Accounting for Time and State-Dependent Vulnerability of ...s-space.snu.ac.kr/bitstream/10371/153549/1/465.pdf · Accounting for Time and State-Dependent Vulnerability of Structural

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13Seoul, South Korea, May 26-30, 2019

Accounting for Time and State-Dependent Vulnerability of StructuralSystems

Maricar L. RabonzaPhD Student, Asian School of the Environment, Nanyang Technological University,Singapore

David LallemantAsst. Prof., Asian School of the Environment, Nanyang Technological University,Singapore

ABSTRACT: The typical process of engineering risk analysis assumes a static state of vulnerabilitythrough the lifespan of the structure. However, many civil engineering systems change states over timecausing significant impact on their vulnerability. Such dynamic changes may involve an increase invulnerability driven by deterioration processes (e.g. corrosion, fatigue, creep, hazard-induced damage,etc.), or a decrease in vulnerability driven by strengthening interventions (e.g. retrofitting, maintenance,building replacement, etc.). Accounting for these dynamics is critical to properly understand hazard-related risk of civil engineering systems over their lifespan. This paper presents a stochastic frameworkfor accounting for time and state dependent vulnerability in risk analysis of civil engineering systems.Time-homogeneous Markov chains are used to model various state change processes, and integratedwithin the risk analysis framework in closed-form expressions. Several applications are demonstrated:(1) quantifying risk of structurally deteriorating buildings and the risk reduction impact of maintenance,(2) urban-scale seismic retrofitting policies based on various retrofit rates, and (3) impact of varying ratesof building replacement to higher design grade. These demonstrate the importance of accounting for timedependent state change as a significant factor in the life-span vulnerability of the built environment. Thestudy further provides a framework to study and compare various risk reduction policies.

1. INTRODUCTION

Developing well-informed and proactive strategiesfor seismic risk reduction requires the abilityto predict risk in constantly changing urbanenvironments, and understand how risk reductionpolicies affect future seismic risk. However,current seismic risk assessment methods focus onunderstanding risk to infrastructure in only theirpresent state.

In this paper, we present a flexible frameworkaccounting for time and state dependentvulnerability in seismic risk analysis. Thisallows for the modelling of processes that canincrease vulnerability (e.g. deterioration, buildingexpansions, cumulative damage), those policies

that mitigate increase in vulnerability (e.g.better maintenance schedule, higher durabilityconstruction) and those policies that improveresilience (e.g. seismic retrofits and buildingreplacement to higher standards).

The framework is applied to hypotheticalcase studies to demonstrate the effects of timeand state dependent vulnerability on a singledeteriorating building and to a neighborhood withbuildings experiencing deterioration, retrofitting,and building replacements over time. Whileit is expected that retrofit policies and buildingreplacements lead to decreased seismic risk,and deterioration leads to increased seismic risk

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over time, this paper demonstrates how riskevolves with time linked to various seismicreduction policies. This methodology is anextension of a time-dependent framework appliedto investigate incremental building expansion asthe significant driver for increasing risk andvulnerability (Lallemant et al., 2017).

The key contribution of this paper is toprovide a tool for stakeholders to investigatethe consequences of various seismic mitigationdecisions to future seismic risk. The case studypresents a proof of concept and a demonstrationrather than actual risk prediction. The frameworkcan be utilized to study a more realistic case onceinformation for transition rates, vulnerability andbuilding stock distribution are available.

2. METHODOLOGY2.1. Proposed framework accounting for time

dependent vulnerability in seismic riskanalysis

The Performance-Based Earthquake Engineering(PBEE) methodology proposes a systematicmethodology to calculate the seismic risk ofstructures through the probabilistic integrationof (1) seismic hazard, (2) seismic demand, (3)damage capacity, combined into a single impactassessment (Krawinkler and Miranda, 2004). Eachof these components are described in Table 1.

Each step in the probabilistic riskanalysis framework comes with inherentuncertainty/variability, thus each associatedvariable is expressed in the form of conditionalprobability of exceedance (Figure 1. With thebasic PBEE framework, the final expression for thedecision variable, λDV ,is obtained by combiningthe conditional probabilities using the totalprobability theorem. The decision variable thenserves as a guide for infrastructure managers tocreate appropriate risk management and mitigationstrategies based on a performance target or riskmetric.

