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Structure and Infrastructure Engineering:Maintenance, Management, Life-Cycle Design andPerformancePublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/nsie20

Life-cycle optimisation in earthquake engineeringLuis Esteva a , Dante Campos b & Orlando Díaz-López aa Instituto de Ingenieria Ciudad Universitaria , 04320, México, DF, Méxicob Instituto Mexicano del Petróleo , Eje Central Lázaro Cárdenas 152, Cd. San Bartolo,Atepehuacan, 07730, México, DF, MéxicoPublished online: 25 Mar 2010.

To cite this article: Luis Esteva , Dante Campos & Orlando Díaz-López (2011) Life-cycle optimisation in earthquakeengineering, Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and Performance, 7:1-2,33-49, DOI: 10.1080/15732471003588270

To link to this article: http://dx.doi.org/10.1080/15732471003588270

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Life-cycle optimisation in earthquake engineering

Luis Estevaa*, Dante Camposb and Orlando Dıaz-Lopeza

aInstituto de Ingenieria Ciudad Universitaria, 04320 Mexico, DF, Mexico; bInstituto Mexicano del Petroleo, Eje Central LazaroCardenas 152, Cd. San Bartolo, Atepehuacan, 07730, Mexico, DF, Mexico

(Received 19 November 2008; final version received 16 December 2009; published online 25 March 2010)

An overview is presented of life-cycle optimisation in the establishment of reliability- and performance-based seismicdesign requirements for multi-storey systems. Alternative approaches are presented for the development of seismicvulnerability functions that do not require the determination of lateral deformation capacities. The influence ofdamage accumulation on the evolution of the seismic reliability functions is discussed, and some results arepresented about the sensitivity of the seismic reliability and performance functions to the contribution of energy-dissipating devices to the lateral strength and stiffness of multi-storey frames. The process of structural damageaccumulation resulting from the action of sequences of seismic excitations is taken into account in the assessment oflife-cycle system reliability and performance, and in the formulation of reliability and optimisation criteria andmethods for the establishment of structural design requirements and for the adoption of repair and maintenancestrategies. Problems related to the transformation of research results into practically applicable seismic designcriteria are briefly discussed. Several illustrative examples are presented.

Keywords: life-cycle optimisation; structural reliability; seismic risk

1. Introduction

For a long time, life-cycle optimisation has beenrecognised as an implicit and essential objective ofengineering design, construction and maintenanceactions. For the establishment of recommended prac-tical criteria and methods for the achievement of thatobjective, it is necessary to adopt a formal decisionframework, based on the identification and evaluationof an adequate set of quantitative indicators todescribe the performance of a given system. In general,this description will be strongly affected by significantuncertainties, associated with (a) the random varia-bility of the times and intensities of future excitationsand of the mechanical properties of the system, (b) ourimperfect knowledge about these concepts, and (c)the limitations of the mathematical models used torepresent them. This leads to the adoption of a pro-babilistic framework to describe the expected per-formance of a system, accounting for two types ofuncertainties: random or aleatory (group a, above) andepistemic (groups b and c). Dealing with the latter twogroups may require the use of subjective probabilities.In earthquake engineering problems, the excitationsinclude the gravitational loads (dead and live) and theseismic events. The expected performance of a systemwhen subjected to any of those events depends on boththe intensity of the latter and the mechanical properties

of the former, which depend in turn on the level anddistribution of damage that may have accumulated asa consequence of the system’s response to previousseismic events or of the action of any other agent, suchas differential settlements due to gravitational loads.

All types of damage costs and consequencesproduced by an earthquake on an engineering systemare strongly related to the levels of physical damagethat the system may experience when subjected toit, including the possibility of partial or total collapse.Therefore, determining the physical vulnerability func-tion of a system, in terms of the criteria and methodsadopted for its design and construction constitutesthe first step in the formulation of the correspondinglife-cycle optimisation analysis.

An important consequence of damage accumulationis the increase of the ordinates of the seismic vulner-ability function of the system. In order to maintain theresulting vulnerability levels within acceptable limits,adequate maintenance and rehabilitation policies mustbe implemented, for which the scope, control variablesand acceptance criteria should be based on a life-cycleoptimisation analysis, consistent with that adopted forthe establishment of optimum seismic design criteriaand quality control requirements.

An important problem in the formulation ofoptimum life-cycle decision criteria is related to thecomparison of costs and benefits that may occur at

*Corresponding author. Email: lestevam@iingen.unam.mx

Structure and Infrastructure Engineering

Vol. 7, Nos. 1–2, January–February 2011, 33–49

ISSN 1573-2479 print/ISSN 1744-8980 online

� 2011 Taylor & Francis

DOI: 10.1080/15732471003588270

http://www.informaworld.com

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different instants in time. The adoption of objectivefunctions based on the present value of expectedutilities, both positive (benefits) and negative (losses,costs) is ordinarily used to cope with this problem.As is well known, very difficult questions arise inconnection with the assignment of quantitative valuesto benefits and costs that cannot be directly expressedin monetary terms. They include the concepts knownas intangible, such as losses of human lives, improve-ments or reductions in the quality of life, social andpolitical problems. Although the formulations pre-sented here avoid dealing explicitly with these con-cepts, it is expected that the general frameworkproposed can serve as a support for the formulationof decision criteria accounting for them.

This article aims at presenting an overview of thegeneral criteria and the available models and tools toformulate a framework for life-cycle optimisation inearthquake engineering. Attention is focused on multi-storey buildings with structural arrangements consist-ing of a rigid frame or a dual wall-frame system. Someproblems associated with the determination of seismicvulnerability functions for those systems are discussed,and some ideas are presented about alternative appro-aches and desirable studies. Optimum maintenanceprogrammes that account for the influence of damageaccumulation are examined. For systems with energy-dissipating devices, these programmes include bothrepair of local damage in members of the structuralsystem, and replacement of energy-dissipating devicesthat have failed or that are suspected to havedeveloped high fatigue levels.

The article ends with several illustrative examples,showing the process and the results of applying theforegoing concepts to the determination of optimumdesign criteria and maintenance strategies to sometypical problems found in earthquake engineeringpractice.

2. Seismic vulnerability and risk functions

2.1. Failure probabilities and expected damage underthe action of a single excitation

In general, the seismic vulnerability function of asystem can be represented as follows:

�d yð Þ ¼ E d yj½ � ¼ pF yð ÞdF þ 1� pF yð Þð Þ�d y Sjð Þ ð1Þ

Here, pF(y) is the probability of ultimate failure(collapse) of the system when subjected to an earth-quake of intensity y, dF is the expected cost of failure incase it occurs, normalised with respect to the initialcost, C0, and �d y Sjð Þ is the expected value of thenormalised cost of damage for an earthquake withintensity y, conditional to survival of the system under

the action of that event. Ground motion intensity isassumed known, but the gravitational, loads and themechanical properties of the structural members aretaken as random.

