International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Disaggregating community resilience objectives to achieve building performance goals Wang, Naiyu; Ellingwood, Bruce R. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Disaggregating Community Resilience Objectives to Achieve Building Performance Goals  Naiyu Wang Assistant Professor, Sch. of Civil Engineering and Env. Science, University of Oklahoma, Norman, USA  Bruce R. Ellingwood Dist. Professor, Dept. of Civil & Envir Eng., Colorado State University, Fort Collins, USA ABSTRACT: Resilience is often regarded as an attribute of communities rather than of individual buildings, bridges and other civil infrastructure facilities.  Previous research to support resilient infrastructure has considered, for the most part, actions and policies to achieve resilience objectives at the community level.  While it is clear that a community cannot be resilient without resilient individual facilities, few attempts have been made to relate the performance criteria for individual facilities to community resilience goals in a quantitative manner.  This paper presents a method for relating risk-informed performance criteria for individual buildings exposed to extreme hazards to broader community resilience objectives and illustrates the application of the method to two residential building inventories.  The paper demonstrates the feasibility of disaggregating broader community resilience goals to obtain performance objective of individual facilities.  1.  INTRODUCTION Resilience is the ability of communities to withstand external shocks to their populations or infrastructure and to recover from such shocks efficiently and effectively (Timmerman, 1981; Pimm, 1984). In the case of civil infrastructure, resilience is often associated with four attributes (Bruneau et al 2003): robustness - the ability to withstand an extreme event and deliver a certain level of service even after the occurrence of that event; rapidity - to recover the desired functionality as fast as possible; redundancy - the extent to which elements and components of a system can be substituted for one another; and resourcefulness - the capacity to identify problems, establish priorities, and mobilize personnel and financial resources after an extreme event.  These attributes are illustrated in Figure 1; all are characterized by considerable uncertainties. Resilience is often regarded as an attribute of communities rather than a property of individual infrastructure facilities (Bruneau et al 2003; McAllister 2013).  Previous research to support resilient civil infrastructure has focused, for the most part, on establishing metrics to measure and quantify resilience at the community level (Miles and Chang 2006; Twigg 2009; Cutter et al. 2010). While it is clear that a community cannot possess these attributes without individual facilities that are resilient, only limited attempts have been made to relate performance-based design criteria for individual facilities to community objectives and goals in a quantitative manner.      Figure 1. Illustration of the concept of resilience  Individual building design is dictated by existing building codes and standards. Yet, the 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 performance levels specified for individual buildings by codes and standards may not be consistent or adequate when viewed from the perspective of the community. Current building codes focus on safeguarding public safety, and generally do not consider or employ an integrated approach to ensure community-level resilience. For example, the performance-based provisions and guidelines for earthquake-resistant design (NEHRP, 2009) allow significant structural and nonstructural damage to occur under rare, intense ground shaking as long as it does not lead to collapse. While this approach may pose minor risk to life safety, when damages are aggregated, the performance of a building inventory may pose an unacceptable financial burden on public and private sector building owners and, ultimately, the community. To address this issue, community stakeholders in San Francisco (Poland, 2013) and the State of Oregon have gone beyond existing minimum standards in attempts to ensure broader patterns of community resilience to earthquakes. More generally, the traditional approach of designing individual facilities must be re-examined to improve the resilience of the built environment, (e.g., Mieler, et al, 2014).  Our research hypothesis expanded upon in this paper, is that it is possible to develop risk-informed performance criteria for design of individual buildings exposed to a spectrum of natural hazards that can be matched to community resilience goals.  