Community‐level post‐hazard functionality methodology for buildings exposed to floods

This paper presents a building‐level post‐hazard functionality model for communities exposed to flood hazards including the interdependencies between the population, buildings, and infrastructure. An existing portfolio of building archetypes is used to model the post‐hazard physical flood functionality of different building typologies within the community with the goal of supporting resilience‐informed decision‐making. Specific fragility‐based flood functionality curves were developed for this portfolio to quantify the exceedance probability of a prescribed set of functionality states. While the physical functionality of buildings is significant to the total functionality of a building and community resilience assessment, the functionality of utilities such as power and water along with the accessibility of the household to essential services such as schools and hospitals is crucial to measure their total functionality. Therefore, functionality models for essential infrastructure were developed to assess housing unit‐ and building‐level functionality following flood hazards. This model also accounts for the functionality of the road network following a flood hazard to identify the level of accessibility of households to different services (e.g., schools, hospitals, gas stations, shopping centers, etc.). The main novelty of this paper is the ability to quantify the total functionality of buildings based on a socio‐physical formulation after including the interdependencies between the functionality of the physical systems and the subsequent functionality of the socio‐economic systems, which is key to measuring resilience at the community level. This was evident from the analysis results using a testbed community of Lumberton, NC, showing that the physical functionality is not sufficient to quantify the total post‐hazard functionality of buildings and that the functionality of other subsystems such as utilities and accessibility to essential services are also needed to quantify the total functionality of buildings and the livability of households.

ability to quantify the total functionality of buildings based on a socio-physical formulation after including the interdependencies between the functionality of the physical systems and the subsequent functionality of the socio-economic systems, which is key to measuring resilience at the community level.This was evident from the analysis results using a testbed community of Lumberton, NC, showing that the physical functionality is not sufficient to quantify the total posthazard functionality of buildings and that the functionality of other subsystems such as utilities and accessibility to essential services are also needed to quantify the total functionality of buildings and the livability of households.

INTRODUCTION
Over the last two decades, the concept of community resilience has been studied in the context of analysis, modeling, measurement, and implementation techniques (Bruneau et al., 2003;Cimellaro et al., 2010;Cutter et al., 2008;Ouyang et al., 2012;Renschler et al., 2010;Sharma et al., 2018Sharma et al., , 2020)).At the same time, understanding and modeling infrastructure interdependencies (Bocchini et al., 2014;Eusgeld et al., 2011;González et al., 2016;Ouyang & Dueñas-Osorio, 2012;Talebiyan & Duenas-Osorio, 2020) offer a stepping stone to more expansive community resilience models with population dynamics and economic activity.The wide array of definitions for community resilience across different disciplines was reviewed in several studies (Bhamra et al., 2011;Koliou et al., 2018;Manyena, 2006;Martin-Breen & Anderies, 2011).In this study, we use the general definition of community resilience introduced by the Presidential Policy Directive (PPD-21, 2013) as ". . . the ability to prepare for and adapt to changing conditions and withstand and recover rapidly from disruptions."McAllister reviewed research needs to achieve community resilience and addressed the current research gaps to measure community resilience using metrics (McAllister, 2016).Quantifying the posthazard functionality of buildings and full communities is the first several steps to measuring community resilience since it provides information on disruption (robustness of systems).Tracking community resilience requires quantitative models for the functionality of the built and social systems, which in turn requires modeling interdependencies between physical, social, and economic systems along with propagating uncertainties in the response of these systems to external stressors.
Decades of disaster research have demonstrated that disasters do not affect all people living in a community equally (Fothergill & Peek, 2004;Sutley & Hamideh, 2020).Other important social, physical, economic, cultural, and political factors drive people, households, and communi-ties to be more or less vulnerable (Cutter, 1996).Research has repeatedly illustrated how people with different socioeconomic status (1) perceive, prepare for, and respond to natural hazard risks; (2) have been impacted differentially both physically and psychologically; and (3) are differentially affected during different stages along the disaster timeline (Fothergill & Peek, 2004;Hamideh & Rongerude, 2018;Peacock et al., 2014;Sutley et al., 2017Sutley et al., , 2020;;Tierney, 2006;van de Lindt et al., 2018).Policy research has also pointed out that existing disaster recovery policies have further exacerbated social inequalities after disasters by setting qualifying criteria that exclude socially vulnerable people, including renters, the poor, and some cultures, from accessing recovery resources (Kamel & Loukaitou-Sideris, 2004;Sutley & Hamideh, 2018).Thus, it is critical to incorporate the differential needs and experiences of the population in community resilience analysis, which in predictive studies has primarily been done through social vulnerability measurements (Berke et al., 2023;Sutley & Hamideh, 2020;Tate et al., 2021).
Population subgroups with higher levels of social vulnerability have a greater probability of experiencing larger reductions in functionality and longer periods of recovery (Van Zandt, 2019).For community-level resilience to be improved, steering assistance to the most socially vulnerable populations may be the most equitable path (Kim & Sutley, 2021;Sutley et al., 2017).The models developed herein will allow for a level of detail that is novel and provide models of post-hazard functionality to explore equitable solutions.Equitable solutions focus on how to create opportunities for populations that have been "denied a full opportunity to participate in aspects of economic, social, and civic life" (Exec. Order No. 13985, 2021).Linking people, employees, children, and households to buildings and infrastructure has produced community resilience planning tools capable of examining differential impacts on population groups (Fereshtehnejad et al., 2021;Guidotti et al., 2017;Mazumder et al., 2023;Roohi et al., 2020;Rosenheim et al., 2021;Wang et al., 2021).Previous studies have explored population impacts by key social vulnerability characteristics such as race (Roohi et al., 2020), income (Wang et al., 2021), and tenure status (Rosenheim et al., 2021).
In recent efforts to integrate the impacts of infrastructure disruptions, socio-economic factors, and building functionality on post-disaster recovery and resilience (Mitsova et al., 2019;Ulak et al., 2018), explored the effects of infrastructure service disruptions on hurricane recovery, with a focus on electric power and cell phone/Internet services, and their relationship with socio-economic and demographic factors.Burton et al. (2016) incorporated the failure of electric power supply and water supply subsystems in the definitions of probabilistic building performance limit states and then aggregated building-level performance metrics to produce community-level performance measures.Lin andWang (2017a, 2017b) considered different building functionality states based on structural and non-structural component damage and utility availability to model post-disaster functionality recovery of building portfolios.Masoomi and van de Lindt ( 2018) accounted for the repair and restoration time of electric and water utilities when assessing the impact of a tornado on community outmigration.Although these studies offer valuable insights, most consider utilities to have binary limit states, neglecting the interdependencies between physical systems and the subsequent functionality of socio-economic systems in the context of post-hazard functionality.Nofal et al. (2022), Nofal, Rosenheim, et al. (2023), and Nofal, Amini, et al. (2023) paved the way for more robust modeling to chain the functionality of the built environment with the socio-economic systems.
In efforts to deploy resilience concepts in practice, Almufti and Willford (2013) developed the Resiliencebased Earthquake Design Initiative (REDi) framework that focused on post-earthquake utility disruption curves, with binary utility states (functional and non-functional) for electrical, water, and natural gas systems.These curves were employed to assess utility-specific and total downtimes for buildings.The REDi rating system supports the implementation of resilience-based design of the built environment.Terzic et al. introduced the F-Rec framework, which explicitly considers this aspect for the probabilistic evaluation of functional recovery in building systems (2021).Physical building and infrastructure recovery do not necessarily imply that the community is functional at pre-disaster levels.This necessitates consideration of socio-economic factors and models, including their impact on infrastructure recovery processes.Ulak et al. examined the functionality of various physical elements of electric power infrastructure, critical facilities, and different socio-demographic segments of the population after Hurricane Hermine in Tallahassee, FL, highlighting the distinct and non-homogeneous performance of different components (2019).Dvir et al. studied the role of access to essential facilities and the status of local infrastructure during hazard weather events in shaping individuals' risk perceptions, emphasizing the importance of evaluating access as an element of public risk perceptions (2022).Esmalian et al. developed a computational multi-agent simulation model to enable the integration of social equity considerations in infrastructure resilience assessments, focusing on hurricane-induced power outages (2022).However, a comprehensive post-hazard functionality model that accounts for all the interdependencies across physical and social systems is still lacking.
This paper provides an overview of a methodology to quantify the post-hazard functionality of buildings following flood hazards by accounting for both physical and social systems.To do this, it utilizes an example community to illustrate the new approach using a flood scenario to demonstrate the key decisions, assumptions, and necessary data for applying the framework.The main contribution of the developed approach is the ability to quantify the post-hazard functionality of each subsystem within the community with a resolution fine enough to allow chaining between the post-hazard functionality of the different subsystems within the community.The proposed analysis resolution is at building-level using high-resolution models that can allow modeling the physical functionality of buildings based on the damage probability to the different components within these buildings.Also, the population model is at the household level, which allows linking households to housing units and buildings.The ability to chain such physical and social models allowed modeling interdependencies between physical systems and social systems through population needs.Through the presented methodology and application in this paper, it became clear that although physical functionality is crucial for building functionality, it is not sufficient to inform the total building functionality and requires a multidisciplinary approach that can integrate physical and social models.