For a multi-storey building, the expected value ofthe damage produced by an earthquake with intensityy is often expressed as a function of the expected valueof the global distortion of the system, C ¼ wN/H,where H is its height of the building above the groundsurface and wN is the peak value of the relative lateraldisplacement of its top with respect to its base. Inmany cases it is convenient to obtain the expected costof damage as the sum of the contributions of severalsegments of the system, as given by Equation (2)(Esteva et al. 2002):

�d y Sjð Þ ¼ lXi

rci�g Ci yjð Þ ð2Þ

In this equation, rci ¼ C0i/C0, C0i is the initial costof the ith segment of the system, g(Ci ) a function of therandom value Ci of the corresponding local distortion,and �g(Cijy) its expected value for intensity y. The initialcosts C0 and C0i, as well as the joint probability densityfunction of the local distortionsCi, are functions of thevector a of structural parameters A factor l, which is afunction of the summation that follows it, is introducedto account for the fact that repair costs include thecontribution of a fixed amount that reflects the costs ofthe logistic arrangements that have to be made beforethe actual repair work starts. As a consequence, l willin general reach its maximum for infinitely small valuesof the summation mentioned above, and it will tendasymptotically to a smaller value as that summationgrows. More details about the determination of theexpected damage function given by Equation (2) havebeen presented by Esteva et al. (2002).

Current approaches for the determination of theseismic reliability of a system subjected to an earth-quake ground motion with a specified intensitypropose to measure that reliability by the probabilitythat C is smaller than the deformation capacity, CC.Approximate estimates of second-moment probabilis-tic indicators ofCC are often obtained with the aid of asimplified reference system, characterised by mecha-nical properties determined by means of a staticpushover analysis of the detailed system. However,these estimates are tied to severe limitations, becauseaccording to this approach it is not possible to accountfor (a) the influence of cumulative damage associatedwith the cyclic response, and (b) the dependence of thelateral deformation capacity on the response config-uration of the system when it approaches failure.

Trying to avoid the introduction of arbitraryassumptions about the determination of the deformation

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capacity of a complex system for the purpose ofobtaining estimates of its seismic reliability, alternativecriteria have been proposed, according to which systemfailure is assumed to take place when the displacementspredicted by the dynamic response analysis becomeindefinitely large and non-reversible. The effective valuesof the elements of the resulting stiffness matrix are theninfinitely small. This condition is described as systemcollapse (Esteva 1992, Shome and Cornell 1999,Alamilla and Esteva 2006). In order to determine thesafety factor of a given systemwith respect to this type offailure for a given ground motion time historey, it isnecessary to obtain the scale factor that has to be appliedto that time historey in order to produce system collapse.The intensity leading to collapse is then denoted as‘failure intensity’. Because the determination of theneeded scaling factor has to be attained by means of aniterative procedure, it may call for excessive demands ofcomputer time.

The method of incremental dynamic analysis (IDA,Vamvatsikos and Cornell 2002) offers both possibilitiesfor the estimation of probabilistic indicators of seismicreliability for given ground motion intensities: eitheron the basis of deformation capacities or using theconcept of failure intensity. However, these advantagesare often tied to excessive computer time demands(Dolsek and Fajfar 2004). This has led Esteva andIsmael (2004) and Esteva and Dıaz-Lopez (2006) toexplore an alternative approach, aiming at estimatingreliability functions relating the reliability index b(Cornell 1969) with the ground motion intensity,including the influence of cumulative damage andavoiding the need to obtain probabilistic definitions oflateral deformation capacities. This approach is basedon the concept of ‘failure intensity’, mentioned above.According to this approach, the collapse condition isexpressed in terms of a secant-stiffness reduction indexDk ¼ (K07K)/K0, where K0 is the initial tangentstiffness associated with the base-shear vs roof dis-placement curve resulting from pushover analysis andK is the secant stiffness (base shear divided by lateralroof displacement) when the lateral roof displacementreaches its maximum absolute value during the seismicresponse of the system. The failure condition isexpressed as Dk ¼ 1.0.

According to the approach proposed by Esteva andDıaz-Lopez (2006) and Esteva et al. (2010), a reliabilityfunction b(y) is obtained from a sample of pairs ofvalues of Dk and y, where b is the safety indexproposed by Cornell (1969) and y is the ground motionintensity. If the sample includes only cases with Dk

smaller 1.0, the reliability function can be obtained bymeans of a regression analysis; if cases with Dk ¼ 1.0are also included, a maximum likelihood analysis mustbe performed. Instead of formulating the problem as

that of obtaining an indicator of the probability thatDk 5 1.0 (survival) for a given intensity, attention isfocused on the determination of second momentindicators of the probability distribution of ZF ¼lnYF, where YF is the minimum value of the intensityleading to the condition Dk ¼ 1.0 (collapse). For anearthquake with intensity equal to y, a safety marginZM can be defined equal to the natural logarithm ofthe ratio of the system capacity to the amplitude of itsresponse to the given intensity; it can also be defined asthe natural logarithm of the ratio YF/y. The reliabilityfunction can then be expressed as follows:

b yð Þ ¼ E ZFð Þ � ln yð Þ=s ZFð Þ ð3Þ

Here, E(�) and s(�) stand for ‘expected value’ and‘standard deviation’, respectively.

For the structural system of interest, a sample ofpairs of random values of Z and the stiffness reductionindex, Dk, can be used to estimate means and standarddeviations of Z(u), the latter defined as the naturallogarithm of the random intensity Y that correspondsto Dk ¼ u. According to the ranges of values of Dk

included in the sample, the estimation can be performedeither by means of a conventional minimum-squaresregression process or through a maximum likelihoodanalysis, as proposed by Esteva et al. (2010).

Figure 1, taken from Rangel (2006), was presentedby Esteva and Dıaz (2006). It shows a plot of thevalues of the normalised earthquake intensities SaM/V�y, leading to different values of the stiffness reductionindex, Dk ¼ (K0 7 K)/K0, for a twenty-storey systemwith hysteretic energy dissipating devices (denoted byEDDs in the following), subjected to a set of syntheticground motion records at a soft soil site in the Valleyof Mexico. Here, Sa is the linear pseudo-accelerationresponse ordinate for the fundamental period of the

Figure 1. Normalised intensity vs Dk for 20-storey systemwith EDDs.