2. BACKGROUND An ideal regulatory framework contains provisions and guidelines which reflect a level of risk that is consistent with what society expects from the system and, as important, what costs society is willing to incur to achieve that level of risk. Mieler et al (2014) employed concepts and procedures from the framework used to design and regulate commercial nuclear power plants to outline a conceptual framework for linking community resilience goals to design targets for individual facilities. Their methodology consists of four steps:  Step 1: Define key community-performance parameters. This step involves identifying undesired outcomes for a community. Step 2: Establish community-resilience goals which should be developed by a group of community stakeholders. These goals usually involve identification of a level of the event that is considered “significant” and a probabilistic assignment to the undesired outcomes in step 1. Step 3: Establish performance objectives for each vital community function, where the vital community functions include services or amenities that are critical to prevent the undesired outcome (e.g. loss of housing, employment, education and other public services). An event tree is constructed for each vital community function; all event trees are then combined into a single “community” event tree to identify the sequences of events that result in undesired community outcomes; probabilities are assigned to each branch of the combined tree in such a way that the community-resilience goal is satisfied; and, branch probabilities are used to establish performance objectives for each vital community function. Step 4: Establish performance targets for important systems and components that support the vital community function. These steps are illustrated in the blue-shaded boxes in Figure 2.   Several assumptions and simplifications were made in the above linkage of the overall community-resilience goals to the performance objectives of facilities within the built environment. For instance, in the example described by Mieler et al (2014), the probability models of the four vital community functions (i.e. housing, employment; education and public services) were assumed to be statistically independent of one another, which greatly simplified the construction of the event tree and the branch probabilities determined for each vital community function. However, these community functions are positively correlated and the impact of interdependences between them on community performance was not considered.    12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3   Figure 2. Illustration of the process of linking community resilience to obtain building performance objective  Moreover, Mieler et al (2014) assumed that if the performance target for the community housing function is determined to be “no more than 5% of the community’s housing stock will become unsafe to occupy after a stipulated event”, then the corresponding performance objective of individual residential building is “less than 5% probability of being unsafe to occupy after the event.” The assumption underlying this reasoning was that the performance of each residential building in the community is mutually statistically independent of all others.  This assumption, as will be shown subsequently, leads to individual building performance requirements that are unconservative with respect those needed to collectively ensure the broader community resilience goal and public welfare.   When a hazardous event affects a complex geographically distributed system like a community, spatial correlations in both demand and capacity must be taken into account.  Hazardous events with large footprints introduce spatial and temporal correlations to the demands on the community infrastructure (Adachi and Ellingwood, 2008; Goda and Hong, 2008; Jayaram and Baker, 2009). Common building practices and code enforcement within a community also introduce positive correlation in structural response above and beyond that introduced by the hazard (Vitoontus and Ellingwood, 2013; Bonstrom and Corotis 2014). For example, increasingly popular modular design and construction allows for greater economies of scale and efficiency, but also increases the vulnerability of the community to the effects of correlated failures.  Previous research on damage and loss estimation for building portfolios (FEMA/NIBS, 2003) has treated individual building damages and losses as if they were statistically independent, which leads to an unconservative estimate of total loss (Vitoontus and Ellingwood 2013).  Such correlations depend on the stochastic variability in the demand from hazardous events over the affected area at both spatial and temporal scales, the number of structures and their locations, and their susceptibility to damage if the hazardous event occurs.  These factors must be taken into account in the last step shown in Figure 2 for disaggregating the community resilience goals to obtain the performance objectives or design criteria of individual structures.  