dur-ing which only structural elements, and potentially stairs and elevators, undergo re-pair. Consequently, intervals where only non-structural elements are being repairedconstitute the second part. Following this interval classification, the resource ad-justment process starts with the lowest floor containing exceeding intervals, withpriority given to its earliest period. Once resource adjustments are completed forall intervals in the first category across all floors, attention shifts to intervals in thesecond category for each floor. As with the first category, priority is given to ad-dressing the earliest exceeding period on the lowest floor, proceeding from bottomto top. This approach maximizes efficiency by ensuring each exceeding periodundergoes resource adjustment only once.95The defined resource limits, criteria for prioritizing exceeding periods, and themethod of adjusting allocated resources within each interval collectively contributeto extending the project’s finish date. In our developed resource adjustment al-gorithm, prioritizing repair activities within exceeding intervals is essential. Theinitial criterion for prioritizing resource changes within each interval is termed theElement Importance Factor (EIF). Therefore, the value of this parameter must bedefined for all components considered within the building. For the case study build-ing model, all structural components are assigned the first rank (EIF = 1), while allnon-structural elements are assigned the second rank (EIF = 2). These values alignwith the defined repair logic network, which emphasizes completing repair worksfor structural elements.4.7.2 Recovery Path ModelGiven the primary focus of this study on enhancing the accuracy of repair timemodeling, time constraints prevented the development of a sophisticated function-ality model that would eliminate the need for selecting a recovery process. There-fore, based on the recommendation of the section “Recovery Path Model” in Chap-ter 3, we adopted the linear function as the representation of the recovery path.4.8 Building Functionality ModelOur developed framework for scenarios lacking detailed building functionalitymodels employs a simplified alternative function. This function necessitates defin-ing two key parameters in terms of cost: Repair Cost and Replacement Cost. TheRepair Cost varies depending on the severity of damage for each scenario, whilethe Replacement Cost remains constant. For our case study building model, basedon an estimated cost of US$216 per square foot by You et al. [60], we calculate atotal replacement cost of as US$53,000,000.4.9 Results and DiscussionsAs previously mentioned, our framework verification on this 20-story case studybuilding model utilized outcomes from performed Incremental Dynamic Analysis96(IDA). The database includes structural responses in terms of EDPs at both di-rections of the building across all floors, for 30 sets of ground motions scaled tovarious seismic intensity levels.Due to variations in key features such as magnitude, frequency content, and du-ration, each ground motion leads to differing outcomes under similar scale factors,and none of them causes collapse within the first six intensity levels:1. Sa=0.0755 g2. Sa=0.1510 g3. Sa=0.2265 g4. Sa=0.3020 g5. Sa=0.3775 g6. Sa=0.4530 gIn this study, we used the EDPs derived from applying all 30 ground motionrecords scaled to these six intensity levels as inputs for our developed resilienceassessment framework. The subsequent sections present final results for all 180considered earthquake scenarios.4.9.1 Component-Level Monterey Loss AssessmentFigure 4.3 provides a comprehensive depiction of how various components withinthe case study building respond to changes in seismic intensity. Each subplot ofthis figure illustrates the mean repair costs across different components at a specificlevel of seismic intensity.As seismic intensity escalates from subplot 4.3(a) to 4.3(f), the mean repaircosts generally exhibit an upward trend across most components. However, themagnitude of this increase varies significantly among different components. Whilesome components experience steep rises in repair costs, others show more gradualincrements.97(a) (b)(c) (d)(e) (f)Figure 4.3: Illustration of the distribution of mean repair costs along differentcomponents across six seismic intensity groups: (a) Sa=0.0755 g ; (b)Sa=0.151 g; (c) Sa=0.2265 g; (d) Sa=0.3020 g; (e) Sa=0.3775 g; (f)Sa=0.4530 g.98At lower Sa values (0.0755 and 0.151 g), repair costs tend to be lower anddistributed more evenly among components, with fewer instances of exception-ally high costs. As seismic intensities reach moderate levels (0.2265 and 0.302 g),the disparity in repair costs between components