Life-cycle analysis (LCA) to restore community building portfolios by building back better I: Building portfolio LCA

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Life-cycle analysis (LCA) to restore community building portfolios by building back better I: Building portfolio LCA Yingjun Wanga, Naiyu Wangb, , Peihui Lind, Bruce Ellingwoodc, Hussam Mahmoudc ⁎

a

School of Civil Engineering and Environmental Science, The University of Oklahoma, USA College of Civil Engineering and Architecture, Zhejiang University, China (Previously, School of Civil Engineering and Environmental Science, University of Oklahoma, USA) c Department of Civil and Environmental Engineering, Colorado State University, USA d College of Civil Engineering and Architecture, Zhejiang University, China b

ARTICLE INFO

ABSTRACT

Keywords: Building back better Building portfolio Community resilience Earthquake Life-cycle analysis Risk-informed decision making

In recent years, the United States and other countries have experienced catastrophic consequences of severe natural hazards, e.g. the 2005 Hurricane Katrina, the 2011 Christchurch Earthquake, and the 2011 Great East Japan Earthquake and Tsunami. These consequences motivate the question: How should a community’s damaged building portfolio be reconstructed efficiently to enhance its performance under future hazard events and meet the resilience goals of the community? The relatively recent notion of Building Back Better requires a quantitative methodology that can address each phase of building portfolio recovery following a severe hazard event and provide decision support to best enhance the performance of the portfolio under future hazards. This study extends notions of life-cycle analysis from individual buildings to building portfolios to support posthazard reconstruction decisions at the community level. The building portfolio expected life-cycle cost and cumulative prospect value are adopted as decision metrics which reflect varying degrees of risk aversion on the part of community decision-makers. The applicability of these building portfolio life-cycle analyses and some key aspects in their implementation are explored using a moderate-sized community that is susceptible to extreme earthquake hazards.

1. Introduction Recent catastrophic natural hazard events around the world, e.g. the 2005 Hurricane Katrina, the 2011Christchurch Earthquake, and the 2011 Great East Japan Earthquake and Tsunami, have highlighted their enormous socio-economic consequences. While these consequences for a community or urban area are determined collectively by the existing condition of its civil infrastructure, socio-economic institutions and governance, the resilience of the community’s building portfolio plays a critical role in mitigating the negative impact of natural hazards, facilitating rapid recovery, and supporting sustainable development of the community [23]. To enhance the resilience of existing building portfolios, communities may implement voluntary or mandatory portfolio-level programs to mitigate the impact of future extreme natural hazards, such as enhanced new construction requirements [34], mandatory seismic retrofit programs [4,16] and non-mandatory programs for hurricane retrofit [12]. In the aftermath of a disaster, communities also may have access

⁎

to previously unavailable resources and opportunities to create more resilient and sustainable infrastructure through well-planned post-event rebuilding activities [6]. However, this recovery process of building portfolios in the past often has been spontaneous and relatively unorganized, e.g. the rebuilding of New Orleans after the 2005 Hurricane Katrina [28]and Banda Aceh and Sri Lanka after the 2004 SumatraAndaman Earthquake and Tsunami [17]. Community reconstruction, for obvious reasons, usually focuses on restoring the prior-event condition as quickly as possible and seldom involves centralized planning toward achieving a long-term vision of a resilient and sustainable community. The concept of capitalizing on the unique opportunities to reshape a community after a disaster is not novel [5,6]; in recent years, resilience and sustainability have advanced as common goals in communities’ recovery efforts that are focused on Building Back Better (BBB), which suggests that the recovery and restoration of communities should incorporate the idea of enhancing future hazard resilience [5,31]. On the other hand, only limited research has attempted to consider BBB in a

Corresponding author. E-mail address: [email protected] (N. Wang).

https://doi.org/10.1016/j.strusafe.2019.101919 Received 30 October 2019; Received in revised form 26 December 2019; Accepted 26 December 2019 0167-4730/ © 2020 Elsevier Ltd. All rights reserved.

Please cite this article as: Yingjun Wang, et al., Structural Safety, https://doi.org/10.1016/j.strusafe.2019.101919

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quantitative analytical context that involves the extended time horizon needed to build a business case for enhancing both resilience and sustainability. Risk-informed BBB strategies should be developed to enhance the resilience of building portfolios within a community with sustainable solutions that allocate public and private investments properly to manage life-cycle costs. This effort requires a life-cycle perspective, considering the future time horizon of hazard exposure and the need to amortize potentially significant investment costs to achieve resilience through sustainable solutions over the projected life of a building portfolio. It is a common practice to utilize life-cycle analysis (LCA) to support decision-making for individual engineering projects over extended time horizons (e.g. [10]). Life-cycle cost (LCC)-based models, which assume decision-makers are risk neural and risk can be converted into money, are most discussed in the literature (e.g. [37,7]). Other decision models have been developed to reflect the fact that decision-makers often find it difficult or impossible to monetize risk. Among these models, Cumulative Prospect Theory (CPT) [30] maps occurrence probabilities and economic consequences into perceived probabilities and values, respectively, thus allowing risk-aversion of decision-makers to be reflected in decision-making (e.g. [15,3]). LCA seldom has been applied to residential building construction because home ownership changes frequently and there are few financial incentives for developers or owners to base their design, repair or retrofit decisions on LCA. However, residential construction represents over 90% of the built environment in the United States (US) [9,21] and [22], and extensive post-event reconstruction of damaged building portfolios should strive to enhance the resilience and sustainability of the community at large. Furthermore, BBB practices must be planned and financed by both public and private stakeholders and justified by LCA if they are to become common in the future. For this to occur, LCA must be extended from individual facilities and projects to a spatially distributed building portfolio to support a community’s investment in BBB (to be discussed in detail in the companion paper [36]). In this paper, a framework is developed that extends LCA from individual buildings to community building portfolios, in terms of portfolio level LCC and CPT-based metrics, to provide an understanding of the potential benefits of BBB and how it might be incentivized within a community.