OVERVIEW OF METHODOLOGY
Community resilience is defined by the Presidential Policy Directive (PPD-21, 2013) as ". . . the ability to prepare for and adapt to changing conditions and withstand and recover rapidly from disruptions."The way to measure and track community resilience is through the functionality of its subsystems.Therefore, in this paper, functionality is defined as "the performance level and ability of a system to provide its intended functions."Functionality can be measured relative to a baseline level of functions, which is usually defined by ideal or pre-event conditions such as occupancy, commercial and organizational activities, and availability of utilities.In the next subsections, flood post-hazard functionality models were developed for the main subsystems that contribute to the total buildings functionality across the community.In this paper, we distinguish between physical functionality and total functionality.The physical building functionality is the functionality of the structure itself in terms of its safety for occupancy.On the other hand, the total building functionality is the functionality of the building as a whole in terms of physical and socio-economic functionality.It includes structural safety, utilities availability, and accessibility to essential services.Two separate quantities were used to calculate the functionality, which is the functionality state (FS) and the functionality ratio (Fr).
The FS is the functional level of a building based on its safety and habitability.The Fr variable is the functionality ratio that represents the percentage of remaining functionality.

Physical building functionality analysis
The physical functionality is introduced herein as the structural and functional integrity of the building to provide safety and habitability to its occupants.The proposed functionality concept is different from the damage concept.This is because buildings might not be damaged but still not be functional because of losing power, water, or accessibility to essential services.With a focus on flood hazards, the proposed methodology was applied to the building sectors within the community to account for their physical post-hazard functionality by transforming damage into functionality.This was done using the relationship between damage and functionality through a conditional probability [P(B_FS i |B_DS i )] where B_FS i is the building functionality state, B_DS i is the building damage state, j is the number of damage states from 1 to 4, and i is the number of functionality states from 0 to 4. A detailed description of the damage and functionality states can be found elsewhere (Nofal & van de Lindt, 2020c;Nofal et al., 2020).Best engineering judgment among the multidisciplinary authors team was used to estimate these conditional probabilities based on the description of each B_DS and B_FS corresponding to each building type as shown in Table 1.However, other conditional probabilities can be used for specific facilities or buildings of interest that are required to perform or provide specific levels of functionality.The probability of exceeding a building functionality state B_FS i for each building at a specific intensity measure (IM) at level x or [P(B_FS i |IM = x)] was calculated using Equation (1), which feeds into the building functionality ratio B_Fr, calculated by multiplying the probability of being in a functionality state by their corresponding functionality ratios (Fr i ) using Equation (2): .(_  | = )] (1) where P(B_FS i |B_DS i ) is the exceedance probability of functionality state FS i given damage state B_DS i , and is the probability of exceeding a building damage state B_DS j at an intensity measure IM = x.Finally, B_Fr(IM = x) is the physical functionality ratio of the building at an intensity measure IM = x, and Fr i is the functionality ratio associated with building functionality state B_FS i .This approach is similar to the approach used by Burton et al. (2018) for earthquake hazards to transform from loss-based limit states to recovery-based limit states.These conditional probabilities can be better evaluated using field observation of the damage states along with their observed post-hazard functionality to correlate both variables empirically.This approach allowed transforming the damage fragility curves into functionality fragility curves.Figure 1 shows a comparison between the resulting damage fragility curves and the functionality fragility curves for archetype F2 (Nofal & van de Lindt, 2020), which is a one-story residential building with a slabon-grade foundation.This was applied to the 15 building archetypes described in Section 2.1.