In the following sections, this paper will explore advantages (and disadvantages) of different mathematical algorithms for disaggregating community risk, demonstrate the feasibility of relating community resilience objectives to building performance goals (expressed in terms of failure probabilities), and 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 suggest possible socio-economic policy incentives to improve resilience of communities in which resilience objectives currently are not met by the existing building inventory. 3. MATHEMATICAL FORMULATION Assume that the performance state (safe, fail) of each component can be described by a Bernoulli random variable.  The probability mass function defining the component performance state is: 𝑃𝑋𝑖 = {𝑝,                   𝑥𝑖 = 11 − 𝑝,            𝑥𝑖 = 0                   (1) where Xi is the performance state of the ith building and xi = 0 and xi = 1 denote the safe state (with probability, p) and failure state (with probability 1 – p) of the component, respectively. If the correlation between failures of two buildings is neglected, the joint probability mass function for the performance state of the two buildings, i and j, can be easily expressed as: 𝑃[𝑋𝑖 = 𝑥𝑖 , 𝑋𝑗 = 𝑥𝑗] = 𝑃[𝑋𝑖 = 𝑥𝑖] ∙ 𝑃[𝑋𝑗 = 𝑥𝑗]                  (2) where 𝑥𝑖 and 𝑥𝑗 are the performance states of building i and j.  As discussed in Section 2, correlation in performance often arises between buildings in a community due to a common building design code, common construction practice, as well as common hazard exposure. Thus, simulation of correlated multivariate distributions is essential to determine the component performance criteria for a given (community) system reliability or to determine the system reliability for (code-stipulated) component failure probability.  Two classic methodologies for simulating multivariate distributions are discussed next.  3.1. Copulas  Copulas are often used to simulate dependent multiple random variables. Given marginal probability information of random variables and the correlation matrix, a copula yields an approximation of the multivariate joint distribution function (Sklar, 1959; Nelsen, 1999).  Copula techniques have been applied to many research fields, e.g., Lee and Kiremidjian (2007) determined the joint probability mass function of earthquake damage states of transportation networks using a copula; Limbourg et al (2007) used this technique for reliability prediction in systems involving correlated component failures.  A copula is a multivariate distribution for which the marginal probability distribution of each variable is uniform  (Nelsen, 1999). Currently, many copula families are available for practical use. In this paper, a classical copula model is adopted, i.e., the Gaussian copula, which has the following definition 𝐶𝐺(𝐮) = 𝚽𝛒(Φ−1(𝑢1),⋯Φ−1(𝑢𝑛) )      (3) where 𝚽𝛒 is a multivariate standard normal distribution with correlation coefficient matrix 𝛒. Φ−1 is the inverse function of standard normal distribution. Equation (3) maps the original space of dependent random variables into a standard normal space. Subsequently, a Gaussian copula model denoting the joint cumulative probability distribution of the original random variables yields the joint normal distribution with a correlation coefficient matrix of 𝛒.  In this paper, the marginal random variables each refer to component performance states which are defined with a Bernoulli distribution. The corresponding cumulative distribution function is expressed in Eq. (4): 𝐹𝑋𝑖 = {0,                       𝑥𝑖 <  01 − 𝑝,       0 ≤ 𝑥𝑖 < 11,                        𝑥𝑖 ≥ 1                   (4) Using the inclusion-exclusion principle, the probability 𝑃(𝑿 = 𝒙)  yields (Limbourg et al, 2007) 𝑃(𝑿 = 𝒙) = 𝐹(𝒙) − 𝑃(𝑿 < 𝒙)= ∑ ⋯ ∑ (−1)𝑗1+⋯+𝑗𝑛𝑗1=1,2𝐶(𝑢1,𝑗1 ,⋯ , 𝑢𝑛,𝑗𝑛)𝑗1=1,2  (5) where:  𝑢𝑖,1 = {𝐹𝑖(𝑥𝑖 − 1),   𝑖𝑓 𝑥𝑖 = 10,                             𝑒𝑙𝑠𝑒              12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5  𝑢𝑖,2 = 𝐹𝑖(𝑥𝑖)                                          (6) where 𝐹(𝒙) is the multivariate distribution and is simulated from the Gaussian copula. The probability that no more than Y % of buildings in the community fail after a particular hazard is: 𝑃 (∑𝑋𝑖 ≤ 𝑌𝑖𝑛𝑖=1) =∑𝑃(∑𝑋𝑖 = 𝑗𝑛𝑖=1)𝑌𝑗=1          (7) Eq. (7) provides an analytical solution to the community resilience objective (see Figure 2) in terms of individual building failure probability. However, this solution is intractable when the number of random variables (components in the system) becomes very large.  3.2. Monte Carlo Simulation  An alternative for simulating the multivariate distribution is Monte Carlo simulation (MCS). It provides a solution for comparison with other approximation approaches if a closed-form analytical solution is unavailable. One advantage of MCS is that its convergence does not depend on the number of random variables in the system, which makes it a practical approach for solving high-dimensional problems.  