λDV =∫im

∫ed p

∫dm

GDV |DM(dv/dm)

× |dGDM|EDP(dm|ed p)|× |dGEDP|IM(ed p|im)|νIM

(1)

The basic PBEE framework is not capableof dealing with time-dependence of each ofthe variables, and does not incorporate otherpotential time-dependent drivers of vulnerabilitythat could affect future seismic risk. Usingthe PBEE framework as a starting point,the proposed framework incorporates a newmodule to account for potential time dependentdrivers of vulnerability. A similar performance-based framework which accounts for timevarying effects of deterioration for bridges wasproposed by Rao et al. (2017). Building onthe PBEE methodology and Rao et al. (2017)’sframework, we propose a flexible frameworkaccounting for time and state dependent seismicvulnerability, incorporating drivers such asstructural deterioration, maintenance schedule,seismic retrofit policies, building replacementrates and other drivers of vulnerability. Thekey component in this proposed framework isthe analysis of the ’time varying vulnerabilitystate’ with associated variable V S and probabilityfvs(vs, t) of being in a vulnerability state vs at timet. This is the mathematical representation of theselected time and state dependent vulnerabilitydriver. An illustration of the proposed frameworkis shown in Figure 1 along with the associatedvariables, stochastic representations and examplesfor each component. Finally, the full equation forthe probabilistic decision variable of interest ispresented which describes the proposed frameworkin detail. The equation in Figure 1 is characterizedby treating the variables as continuous. However,the equation can easily be rewritten to handlediscrete states by replacing the integration by asummation over the discrete vulnerability states.

2.2. Mathematical representation of vulnerabilitystates and corresponding transition scenario

Along with the proposed framework shown inFigure 1, we illustrate in Figure 2 several potentialtime-dependent vulnerability drivers for seismicrisk.

A certain vulnerability state can be dependenton the building’s degree of deterioration, stateof expansion, current structural condition, orthe developed retrofit standard depending on

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Table 1: Four components and associated variables of the Performance-Based Earthquake Engineeringframework. Descriptions based on explanations by Krawinkler and Miranda (2004); Deierlein (2004); Moehleand Deierlein (2004)

Framework component Variable name Description of variable

Hazard analysis Intensity measure (IM) Earthquake induced shaking at study siteDemand analysis Engineering demand parameter (EDP) Response of structure to earthquake loadingDamage capacity modelling Component damage measure (DM) Seismic-induced damage sustained by structureImpact assessment Decision variable(DV) Performance-related variables for decision making

Figure 1: Proposed framework accounting for time dependent vulnerability in seismic risk analysis

the significant vulnerability drivers of interestto a certain structure. Given in Figure 2 aresuggested relative vulnerability states exhibitingincreasing seismic vulnerability (for deteriorationand building replacement) and decreasingvulnerability/increased resilience states (forseismic retrofitting).

Using Markov Chains is a simple approach torepresent the transition of these discrete statesover time. Markov chains are used to mapthe probability of transitioning from one state toanother state in a specific time interval. Markovmodels are “memoryless”, such that the new state issolely dependent on the current state; not on the set

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Figure 2: Potential time-dependent vulnerability drivers for seismic risk and associated transition scenario

of events preceding it (Agresti, 2003). The columnshowing the ’Transition Scenario’ provides aschematic of the types of transition processes linkedto each time-dependent vulnerability driver, andcan then be mapped into corresponding transitionprobability matrices.

For instance, a non-retrofitted building can eithertransition into a retrofitted state with a certainprobability within a year, or it can stay in its currentstate. Similarly, a building can also transition into amore deteriorated state over time with an associatedtransition probability. A common assumption foranalysis incorporating structural deterioration orseismic retrofitting is that once a certain state isreached, it cannot go back to a previous state.For example, we assume that once a building isretrofitted, it is not possible for the building to goback to its unretrofitted state. In the schematic forthe transition scenarios (Fig. 2) of deteriorationand seismic retrofitting, this is shown by having alltransition arrows going to the right only. On thecontrary, possible transition patterns for buildingreplacements can bring back a building to itsoriginal (as-new) state as time goes by as presentedby the arrows going to the left in its transitionscenario diagram (Figure 2). Alternatively, abuilding could be replaced by another built tohigher standards, as is often the case resultingfrom building code improvements. The same

figure also shows a sample transition probabilitymatrix for each vulnerability driver using the giventime-dependent vulnerability states. The size ofthe transition probability matrix depends on thepotential vulnerability states considered in theanalysis, but they should exhibit similar pattern.