Functionality analysis of networks
To assess the power availability at the building level using the power distribution network (PDN) model, first, the direct damage to the point of delivery (POD) substations is assessed.The level of power outage depends on both the intensity of the hazard and the physical characteristics of the substation components.For the flood hazard, the elevation of the control room and other components is used in conjunction with a step fragility function to determine if a substation is functional or not.If the equipment height from the ground level is unknown, it may be assumed to be either at 3 ft for risk-managed electric power facilities or at the ground level for low, medium, and high voltage substations as per the Electric Power System Classifications, Functionality Thresholds, and Flood Model Default Parameters in (FEMA, 2021).Using the step function as the substation fragility function for the  flood hazard implies that all the buildings in the service area corresponding to an affected substation experience a power outage.The graph model of the PDN contains the distribution poles and distribution lines.After assessing hazard-induced damage to the PDN components, the connectivity of the PDN model can be checked using either breadth first search or depth first search (DFS) algorithms.
The distribution poles may be vulnerable to hydrodynamic pressure and water-borne debris as well.However, in this study, direct damage to the distribution poles due to flooding is not considered.
Regarding water distribution networks (WDNs) functionality, it consists of three correlated measures: quantity, pressure, and quality.In this study, the functionality focus is on water quantity because it is often the most basic requirement for the stability of a community.The systemwide water service availability () of a WDN at time t is defined as where N* is the set that contains all the nodes that have water demands;    () and    () are the delivered and expected demands at node  at time , respectively.The expected demand    () is estimated using household allocation data, an assumed average demand per person per day, and a general hourly demand time series.Node  is mapped to specific households based on the shortest distance rule, and the expected demand    () reflects the cumulative estimated demands of these households.The delivered demands are obtained from hydraulic analysis or simplified network flow analysis if the hydraulic analysis does not converge.In the context of flood hazards, system failures include inundation of facilities and loss of electricity, which are modeled in the post-hazard functionality analysis.The inundation of facilities such as the water treatment plants and pump stations is modeled by turning off the corresponding nodes or links in the EPANET hydraulic model.The loss of electricity is modeled through the partial functionality of the pump link or the water treatment plant node.
The functionality of the roadway system was assessed by estimating the travel time to the nearest hospital, TA B L E 2 Qualitative description of the functionality states for residential buildings in terms of the functionality of each subsystem.grocery store, gas station, and school (if applicable) for each building before and in the aftermath of a hazard event, herein floods.For this purpose, the road network was modeled using a graph theoretic approach where intersections were modeled as nodes and the roads were modeled as links.Bridges can be considered to be a part of roads (i.e., links) or they could be modeled separately as nodes or links.To estimate realistic travel times between any two locations in the network, first, travel times on each roadway link were obtained using OpenStreets maps and their travel distance matrix data (Luxen & Vetter, 2011).These travel times were used in a weighted Dijkstra's algorithm (Broumi et al., 2016) to estimate the travel time between any two locations in the network.In this process, during hazard conditions, damaged or impassable roads were identified by determining the maximum inundation depth for each link.Following recommendations from the Federal Emergency Management Agency, roads with more than 6 inches of inundation depth were considered to be impassable and were removed from the road network.Travel times were re-estimated for the flooded road network.Herein, the percentage change in the travel time was used for determining the functionality of the road network for each building.Furthermore, to assess connectivity, a DFS was used, which can be used to determine the existence of a path between two nodes in a road network.

Total building functionality analysis
A total building socio-physical post-hazard functionality model was developed to capture the functionality contributions from the different subsystems (Ω j ), with j belonging to set J = {1, . . ., 6}, including buildings, networks, and services.In particular, the total post-hazard functionality Ω t calculation is based on the contribution of the physical functionality of the building (Ω 1 ), utilities functionality (power [Ω 2 ] and water [Ω 3] ), accessibility to services (shopping centers and gas stations [Ω 4 ]), and accessibility to social institutions (educational [Ω 5 ] and healthcare services [Ω 6 ]).Tables 2 and 3 show a qualitative and quantitative description of the functionality states (FS) of each physical and social system that contribute to the total post-hazard functionality of a residential building.Each table is divided into physical functionality and socio-economic functionality.The physical functionality includes the functionality of buildings and the availability of utilities including power and water.The socio-economic functionality includes the accessibility of households to gas stations, shopping centers, schools, and hospitals.The assumed functionality weight C j for each system j is listed in the last row of Table 3, which represents the contribution of each system to the total post-hazard functionality of the building.The summation of the C j terms for all contributing systems should be equal to 1.0 assuming that all the considered systems will make up the 100% functionality of a building.The proposed model can be parameterized to handle more systems by adjusting C j and the cardinality of set J accordingly.These functionality weights are based on the best judgments from the multidisciplinary team to represent the contribution of each system.However, these judgment-based weights can be replaced with fielddriven (data) weights based on community engagement projects.
The total functionality Ω j of each subsystem j is quantified using Equation (4).The quantified value is then compared to the performance criteria defined in Tables 2  and 3 to assign a functionality state.For example, the physical functionality of a building is Ω 1 = B_FR as calculated from Equation (2), and similarly done for other Ω j for considered subsystems j to build the functionality vector in Equation ( 5).The functionality vector Ω contains the functionality ratio for each considered subsystem within the community, obtained by multiplying the exceedance probability of each functionality state by the functionality ratio R corresponding to each system j associated with each damage state i.Then, Ω t , which is the total functionality ratio of the building after including the functionality of each subsystem, is calculated by multiplying the subsystem's functionality vector Ω by the functionality weight vector C using Equation ( 6).The same approach can become fully probabilistic by using Equation ( 7), where the exceedance probability functionality matrix (with elements P(FS i ) is weighted by the functionality TA B L E 3 Quantitative description of the functionality states for residential buildings in terms of the functionality of each subsystem (T is the actual travel time, and T n is the average normal travel time).