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system of interest, M the mass of that system and V�ythe yield value of the base shear force determinedthrough a pushover analysis using the expected valuesof the mechanical properties of the structural membersthat constitute the system. The records were simulatedwith the aid of the hybrid algorithm presented byIsmael and Esteva (2006). As shown in the figure, theresults for values of Dk smaller than 1.0 were used toestimate the mean and the standard deviation of thenatural logarithm of the failure intensity. The resultingreliability function is one of the three curves shown inFigure 2. The values of b(y) obtained following thisapproach were slightly higher than those obtainedusing the method of incremental dynamic analysisproposed by Vamvatsikos and Cornell (2002).

In spite of the limitations of the criteria andmethodsused to define a lateral deformation capacity for astructural system, it is very likely that the approach ofassessing the seismic reliability of nonlinear systems bycomparing lateral displacement demands with thecorresponding deformation capacities will offer signifi-cant conceptual advantages for the practice of earth-quake resistant design. Currently, acceptable values oflateral deformation capacities recommended in norma-tive documents are estimates based on engineeringjudgment. This fact points at the convenience ofdeveloping studies oriented at deriving seismic failureprobability functions in terms of probabilistic estimatesof dynamic response demands of different types ofstructural systems and arrangements.

2.2. Damage accumulation

It is well known that modern methods of earthquake-resistant design lead to structural systems that develop

considerable nonlinear behaviour when subjected tohigh-intensity earthquakes occurring at intervals of theorder of a few tens of years. Thus, the possibility ofsignificant damage is implicitly accepted, provided theprobability of collapse under the action of futureearthquakes is kept sufficiently low. Design intensitiesand safety factors are determined on the basis of anoptimisation analysis (either formal or informal)aiming at obtaining a balance between the expectedvalues of repair costs – and other consequences ofdamage – and the construction cost increments neededto reduce the former.

Some exploratory studies about the influence ofinitial damage on the seismic reliability of multi-storeyrigid-frame buildings were made by Esteva and Dıaz(1993), and Dıaz and Esteva (1996, 1997). In the initialstudies, the multi-storey frames were replaced byequivalent single-degree-of-freedom (sdof) systems,with constitutive functions for storey shear vs lateraldisplacement similar to that shown later in Figure 7;they were followed by studies on detailed models ofthe frames, assuming the nonlinear behavior to beconcentrated at nonlinear hinges at the ends ofbending members, with moment-rotation constitutivefunctions also similar to the mentioned figure. Sometypical results for the sdof systems are shown inFigures 3 and 4, which correspond to a large-spanreinforced concrete single-storey frame, designed inaccordance with Mexico City seismic design regula-tions of 1987. Its natural period is equal to 0.43 s; itsmechanical properties were taken deterministicallyequal to their expected values.

Figure 3 shows values of the damage index in theinitially undamaged system subjected to an ensembleof simulated ground motion records of differentintensities, which are represented in the figure in termsof their normalised values, obtained in each case as the

Figure 2. Reliability function in terms of normalisedintensity for several 20-storey frame buildings with differentcontributions of EDDs to system lateral stiffness andstrength.

Figure 3. Expected damage function for conventionalreinforced concrete frame.

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ratio of the ordinate of the linear pseudo-accelerationresponse spectrum for 0.05 damping, for the naturalperiod of the system, divided by the lateral strength ofthe system, expressed in the same units as the intensity.Here, damage was measured by the ratio (K0 7 K)/K0,where K0 is the initial value of the tangent storeystiffness of the undamaged system, and K is the secantstorey stiffness computed as the ratio of the storeydeformation to the storey drift at the instant when thelatter reaches its peak absolute value; rK and rR are,respectively, the ratios of the lateral stiffness andstrength of the EDDs to those of the combined system.The figure shows a low damage level for a normalisedintensity equal to unity, with fast growing valuesbeyond it.

The influence of initial damage on the damagingpotential of new earthquakes is illustrated in Figure 4,which shows the form of variation of the expectedvalue of the final damage after an earthquake, in termsof the initial damage and the intensity of the earth-quake. The horizontal axis represents initial damage,the vertical axis represents final damage, and theindividual points in the graph represent final damagelevels corresponding to different values of the intensity;the latter is normalised with respect to that assumed toact initially on the undamaged structure. The contin-uous curves shown are minimum square fittings tothe original data; they are intended to representthe expected values of the final damage. The largeuncertainties associated with the prediction of finaldamage indexes are evident, even for such a simplesystem.

According to Figure 4, the expected damageproduced by a normalised intensity equal to 1.00 issmaller than 0.1 for an initially intact structure;however, it grows very fast with the initial damage,reaching a final value of 0.7 for an initial value of 0.1.This calls the attention for the high reductions in the

reliability indicators of the system that may beassociated with initial damage levels. Optimum main-tenance and rehabilitation policies and programs tocontrol these reductions are studied in the followingsections.

2.3. Mathematical models vs empirical information

Modern computational tools and systems allowprofessionals and researchers in earthquake engineer-ing to generate simulated samples of the gravitationalloads acting on large complex structural systems, aswell as of the mechanical properties of their structuralmembers; including models of the constitutive func-tions representing the relations between local internalforces and deformations at critical sections of mem-bers. This capability has been used to make estimatesof the dynamic responses and hence the vulnerabilityfunctions of complex nonlinear systems, accordingto the concepts discussed in Sections 2.1 and 2.2.However, even though the uncertainties associatedwith the forms and parameters of those functions forindividual members and critical sections have beenrecognised, very little attention has been paid to theevaluation of their impact on the uncertainties affect-ing the probabilistic estimates of responses andvulnerabilities of the full systems.

Two complementary approaches may be envisagedto account for the uncertainties mentioned above:one is to consider a set of assumptions related to themodels of the constitutive functions and assign aprobability value to each of them; another consists incomparing values of predicted and actual (measured)responses of laboratory specimens or real systems.Bayesian probability methods might be applied to useavailable empirical results to update the prior prob-ability distribution assigned to the mentioned set ofassumptions and to determine probability distributionsof actual versus computed responses for each of them.Uncertainties associated with the properties of themodel should be handled as epistemic uncertainties, asdescribed in the following section; those related to theratios of observed versus computed responses for eachmodel assumption should be handled as random.

3. Life-cycle decision framework

3.1. Seismic hazard models

The seismic hazard at a site can be described by astochastic process model of the occurrence of seismicevents and of the conditional probability densityfunction of the intensities; the latter may be describedby a ground motion indicator showing significantcorrelation with the peak values of the structuralresponse. Peak ground accelerations or velocities, or

Figure 4. Influence of initial conditions on the damagingpotential of new earthquakes.