becomes more pronounced. Athigher Sa values (0.3775 and 0.453 g), repair costs rise significantly for most com-ponents, with certain elements exhibiting notably elevated costs, highlighting theirsusceptibility to intense seismic events.Among structural components, Steel Coupling Beams consistently register higherrepair costs across all seismic intensities. In contrast, BRB Hold-downs initiallyshow minimal repair costs, often near zero, at lower seismic intensities, but thesecosts escalate as intensities increase. Balloon-type CLT Walls exhibit relativelystable repair costs across all intensity levels.Notably, partitions emerge as the most consistently vulnerable non-structuralcomponent across all seismic intensity levels, incurring the highest repair costs.Their associated finish (wallpaper), also rank second in repair costs across mostintensities, except at the two highest Sa values. At these levels, repair costs for ex-terior windows exceed those of wall finishes, positioning exterior windows as thesecond most sensitive non-structural element, with wall finishes subsequently rank-ing third in terms of repair cost. In contrast to these three groups of non-structuralcomponents with significant repair costs, a distinct set of ten other components usu-ally exhibit negligible repair costs across all seismic intensities. This trend likelystems from their lower quantities compared to partitions and windows.To determine which elements’ repair costs play a vital role in the total mone-tary loss, and to identify components whose exclusion from the case study buildingmodel results in a considerable underestimation of repair costs, the mean repaircost for each type of component, regardless of the applied intensity level, has beencalculated as depicted in Figure 4.4. Structural elements, as illustrated, gener-ally incur substantial monetary losses. Specifically, Steel Coupling Beams exhibitthe highest mean repair cost, significantly surpassing all other components. Par-titions rank second overall, following Steel Coupling Beams, and first among allnon-structural elements. Approximately half of the non-structural elements havesignificant repair costs, suggesting that excluding the others from further analysiswould minimally impact the total calculated repair cost of the building.99Figure 4.4: Illustration of the distribution of mean repair costs along differentcomponents.4.9.2 Floor-Level Monterey Loss AssessmentTo analyze the distribution of repair costs across different floors at various seismicintensity levels, Figure 4.5 has been prepared. Each subplot corresponds to a spe-cific seismic intensity level, depicting the mean repair costs for both structural andnon-structural elements across all floors, including the roof.At the lowest seismic intensity level (Figure 4.5(a)), repair costs for both struc-tural and non-structural elements are relatively low, particularly for structural el-ements on the first five floors. As seismic intensity increases, the disparity be-tween the repair costs for structural and non-structural elements becomes more pro-nounced. Excluding the results for the two lowest Sa values (Figure 4.5(a), 4.5(b)),he first floor consistently exhibits higher mean repair costs for structural compo-nents compared to non-structural components, with this gap widening at higherintensity levels (Figure 4.5(e)- 4.5(f)). This trend is consistent with the findingsfrom the previous section, indicating that the repair costs for BRB Hold-downs,which are located solely on the first floor, significantly increase with higher seis-mic intensities.In contrast, for all other floors and across all earthquake intensities, the re-pair costs for non-structural elements consistently exceed those for structural ele-100ments. This pattern suggests that, except for the first floor, non-structural elementsgenerally incur higher repair expenses throughout the building. At the last threehigher intensity levels, there is no observable increase in the required repair costsfor structural elements across all floors except for the first floor. This observationaligns with Figure 4.3 for similar intensities, where the required repair costs for allconsidered structural elements remain unchanged. This can be explained by thefact that these elements have likely reached their maximum possible damage statesafter specific intensity levels, beyond which no further damage occurs.Figure 4.6 depicts the mean repair costs for both structural and non-structuralelements at all floor levels, irrespective of seismic intensity. It is evident that,except for the first floor, non-structural elements consistently incur higher repaircosts than structural elements on all other floors. Additionally, the repair costsfor structural