The procedures for generating samples and perform MCS are as follows: 1) Generate n×N (n is the number of random variables and N is the sample size) random samples that following the multi-dimensional standard normal distribution with correlation matrix 𝛒; 2) Transform the normally distributed random samples generated in the previous step into uniformly distributed samples by using normal distribution function 𝑢 = Φ(𝑧); 3) Assign each sample a weight 1/N.  Judge the performance function in a way that if 𝑢 < 𝑝 (p is component failure probability of each building), then this sample fails. Eventually, the system failure probability equals the number of failed samples divided by total sample size N.  We compared this MCS procedure with the closed-form solution by copulas, and found that the two methods yielded practically the same answers when the systems of 5 or 10 components were considered. Thus, MCS is adopted to perform the probability transformations in the Gaussian copulas and to disaggregate the community resilience objectives.  4. NUMERICAL ILLUSTRATION Assume that the predefined performance goal for the community residential building inventory is: 95% probability that no more than 5% of the housing inventory becomes unsafe to occupy after a stipulated hazard event with a certain intensity level (e.g., an earthquake with a 2,475-yr return period, termed the MCE in ASCE Standard 7-10).  The objective is to assign failure probabilities to individual buildings in such a way that the predefined community resilience goal above is satisfied.   4.1. Homogeneous building inventory We first assume that the building inventory is homogeneous, containing nominally identical modular residential buildings. The coefficient of correlation between two buildings, i and j, is assumed to decrease exponentially with the increase of the separation distance between buildings, i.e. 𝜌𝑖𝑗 = exp (−|𝑖 − 𝑗|/𝐿𝑐) , where 𝐿𝑐 is the correlation length which is taken in the subsequent analysis as the maximum distance between two residential buildings in the inventory.  This analysis identifies the failure probability of individual residential buildings (𝑃𝐹) such that there is 95% probability (resilience objective of the housing stock, 𝑅𝐻𝑆) that no more than 5% of inventory will become unsafe for occupancy after the stipulated hazard event.  Figure 3 shows the resilience objective of the building inventory as a function of the failure probability of individual buildings for homogenous housing stocks of different size 𝑛. The analysis shows that in order to meet the performance objective of the housing stock as a whole (i.e. 𝑅𝐻𝑆 = 0.95), the failure probability 𝑃𝐹  of individual residential buildings should be be less than approximately 0.012 when correlation between buildings as stated above are 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 considered, which is more conservative than the 𝑃𝐹  calculated otherwise if the correlation is neglected ( 𝑃𝐹 = 0.05  all buildings in the inventory are assumed statistically independent).   To investigate the impact of the community resilience goal on the direct financial loss to the housing inventory, we introduce a financial index, γ, characterizing the overall financial risk to the housing stock:  𝛾𝐻𝑆 = ∑ 𝑃𝑚 ∙ 𝑚 ∙ 𝐶𝐵𝑀𝑚=0                       (8) where 𝑚 = the number of buildings that become unsafe to occupy after the hazard event; 𝑀 = the maximum expected number of building allowed to fail as determined by the predefined community resilience goal of the entire housing stock, here 𝑀 = 𝑛 × 5% ; 𝑃𝑚 = the probability that 𝑚  buildings become unsafe to occupy following the event; 𝐶𝐵 = damage cost to individual buildings. Since the damage level of an individual building is inversely proportional to its reliability, we assume the damage cost is proportional to 1/Φ−1(𝑃𝐹).  Figure 4 shows that, for a homogenous inventory with n =100, the financial risk to the housing stock decreases significantly when the individual building failure probability 𝑃𝐹 decreases, which suggests that adjusting 𝑃𝐹  could be an efficient method to ensure new construction meets the overall community resilience objective (which addresses economic concerns).  The 𝑃𝐹 that is necessary to meet community resilience goals could be more stringent than the failure probability threshold stipulated in the current performance based design guidelines (which addresses mainly life safety).  4.2. Inhomogeneous building inventory A residential building inventory in a community is likely to be inhomogeneous because houses may have been constructed at different times according to different building standards. It is very likely that not all the buildings in the inventory will meet a performance goal that ensures the resilience objective of the community; in fact, some buildings may not be compliant with the current building code. In order to investigate the financial risk to such a community, we model the existing inventory as a series of development areas or “zones” that are related to the structural characteristics of the dominant buildings found in each zone (Mahsuli and Haukaas, 2013).     