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ordinates of response spectra for the fundamentalperiod of the system of interest are often used forthis purpose. Poisson and renewal stochastic processmodels with random selection of magnitudes are oftenused to represent the generation of seismic events ata source. These models have to be combined withadequate intensity attenuation functions in order togenerate stochastic process models of the occurrence ofearthquakes of different intensities at a site near one ormore active sources.

A source-activity model considering the super-position of a renewal process of the occurrenceof characteristic earthquakes and a Poisson processrepresenting background activity was used by Singhet al. (1983) for the development of a probabilisticseismic hazard model at sites near the subduction zonethat runs along the southern coast of Mexico. Markovprocess models have been used by Lutz et al. (1993) torepresent seismic hazard in the vicinity of strike-slipfaults near the Western coast of the United States.

For the Poisson model, the seismic hazard remainsconstant, regardless of the time elapsed since thelast previous event; however, this is not true for therenewal and the Markov process models, whichare characterised by the steady increase of the hazardlevel with the time without activity at a source.

3.2. Optimisation analysis

Let us first consider the simple case of a structuralsystem to be built at instant t0 ¼ 0, and exposedto a sequence of seismic events with uncertainlyknown times of occurrence T1, . . . Tn, and intensitiesY1, . . . Yn, during the time interval 0 5 t 5 tL,assumed here to correspond to the useful life-time ofthe system. Let C0 denote the initial construction cost(which is a function of the design parameters), Di thecost of damage and maintenance actions occurring atthe time of the ith event, or immediately after it, and ga ‘discount rate’ used to transform values of benefitsor losses that may be produced in the future intoequivalent values assumed to be generated at instantt0 ¼ 0. The transformation would adopt the formU0 ¼ Ut � exp (7gt), where Ut and U0 are respectivelythe nominal value of the utility (a benefit, if positive; aloss if negative) generated at instant t, and its presentvalue, at instant t0 ¼ 0. In the following only thespecial case, when tL ! ? and the system is system-atically rebuilt after failure, will be considered,assuming that the vulnerability function of each newsystem will be equal to that of its predecessors. It isalso assumed that the expected value of the benefits perunit time will remain constant while the systemremains in operation. Thus, the optimisation analysiswill consist in minimising the absolute value of an

objective function constituted by the addition ofthe initial cost (which is an increasing function ofthe strength and stiffness values determined by thestructural design requirements) plus the present valuesof the expected losses, Di. Under these assumptions,the objective function to be minimised is U, given byEquation (4), where E stands for the expected valuefunction.

U ¼ C0 þ EX1i¼1

Die�gTi

" #ð4Þ

The probability distribution of Di depends on theintensity Yi of the earthquake occurring at instant Ti.The joint probability distribution of the latter twovariables is the subject of the seismic hazard modeladopted. Rosenblueth (1976) and Rackwitz (2000)present analytical models based on the assumptionsthat earthquakes with intensities above a given thresh-old value occur in time as a Poisson process or as arenewal process, where the intensities are independentrandom variables with identical probability distribu-tion. They also study more general cases than thoseconsidered in the preceding paragraph. For theparticular case where earthquakes occur as a Poissonprocess, the last term in Equation (4) adopts the forml � E[D]/g, where l is the mean rate of occurrence perunit time of earthquakes with intensities above aspecified threshold value (sufficiently low to include allintensities significant for engineering applications) andD is the expected value of damage and maintenanceactions given the occurrence of any earthquake withintensity above the mentioned threshold value; itsexpected value is obtained by integration of the jointprobability density function of the mechanical proper-ties of the system under study and of the intensitiesgiven the occurrence of an earthquake.

Equation (4) does not explicitly take into accountthe epistemic uncertainties mentioned in the Introduc-tion, and included in the analytical models presentedby Rosenblueth (1976) and Rackwitz (2000); however,it can be easily modified as follows, to consider them:

U ¼ C0 þ Ee EX1i¼1

Die�gTi ej

" #" #ð5Þ

Here, e is a vector used to represent the set ofassumptions about the possible models used toestimate the last term in the second member inEquation (4), or about the values of the parametersadopted for those models, and Ee is the expected valuewith respect to the probabilistic weights assigned toeach of those assumptions. Assigning a subjectiveprobability distribution to e is not an easy task: it mayrequire at least a rough assessment about alternative

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forms of the models used, uncertainties about theirparameters, and the potential impact of all theseconcepts on the probabilistic models of Di.

Closed form solutions for some particular casesincluded in those represented by Equations (4) and (5)were proposed by Rosenblueth (1976); they can beused for the determination of present values ofexpected costs of the possible failure events of a systemsubjected to seismic hazard conditions described byrenewal process models of earthquake occurrence, withthe intensities of the different events considered asstochastically independent, identically distributedrandom variables. He assumes that the system isimmediately rebuilt after each failure event, with itsloads and mechanical properties being random vari-ables with the same probability distributions as for theoriginal construction. Because of the latter assump-tion, the resulting equations proposed to estimatepresent values of expected costs cannot be applied togeneral cases of maintenance and repair strategies, aswill be shown later.

For purposes of obtaining life-cycle indicators ofsystem safety and vulnerability, three groups ofuncertainly known variables are identified; they aredescribed in the following, after introducing someslight changes:

Type 1: Random fluctuations in the response of thesystem, resulting from the random characteristics ofthe earthquake ground motion. These fluctuationsvary from one disturbance to another and arecompletely uncorrelated.

Type 2: Random values of the gravitational loads andthe mechanical properties of the system. They remainconstant between reconstructions and are stochasti-cally independent from one structure to the next

Type 3: Epistemic uncertainties, associated withdifferent concepts, such as our imperfect knowledgeabout the seismic hazard at the site or about themodels used to represent the constitutive functionsthat describe the cyclic behavior of structural elementsor critical sections.

Under the assumptions that earthquakes with inten-sities above a specified threshold value occur inaccordance with a stochastic renewal process, andthat the system is rebuilt immediately after eachfailure, the probability density functions of the timesto the first and to the nth failures are given byEquations (6) and (7), respectively:

g1 tð Þ ¼X1i¼1

fi tð ÞP 1:0� Pð Þi�1 ð6Þ

gn tð Þ ¼Z t

0

gn�1 t� tð Þg tð Þdt ð7Þ

Here, fi(t) is the probability density function of thewaiting time to the first seismic event, P the failureprobability given the occurrence of that event (for asystem with deterministically known properties), andg(t) the probability density function of the timebetween successive failures. The present value of theexpected loss for the rebuilding policy adopted is

D ¼ DF

X1n¼1

Z10

gn tð Þe�gtdt ð8Þ

where DF is the expected cost of failure in case itoccurs. Replacing the values of gn(t) in Equation (8) bythose given by Equations (6) and (7), the followingexpression is obtained for D:

D ¼ DFPf1 � gð Þ1� f � gð Þ ð9Þ

Here, f*(g) and f1*(g) stand for the Laplacetransforms of f(t) and f1(t), respectively.