elements remain approximately constant from the second floor to thetop floor, whereas the costs for non-structural elements exhibit a smooth, gradualincrease across these floors.101(a) (b)(c) (d)(e) (f)Figure 4.5: Illustration of the distribution of mean repair costs along elevationacross six seismic intensity groups: (a) Sa=0.0755 g ; (b) Sa=0.151 g;(c) Sa=0.2265 g; (d) Sa=0.3020 g; (e) Sa=0.3775 g; (f) Sa=0.4530 g.102Figure 4.6: Illustration of the distribution of mean repair costs along eleva-tion.4.9.3 Intensity-Level Monterey Loss AssessmentFigure 4.7 illustrates the correlation between seismic intensity measures and in-curred monetary losses in the case study building model. This figure enables eval-uation of the economic consequences of higher seismic intensity levels from twoperspectives. The first approach examines the trend in total repair costs due toseismic events across varying spectral accelerationSa values.The second approach,inherently nested within the first, breaks down these total costs into two segmentscorresponding to structural and non-structural components. This method allowsfor a more precise comparison of how repair costs for each component type changewith increasing seismic intensity levels.Observably, both Sa levels and the total repair costs move in tandem. Increasesin costs correspond to higher intensity levels, reflecting escalating damages due tomore severe seismic events.Notably, at the lowest Sa level (0.0755 g), the cost associated with structuraldamages is negligible. However, it becomes considerable as Sa values increase.This trend highlights a critical transition in damage severity, where challenges tostructural integrity become more pronounced, particularly evident at higher Sa val-ues (0.2265, 0.302, 0.3775, and 0.453 g). The rapid escalation of structural repair103cost in Sa= 0.302 g indicates an exponential increase in structural damage, ne-cessitating more intensive repairs. Subsequently, after reaching this third seismicintensity level, the monetary losses associated with repairing damaged structuralelements remain relatively constant. This observation aligns with the findings inthe “Component-Level Monetary Loss Assessment” section, indicating minimalchange in the mean repair costs for the three types of structural elements at theseseismic intensities.The repair costs of non-structural components have consistently increased acrossall considered Sa surpassing the costs of structural parts. The notable shift in theirrepair costs becomes more pronounced at Sa= 0.302 g, similar to the trend observedin structural components.Figure 4.7: Contribution of structural and non-structural elements to total re-pair costs at various spectral acceleration levels.4.9.4 Resilience AssessmentThis section offers a comprehensive evaluation of the resilience of the case studybuilding model, directly addressing the primary research objective outlined pre-viously. The results presented in preceding sections were obtained to fulfill thisobjective and served as secondary outcomes within this framework.To evaluate the resilience of the case study building model under various seis-104mic intensities, plotting the corresponding resilience curves is a crucial step. Thesecurves are derived by assessing functionality changes during the recovery process.As the first step, functionality loss is calculated based on repair costs and the esti-mated replacement value of the building. This metric represents the decline fromfull functionality at time zero, marking the onset of the earthquake hazard. Duringthe mobilization period, fixed at 30 days in our study, functionality losses remainconsistent and are depicted as horizontal lines. As repairs commence, system func-tionality gradually restores. We employ a linear recovery model, connecting repairinitiation to full functionality by the end of the repair period, resulting in a triangu-lar shape that depicts the second part of the recovery process. Finally, the derivedresilience curves for each of the 30 sets of ground motions at the same intensitylevel are illustrated in a single subplot in Figure 4.8.Once these resilience curves are obtained, the next step is to calculate the build-ing’s resilience index by determining the area under each curve. The initial dropafter the earthquake and the calculated repair time are significant factors in defin-ing this area. Therefore, accurately calculating these factors, particularly the repairtime, is crucial to avoid underestimating or overestimating the resilience index.The plotted resilience curves for the case study building model clearly illus-trate how severe seismic events impact the building’s recovery time. For low-intensity events, the recovery period is