extent, it accelerates the renewal of building portfolios, and well-designed building-back strategies could even help enhance future hazard preparedness of portfolios. In this section, two key ingredients for the building portfolio life-cycle analysis (BPLCA) framework are introduced: (1) the “life-cycle” of a building portfolio (Building Portfolio Life-cycle, BPLC), and (2) its renewal rate (Building Portfolio Renewal Rate, BPRR). 2.1. Estimating BPLC of an existing portfolio Conceptually, the BPLC is defined as the time-span during which a building portfolio has been renewed completely through new construction, retrofit and rebuilding, as shown in Fig. 1. The renewal process is stochastic and affected by many variables, including the lifecycle (service life) of individual buildings (Building Life-cycle, BLC), the community’s economic development, demographics, hazard characteristics, etc. The life-cycle of a building portfolio (BPLC), TP , is assumed to be a function of the BLC, Tl , l (1, 2, , L) , in which L is the number of buildings in the portfolio. Further, let TP, r denote the time during which r% of the buildings are renewed (e.g., the time required for 90% of the portfolio to be renewed is TP,90 ). The probability distribution function (PDF) of TP, r for a portfolio containing L buildings, each with identical BLC, is [18]:

(

fTP ,r (t ) = L L rL

)

1 f (t )[F (t )]r·L 1 1 1

1 [1

F1 (t )]L

r·L

(1)

in which, r L denotes the number of buildings that are renewed; F1 (t ) and f1 (t ) denote the cumulative distribution function (CDF) and PDF of each BLC, respectively. For realistic communities with a complex mix of buildings with different BLCs, Monte Carlo Simulation (MCS) is generally needed to obtain the BPLC. The BPLC is defined as tpc, r = E [TP , r ] in the remainder of this paper. For example, consider a portfolio with 2 types of BLCs (BLC1 and BLC2, tabulated in Table 1) in which the ratio of the two types is n1/ n2 = 2.0. Suppose the BLC of both types follow exponential distribution. MCS is employed to obtain tpc, r and its sensitivity to different portfolio sizes. Fig. 2 shows the mean and coefficient of variation (c.o.v.) of tpc, r , the time for 90% of the BP to be renewed, from multiple groups of MCS. As the portfolio size increases, the mean of tpc,90 is quite stable while its c.o.v. decreases monotonically. Thus, the tpc,90 obtained from a building portfolio of small size is also applicable to portfolio of large size if n1/ n2 is kept the same. At the end of the portfolio LC, many buildings may have already been reconstructed for two times or more, since the mean LC of individual buildings is much shorter than that of the building portfolio as given in

2. The Life-cycle of a building portfolio Buildings in a community are continuously ongoing a renewal process, i.e. constantly being constructed, maintained, repaired, retrofitted, damaged, demolished and rebuilt, as illustrated in Fig. 1. A natural disaster is a temporary setback in this long-term process - to an

Fig. 1. Illustration of portfolio renewal under natural hazard events over a long-time horizon. 2

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of a building portfolio, which is an extended methodology from the concept of life-cycle analysis (LCA) for individual buildings [37]. It may include not only the direct economic impact from post-hazard reconstruction and renewal, but also indirect impact from the loss of building functionality, fatality, population dislocation, and reduced economic activities etc. [23]. In the following, two BPLCA formulations are introduced with the objective of facilitating the development of post-event BBB decisions for building portfolios following a severe hazard event in the companion paper [36].

Table 1 Statistics of two BLC types. BLC type

Mean BLC

BLC1 BLC2

50 30

Eq. (1). 2.2. The renewal rate of a building portfolio (BRR)

3.1. Expected building portfolio Life-cycle cost

Let N (t ) = the number of buildings in the portfolio at time t, N (t ) = growth rate of N (t ) at time t, and NN (t ) and ND (t ) are the rates of new construction (renewal rate) and demolition, respectively, within the portfolio. The portfolio growth rate is

N (t ) =

N (t + t ) t

N (t )

= NN (t )

ND (t )