If uncertainties of type 2 are introduced, the valueof P in Equation (6) is a function of the (random)vector of system properties; the values of gi(t) inEquations (7) and (8) have to be replaced by theirexpected values with respect to the joint probabilitydensity function of those properties, and D is given byEquation 10, where g* (g) and g1(g) are the Laplacetransforms of g(t) and g(t), respectively.

D ¼ DFE2 g1 � gð Þ½ �

1� E2 g � gð Þ½ � ð10Þ

For the particular case where the occurrence ofseismic events can be represented by a Poisson processwith annual rate l, Equations (9) and (10) lead toEquations (11) and (12), disregarding and includinguncertainties of type 2, respectively:

D ¼ DFPlg

ð11Þ

D ¼ DF

E2Pl

Plþg

h i1� E2

PlPlþg

h i ð12Þ

In most cases of interest in engineering applica-tions, E2[Pl] � g. In this case, Equation (12) can bereplaced by the following approximation:

D ¼ DFlg

E2 P½ � ð13Þ

For any of the cases considered in Equations (9) to(13), epistemic uncertainties can be taken into account

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by obtaining the expected values of D with respect tothe probability density functions of the variables usedto describe those uncertainties.

Equation (13) can be extended to account forstructural and non-structural damage covering thepossible conditions of failure and survival for eachseismic event:

D ¼ lgE2 DFPþDS 1� Pð Þ½ � ð14Þ

Here, DS is the expected cost of damage given thatthe system survives when subjected to an earthquakewith a randomly chosen intensity.

Cornell et al. (2002) and Jalayer and Cornell (2003)have presented approximate expressions to estimatethe expected failure rate l � E2[P], as well as itsexpected value with respect to the probability densityfunction of epistemic uncertainties (type 3), usingsimplified models to describe the probability densityfunctions of earthquake intensities and dynamicresponses given the intensity. Because of the assump-tions implicit in their approach, the resulting failurerates cannot be used to obtain present valuesof expected costs for general assumptions aboutrepair and maintenance strategies, as was mentionedfor Rosenblueth’s closed form solutions given byEquations (9) to (14).

3.3. Risk control considerations

Decisions made in accordance with the optimisationanalysis just presented may lead to excessively high,perhaps unacceptable, risk levels. This may occur asa consequence of adopting a structural system ofarrangement that is not sufficiently efficient for theconditions considered. In those cases, two alternativesare available to the decision maker: look for a moreefficient system or design for a pre-established risklevel, even if that is not optimum on the basis of purelyeconomic considerations.

It may also happen that an optimum designresulting from the minimisation of the correspondingobjective function presented in one of Equations (4) or(5) (or in their counterparts where the present value ofexpected cots is evaluated in accordance with one ofEquations (10) to (14)), leading to acceptable risklevels at the moment of making the decision, may leadto unacceptable risk levels in the future, either becauseof an increase in the vulnerability that may arise as aconsequence of damage accumulation on the system,or because of the steady increase in seismic hazardmentioned at the end of the section on seismic hazardmodels. The first source of risk enhancement can becontrolled through the adoption of adequate repair

and maintenance strategies and programs. This ap-proach could also be applied to control the processassociated with the increase in seismic hazard; how-ever, this would be questionable because it wouldimply leaving to future generations risks that wouldnot be acceptable to us. Therefore, the only optionleft would be to design a system capable of ensuringacceptable risk levels, provided adequate repair andmaintenance policies are applied to prevent thevulnerability function of the system to raise higherthan that corresponding to the initial, undamagedsystem.

4. Maintenance and rehabilitation

4.1. Evolution of risk functions with time

As a consequence of damage accumulation, seismicrisk (expected cost of damage per unit time) grows withtime after each rebuilding or maintenance action, evenif seismic hazard remains constant, as assumed byPoisson process models. For those cases when seismichazard is represented by renewal or Markov stochasticprocess models, the systematic increase in hazardassociated with these models constitutes an additionalsource for risk increase. Extended forms of Equations(9) to (14) can be obtained to account for these effects.Alternatively, Monte Carlo simulation may be used toevaluate D as given by the last term in either Equation(4) or (5). For simplicity, this approach will be adoptedin the following.

4.2. Repair and maintenance policies based onoptimisation analysis

For the purposes of illustration, consider the case of asimple (single-bay, single-storey) system shown inFigure 5 (taken from Esteva and Dıaz 1993), con-stituted by a conventional frame (CF) and an energy-dissipating device (EDD); the nominal design valuesof its mechanical properties are summarised by thelateral strength (RF, RD) and the initial stiffness(kF, kD) of each of its components, where subscriptsF and D denote the CF and EDD, respectively. Typicalidealised load-deformation curves for these elementsare shown in Figures 5(b) and 5(c), which distinguishbetween the deterioration of stiffness and strengthtypical of reinforced concrete frames and the stabilityof the hysteretic cycles of the EDD. Every time that amoderate or high intensity earthquake occurs, damageincrements take place on both elements. These incre-ments are in addition to those accumulated as aconsequence of previous events, provided the CF isnot repaired and the EDD is not replaced. It isassumed that damage on the CF is visually identified,thus leading to repair actions when it reaches a

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pre-established level, and that the EDD is replacedwhen it breaks or, according to a preventive strategy,when it has been subjected to a number of highintensity earthquakes such that the probability offailure in the event of the next potential earthquake isunacceptably high. An optimum strategy for thedesign, construction and maintenance of such a systemimplies selecting the structural design parameters andthe threshold values for repair of the CF and forpreventive replacement of the EDD, in such a way asto minimise the objective functions given by Equations(4) or (5).

The indicators of the damage accumulated in theCF and the EDD up to and including the jth event aredFj and dDj, respectively. If a repair or replacementaction takes place after the jth earthquake, dFj and dDj

are transformed into their updated values, d0Fj and d0Dj.The increments corresponding to the (j þ 1)th earth-quake are DdF(j þ 1) and DdD(j þ 1), respectively. In thecase of collapse of the structure during the ith event,followed by immediate reconstruction, the value of Di

to be substituted into Equation (4) or (5) equalsC0 þ CF, where C0 is the initial construction cost andCF includes all other related failure costs, includingdirect and indirect costs, as well as those associatedwith human losses or social impact. If collapse is notreached, Di includes the repair costs of structural andinfill elements, the replacement of those EDD thathave failed or are estimated to have attained significantfatigue damage, and the losses inflicted by the eventualinterruption of the normal operation of the system.