Depending on the transition rates, di,i, bi,i, ri,i,the transition between the vulnerability states cango either slower or quicker based on differentfactors. For example, building deterioration ratesare usually affected by the level of environmentalexposure of the structure, its initial structuralquality, or the implemented maintenance schedulewhich could potentially mitigate the seismic riskover time. It should be noted that the factorsaffecting transition rates don’t necessarily affect theactual fragility curve for each state.

Fragility curves define the state of vulnerabilityof a building or structure. These curves showthe relationship of the earthquake intensity and theprobability of exceeding a particular damage level.

Numerous methods exist to derive these curves:(1) analytical (Singhal and Kiremidjian, 1996;Lallemant et al., 2015), (2) empirical (Sanchez-Rodriguez et al., 2005; Noh et al., 2015) orheuristic/ based on expert opinion (Jaiswal et al.,2012).

Given a hazard curve at a study site, theannual collapse rate of a building is calculated by

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integrating the fragility curve over the hazard curvein Equation 2.

λCollapse =

IMmax∫IMmin

P(Collapse|IM = im)|dλim(im)|

(2)

where λim(im) is the seismic hazard curve and|dλim(im)| is the absolute value of the derivative ofthe hazard curve.

For a portfolio of buildings the annual expectednumber of building collapse at a time t can becalculated using Equation 3.

λCollapseTotal(t) =Staten

∑State1

IMmax∫IMmin

P(Collapse|IM = im)|State = Statei)

× P(State = Statei|Do = do,P = p)(t)× |dλIM(im)|

(3)

where P(Collapse|IM = im)|State = Statei isthe fragility curve for each building state atcollapse, P(State = Statei|Do = do,P = p)(t) is theprobability of being in a building state at time t fora given transition probability matrix p and initialstate distribution do.

It can be shown that the expected vulnerabilitystate distribution Dt at time t is E(Dt |Do = do) =doPt where P is the transition probability matrix ofvulnerability states. Therefore 3 can be re-writtenas:

λCollapseTotal(t) =Staten

∑State1

IMmax∫IMmin

P(Collapse|IM = im)|State = Statei)

× doPt |dλIM(im)|

(4)

3. HYPOTHETICAL CASE STUDY3.1. Building-level deterioration analysisWe apply the methodology discussed previouslyto model the changing risk of a hypotheticaldeteriorating building over time, we focus onthree (3) vulnerability states ranging from as-new,heavily deteriorated and very heavily deteriorated

for a hypothetical reinforced concrete (RC)building.

A hypothetical vulnerability curve is generated torepresent the non-deteriorated state. For simplicity,we only use fragility curves corresponding to"Extensive/ Complete Damage.”

Vulnerability curves for the three (3) assumedvulnerability states corresponding to each degree ofdeterioration,w, are obtained using Equations 5 and7 (Rao et al., 2017).

m(w) = moe−αmw (5)

ξ (w) = ξo(1−αξ w) (6)

wherem(w), ξ (w)= median and dispersion of fragilityfunction at level of deterioration w consecutively ,mo, ξo = median and dispersion of the fragilityfunction for the column in its non-corroded stateconsecutivelyαm= exponential decrement function for the medianandαξ = coefficient of the linear decrement function forthe dispersion of the fragility function.

The coefficients of the decrement function areadopted from estimates by Rao et al. (2017) for ahypothetical RC column built in 1960 to pre-1971design standards: m = 1.43, αξ =−0.18

Using the framework for this application,we compare the impact of different levels ofmaintenance on seismic risk over time. Threemaintenance schemes are used to demonstrate thediversity of maintenance options for seismic safety:(1) Low, (2) Medium, and (3) High Maintenance.Transition probability matrices are assumed basedon a Markovian model developed by Duling (2006)for an RC building constructed with pre-1971design standards to predict the building service lifegiven three varying levels of maintenance. Thepercent change in annual collapse risk normalizedto baseline risk at t=0 is calculated using Equation2 and presented in Figure 3.