Suppose that the set of reconstruction actions at time, t 0 , is represented by X ; then the building portfolio life-cycle cost (BPLCC) may be expressed as a function of X :

in which CRe (X ) denotes the cost of reconstruction of the damaged portfolio at t 0 (time to last characteristic earthquake), which depends on the number of buildings of each type (defined by structural system, occupancy type and number of stories) in each of damage state, DS (total of four damage states: minor (DS = 1), moderate (DS = 2 ), extensive (DS = 3), complete (DS = 4 )) following the disruptive event. The superscript C on the remaining terms indicates that the corresponding cost is accumulated throughout the entire BPLC. Term C CNew (X ) is the cumulative construction cost of new buildings during the C C (X )andCCas (X ) are the cumulative costs of building damages BPLC; CDam C (X ) is the and casualty due to future hazard exposure, respectively; CInd cumulative indirect loss due to disruptions of local economy and social well-being caused by functionality loss of buildings following future disasters. The expected BPLCC is:

If the population of the community remains relatively stable over time, the size of the portfolio is nearly constant over time, which imi plies N (t ) = 0 and NN (t ) = ND (t ) . The renewal rate NN (t ) of buildings of Type i is: i

NN (t ) = ni

0

hi (t , ) f ai (t , ) d

(3)

where hi (t , ) = fi (t , )/(1 Fi (t , )) is the probability of demolition (due to natural replacement) of building Type i at time t (at age ) on the condition that it has survived prior to time t , where fi (t , ) and Fi (t , ) are the PDF and CDF of Type i BLC evaluated at time t with an age of ; f ai (t , ) is the PDF of age for Type i BLC in time t ; and ni is the number of buildings with Type i BLC in the portfolio. For a portfolio with total I types of BLC, the total renewal rate of the portfolio is: I

NN (t ) =

i

NN (t )

P C C E [CLCC (X )] = E [CRe (X )] + E [CNew (X )] + ( + 1) E [CDam (X )] + E

(4)

i=1

(5)

P C C C C CLCC (X ) = CRe (X ) + CNew (X ) + CDam (X ) + CCas (X ) + CInd (X )

(2)

(6)

C [CCas (X )]

To illustrate the proposed approach, it is assumed that the renewal process of individual buildings is a Poisson process. Thus, the BLC for buildings of Type i follows an exponential distribution with hi (t , ) = 1/ i , here i is the mean value of BLC of Type i. Eq. (4) now becomes i NN (t ) = ni (1/ i ) 0 f ai (t , ) d = ni / i if f ai (t , ) is independent of time (i.e. fi (t , ) = fi ( ) ).

in which

C E [CInd (X )] I

4

E [CRe (X )] =

is assumed to equal

i, j ni, j, x CRe (x )

i=1 j =1

times

C (X )] E [CDam

[26],

(7)

where ni, j, x = the number of Type i buildings in the j-th damage state at t 0 with reconstruction action x (for simplicity, i and j are dropped in i, j (x ) = the reconstruction (or structural repair) cost of subscript of x ); CRe the Type i buildings in the j-th damage state with reconstruction action C x , and E [CNew (X )] in Eq. (6) is:

3. Life-cycle analysis of building portfolios Building portfolio life-cycle analysis (BPLCA) is a technique to quantify a specific metric (e.g. monetary cost) associated with all stages

Fig. 2. The (a) mean and (b) c.o.v. of BPLC tpc,90 with different portfolio sizes. 3

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little or no experience [30]. In CPT, total of NC consequences first are ranked in ascending order 0 y +1 < < yNC , where yi is the consequence of the (i.e. y1 < y ith event). Then, the monetary cost yi and probability pi of the ith event are converted into values v (yi ) and decision weights i , respectively, (Eqs. (10) - (12)) to reflect the subjective utility and perception of probability on the part of decision-makers, which is a generalized form of expected life cycle cost or utility theory [32]. If the status quo is selected as the reference point, the total value of NC events [15]is: NC

V = V + V+ =

v (yi )

+

i

+ i

v (yi )

i=1

(10)

i= +1

where v (yi ) is the value of the consequence of i-th event yi ; i and i+ are the decision weight for the ith event with loss and gain, respectively. To reflect the loss-averse nature of the decision-maker, the value function v ( ) (Eq. (11)) is employed (different for gain ( y 0 ) and loss ( y < 0 )), with parameter controlling the difference and and being exponent parameters: Fig. 3. The decision weight of cumulative prospect theory (CPT) when = 1.0 and = 0.8.

C E [CNew (X )]

=

t0+ TP t0 I

= i=1

I

e

rd (t t0 )

i ni, x NN

i CCon (x )

i=1 i CCon (x ) ni, x (1 i rd

(8)

i (x ) = the construction cost of Type i building with in which CCon action x ; ni, x = the number of Type i buildings in the portfolio with i reconstruction action x ; NN (t ) = the renewal rate of building Type i at time t (Eq. (4)); rd = the discount rate; and e rd (t t0) = the discounting of all future new construction cost to the current cost at t 0 (which is the time immediately following current disaster, as shown in Fig. 1). C C The last two terms in Eq. (6), CDam and CCas , depend on the hazard exposure of the community. This hazard is modeled by discretizing the mean annual frequency vs intensity into K frequencies and intensity levels, with each level k (1, 2, , K ) represented by the median value C (X )] can be obtained by calculating of the interval. Accordingly, E [CDam the discounted annual building damage cost over the BPLC, TP (defined in Section 2.1): K C E [CDam (X )] =

E [CDam (k, X )] k=1 K

t0+ TP t0

e

rd (t t0 )

k=1

0 (11)