negligible. However, for higher intensitylevels, it typically takes approximately one year for this tall building to regain itsfull functionality. Consequently, the greater reduction in the area under the plot-ted resilience curve at higher intensity levels is not unexpected, as observed fromFigure 4.8(a) to 4.8(f).Three significant outcomes of resilience assessments for this building undervarious seismic intensity levels, as illustrated in the plotted resilience curves inFigure 4.8, are reported in Table 4.6. These outcomes include the mean, minimum,and maximum repair times, functionality losses, and resilience indices for eachintensity level.To calculate the resilience index, the third reported parameter in the table, it isnecessary to post-process the calculated area under each plotted resilience curve.This is because, according to the literature [9, 11], the resilience index shouldbe a unit-less parameter, while the calculated areas are measured in units of days.105(a) (b)(c) (d)(e) (f)Figure 4.8: Illustration of recovery curves across six seismic intensity groups:(a) Sa=0.0755 g ; (b) Sa=0.151 g; (c) Sa=0.2265 g; (d) Sa=0.3020 g; (e)Sa=0.3775 g; (f) Sa=0.4530 g.Therefore, in this study, we used the building’s replacement time of 600 days,as suggested by You et al. [60], as the standardized function for determining allresilience indices.The reported mean values show that as seismic ground motion intensities in-crease, the mean repair time nearly triples, while the functionality loss increasestenfold. Also, the observed change in the mean resilience index from the lowest tothe highest applied intensity levels is 6%, which is a significant amount.106Table 4.6: Seismic resilience indices across various intensity levelsIntensity (g) Repair Time (days) Functionality Loss ResilienceMean Minimum Maximum Mean Minimum Maximum Mean Minimum Maximum0.0755 115.83 84 145 0.02576 0.01380 0.04341 0.99618 0.99319 0.998340.151 198.53 166 230 0.07562 0.03984 0.11101 0.98336 0.97391 0.992500.2265 243.87 218 264 0.12186 0.07740 0.16207 0.96910 0.95746 0.982070.302 271.57 245 306 0.15779 0.09740 0.21143 0.95628 0.93794 0.972810.3775 291.87 268 387 0.18537 0.12862 0.25393 0.94526 0.90584 0.963130.453 313.00 262 414 0.20739 0.14672 0.28676 0.93486 0.88779 0.95574107Chapter 5Conclusions andRecommendations for FutureResearch5.1 ConclusionThis thesis introduces a comprehensive repair time model, which serves as a criticalparameter in enhancing the accuracy of seismic resilience assessments. While theproposed model aligns with the framework suggested by Terzic et al.[50] (2016), itadvances their approach by addressing several critical gaps. The repair time modeldeveloped in this thesis operates through three sequential steps:1. Crew Schedule Model: This step determines the repair duration of each ac-tivity by allocating the appropriate number of workers based on the numberof damaged units and their average damage condition. Our model introducesa novel formulation for calculating the average damage state across all unitsof an element, which reduces uncertainty in calculations and decreases com-putational costs compared to other existing models.2. Repair Schedule Model: To calculate the total repair duration of the build-ing based on the repair activity durations obtained in the previous step, our108model, similar to the approach suggested by Terzic et al.[50], utilizes a flex-ible repair logic network. This network considers all critical factors thatshould be taken into account when planning the repair process.3. Resource Restriction Model: Recognizing the limitations on worker alloca-tion due to safety concerns and resource constraints in the aftermath of anearthquake, our model offers an advanced solution for optimizing resourceallocation. The key innovation in this thesis is the proposed framework’sability to dynamically adjust the number of resources to meet daily restric-tions, surpassing the capabilities of existing models.This model has been subsequently integrated into a probabilistic seismic re-silience assessment framework. To demonstrate the applicability of our developedframework and the proposed repair time model across various building types, weapplied it to a 20-story case study building. The case study results highlighted thatneglecting resource congestion under high seismic intensities could lead to a sub-stantial underestimation of total repair time—by an average of 30%. Although thebuilding exhibited a significant resilience index, the findings indicate that account-ing for resource constraints can reduce the resilience index by an average of 2.6%at the highest intensity levels.5.2 Chapter SummariesChapter 2: In this chapter, a comprehensive literature review was provided,identifying all