Figure 3. Component failure probability versus system reliability   Figure 4. System reliability and financial risk versus individual building failure probability for a building inventory (n=100)  For illustration, we assume an inventory of 100 houses with two types of construction: 50 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 Type I buildings and 50 Type II buildings.  Assume that Type I meets the performance objective identified in §4.1, i.e. if all buildings in the community are Type I, the inventory meets the resilience objective of the community.   Assume further that Type II buildings represent earlier construction, which do not comply with the current code and consequently are more vulnerable than Type I buildings to potential natural hazards. To measure this additional vulnerability, assume that the reliability index of the Type II building β2 equals β1-0.5, where β1 is the reliability of building Type I.  Correlations proportional to separation distance are considered among building of the same type, as before, while correlations between Type I and Type II buildings are negligible.  Figure 5 compares the financial risk of this building inventory (50 Type I bldgs. and 50 Type II bldgs.) to the homogenous building inventory (100 Type I bldgs.) that meets the community resilience objective. It is clear that the inhomogeneous building inventory bears higher financial risk. With the failure probability of Type I buildings being 0.012, the system reliability of inhomogeneous building inventory is 89% (as opposed to the target of 95%). In order to increase the system reliability of the inventory from its current value to 95% in 25 years (a reasonable community resilience objective), several risk mitigation measures may be employed to stimulate public actions towards the desired level of community resilience. Such measures may include structural retrofit stimulated by government subsidies or tax credit, or socializing the financial risk of the community through insurance incentives.    5. IMPLIMENTATION OF THE DISAGGREGATION METHODLODY  The significance of the risk disaggregation methodology illustrated above is two-fold. First, it provides a quantitative approach for explicitly linking community resilience goals to the performance targets for individual constructions. Performance-based design criteria for individual facilities can be derived subsequently to facilitate the broader community resilience goals. Second, it provides a means to measure the resilience of existing communities against the resilience objectives toward which they wish to strive. The methodology can facilitate policies and decision-making to establish appropriate resilience-based risk mitigation strategies to improve community resilience and to measure the effectiveness of alternative strategies that are under consideration for adoption.   Figure 5. Comparison of financial risks between the existing inventory (50 Type I bldgs. and 50 Type II bldgs.) and an equal-size resilient inventory (100 Type I bldgs.)   6. CONCLUSION  The new paradigm of performance-based engineering is taking hold in the structural design, which is likely to be more closely tied to broader community performance objectives during the next decade. Thus, the concept of community resilience is likely to impact the development of building codes, regulations and standards. In such circumstances, individual designers are more likely to be required to take community resilience indicators and goals into account in the design and construction of individual facilities or in the retrofit of existing ones. This paper has demonstrated the feasibility of disaggregating broader community resilience goals to obtain the performance objective of individual facilities.  In turn, the methodology can also be used to measure the effectiveness of 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 alternative resilience-based risk mitigation strategies.   7. REFERENCES Adachi, T. and B. R. Ellingwood. 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Natural Hazards 53(1); 1-20. doi: 10.1007/s11069-009-9408-x.  Poland, C.D. (2013). “SPUR Resilient City Goals.”  Roundtable on Standards for Disaster Resilience for Buildings and Infrastructure Systems.” September 26, 2011.  Sklar, A. (1959) Fonctions de repartition ?̀?  n dimensionset leurs marges, Publ. Inst. Statis. Univ. Paris, 8, 229-231. Timmerman, P. (1981). “Vulnerability. Resilience and the collapse of society: A review of models and possible climatic applications.” Environmental Monograph, Institute for Environmental Studies, Univ. of Toronto, Canada. Twigg J. (2009). “Characteristics of a Disaster-Resilient Community: A Guidance Note,” 2nd ed., Disaster Risk Reduction Interagency Coordination Group, London. Vitoontus, S. and B.R. Ellingwood. (2013). “Role of Correlation in Seismic Demand and Building Damage in Estimating Losses under Scenario Earthquakes.” Proc. Int. Conf. on Struct. Safety and Reliability (ICOSSAR 2013), New York, NY, Taylor & Francis, A.A. 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