( y ) , if y < 0

i

=w

i

i 1

pj

w

+ i

pj ,

j=1

= w+

j=1

NC

w+

pj j=i

NC

pj j=i+1

(12a) i

w

vkH (t ) dt

C E [CDam (k, X )]

=

y , if y

The decision weight (subjective probability) of the i-th positive or negative event ( i+ and i ) is calculated by Eq. (12a), where w and w+ are nonlinear subjective probability functions for loss and gains (obtained from Eq. (12b) and Eq. (12c)). Parameters , + , and control the shape of the curve mapping from real probability p to nonlinear subjective probability function w and w+ [15]. The contribution of rare events to the value function, adjusted by the subjective probability function, not only depends on the magnitude of the impact but also the occurrence probability. For typical risk-averse decision makers, an in= 1.0 , and verse-S shaped transformation function ( + = 1.0, 0 < < 1.0 , e.g. as shown in Fig. 3; see Cha & Ellingwood [3] implies that the decision maker elevates the importance of LPHC events subjectively, which is the case for decision-making under earthquake hazards and many other extreme natural events.

(t ) dt

rd TP )

e

v (y ) =

i

pj = exp

ln

j=1

(9)

w+

NC

pj = exp j=i

where CDam (k , X ) denotes the building damage due to hazard level k with reconstruction action X ; vkH (t ) denotes the occurrence rate of hazard level k at time t . Eq. (9) implies that the CDam (k , X ) is timeindependent (implying complete repair after each damage). CalculaC (X )]in Eq. (6) are similar. The cost from demolition has tions for E [CCas not been considered due to lack of available data but can be added to the framework easily when data becomes available.

pj

(12b)

j=1

+

NC

ln

pj

(12c)

j=1

In this paper, we assume the value function v (y ) = y and the bias in risk perception is taken into account through the assignment of πi to emphasis the decision-maker’s subjective probability perception on low probability/high consequence events. Thus, the Expected Building Portfolio Cumulative Prospect Value (EBPCPV) under decision X is: P E [VCPV (X )]

3.2. Expected building portfolio cumulative prospect value

C C C = E [CRe (X )] + E [CNew (X )] + (1 + ) E [V Dam (X )] + E [VCas (X )]

(13)

Minimum expected life cycle cost analysis is based on the assumption that all risks can be monetized and that the decision-makers are risk-neutral [24,30]. However, decision-makers often value monetary consequences nonlinearly according to their risk tolerance. In addition, low-probability/high-consequence (LPHC) events pose difficulties for decision-makers in designing post-hazard reconstruction strategies because estimates of likelihood and consequences for LPHC events are biased. Cumulative Prospect Theory (CPT) can address the difficulties encountered in perception when dealing with events for which there is

C (X ) - will definitely incur in the - CRe (X ) and CNew C C V . = p = 1 Dam (X ) and V Cas (X ) denote cumui i

In Eq. (13), the cost rebuild process, thus lative prospect value (CPV) of building damage and casualties, respectively. Since only loss is considered in Eq. (13) [3], we will neglect P (x )] for simplicity. Thus, the EBPCPV of the negative sign “ ” in E [VCPV C (k , X )], is: damage due to a future level-k hazard, E [V Dam C E [V Dam (k , X )] = E [CDam (k, X )]

4

t0+ TP t0

e

rd (t t0 )

[

k

vkH (t )] dt

(14)

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where k is the probability adjustment factor for a level-k hazard, and C (k, X )]. Compared to Eq. (9), the original vkH (t ) is similarly for E [VCas replaced by k vkH (t ) but everything else is the same. k is calculated by: k

where

pk

vkH k

(t )

=

(15)

=

E [CDam (mk , X )] k=1 K

t0

e

rd (t t0 )

vkE

E [NE (mk )] =

t0

e

rd (t t0 ) v E (t , t ) dt 0 k

=

t0 + TP t0

e

rd (t t0 ) q=1

I

t

fT2+

+ Tq

(t

ni, x i=1

i CCon (x )

4

i, j i, j DV Item pItem (mk , s, x )

j=1

NOi

4

i, j i, j DVcas pST (mk , s, x )

j=1

(23)

where denotes the casualty loss in a Type i building due to physical damage in the j-th damage state and NOi is the number of occupants in Type i building. Thus, the mean damage and casualty losses due to earthquake mk is i, j DVcas

E [CDam (mk , X )]=

fWq (t , mk |W1 > t 0 ) dt

1 NS

Ns

CDam (mk , s, X ) s=1

(24)

in which NS denotes the total number of MCS for each magnitude mk . The mean casualty loss due to earthquake mk , E [CCas (mk , X )], can obtained similarly. Similarly, the EBPCPV can be estimated by employing Eq.(13). As introduced in Section 3.2, the only difficulty in the cumulative prospect-based methodology lies in the quantification of k , which relates to vkE (t ) . For CE mK , vKE (t , t0 ) is time-dependent, which causes difficulties in defining K . In this study, the mean occurrence rate vkE for both NCE and CE, k = 1, 2, , K , is employed, i.e. vkH (t ) = vk =pk , when quantifying k . Then k can be obtained by Eq. (15).