Every time that the damage accumulated on the CFexceeds a pre-established value dR, the repair must

eliminate the accumulated damage; that is, it mustrestore the initial properties RF and kF of the structuralframe. The replacement of the EDD follows apreventive strategy, based on an index of calculateddamage, dD, to be defined later. Consequently, d0Fi ¼dFi if dFi 5 dR; d0Fi ¼ 0, otherwise. Also, d0Di ¼ dDi ifdDi 5 1.0 and no preventive replacement takes place;d0Di ¼ 0, otherwise.

Because DdF(j þ 1) and DdD(j þ 1) depend on dFj anddDj, damage accumulation occurs as a Markov process,regardless of whether earthquakes are assumed tooccur in accordance with a Poisson, a renewal or aMarkov process. The transition probability matricesbetween the states of damage in consecutive events areobtained from the probability density functions ofDdF(j þ 1) and DdD(j þ 1) conditional to dFj and dDj. Thestate of the system immediately after the jth earth-quake is expressed in terms of two sets of variables,describing respectively the state of damage on thesystem and the stage of the seismic process: dFj and dDj

belong to the first set; the second is assumed to bedescribed in terms of the vector Sj of seismologicalvariables, if the characteristics of consecutive earth-quakes are not stochastically independent. In thegeneral case, in order to determine the conditionalprobability distribution functions of dF(j þ 1), dD(j þ 1)

and Sjþ1 given their values just after the occurrence ofthe jth event and the resulting maintenance actions, itis also required to determine the joint p.d.f. of Yjþ1 andTiþ1, the intensity of the next earthquake and thewaiting time for its occurrence, respectively. Followingthe Monte Carlo simulation process proposed above,the damage values after the jth earthquake, dF(j þ 1)

Figure 5. Single-storey frame with energy-dissipating device.

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and dD(j þ 1), would be obtained by step-by-stepresponse analysis of the system, taking into accountthe information about accumulated damage (includinglocal deformation history) at all critical members andsections. Alternatively, transition probability matricesfor system damage may be previously generated fordifferent values of the intensity of the new earthquakeand used to determine joint probability densityfunctions of dF(j þ 1) and dD(j þ 1), conditional to dFj,dDj and Yj þ 1.

Closed-form expressions to determine functionfFD(j þ 1)(u,vjd0Fj, d0Dj), the joint probability densityfunction of dF(j þ 1) and dD(j þ 1) conditional to thevalues of the damage indicators d0Fj and d0Dj after thejth earthquake and the maintenance actions followingit, have been proposed by Esteva and Dıaz (1993).More general versions of those equations are presentedin the Appendix; they will be used for the Monte Carlosimulation of the damage accumulation process in theillustrative examples presented in Sections 5.2 and 5.3.They take into account the uncertainty about theintensity of the (j þ 1)th earthquake.

Unlike Equations (11) to (14), where the expectedfailure rate per unit time remains constant, this riskvariable keeps growing steadily after each repair ormaintenance action, as a consequence of the incre-ments in failure probabilities resulting from damageaccumulation. This increasing trend appears, even forthe cases of constant seismic hazard corresponding toPoisson seismicity models. Instead of Equation (14),the present value of expected damage is now deter-mined as follows:

D ¼Z

D0 tð Þexp �gtð Þdt ð15Þ

In this equation,

D0 tð Þ ¼ DC0 ¼ DS tð Þ þ DF tð Þð ÞC0 ð16Þ

C0 is the initial construction cost, DF (t) and DS (t) arethe values of expected damage costs per unit time, atinstant t, for the conditions of survival and collapse,respectively; they are normalised with respect to theinitial construction cost. They are calculated in termsof the seismic hazard rates (assumed constant in thiscase) and the instantaneous values of the failure anddamage probabilities, which are functions of t, as aconsequence of damage accumulation.

DF tð Þ ¼Z

dnY yð Þdy

��������dFpF y; tð Þdy ð17aÞ

DS tð Þ ¼Z

dnY yð Þdy

��������d y; t Sjð Þ 1� pF y; tð Þð Þdy ð17bÞ

Here, dF and d(y, tjS) are the expected cost of collapseand the expected cost of damage conditioned tosurvival, both normalised with respect to C0. It iseasy to understand that the influence of damageaccumulation leads to the dependence of d on t, for agiven value of y; this dependence does not occur for dF.

Direct use of the closed-form expressions presentedabove may ordinarily call for excessive computer time,which can be circumvented through the use of MonteCarlo simulation. This approach was followed in thelife-cycle optimisation studies described in Sections 5.2and 5.3.

5. Illustrative examples

5.1. Limitations of criteria to determine deformationcapacity

Consider a twelve-storey tall building with the generalgeometrical configuration shown in Figure 6. Severalalternative reinforced-concrete structural systems wereexamined: FS, which is made exclusively by rigidframes, and systems D3, D4 and D6, with shear walls3m, 4m and 6m wide, respectively. Their dynamicresponse was studied for a set of horizontal single-component ground motion acceleration time historiesacting in the direction parallel to the shear walls. Forthis purpose, the nonlinear behaviour of each structur-al member was assumed to be concentrated at flexuralplastic hinges at each of its ends, characterised by aconstitutive function similar to that shown in Figure 7,proposed by Wang and Shah (1987) and modified byCampos and Esteva (1997). Here, the influence of theaxial load on the moment-rotation constitutive func-tion was not taken into account.

The four structural systems considered weredesigned in accordance with the 2004 edition ofMexico City earthquake resistant regulations (NTC-DS DF 2004). Figure 8 shows the results of the

Figure 6. Frame and dual systems with different shear-wallwidths.

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pushover analysis performed on each system, takingits gravitational loads and mechanical properties(strength, stiffness) equal to their correspondingexpected values, and considering the horizontal dia-phragms as infinitely rigid. Using this information, thelateral deformation capacity was determined for eachsystem, according to the conventional practice oftaking it equal to that corresponding to a reductionof 20% in the base shear force, with respect to itsmaximum value reached during the analysis. It caneasily be observed that the pattern of the residuallateral capacity (and hence the residual energy-dissipation capacity) may be very sensitive to the rateof decrease of the base shear with respect to the lateraldeformation; this variable is highly influenced by thepattern of load redistribution associated with thedevelopment of nonlinear behaviour at each memberend. In particular, the sharp decrease in the base shearforce in system D6 can be associated with the highproportion of the lateral force capacity that is provided

Figure 7. Constitutive function for moment vs plasticrotation at flexural member ends.

Figure 8. Pushover curves, deformation capacities and residual energy dissipation capacity for several frames and dualsystems.