Physical functionality
Socio-economic functionality Buildings Utilities Provided services (Ω 4 ) No access (0%) Functionality weight C j 0.45 0.2 0.15 0.1 0.1 weight vector C j to calculate the overall exceedance probability of the building after including all the contributing subsystems.The final calculated functionality Ω p is in terms of a probabilistic functionality vector with the exceedance probability of each functionality state in each row PΩ(FS i ). (5) Here, Ω p is the total functionality vector in a probabilistic format.P(FS i j ) is the exceedance probability of each functionality state i corresponding to each subsystem j in the probabilistic setting, P Ω (FSi) is the probability of exceeding a functionality state i for the entire building as a system of systems, which represents the total functionality of a building,    is the functionality ratio corresponding to each functionality state i associated with each system j, Cj is the functionality weight corresponding to each system j, m is the number of systems, and n is the number of functionality states.

IN-CORE) model
The data used to model the community components and the computational models developed to link between these components were made The proposed computational model can handle different types of data at different levels of resolution with varying levels of complexities between the linkages that connect the components/systems.Figure 2 shows how the data were organized and linked before applying the hazard, how the damage is transformed into functionality, and how the functionality of each component/system is linked to its supporting elements and systems.The flow chart starts by showing how the population information (household size X1(i,u), number of labor X2(i,u), students X3(i,u), etc.) are linked to the different building sectors including each housing unit within each residential building Y1(i,u), each business unit within each commercial building Y2(j,b), and each social institution Y3(k,s) where i, j, and k are the buildings and u, b, and s are the units within the buildings where i: residential building, j: commercial building, k: social building, u: housing unit, b: business unit, and s: social institution.Also, the service area associated with each network was specified (Z1, Z2, Z3).Using housing unit identities (IDs) and building IDs facilitated these linkages between the population, buildings, and infrastructure.Then, the hazard was coupled with the linked/interdependent components/systems for the community.Finally, the damage to buildings and other physical infrastructure was transformed into functionality estimates.

BASIC STRUCTURE OF MODELS AND DATA
The proposed post-hazard functionality method uses highresolution community-level models for population allocation, building functionality, and infrastructure functionality.A computational environment named IN-CORE served as the modeling technology necessary for studying the framework (van de Lindt et al., 2023).The application discretizes the built environment by performing spatial analysis of households, buildings, and networks to identify the social and physical exposure of the community from the spatial distribution of hazard intensity across the community.This was done by modeling of the households' synthesis (e.g., size, race, ethnicity, etc.), different building types (e.g., residential, commercial, social, etc.) along with essential networks (e.g., power, water, roadway, etc.) within the community.

Household, population, and building model
Usually, household information is restricted at the housing unit level to protect privacy.The US Census Bureau provides information about the population at higher levels of geography such as blocks, block groups, and census tracts.Household-level population characteristics are necessary, however, to get a more accurate depiction of equity or inequity.To enable a more accurate analysis, Rosenheim (2021) developed a housing unit inventory methodology with person record files that estimates sociodemographic characteristics at the household level.The estimated population data include each household member's race, ethnicity, sex, age, and school grade level; the model also estimates each household's tenure status and income level and assigns each household a physical home based on geospatial housing unit data.Readers are referred to Rosenheim (2021) for more details about the household and population characteristics methodology and Rosenheim et al. (2021) for specifics on the household allocation method.The individual and household estimates are based on de-aggregated Census data and thus are accurate at the selected aggregate level, that is, Census block.The population data are designed to reflect the community as a whole and provide a tool to connect impacts to individual structures and infrastructure on the people that live in the community, and vice versa.
Similarly, while many datasets exist (Data-Axel, 2020; Geverdt, 2018; U.S. Census Bureau, 2021) that link businesses to jobs and schools to staff, there is only limited data that provide the link between where workers live and where they work within large groups of industry sectors.To overcome this limitation and provide details on industry sectors such as education and health care, an algorithm was developed to link jobs to people in the person record file (Rosenheim, 2021).While the labor allocation data presented in this paper apply to the education sector, the labor market allocation model can be extended to apply to any economic sector in a community.The labor data consist of details on each job's industry and wage level, as well as the employee's race, ethnicity, sex, and age level.The labor data are designed to reflect the economic and social institutions (schools and hospitals) within a community and link impacts to businesses with the employees as well as impacts population disruptions may have on business and social institutions.
To model the building inventory within the community, the building portfolio developed by Nofal and van de Lindt (2020c) was used to model the different building typologies within the community.This portfolio consists of 15 building archetypes with a number of residential, commercial, and social institution buildings as shown in Table 4. Readers are referred to Nofal and van de Lindt (2020c) for more details about these archetypes.This suite of 15 archetypes is believed to be diverse enough such that they can represent small-to mid-size communities with an appropriate resolution.These archetypes can then be mapped to any community of interest and linked with the population model as shown schematically in Figure 3, which shows a schematic representation of how the household data get linked with building data through a household allocation algorithm.