(18)

t0

(21)

(22)

where fWq (t , mk |W1 > t 0 ) = the PDF of Wq at time t , given there is no event of magnitude mk between t = 0 and t 0 . The summation of fWq (t , mk |W1 > t 0 ) over all q, q (1, 2, ) yields vkE (t , t 0) . For the CE of magnitude mK , according to Takahashi, et al. [27],

fWq (t , mK |W1 > t 0 ) =

(20)

i, j (mk , s , x ) denotes the probability of the j-th damage In Eq. (22), pItem state for component item (ST , NA, ND , CT ) of a Type i building under the s-th scenario of earthquake magnitude mk under decision x ij (for i, j simplicity, i and j are dropped in subscript of x ); and DV Item denotes the damage to Type i building with respect to component item due to physical damage in j-th damage state[20,34]. Similarly, for CCas (mk , s, X )

Let Tq, (q = 1, 2, ) be the inter-arrival time between the (q 1) th and qth events of magnitude mk ; for simplicity, k is omitted. Thus T1, T2, , Tq, are independent and identically distributed (i.i.d.) variq ables, and Wq = 1 Tq is the waiting time to the qth earthquake. From Takahashi et al. [27], t0+ TP

d

item i = 1

(17)

allsources k = 1

)

rd TP )

ni , x

CCas (mk , s, X ) =

C E [CDam (mk , X )]

e

I

t 0 is considered in to designate the time of occurrence of the current earthquake, and t = 0 denotes the time of the most recent CE. c (X)] becomes: When all seismic sources are considered, E [CDam c E [CDam (X)] =

vkE (1 rd

CDam (mk , s , X ) =

vkE (t , t 0)

K

(t

q 1 K

To quantify E [CDam (mk , X )] in Eq. (16), we must consider uncertainties in magnitude M, source to site distance R, ground motion attenuation, local soil condition, and building fragilities (related to X ) (e.g. [19]). For each fault, we assume the location of future earthquake epicenters is uniformly distributed along the fault. To reduce the sample points, we employ Latin Hypercube Sampling [25] with a sample size of 100 for R. Further, for each hazard scenario (in combination of M and R), 1000 simulations are employed to consider the uncertainty in ground motion attenuation at each building site [2], considering the correlation between sites by an exponential function [33]. Thus, the mean loss is calculated at a given hazard demand from a total of K × 100 × 1000 simulations. CDam (mk , X ) = The total damage cost is CST (mk , X ) + CNA (mk , X ) + CND (mk , X ) + CCT (mk , X ) , where ST, NA, ND, and CT denote structural, non-structural acceleration sensitive, non-structural drift sensitive component, and contents damage, respectively. CDam (mk , X ) in the s-th scenario is:

(16)

k=1

K

)

) q 1 K

For the NCEs, mk , k = 1, 2, , K 1, with exponential inter-arrival times, the term in Eq. (18), q = 1 fWq (t , mk |W1 > t0) = vkE [27]; thus, Eq. (18) is simplified:

(t , t 0 ) dt

C E [CDam (mk , X )]

=

K

K

ln(t

1

t t0

K

The BPLCA framework in this paper is illustrated considering seismic hazards. The seismic hazard was selected because its impact is felt over large areas, and portfolio recovery will require coordinated action on the part of the community. The occurrence of seismic events usually has been modeled as a Poisson process, implying that the earthquake inter-arrival time can be described by an exponential distribution. However, analysis of historical records [27] suggests that the occurrence of earthquakes at certain seismic sources may be related to previous seismic history, and that the Poisson occurrence model may not be appropriate in such cases. Optimal post-hazard portfolio-level BBB decisions should be conditioned on the characteristics of the event that has just occurred; thus, the more sophisticated non-Poisson occurrence model developed by Takahashi et al. [27] is adopted, in which the inter-arrival time for characteristic earthquakes (CEs) (defined as near-periodic occurring earthquakes of magnitude equal to or near the maximum magnitude of a seismic source, as compared to smaller and near-random occurring non-characteristic earthquakes (NCEs)) is modeled by a log-normal distribution. A comparison between nonPoisson Hazard Models, NPHM (CEs are modelled by non-Poisson process while NCEs are modelled by Poisson process) and Poisson hazard models, PHM (both CEs and NCEs are modelled by Poisson process) will be made in Section 5. Consistent with Section 3, the magnitude of earthquakes M is discretized, into K intervals, k (1, 2, , K 1) for non-characteristic earthquakes (NCE), and k = K for characteristic earthquakes (CE) [27]. Eq. (9) is re-written as t0+ TP

1 ln(t0)

ln( )

4. Building portfolio LCA considering seismic hazard

K

(

1

is obtained by substituting the pk in Eq. (15) into Eq. (12).

C E [CDam (X )]

, and LN stands for the log-

fWq (t , mK |W1 > t 0 )

k

=

pk =

in which Tq ~ LN ( K , K ) , q = 1, 2, normal distribution. Eq. (19) becomes:

, mK ) fW1 ( , mK |W1 > t 0 ) d (19) 5

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Fig. 4. Centerville zoning map and location of seismic fault.