The trends shown in the figure highlight theimpact of building deterioration on the seismicrisk of buildings, and the benefits of maintenance;it demonstrates the importance of accounting

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for time-dependent vulnerability drivers such asdeterioration in studying future seismic risk ofa building. This demonstration shows thatthe proposed framework enables the testing ofimpact of maintenance or other building mitigationstrategies on seismic risk over time. If linked withfinancial loss information, the framework could beused for cost benefit analysis for mitigation.

Figure 3: Percent change in annual collapse risknormalized to baseline risk at t=0 for a hypotheticalbuilding

3.2. Policy analysis for community level seismicrisk reduction

Using the proposed framework, we can alsotest the impact of various seismic risk reductionpolicies at a regional level. A hypotheticalurban community was simulated, consisting offour districts each having their own buildingtype distribution and seismic hazard curve. Forsimplicity of demonstration, the design of buildingsare either high grade or low grade, and eachcan transition to deteriorated states, retrofittedstates (for low-grade buildings), or get replacedover time. Hypothetical fragility curves aredeveloped to represent each of these states:(1) Undeteriorated/unretrofitted state (2) Heavilydeteriorated state (3) Very heavily deteriorated state(4) Retrofitted state with low standard and (5)Retrofitted state with high standard. The fragilitycurves for each vulnerability state are shown infigure 4.

Hazard curves for each district are syntheticallygenerated as idealized power-law hazard curves of

Table 2: Indices of Transition Probability Matrices.(Abbreviations: HD = Heavily deteriorated, VHD =Very heavily deteriorated, n/a = Unretrofitted building

Index P1 P2 P3 P4 P5 P6 P7 P8

Buildingdesign grade high high high low low low any any

Degree ofdeterioration as-new HD VHD as-new HD VHD as-new as-new

Retrofitstandard n/a n/a n/a n/a n/a n/a low high

the following form:

λim(IM) = koIM−k (7)

Parameters used for the four districts are k0 =0.0002, 0.0003, 0.00022, 0.00035 and k = 2, 2.1,2.2, 2.5 for districts 1, 2, 3 and 4 respectively.

Figure 4: Fragility curves for assumed vulnerabilitystates at the hypothetical building stock

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Figure 5: Transition probability matrix calibrated for alow quality seismic risk reduction scheme described inTable 3

Figure 6: Transition probability matrix calibrated for ahigh quality seismic risk reduction scheme described inTable 3

The purpose of the framework developed isto compare the impact of various policies onseismic risk over time. The types of decisionsin the policy-making space includes the levelof retrofit standards used, maintenance scheduleand rate of development (building replacement) ineach district. All these are being implementedwhile taking into account deterioration rate.Mathematically, the transition matrix for this typeof problem is represented by a combination ofthe typical transition probability matrices shownin Figure 2. Two community-level policiesare simulated using the proposed framework asdescribed in Table 3. Corresponding transitionprobability matrices for each policy are shown inFigures 5 and 6. Each value corresponds to theprobability of each state described in Table 2 totransition to the next state.

Using Equation 3, the change in risk linked tothese two policies implemented on the hypotheticalbuilding stock is demonstrated in Figure 7. Asexpected, mandatory retrofit schemes with shorter

time frames result in early and rapid reduction inseismic risk (Figure 7). Also, better maintenanceschedules significantly slow down the deteriorationof a building portfolio thus reducing seismic riskover time. Demonstrated in the hypotheticalbuilding stock case as well is that encouragingdevelopment or high building replacement rates tobetter code standards contributes to seismic riskreduction over time. This demonstration of a policyanalysis for community level seismic risk reductiondemonstrates the capability of the proposedframework to compare various policy choicesrelated to different standard of improvements suchas building codes or time frame for which thesepolicies are enforced/implemented.

Note that the framework can be used totest complex combinations of policies, includingencouraging development in lower-hazard districts,different retrofit time-frames, retrofit standards,new building codes, and much more.