Network models
The scale of the electric power system model to be used in the analysis is primarily determined by the geographical extent of the hazard.As the fluvial flood hazard is limited to the community, phenomena at the power transmission network scale are minimal thus mainly requiring community-level PDN models.The PDN model consists of POD substations and a distribution network layout or graph.The POD substations receive bulk power from the transmission network and distribute it in the corresponding service areas through radially operated main feeders and laterals.When high-resolution data for the community PDN is unavailable, the service areas of individual substations can be estimated using Voronoi polygons or cellular automata (Pala et al., 2014).If the locations of utility poles are known, the topology of the PDN can be modeled either as a minimum spanning tree (Mensah & Dueñas-Osorio, 2016;Montoya & Ramirez, 2012) or it can be modeled using constraints based on road infrastructure (Meyur et al., 2020), where damage to substations or poles leads to downstream user outages.If the locations of utility poles are unknown, nodes representing aggregate loads may be assumed to be distributed in the service area (Mensah & Dueñas-Osorio, 2016) or inferred based on load models.
The WDN of a community should deliver water flows of adequate quantity, pressure, and quality to end consumers.In this study, the asset and topological data provided by the local government, the building, and household alloca-tion data in IN-CORE, as well as the operation information shared by the community, is used as inputs to build an integrated WDN model, where an EPANET (2017) hydraulic network model is integrated with the building and household allocation model.With this integrated model, each household's water demand is mapped to its nearest node in the hydraulic network.Therefore, this approach can simulate the water service disruption to any what-if hazard scenario and map its impact on the community's households, revealing how water infrastructure functionality is connected with the community functionality.
Road networks facilitate the movement of goods and people and can be modeled in IN-CORE using a graphbased approach as described in Section 2.2.Locations of interest such as buildings, essential facilities, and other infrastructure components (e.g., water and power assets) were linked to the road network.Furthermore, the population data from buildings were associated with the road network to determine the number of people that can access different essential service facilities (ESFs) and other social institutions such as schools.For this purpose, the locations of buildings, schools, ESFs, grocery stores, and other locations were mapped to the nearest link or node.Herein, for analysis, all major roads were included in the road network.Simplifications were made to remove roads ending in dead ends and cul-de-sacs.Additionally, two-way streets were converted to two one-way roads to preserve the two directions.During a given hazard event (floods), the resulting network was used for assessing connectivity and travel time before and in the aftermath of a hazard using the procedure mentioned in Section 2.1.

ILLUSTRATIVE EXAMPLE OF METHODOLOGY: COMMUNITY OF LUMBERTON, NC
Lumberton is located in Robeson County in southeast North Carolina, US, about 130 km (80 miles) from the Atlantic Ocean.The Lumber River flows from west to east, through the middle of the community.Lumberton is the site of a longitudinal field study following flooding caused by 2016 Hurricane Matthew and 2018 Hurricane Florence (Helgeson et al., 2021;van de Lindt et al., 2018;van de Lindt et al., 2020).Lumberton is selected for the illustrative example here due to its repeated flooding exposure and the data available from the longitudinal study and other testbed analysis (Crawford, 2022;Nofal & van de Lindt, 2020a, 2020b, 2021).Figure 4 shows a map of Lumberton color-coding building occupancies, including residential buildings in green, commercial buildings in red, social institutions in yellow, and the city boundaries in magenta.The map depicts 20,000 buildings within and around Lumberton that share the city's facilities and networks.The Lumberton virtual building inventory includes 16 school campuses with 10 elementary schools, three middle schools, and three high schools along with one main hospital and three medical clinics.The location of these social institutions with respect to Lumberton is also shown in Figure 4. Detailed information about the buildings and networks was provided by the North Carolina spatial data download website along with other field and virtual reconnaissance studies (Helgeson et al., 2021;Nofal & van de Lindt, 2020b;Sutley et al., 2021;van de Lindt et al., 2020van de Lindt et al., , 2018)).
The data needed to run the models proposed herein includes buildings data, infrastructure data, and population data.A shapefile format for these data is adopted across the entire model.Buildings shapefile has the essential data about the buildings within the community of interest including first-floor elevation (FFE), building archetype, construction material, and foundation type.Also, a shapefile for each lifeline including power, water, and transportation network is used to extract the essential data about each infrastructure.This shapefile provides a topology, connectivity, and the characteristics of each network (pipe diameters, road speed, number of lanes, etc.).The population data were in the form of an Excel file downloaded from the US Census data and are then linked with building data using unique IDs.A complete description of data can be found in van de Lindt et al. ( 2023), which summarizes the theory behind the IN-CORE modeling environment and includes a section on data.

Buildings data
Detailed building data were provided by the NC OneMap ( 2023), which includes building occupancy, number of stories, foundation type, and construction material.Then, the aforementioned portfolio of 15 building archetypes along with their associated fragility functions was mapped to the buildings within the community as shown in Figure 4a.
The mapping process is based on a mapping algorithm that uses the collected detailed building information (field data, Google Street Map View data, and NC OneMap data) to assign a specific archetype to each building.More information about the mapping process for Lumberton can be found in Nofal et al. (2021) and Nofal andvan de Lindt (2020a, 2021).Additionally, the housing units for each building along with addresses are provided to the household allocation model.