5. Case

Wang [35]. Generally, the same fragility and capacity parameters of W1-W6 building types is the same as those in Lin and Wang [20]; To reduce computational time, 1,500 buildings are sampled from the portfolio according to their proportion in each zone. Results in this case study are all based on this reduced sample. The characteristic BPLC, tpc,90 is calculated first, assuming that the mean individual BLC, relates to the occupancy type. Single-family (SF) dwelling units and multi-family (MF) dwelling units are assumed to have the same mean BLC - E [T1]=50 years - while mobile homes (MH) have mean BLC E [T2]=30 years. Using MCS and the procedure described earlier, tpc,90 = 115 years. Since the portfolio is dominated by SF and MF buildings, the portfolio lifetime is almost identical to the BPLC of a portfolio with one type of BLC (T1). In rest of this study, TP = tpc,90 = 115 years is employed for BPLCA by default. Suppose that the building portfolio is severely damaged by a destructive Mw = 8.0 earthquake with an epicenter (Point A in Fig. 4) located southwest of Centerville at a distance of 22.5 km to the centroid of Centerville and that Centerville is situated on Site Class B soil [1]. Table 3 summarizes the post-hazard damage states of the residential

5.1. Study – Centerville building portfolio LCA The community investigated in this study is Centerville, a virtual community [8,20]) chosen to represent a typical mid-size community in the Midwest United States. As shown in Fig. 4, Centerville includes 7 residential zones (Z1-Z7) with approximately 13,500 single and multifamily residential units, summarized in Table 2. Z1 is a high-income/ low-density (HI/LD) zone near the western hills, Z2-Z4 are zones with moderate income (MI), Z5-Z6 are low-income (LI) residential areas close to the business/retail district, and (Z7) is a sizeable mobile home park adjacent to one of the industrial facilities. All residential buildings in Centerville are light frame wood structures with different occupancy types, stories, and year built (denoted as W1–W6 in Table 2). The seismic capacities of these buildings due to the ground motions described in the previous section are characterized by the capacity spectrum approach [11]; this approach, along with the building fragility models that model uncertainty in building behavior, are summarized in Table 2 Building characteristics. Bld. ID

Occup. class

# of occupants

Story

Year built

Area (ft2)

Building value (2003 estimate)

# in Centerville

BLC, Tl

W1 W2 W3 W4 W5 W6

SF1 SF1 SF1 SF1 MF2 MH3

2 3 5 3 90 2

1 1 2 1 3 1

1945–1970 1985–2000 1985–2000 1970–1985 1985 NA

1400 2400 5200 2400 36,000 NA

$139,426 $239,016 $318,816 $239,016 $3,918,960 $61,800

6190 4000 50 3196 102 1352

50 50 50 50 50 30

1. 2. 3. 4.

SF: Single-family dwelling. MF: Multi-family dwelling. MH: Mobile home. Assume content value is 50% of building value according to HAZUS [13]. 6

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Table 3 Number of sampled building in each damage state after a Mw = 8.0 earthquake event. Bld. ID

DS0

DS1

DS2

DS3

DS4

Total

W1 W2 W3 W4 W5 W6

11 9 1 8 0 0

76 80 2 49 1 6

207 166 2 109 5 35

192 133 0 93 2 56

122 40 0 55 1 39

608 428 5 314 9 136

Table 4 Mean annual occurrence rate of each magnitude representative vkE [29] and amplification factor k in BPCPV model. Item

vkE k

= = = = =

1, 1, 1, 1, 1,

= = = = =

0.9 0.8 0.7 0.6 0.5

m1 = 5.5

m2 = 6.5

m3 = 7.5

0.033

0.0033

0.002

1.31 1.60 1.82 1.93 1.91

1.86 2.98 4.16 5.14 5.66

2.82 6.70 13.77 25.08 41.33

P P (t 0, X )] grow with time under different dis(t 0, X )] and E [VCPV Fig. 5. E [CLCC count rates for economic loss.