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by the shear wall, as compared to the contribution ofthe rigid frame. The impact of these concepts on theseismic reliability levels attained by the systems shownfor a given value of the normalised intensity, expressedas the ratio of the ordinate of the linear displacementresponse spectrum to the deformation capacity can beappreciated in Figure 9, where the vertical ordinateshows values of Cornell’s reliability index b, as afunction of the normalised intensity Z ¼ Sd/DC, whereSd is the ordinate of the linear displacement responsespectrum for the simplified reference system (withproperties derived by means of a pushover analysis)and DC is the deformation capacity of that system,determined in accordance with generally acceptedcriteria (Qi and Moehle 1991). The wide variabilityof the results justifies the comments in the section onseismic vulnerability and risk functions, about thelimitations of the concept of deformation capacity inthe assessment of seismic reliability functions.

5.2. Life-cycle present values of expected costs onsystems with energy-dissipating devices

Campos (2005) made systematic studies about thevalues adopted by the utility function U defined inEquation (4). For this purpose, he considered severalstructural systems, as shown in Figure 10, with theproperties summarised in Table 1, and examined theinfluence of the following variables, among others:CC ¼ deformation capacity, measured as the lateraldistortion corresponding to the ultimate failure condi-tion; r1 ¼ ratio of indirect to direct repair costs;aC ¼ expected cost of collapse divided by the initialconstruction cost; dR ¼ threshold damage ratio forrepair of a member of the conventional system;dD ¼ threshold damage ratio for replacement of anenergy-dissipating device (EDD). The responses ofeach system to a set of simulated ground motion timehistories with amplitude-frequency evolutionary

properties similar to those of the E-W component ofthe ground acceleration recorded at the SCT station(soft clay in Mexico City) during the 19 September1985 earthquake. Mean values of the correspondingacceleration response spectra for elasto-plastic systemsare shown in Figure 11 for different ductility levels.

Mean values of the physical-damage indicator Dk

defined in the section on seismic vulnerability and riskfunctions, attained at different stories of the systemsshown in Figure 10 and Table 1, are presented inFigure 12, taken from Campos (2005), which shows thesensitivity of the mentioned indicator to the presenceof the EDDs and to the value of the nonlinear-response behaviour factor Q assumed for design: forsystems a, b and d, the value of Dk decreases withincreasing values of the ratio rK ¼ KD/K0, where KD/

Figure 9. Reliability functions in terms of Z ¼ Sd/DC.

Figure 10. Systems considered in parametric study(Campos 2005).

Table 1. Cases considered in parametric study.

Type

Storieswith

EDD’sPeriod(s)

rk ¼ KD/K0

l ¼ dyd/dyc Q

Safetyfactor

a None 1.46 – 4 1.0b All 1.48 0.25 0.50 4 1.0c All 1.49 0.25 0.50 5 1.0d All 1.48 0.50 0.50 4 1.0e All 1.55 0.50 0.50 5 1.0f All 1.44 0.50 1.00 5 1.0g 1–4 1.41 0.50 1.00 5 1.0h 1–4 1.41 0.50 1.00 5 1.1i 1–4 1.41 0.50 1.00 5 1.2

K0 ¼ total story stiffness of the dual system, accounting for thecontributions of conventional elements and energy-dissipatingdevices.

KD ¼ story stiffness of the energy-dissipating elements.

dyd ¼ lateral story yield displacement of energy-dissipating elements.

dyc ¼ lateral story yield displacement of conventional system.

Q ¼ nonlinear-response behavior factor assumed for design inaccordance with Mexico City building code (NTC-DS DF. 2004).

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K0 are the values of the global lateral stiffness of theenergy dissipating and of the combined system,respectively. This trend is also evident for systems cand e, where Q was taken equal to 5 during the designprocess; however, for these cases the values of Dk arelarger than those corresponding to their counterpartswith the same values of rk, but with Q ¼ 4. It can alsobe seen that a reduction of Dk is achieved by increasingthe value of l ¼ dyd/dyc.

Introduction of the EDDs only at the first fourstoreys produces drastic reductions in the values ofDk at those stories, at the expense of generating asignificant increase in the responses of the upper stories

(system g); in order to reduce these responses to valuessimilar to those experienced by system a it is necessaryto increase in 20% the storey safety factors for lateralshear force.

Life-cycle studies about the long term performanceof systems during a time interval of 100 years, withoutany repair or maintenance action (with the exceptionof systematic re-construction in case of collapse), weredeveloped for systems a, d, f and i. For this purpose,five sequences of seismic events with randomly chosenintensities were generated by Monte Carlo simulation,taking the occurrence of earthquakes at the site as aPoisson process with independent sampling of theintensity of each event; both the rate of occurrence ofseismic events and the probability distribution of theintensities were derived from the seismic hazardfunction at the site, as proposed by Alamilla andEsteva (2006). The peak amplitudes of the storey driftsexperienced by each system during all events in eachearthquake sequence are shown in Figure 13. The largevalues of the storey drifts in the upper portion offrames a and i for sequences a and d are consistent withthe high intensities of the last seismic events in thosesequences; however, this effect was not observed inframes d and f.

The mean values of the peak storey distortionsreached during all the seismic sequences included in thesample studied are shown in Figure 14. The meanvalue and standard deviation of Dk for each systemduring the 100 years interval considered in the studywere used to estimate the corresponding values ofCornell’s reliability index b during the cycle; forsystems a to i. These values were 1.137, 1.694, 1.900and 1.193, respectively; the corresponding failureprobabilities are approximately equal to 0.128, 0.045,0.029 and 0.116.

The results of examining several alternative repairand maintenance strategies are presented in Figures 15and 16 in terms of rI, the ratio of indirect to directrepair costs. All costs are normalised with respect tothe initial construction cost of system a defined inTable 1. The lowest values correspond to case d,followed by case f, which correspond to a value ofrK ¼ 0.5, constant along the building height; thehighest values correspond to case i, where EDDs areinstalled only at the first four storeys and the safetyfactor at the upper storeys is largest; as expected, thelong-term costs increase with the indirect repair costs.The influence of the threshold values for the replace-ment of EDDs on the expected costs is negligible.

5.3. Optimum repair and maintenance policies

The concepts introduced in the section on maintenanceand rehabilitation has been applied by Esteva et al.

Figure 11. Mean values of the acceleration response spectrafor elasto-plastic systems corresponding to the groundacceleration, E-W component, recorded at the SCT stationduring the 19 September 1985 earthquake for differentductility levels (Campos 2005).

Figure 12. Mean value of damage index for each frame type(Campos 2005).