Figure 7: Percent change in annual collapse risknormalized to baseline risk at t=0 for a hypotheticalbuilding stock. Refer to Table 3 for descriptions of eachpolicy

4. CONCLUSION

This paper presents a flexible frameworkaccounting for time dependent vulnerability inseismic risk analysis. This enables modellingboth of those processes that increase vulnerability(e.g.deterioration, building expansions, cumulativedamage), those policies that mitigate increase invulnerability (e.g. better maintenance schedule,higher durability construction) and those policies

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Table 3: Features of seismic reduction policies tested on a hypothetical community

Policy feature Low quality seismic reduction scheme High quality Seismic reduction scheme

Retrofit policy Voluntary, long time frame, low standard Mandatory, short time frame, high standardBuilding replacement policy Low rate of replacement High rate of replacementMaintenance schedule Low level maintenance schedule High level maintenance schedule

that improve resilience (e.g. seismic retrofits andbuilding replacement to higher standards).

The methodology builds on the fundamentalprobabilistic performance based framework byadding a component for time-varying featureswhich affect vulnerability. Overall, the frameworkallows stakeholders to study the consequencesof different mitigation schemes to future seismicrisk, and analyze its sensitivity to the initialbuilding stock quality, the structural deteriorationrate, maintenance schedule, features of mandatoryretrofit policies in terms of their timeframeand standard, building replacement rate, urbandevelopment rates and pattern, and other drivers ofchanging risk.

5. ACKNOWLEDGEMENTS

This research is funded through a NationalResearch Foundation (Singapore) Fellowshipgrant (NRF–NRFF2018–06), along with an EarthObservatory of Singapore scholarship.

6. REFERENCESAgresti, A. (2003). Categorical data analysis, Vol. 482.

John Wiley & Sons.

Deierlein, G. G. (2004). “Overview of a comprehensiveframework for earthquake performance assessment.”Performance-Based Seismic Design Concepts andImplementation, Proceedings of an InternationalWorkshop, 15–26.

Jaiswal, K., Aspinall, W., Perkins, D., Wald, D., andPorter, K. (2012). “Use of expert judgment elicitationto estimate seismic vulnerability of selected buildingtypes.” Proc. 15th World Conference on EarthquakeEngineering, Lisbon, Portugal, 24-28 Sep 2012.

Krawinkler, H. and Miranda, E. (2004). “Performance-based earthquake engineering.” Earthquakeengineering: from engineering seismology toperformance-based engineering.

Lallemant, D., Burton, H., Ceferino, L., Bullock, Z.,and Kiremidjian, A. (2017). “A framework andcase study for earthquake vulnerability assessmentof incrementally expanding buildings.” EarthquakeSpectra, 33(4), 1369–1384.

Lallemant, D., Kiremidjian, A., and Burton, H. (2015).“Statistical procedures for developing earthquakedamage fragility curves.” Earthquake Engineering &Structural Dynamics, 44(9), 1373–1389.

Moehle, J. and Deierlein, G. G. (2004). “A frameworkmethodology for performance-based earthquakeengineering.” 13th world conference on earthquakeengineering, Vol. 679.

Noh, H. Y., Lallemant, D., and Kiremidjian, A. S.(2015). “Development of empirical and analyticalfragility functions using kernel smoothing methods.”Earthquake Engineering & Structural Dynamics,44(8), 1163–1180.

Rao, A. S., Lepech, M. D., and Kiremidjian, A. (2017).“Development of time-dependent fragility functionsfor deteriorating reinforced concrete bridge piers.”Structure and Infrastructure Engineering, 13(1), 67–83.

Sanchez-Rodriguez, R., Seto, K. C., Solecki, W. D.,Kraas, F., and Laumann, G. (2005). “Scienceplan: urbanization and global environmental change.”Science plan: urbanization and global environmentalchange.

Singhal, A. and Kiremidjian, A. S. (1996). “Methodfor probabilistic evaluation of seismic structuraldamage.” Journal of Structural Engineering, 122(12),1459–1467.

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Accounting for Time and State-Dependent Vulnerability of ...s-space.snu.ac.kr/bitstream/10371/153549/1/465.pdf · Accounting for Time and State-Dependent Vulnerability of Structural