CReb (X ) (given in Table 7). Both sets of curves diverge as time increases and the curves from the EBPCPV-based methodology, which incorporates risk aversion, are greater than those from the EBPLCC-based methodology for comparable discount rates. In other words, the risk aversion (EBPCPV-based) leads to higher perceived cost from future hazard exposure, which will relate to more conservative reconstruction decisions. By contrast, risk neural decision-makers (EBPLCC-based) tend to rebuild to lower level due to their lower perceived future cost (More formal discussion see the companion paper [36]. When the disP P (X )] be(X )] and E [VCPV count rate is relatively high (e.g. 5%), E [CLCC come almost constant after t = 60 years while for low discount rates (e.g. 1% or less), the Expected Building Portfolio Life-cycle analysis (EBPLCA) value increase even after TP = 115 years, indicating that more risk is being assumed by the present generation. Table 7 summarizes the BPLCA using different hazard models and values of t 0 when the discount rate on economic losses and human casualties are 3% and 1%, respectively. The EBPLCA value and the contribution from each component are greatly affected by t 0 under the NPHM for both methodologies. For instance, if t 0 = 0 (the current C ] from the m3 = 7.5 interval is zero, which earthquake is a CE), E [CDam can be explained as the periodic characteristics of CE and zero value obtained from Eq. (18). When t 0 increases, the contribution of m3 = 7.5 becomes higher. For instance, when t 0 = 500 , the contributions to C C E [CDam (X )] and E [V Dam (X )], from m3 = 7.5 are 56.4% and 81.0% reC C (X )], are (X )] and E[VCas spectively, while the contributions to E [CCas 82.7% and 92.52%, respectively. For the remainder of the case study, unless otherwise indicated, we employ the NPHM with t 0 = 0 as well as 3% and 1% discount rate on economic loss and human casualty, respectively. Next, the effect of portfolio life-cycle (expressed in tpc,90 ) on P P (X )] is investigated. Since tpc,90 depends on BLC Ti E [CLCC (X )] and E [VCPV and the corresponding number, ni , an array of mean BLCs (E [T1], E [T2]) are assigned for single-family and multi-family dwellings and the mean BLC of MH is fixed as E [T2] = E [0.6T1] to obtain an array of tpc,90 . Fig. 6 C C (X )] andE [V New (X )] under PHM illustrates the effect of tpc,90 on E [CNew P (X )] and and NPHM (with different values of t 0 ), showing that E [CLCC P E [VCPV (X )] both decrease monotonically as tpc,90 increases. While this behavior seems counterintuitive, it results from the cumulative new C C (X )orV New (X )) , as the other components either construction cost (CNew C C (X ) , (X ) or V Dam remain the same (CReb (X ) orVReb (X ) ) or increase (CDam C C C C (X ) ) when tpc,90 increases. Eq. (8) (X ) or V Ind CCas (X ) or VCas (X ) , and CInd C C (X )]) is proportional to 1/ i , (X )] (or E [V New shows that the E [CNew where i is the mean BLC of a Type i building; when the mean BLC is short, the cumulative cost due to natural updating within tpc,90 ,

Table 5 Structural reconstruction cost1 in US Dollars of each building in each post-hazard damage state [13]. Bld. ID

W1 W2 W3 W4 W5 W6

Post-hazard Damage State DS1

DS2

DS3

DS4

697 1,195 1,594 1,195 11,757 247

3,207 5,497 7,333 5,497 54,865 1,483

16,313 27,965 37,301 27,965 270,408 4,511

32,626 55,930 74,603 55,930 540,816 15,079

1. The actual reconstruction cost for building in specific damage states are calculated as a percentage of the full cost in the table Table 6 Monetary values in US Dollars of human casualties [14]. Severity Level

S1

S2

S3

S4

Cost

1000

5000

100,000

6,600,000

buildings, revealing that most are damaged to some extent and require either repair or reconstruction. The location of seismic fault for future earthquakes is also illustrated in Fig. 4 (fault AB). The mean occurrence rate of future earthquakes is tabulated in Table 4 [29]. It is assumed that the c.o.v. in inter-arrival time = 0.3 for CE [38,27]. The low mean hazard occurrence rate is meant to represent the LPHC nature of earthquakes in mid-America (e.g. sites near the New Madrid Seismic Zone)[29]. Since the optimal reconstruction decisions are unknown, all the decisions are to rebuild to current performance levels regardless of their post-hazard damage states. More discussion on optimal reconstruction decision based on BPLCA is in Wang et al. [36]. The costs of rebuilding and casualties are tabulated in Tables 5 and 6. With TP = tpc,90 = 115 years and the portfolio post-hazard damage P P (X )] are calcu(X )] and E [VCPV states summarized in Table 3, E [CLCC lated based on different discount rates rd for economic losses and a constant discount rate of 1% for casualties. A NPHM with t 0 = 0 , and x ij = 0 as the default condition is employed. Fig. 5 shows the curves of P P (X )] when the time increases from 0 to TP = 115 E [CLCC (X )] and E [VCPV years for different discount rates of economic losses. When t = 0 , P P E [CLCC (X )] = E [VCPV (X )] = $23.41M , which is the reconstruction cost 7

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Table 7 The EBPLCA decomposition from different hazard models (when x ij = 0 , for Hazard Model

Metric

Poisson hazard model (PHM) Non-Poisson hazard model (NPHM)

t0 = 0 t0 = 300 t0 = 500

EBPLCC EBPCPV EBPLCC EBPCPV EBPLCC EBPCPV EBPLCC EBPCPV

CReb orVReb

23.41 23.41 23.41 23.41 23.41 23.41 23.41 23.41

C C orV New CNew

202.04 202.04 202.04 202.04 202.04 202.04 202.04 202.04

i and j ) in million dollars.