the necessary steps and models essential for achieving an accurateresilience factor. It concluded by highlighting existing gaps in the proposed repairtime models within the literature.Chapter 3: This chapter centered on our proposed resilience framework, in-cluding a detailed exploration of each component, with particular focus on thecornerstone of our framework: the repair time model.109Chapter 4: In this chapter, the application of the proposed framework detailedin Chapter 4 was illustrated using a 20-story timber building as a case study. Thisapplication clarified the framework by discussing the methodologies used to obtainall relevant input data. Ultimately, the resilience index was calculated for this build-ing. The outcomes indicated that the building had a significant level of resilience,with the minimum resilience index under the worst-case scenario remaining above0.85. The results also showed that at the highest seismic intensity levels, consider-ing resource congestion of 13 workers per day could extend the repair time by anaverage of 30% over the initial estimation. In one scenario, this prolongation wasequal to 123 days, or 40%.5.3 ContributionsThe resilience index has been introduced to encapsulate the capacities of systems,buildings, and communities not only to withstand sudden, severe events like earth-quakes but also to rapidly regain their functionalities. Precise calculation of thisindex requires comprehensive modeling of the recovery process. Without suchmodeling, the reported parameters can be misleading, potentially causing decision-makers to misallocate critical resources in post-earthquake scenarios among af-fected systems. Despite significant advancements in this field through high-qualityresearch, considerable gaps remain that necessitate further investigation and devel-opment.One crucial aspect of accurately evaluating seismic resilience is the estimatedrepair time of damaged buildings, which directly impacts the certainty of the re-silience index. To address this, various models have been proposed in the literaturefor calculating repair time, with three frequently utilized models being FEMA P-58[18], the Resilience-based Earthquake Design Initiative (REDi™) Rating System[2], and Terzic et al. [50]. Each of these models builds upon its predecessor, withthe most advanced being the model by Terzic et al. (2016). However, even thismodel contains simplifications that, if overlooked, may result in potential scenar-ios being inadequately addressed during the recovery process.In this thesis, we introduce a modest yet significant enhancement to existing re-pair time models. By developing an algorithm that removes the oversimplifications110found in current models, we aim to reduce the uncertainty in estimating necessaryrepair times—a crucial element in achieving accurate resilience indices.5.4 LimitationsDespite our diligent efforts to refine the recovery time model, it was necessary toincorporate some simplifications, given the constraints of the defined time-framefor this thesis. One critical aspect that requires further exploration is the variableof mobilization time, which should ideally depend on the level of damage. In thisstudy, we assumed a fixed value for this parameter across all scenarios, regardlessof the level of damage inflicted on the building. However, by employing proba-bilistic models, this parameter could be determined more realistically, allowing fora more reliable calculation of the final recovery time.5.5 Recommendations for future research1. Utilizing comprehensive functionality models that can accurately trace therecovery process is recommended for future research. This approach notonly reduces the uncertainty in final results by mitigating the impact of sim-plified functionality functions but also eliminates the need to define a sep-arate recovery function, thereby avoiding the introduction of additional un-certainties.2. The consequence function models in FEMA P-58 [19] use repair cost datalast updated in 2019. For more current and accurate repair cost estimations,utilizing updated resources can significantly enhance the precision of thefinal results.3. Extend the developed framework to include considerations for the effectsof aftershocks during the recovery process. This addition is critical, as af-tershocks can significantly decrease functionality and extend the calculatedrecovery time.111Bibliography[1] N. Agrawal. Natural disasters and risk management in Canada. Springer,2018. → page 1[2] I. Almufti and M. 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Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Tesfamariam, Solomon
Development of an advanced repair time model for assessing the seismic resilience of buildings
Text