Population data
Lumberton is a multiracial community with a population of 21,542 people according to the 2010 Census data (U.S. Census Bureau, 2010).The community's population has significant racial diversity, including 36% African American, 13% Native American, and 39% White.The median household income in Lumberton is $32,383, which is significantly lower than the median income for the United States ($53,046), and North Carolina ($46,450; U.S. Census, 2012)., 2012).These data provide a means to link detailed socio-economic characteristics to the building data described in the previous section and the network data described in the next section.
In addition to the general population, this paper expands the community model to include the education system, which includes students and school staff.While the building inventory identified 16 possible school locations, the National Center for Education Statistics identifies eight schools within the City of Lumberton, five primary schools, two middle schools, and one high school.The schools are part of the Robeson County School District.Within the city of Lumberton, there were no private schools identified.During the 2009-2010 school district (selected to match the 2010 Census data), there were 4758 students; 21% American Indian alone, 30% White alone, and 39% Black alone (National Center for Education Statistics, 2022).For the labor market, this analysis focuses specifically on the education sector, which in 2010 included 2915 jobs or 15% of the total jobs located in Lumberton (U.S. Census Bureau, 2021).The education sector includes public schools, secondary schools, and private education-related businesses.For this study, we focused specifically on the eight public schools that serve students within the city of Lumberton.The labor market allocation uses origin-destination data on where workers live and work (U.S. Census Bureau, 2021).These data show that only around 16.5% of workers live and work within Lumberton, and 83.5% of workers commute into the study area.For the workers commuting in, 60% have commutes of greater than 10 miles.Knowing where workers live helps inform the extent of the building inventory and networks to accurately predict the community-level needs and functionality.

Networks data
Information about the water and roadway networks was provided, among other sources, by the North Carolina spatial data download website (State of North Carolina, 2019).These datasets include network topology, connectivity and a number of specific characteristics for each network (e.g., pipe size for water network, number of lanes for roadway network), complemented by Public Works information.As a detailed model for the PDN is unavailable publicly, the model was created using the techniques described in Section 2.3, particularly the locations of substations from OpenStreetMap and HIFLD, and the service-area approach for power distribution.Figure 4b shows a layout for the essential networks within Lumberton, NC, including power, water, and roadway.The roadway network is green, the power network is red, and the water network is blue.There is one water treatment plant within Lumberton, and the intake source of this plant is at the Lumber River along with other eight deep ground wells to provide around 4 to 6 million gallons of water per day to 25,000 people within and around Lumberton.

Power network
While the Eastern Interconnection has thousands of substations, in this study, we only identify the three POD substations that provide electric power to all the buildings in Lumberton.The service areas were delineated for each substation in Lumberton using the Voronoi polygons technique (Pala et al., 2014).Additionally, as the locations of utility poles are not available, a uniform grid of nodes representing aggregate loads was used and connected using Kruskal's algorithm to create a minimum spanning tree.This is a reasonable assumption to model the PDNs, as they are radially operated and aggregate customers upstream to their respective substations (Mensah & Duenas-Osorio, 2016;Montoya & Ramirez, 2012).Figure 5 shows the development of the PDN for building functionality assessment in this study.

Water network
To build the hydraulic WDN model, data from different sources was used, namely: (1) publicly available Geographical Information System (GIS) infrastructure data that describes the topology and the structural attributes of physical components (NC OneMap, 2023); (2) public digital elevation maps that provide node elevations (NC Floodplain Mapping Program); (3) IN-CORE's synthetic household unit allocation data that produce realistic water demands at nodes; and (4) operational data shared by the local utility, Department of Public Works, Lumberton, to define system settings (e.g., pump curves, water level ranges in tanks); and modeling knowledge based on experience by the networks team.The hydraulic model can easily be updated when new information is available.Lumberton's WDN includes one water treatment plant, three elevated tanks, each having 1-million-gallon capacity, and one operating pump at the water treatment plant.Of its average daily supply between 4 and 6 million gallons, 60% comes from the Lumber River and 40% from the ground wells.

Road network
Although Lumberton is considered for the analysis, the road network model extends beyond Lumberton and spans the entire Fayetteville-Lumberton-Laurinburg combined statistical area, which includes Robeson, Scotlan, Hoke, and Cumberland counties.A larger extent of the road network is necessitated due to commuters working and living The road network for the region within and around Lumberton.
in different counties.Since Lumberton is the region of interest, all types of roads are considered in the Lumberton urban area (as per the census).The data on all the roads in the region were obtained from NC OneMaps.
Only interstate and arterial roads were considered outside Lumberton.Figure 6 shows the road network for the region along with locations of ESFs such as hospitals, grocery stores, gas stations, and schools to which connectivity is assessed.These facilities and building locations are mapped to the nearest nodes in the road network.
The flood hazard map is overlaid on the road network, and when there was more than 6 inches of inundation, it was assumed to be impassable to traffic.Correspondingly, travel time analysis is performed using travel times on individual links obtained from OpenStreet maps and its distance matrix Application Programming Interface (API).

Flood scenario data
A scenario-based flood hazard map for Lumberton was used herein to investigate the developed post-hazard functionality approach.The flooding event after Hurricane Matthew 2016 was used to model the functionality of the built environment.A high-resolution flood hazard map developed by Nofal and van de Lindt (2020c, 2020d) was applied as hazard input.Figure 7 shows the simulated flood hazard map.The exposure analysis results showed that almost all of the buildings and networks along with their associated facilities in the southwest side of the city got flooded as shown in Figure 7a shows the flooded buildings, which are color-coded based on their archetypes.
Figure 7b shows a close-up view of the neighborhoods that were severely flooded by Hurricane Matthew in 2016.The analysis shows that there are 2857 exposed buildings in the flood model with 2400 buildings being inundated, that is, water levels above the FFE.The main water treatment plant that provides potable water to the entire city was also severely flooded as shown in Figure 7c.The main electrical power substation that provides power to the majority of buildings in Lumberton was also flooded as shown in Figure 7e.