C C orV Dam CDam

C C orVCas CCas

m1 = 5.5

m2 = 6.5

m3 = 7.5

Sum

m1 = 5.5

m2 = 6.5

m3 = 7.5

Sum

10.84 17.39 10.84 17.39 10.84 17.39 10.84 17.39

4.92 14.66 4.92 14.66 4.92 14.66 4.92 14.66

7.61 7.61 0.00 0.00 7.19 48.19 20.41 136.77

23.38 39.65 15.77 32.04 22.96 80.23 36.18 168.81

0.19 0.31 0.19 0.31 0.19 0.31 0.19 0.31

0.43 1.27 0.43 1.27 0.43 1.27 0.43 1.27

1.16 1.16 0.00 0.00 1.27 8.53 2.92 19.54

1.78 2.73 0.62 1.57 1.89 10.11 3.53 21.12

C orV C CInd Ind

P P or VCPV CLCC

23.38 39.65 15.77 32.04 22.96 80.23 36.18 168.81

273.98 307.49 257.60 291.11 273.25 396.02 301.34 584.19

P (X )] to the CPT parameters is considered in The sensitivity of E [VCPV Fig. 7, in which the contributions from five components are illustrated (see Eq. (13)). Note that in Fig. 7(a) the second and third line from bottom up are almost together on top of each other, and can be barely C ] is very small when t 0 = 0 ). Lower distinguished in the left side (E [CCas P values of lead to higher E [VCPV (X )] since the decision-maker is overestimating the probability of an extreme event and is more riskC (mk , X )] from the CE is averse. Furthermore, the contribution to E [V Dam more sensitive to the than that from the lower magnitude NCE. A comparison of Fig. 7(a) and 7(b) reveals that the effect of different on C = 1, E [V Dam ] also depends on t 0 . Note that when P P E [VCPV (X )] = E [CLCC (X )] because with that value, the probability mapping function is linear.

6. Summary and conclusions In this paper, life-cycle analysis for individual buildings was extended to a community building portfolio consisted of wood frame residential buildings to provide support for policies that might ultimately form the basis for optimal building reconstruction decisions following severe earthquakes. The duration of the building portfolio life-cycle and the building portfolio renewal rate were key factors in this extension. The building portfolio life-cycle was defined as the time during which 90% of the buildings within a community are demolished and reconstructed at least once due to natural building turnover, while the building portfolio renewal rate reflects the annual rate at which old or

P P (X )] under different tpc,90 for different hazard (X )] and E [VCPV Fig. 6. E [CLCC models and values of.t0 .

C E [CNew (X )]is high since demolition and re-construction occur in a short period with high discount factor (larger e rd (t t0) ), and vice versa. This supports the idea that individual buildings and portfolios with longer life-cycles will lead to lower LCA values and thus are more sustainable.

P (X )] under different Fig. 7. E [VCPV

by using Non-Poisson hazard model with (a), t0 = 0 and (b)t0 = 300. 8

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obsolete buildings are replaced by new buildings under normal conditions. Two life-cycle analysis methodologies - expected building portfolio life-cycle cost and expected building portfolio cumulative prospect value – enable both risk-neutral and risk-averse decision stances. Both methodologies were illustrated using a typical mid-size community exposed to seismic hazards. The life-cycle duration of a building portfolio is much longer than that of individual buildings but can be obtained from the life-cycle of buildings within it. Building portfolios with longer life-cycle durations (individual buildings within relate to longer life-cycle) have lower lifecycle impact and are more sustainable. Further, the temporal characteristics of the seismic hazard have a significant impact on the result of building portfolio life cycle analysis, given the extended life-cycle length of portfolios. The expected cumulative prospect value model permits risk-averse decision-makers to focus on the risks from extreme events over the life-cycle of building portfolio and further enhance the resilience of community. While the case study presented in this paper focused on seismic performance, the methodology is sufficiently general to apply to other hazards. One limitation of the current portfolio life-cycle analysis is that it assumes that the number of buildings within the portfolio is relatively stable; however, communities in real world trend to expand in size due to economic growth or shrink due to decline in local economy or population out-migration after catastrophic hazards. Further, it is assumed the long-term mean occurrence rate and intensity of hazards to be stable, however, weather related hazards (e.g. hurricanes) may change in frequency and intensity. These limitations provide directions for future studies. Nonetheless, the building portfolio life-cycle analysis framework developed in this paper can systematically quantify a specific metric associated with a building portfolio over long time under the post-hazard rebuilding context and form the basis for developing optimal post-hazard rebuild decisions in the companion paper [36] to support building back better.

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Acknowledgment This research is supported by the National Key R&D Program of China (Grant No. 2017YFE0119500), and the US National Science Foundation under Collaborative Research (Grant Nos. CMMI-1452708 to the University of Oklahoma and CMMI-1452725 to Colorado State University). The supports are gratefully acknowledged. The authors would also like to thank Kate Simonen at University of Washington for providing insights on the subject matter. References [1] ASCE. Minimum Design Loads for Buildings and Other Structures. Reston, VA: American Society of Civil Engineers; 2016. p. 2016. [2] Campbell KW. Prediction of strong ground motion using the hybrid empirical method and its use in the development of ground-motion (attenuation) relations in eastern North America. Bull Seismol Soc Am 2003;93(3):1012–33. [3] Cha EJ, Ellingwood BR. Risk-averse decision-making for civil infrastructure exposed to low-probability, high-consequence events. Reliab Eng Syst Saf 2012;104:27–35. [4] City and, County of San Francisco. Ordinance 66–13, 2013: Building Code – Mandatory Seismic Retrofit Program – Wood-Frame Buildings; 2013. [5] Clinton, W.J. Lessons learned from tsunami recovery: Key propositions for building back better. New York: Office of the UN Secretary-General's Special Envoy for Tsunami Recovery; 2006. [6] Czajkowski J. Moving from risk assessment to risk reduction: an economic perspective on decision making in natural disasters. The Bridge 2015;45(4):52–9. [7] Ellingwood BR, Wen YK. Risk-benefit-based design decisions for low-probability/ high consequence earthquake events in Mid-America. Prog Struct Mat Eng

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