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(1999) to the study of the time-dependent process ofdamage accumulation and reliability evolution inbuilding frames. They were also employed by thesame authors for the study of optimum design criteriaand maintenance strategies for structural frames withhysteretic energy-dissipating devices. Because of thecomplexity of the probability transition matricesinvolved, extensive use has been made of Monte Carlosimulation. One of the cases studied corresponds to atwo-bay, fifteen-storey frame system with hystereticenergy dissipating devices. The system was supposed tobe built at a site in Mexico City where seismic hazardwas represented by a Poisson process characterised bythe function relating seismic intensity with annual rateof being exceeded. The state of damage at the end ofeach earthquake was measured by the maximum valueattained at any storey by the index Dk defined above.

Searching for a life-cycle optimum solution, severaloptions were explored regarding the seismic designcoefficient c and the threshold the values of Dk adopted

as a condition for repair of the main frame and forreplacement of the energy dissipating devices (Drc andDrd, respectively). Values of a negative utility function

Figure 13. Storey drift in the frames at the end of each seismic history (Campos 2005).

Figure 14. Mean values of the peak story distortionsreached during all the seismic sequences included in thesample studied (Campos 2005).

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U, calculated as the sum of initial construction cost,C0, and expected present values of future expenditures,were obtained for each option. The resulting values ofU, normalised with respect to the initial construction

cost for the main frame designed for gravitationalloads only, are depicted in Figure 17. The first sectioncorresponds to the plain conventional frame, while theother three sections contain energy dissipating devices

Figure 15. Present values of expected costs of systems, normalised with respect to initial cost of system a, cC ¼ 0.04.

Figure 16. Present values of expected costs of systems, normalised with respect to initial cost of system a, cC ¼ 0.06.

Figure 17. Life-cycle utility values for 15-storey system: influence of design criteria and maintenance policies.

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that contribute 75% of the lateral strength and stiffnessof each storey. It can be observed that the negativeutility function is sensitive to both the seismic designcoefficient and the repair and replacement strategies.The large values that resulted for the initial and longterm costs of the systems with EDDs are probablydue to the excessively large values assumed for thecontributions of those devices to the lateral strengthand stiffness of the combined system. This fact mayalso be responsible for the high values shown by theoptimum threshold value of the damage level for repairof the conventional frame. Because no analysis hasbeen made of the sensitivity of these results to theconstitutive functions, the damage–response models,the repair and replacement costs, and the consequencesof ultimate failure, their value is limited to their rolefor the purpose of illustrating the application of theproposed life-cycle optimisation criteria.

6. Concluding remarks

Life-cycle optimisation provides an adequate frame-work for the establishment of seismic design require-ments and maintenance programs for systems built atsites with significant seismic hazard. Optimum designcriteria formulated within that framework explicitlyaccept that under the action of each seismic excitationa system affected may experience one or more excur-sions into its range of nonlinear behaviour; this impliesthe occurrence of damage, as well as its accumulationduring the life-cycle of that system. Hence the need toformulate damage control strategies and programsintended to keep failure probabilities and expectedlosses within tolerable limits. It is not sufficient todesign and build a system capable of resisting a highintensity earthquake associated with a sufficiently longreturn interval: it is also necessary to prevent theconsequences of damage accumulation on the increaseof the system vulnerability for future earthquakes andto make repair actions easy, effective and inexpensive.

Planning for this includes actions such as concen-trating the expected nonlinear behaviour and damageaccumulation at easy-to-replace replace energy-dissi-pating devices or at easy-to-repair small segments atcritical sections of structural members.

Modern computational tools permit the evaluationof expected performance of complex systems undersequences of earthquakes of different intensities.However, their results may be too sensitive to themodels used to represent cyclic behaviour and damageaccumulation on the critical members and sections ofthose systems. Significant research efforts are desirablein the calibration of those models using experimentalinformation from laboratory tests and from actualseismic events.

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Appendix. Transition probabilities between damage

states, accounting for the influence of damage

accumulation

Let us consider a multi-storey frame-building structureprovided with a system of hysteretic energy-dissipatingdevices; subscripts F and D will be used to denote the main

frame and the energy-dissipating system. It will be assumedthat both may be affected by the process of damageaccumulation. Let the random variables DFi and DDi

represent the corresponding damage levels at the end of theith earthquake, assuming that no repair or maintenanceactions are taken after its occurrence. The initial damagelevels at the start of the (i þ 1)th event will be equal to D0Fiand D0Di, which will depend on the repair or replacementactions at the end of the i–th earthquake, as follows:

D0Fi ¼ DFi; ifDFi < DrFðA1a; bÞ ¼ 0; otherwise ðA1a; bÞD0Di ¼ DDi; ifDDi < DrDðA1c; dÞ ¼ 0; otherwise ðA1c; dÞ

In these equations, DrF and DrD are the damage thresholdvalues for repair and replacement of the main frame elementsand the energy–dissipating devices, respectively. ForD0Fi ¼ d0Fi and D0Di ¼ d0Di, the joint probability densityfunction of the damage levels DF(i þ 1) and DD( i þ 1) on themain frame and the energy–dissipating system at the end ofthe (i þ 1)th earthquake will be the following:

fFD iþ1ð Þ u; v d0Fi; d0Di

��� �¼Z

fFD iþ1ð Þ u; v d0Fi; d0Di; y

��� �fY iþ1ð Þ yð Þdy

ðA2Þ

The integral in the second member of this equation serves toaccount for the uncertainty in the value of the intensity of the(i þ 1)th earthquake.

The transition probabilities for the damage states onboth systems at the end of the (i þ 1)th earthquake, startingfrom that at the end of the ith earthquake, prior to any repairor maintenance action, is determined taking into account thefour alternative combinations of possible actions, whichdepend on DFi and DDi, as expressed in Equations (A1a) to(A1d). Taking into account the joint probability densityfunction of the lateral variables, the following expression isfound for the joint marginal probability density function ofDF(i þ 1) and DD(i þ 1):

fFDðiþ1Þðu; vÞ

¼Z DrD

0

Z DrF

0

fFDðiþ1Þðu; vju0; v0ÞfFDiðu0; v0Þdu0dv0

þZ DrD

0

Z 1DrF

fFDðiþ1Þðu; vj0; v0ÞfFDiðu0; v0Þdu0dv0

þZ 1DrD

Z DrF

0

fFDðiþ1Þðu; vju0; 0ÞfFDiðu0; v0Þdu0dv0

þZ 1DrD

Z 1DrF

fFDðiþ1Þðu; vj0; 0ÞfFDiðu0; v0Þdu0dv0

ðA3Þ

For the particular case when no repair or maintenanceactions are taken after the ith earthquake, this equation isreplaced by the following:

fFDðiþ1Þðu; vÞ

¼Z 10

Z 10

fFDðiþ1Þðu; vju0; v0ÞfFDiðu0; v0ÞðA4Þ

Structure and Infrastructure Engineering 49

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2014