Building post-hazard physical functionality analysis results
The physical functionality of each building was quantified based on the methodology in Section 3.1.The fragility functions developed in this paper provide the exceedance probability for each functionality state using the hazard intensity.For example, a building exposed to a specific flood depth has a specific exceedance probability of each FS.Since it is not feasible to present these results for every building in the community, Table 6 summarizes the physical functionality of all the buildings considered in the study area in terms of the probability of being in each functionality state.The exceedance probability of each functionality is divided into ranges from 0% to 100% exceedance probability.The number of buildings within each range is determined and listed as shown in Table 6.If a building is not exposed to flooding, the probability of losing physical functionality is 0%, and if a building is submerged in water (e.g., 4.0-m flood depth), the probability of losing functionality is 100%.It is clear that there are a large number of buildings that remain fully functional (FS4) since the considered buildings within the study area are 20,085 buildings, and the flood-exposed buildings are 2857 buildings.On the other hand, 393 buildings are identified as restricted occupancy (FS2), and 10 buildings are identified as restricted entry (FS0).Figure 8a,b shows a color-coded map for the functionality state of each building, which shows the spatial variation in the functionality state across the entire community for the building directly exposed to flood hazards.These physical functionality states served as an input for the total functionality of each building.The last row in Table 6 (Building FS) provides the number of buildings designated by their most probable functionality state.

5.2
Post-hazard functionality analysis results for networks

Power network
Of the three substations in Lumberton, the central substation located near the Lumber River was severely affected by flooding.Based on the flood hazard scenario data, the maximum water depth at this substation had reached 8.68 ft (2.65 m) thus warranting the de-energizing of the substation that ultimately affected its service area.This is coherent with the findings of the community resiliencefocused technical investigation of the 2016 Lumberton, NC, flood (van de Lindt et al., 2018).Although the disruption due to the substation flooding was alleviated to some extent using power rerouted from the other two substations, the rerouting was not modeled in this study because the exact technical details are not public.Electric power was also disrupted for many households throughout Lumberton due to downed trees (van de Lindt et al., 2018).However, these outages resulted from wind, which was not modeled in this study.During the hurricane, a large majority (99.4%) of the surveyed households in Lumberton reported loss of power due to cumulative effects from flood and wind (van de Lindt et al., 2018).Figure 8c shows the resulting electric power outage within and around Lumberton.

Water network
According to the technical investigation following the Hurricane Matthew flooding (van de Lindt et al., 2018), Lumberton's water supply system was disrupted on October 10, 2016, after failures of the river intake pump, the treatment plant generator, and the water treatment plant due to inundation.Limited service resumed by October 15, when portable treatment units were available to treat water from the ground wells, with water conservation notices launched to save water to backwash the system.Many households reported days to weeks of water supply outage.
Based on the information from the technical investigation, as well as from the simulated water depth, it is inferred that the pump as well as the water treatment plant was offline starting from 10:00 p.m., October 10; the system recovered 50% of total water supply and pumped associated amounts to the system starting at 2:00 p.m., October 15; and  Simulation results match empirical data and show that nearly all the households lost water around 2:00 p.m., October 10, 16 h after the system failure, when the elevated tanks were drained.When the tanks supplied water for the system, the water pressure at all nodes was satisfied as enough potential energy was stored in them.Afterward, the WDN remained empty until 2:00 p.m., October 15, after which a fraction of households got access to water of adequate quantity and pressure, while some households could not get water during the peak demand period (6:00 to 10:00 p.m.). Figure 8d shows the water service availability results for buildings (at 8:00 p.m., October 19).In the current simulation for Hurricane Matthew, water quantity availability dominates the post-hazard functionality because there were no significant structural damages, which can dissipate energy and reduce pressure.The water service availability for each household from 10:00 p.m., October 10, to 2:00 p.m., October 20, is obtained from joint hydraulic and network flow analysis.

Road network
Immediate post-flood connectivity to gas stations, grocery stores, and schools was evaluated herein.Figure 9 shows the analysis results in terms of the accessibility of each household within each building to reach hospitals, gas stations, and shopping centers with and without flood.For comparison, travel times to these locations were also calculated.Without floods, the travel time to the hospital ranged from 0 to 17 min, with an average travel time of approximately 9 min.Considering the loss of connectivity caused by floods, over 17% of the buildings lost connectivity to the hospital.Furthermore, the travel times increased up to 80 min for some buildings, and the average travel time increased to 20.5 min.Without flood-induced inun-dation, the travel times to the nearest gas station and grocery store ranged up to 6 and 10 min, respectively.The average travel time to both of these facilities was a few minutes.While the consideration of floods did not change the average travel time to the gas station and grocery stores, the maximum travel time doubled, and approximately 15% of the buildings lost access to grocery stores and gas stations.
A similar analysis was conducted for schools.For this purpose, students attending public schools in Lumberton were assigned to one of the buildings in Lumberton (see Rosenheim, 2021).Correspondingly, the travel time for each student to their school with and without flooding was evaluated.Since more than one student was assigned to some of the buildings, the average travel time to school for all the students in a building was calculated along with the number of students who can reach the school.The average travel time to school without flooding was approximately 3 min, with a maximum of 11 min.Considering flooding, the average travel time for students who could access the school increased to 11 min, and the maximum time increased to 66 min.Over 22% of the students lost connectivity to schools.

Total building post-hazard functionality analysis results
The total building functionality for each building was calculated using the proposed total post-hazard functionality approach.This was done by including the impact of the building integrity, utilities functionality, and the accessibility to schools, hospitals, and essential services.Figure 10 shows the buildings with Lumberton color-coded based on their total functionality ratio.This map shows that most of the flooded buildings have a total functionality ratio in the range of 50%-75% or less due to the large contribution of the loss of functionality from physical damage (color-coded in red and orange).On the other hand, many of the non-flooded buildings have total functionality ranging from 70% to 95%, and the major loss of functionality comes from losing utilities and accessibility to essential services (color-coded in green).The buildings color-coded in blue have a functionality ratio ranging from 95% to 100%, which can be explained because they are far away from the impacted locations and supported by power from other non-impacted substations.
The total functionality for buildings in community.The major observation from Figure 10a, although most of the buildings color-coded with green were not directly exposed to flooding, is that they still have functionality level from 86% to 95%.This can be due to a number of reasons mentioned in the methodology, which include utilities availability and accessibility to essential services including gas stations, shopping centers, schools, and hospitals.Figure 10b shows a close-up view of a part of the impacted locations west side of Lumber River with some highlights on the total functionality of some buildings.Figure 10b shows the variation in the total functionality ratio across the buildings within the community.Such results can help policymakers make more informed decisions based on the holistic performance of the coupled built and social systems.

SUMMARY AND CONCLUSION
The proposed approach advances the state of the art in community resilience modeling by capturing the performance of the interdependent social and physical systems within the community and their impacts on the total posthazard functionality of the buildings themselves.This was done by linking the physical functionality models of buildings and infrastructure with social science models for the population to calculate the post-hazard functionality of each building.Although the approach was applied to a specific hazard for a specific community, the approach is general and can be applied to any community of interest where data are available and for other hazards or as many hazard scenarios as possible to span the distribution of the hazards if return period-based hazard model is considered.The example demonstrated that the total post-hazard functionality of a specific building relies on the physical components and key social science components.This socio-physical dependency became apparent from the analytical results that showed that buildings that were not located in the flood exposure zone and did not sustain any direct physical damage lost some measure of functionality (around 12%−20% of its total functionality from the presented example) from indirect sources such as loss of power and water along and/or loss of accessibility to essential services.The ability to quantitatively integrate the functionality of social and physical systems in the functionality assessment process can help enable community resilience analysis research to be integrative and more comprehensive.The analysis resolution proposed in the study is at the housing unit-and building-level, which enabled tracking the performance of different parameters that can potentially impact the post-hazard functionality of a specific housing unit or a specific building.This includes the acces-sibility of the households occupying a specific building to a number of services such as healthcare and educational services.One of the main contributions of this research is the ability to quantitatively include non-physical aspects in the post-hazard functionality of buildings and integrate the community-level performance of the infrastructure such as power and water network in the functionality assessment of buildings.The output from this study can be used to initiate a number of studies such as investigating the impact of the calculated post-hazard functionality on population dislocation, and socio-economic activity.The proposed approach can be further enhanced by including other subsystems (e.g., fire and police stations, recreational centers, etc.) and infrastructure functionality (e.g., telecommunication network, sewer network, gas network, etc.).Finally, this approach can provide a robust means for policymakers to make resilience-informed decisions for communities impacted by flooding.

R E F E R E N C E S
Comparison between the flood fragility curves of damage and physical functionality for a one-story residential building on a slab-on-grade foundation: (a) Fragility curves for the exceedance probability of each damage state and (b) fragility curves for the exceedance probability of each functionality state.

F
Post-execution of the post-hazard functionality assessment framework for the built environment.

F6
single-family residential building on a crawlspace foundationF2One-story multifamily residential building on a slab-on-grade foundation F3 Two-story single-family residential building on a crawlspace foundation F4 Two-story multifamily residential building on a slab-on-grade foundation F5 Small grocery store/gas station with a convenience store 3 A schematic representation of the linkages between populations and the building inventory.

F
I G U R E 4 Color-coded map for the built environment: (a) Buildings color-coded based on their archetypes and (b) transportation, power, and water networks within Lumberton, NC.TA B L E 5 Summary of housing unit, population, and socio-economic characteristics of Lumberton, 2010.

F
Development of power distribution networks for Lumberton.(a) determine location of point of delivery substations, (b) estimate service area for each substation using Voronoi polygons, (c) example substation and corresponding service area, (d) use Kruskal's algorithm for minimum spanning tree to create a graph model of uniform-grid aggregated loads.

F
I G U R E 7 A scenario-based flood hazard map based on Hurricane Matthew 2016: (a) The state of flooding across Lumberton, NC; (b) a close-up view on the west side of Lumberton where most buildings got flooded; (c) the state of flooding for the water network; (d) the state of flooding for the transportation network; and (e) the state of flooding for the power network which affects the central region.

F
I G U R E 8 Color-coded maps for the performance of the infrastructure in terms of their functionality: (a) Buildings physical functionality ratio; (b) buildings physical functionality states; (c) electrical power network functionality; and (d) water network functionality.

F
The analysis results for the transportation network in terms of the accessibility time in seconds for each household to the different facilities with and without flooding including: (a,b) Hospitals; (c,d) gas station; (e,f) shopping centers.
Figure 10a reflects many functionality parameters related to the contribution of the different subsystems within the F I G U R E 1 0 Color-coded buildings based on the total post-hazard functionality for buildings: (a) The total functionality of all buildings within and around Lumberton and (b) a close-up view of a part of the city.
This research was conducted as part of the NIST Center of Excellence for Risk-Based Community Resilience Planning under Cooperative Agreement 70NANB20H008 and 70NANB15H044 between the National Institute of Standards and Technology (NIST) and Colorado State University.The content expressed in this paper is the views of the authors and does not necessarily represent the opinions or views of NIST or the US Department of Commerce.

TA B L E 1
Conditional probability of the occurrence of a certain building functionality state B_FS i given the occurrence of a certain building damage state B_DS j [P(B_FS i |B_DS j )] for residential buildings.

Table 5
provides intersectional socioeconomic characteristics of Lumberton based on data from the 2010 US Census.Although 13 years old at the time of this writing, the 2010 Census data were used because it is the most detailed and spatially refined data available to represent the community's baseline prior to the 2016 flooding scenario modeled herein.At the time

probability of a functionality state Number of impacted buildings (total = 20,085)
The number of buildings within each probability range of exceedance of a functionality state.
TA B L E 6