Engineering Structures 29 (2007) 3197–3209www.elsevier.com/locate/engstruct

An analytical framework for seismic fragility analysis of RChigh-rise buildings

Jun Ji, Amr S. Elnashai, Daniel A. Kuchma∗

Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, IL 61801, USA

Received 20 January 2007; received in revised form 24 July 2007; accepted 30 August 2007Available online 13 November 2007

Abstract

An analytical framework and sample application are presented for the seismic fragility assessment of reinforced-concrete high-rise buildings.Since no probabilistic fragility relationships exist for this class of structure, the work fills an important void in regional earthquake impactassessment. The key element of the presented framework is the methodology for the development of a simple lumped-parameter modelrepresentative of the complex high-rise building system. This model was created in the ZEUS–NL environment to enable computationally efficientdynamic response history analyses of high-rise structures that were previously not possible and that can accurately account for the complexbehaviour and interactions predicted by more detailed analytical models. The parameters for this model were selected using genetic algorithms.The development of a simple lumped-parameter model is presented for an existing high-rise structure with dual core walls and a reinforcedconcrete frame. The accuracy of the individual components of this model is compared with the predictions of more detailed analytical models andsample fragility curves are presented. The proposed framework is generally applicable for developing fragility relationships for high-rise buildingstructures with frames and cores or walls.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: RC high-rise buildings; Seismic fragility; Lumped modelling; Genetic algorithm

1. Background and motivation

Urbanization has led to a dramatic increase in thenumber and variety of high-rise building structures. Theseismic vulnerability of these high-rise infrastructures ispoorly understood and probabilistic assessment tools of theirperformance is lacking. Reinforced concrete (RC) is now theprincipal structural material used to provide lateral resistancein high-rise structures. The tendency to use RC systems isexpected to continue due to the development of commercialhigh-strength concretes up to 170 MPa, the advent ofadmixtures that can provide high-fluidity without segregationand advances in construction techniques in both pumping andformwork erection (commented by Ali [1]).

Due to the significance of wind forces on the lateral loaddemands in high-rise structures, the effects of lateral loads

∗ Corresponding address: Department of Civil and Environmental Engineer-ing, University of Illinois at Urbana-Champaign, 2106 Newmark CE Lab, MC-250, 205 N. Mathews Ave., Urbana, IL 61801-2352, USA. Tel.: +1 217 3331571; fax: +1 217 265 8040.

E-mail address: kuchma@ad.uiuc.edu (D.A. Kuchma).

0141-0296/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2007.08.026

from seismic action are often not separately considered. Thisis often inappropriate as the wide frequency content in realground motions can significantly excite both lower and highermodes to a degree that is not excited by wind forces. Thisresults in a complex seismic response and leads to demandsin some parts of the structure that can be significantly largerthen the lateral forces from wind actions Brownjohn [2]. Inaddition, the imposed displacements in earthquakes may bevery substantial since the standard earthquake displacementspectrum peaks in the period range of 3–6 s (Bommer andElnashai [3]), corresponding to the fundamental modes of manyRC high-rise structures.

For the reasons given above, there is the need for animproved understanding of the inelastic non-linear dynamicresponse of RC high-rise structures subjected to realisticseismic records representative of near and far field earthquakes.Moreover, and motivated by the increasing interest in obtainingmore accurate assessments of earthquake losses, there is theneed for deriving fragility relationships for various forms ofhigh-rise structures from these analyses.

3198 J. Ji et al. / Engineering Structures 29 (2007) 3197–3209

Fig. 1. Proposed fragility assessment framework.

Fragility is the conditional probability of attainment orexceedance of multiple damage states for a given intensity ofground excitation, as shown in Eq. (1).

P(fragility) = P [LS|IM = x] ,

P(LS) = P(C = D)(1)

where C is the capacity, D is the demand, and IM represents theintensity measure of input ground seismic hazard with intensitylevel x . LS refers to the limit state.

There are multiple proposed fragility relationships forreinforced concrete structural systems that were developedusing different methodologies and parameters for represen-tation of seismic demand and damage. These relationshipsare derived using a few different types of approaches, whichare summarized by Rossetto and Elnashai [4] as follows:(i) Empirical Fragility Curves based on field data. These arederived through statistical analysis of how real buildings per-formed in past earthquakes. Examples of such fragilities arethose proposed by Miyakoshi et al. [5] and Orsini [6]; (ii) An-alytical Fragility Curves. This approach uses numerical tech-niques to simulate the behaviour of systems including variationsin structural capacity and seismic demands. Studies done inthis category include those by Singhal and Kiremidjian [7] andMosalem et al. [8]; (iii) Judgemental Fragility Curves. Theseare fragility curves that are based partially or wholly on expertopinion. With this approach a wide range of structure types aredealt with in the same manner treating the level of uncertaintyuniformly; (iv) Hybrid Fragility Curves, which are constructedthrough a combination of more than one of the other threeapproaches.

The derivation of fragility relationships for high-risestructures requires a thorough evaluation of the factors that

influence the response of these complex and highly variablestructural systems and the predictions of their response usinga range of realistic input motions. There is a dearth of fragilityrelationships for high-rise buildings due to limited field dataand absence of efficient analytical approaches for conductingdynamic response history analyses (DRHA) of these structures.Of particular importance for predicting the response of RChigh-rise buildings to seismic actions is the complex behaviourof RC structural walls that can dominate the response ofthese structures to lateral loadings. Unfortunately, previousexperimental studies on different configurations of complexwalls, such as those done by Salonikios et al. [9], Thomsenand Wallace [10] and Riva et al. [11], are limited either inloading and boundary conditions or in the provided high-resolution test data. One ongoing work by Lowes et al. [12]is considering complex wall configurations including C-shapeto double C core sections and employing high-end technologiesfor high-resolution data acquisition; however, this project hasnot yet provided any experimental data. Consequently, fragilityrelationships for high-rise RC buildings must at this time bederived by purely analytical methods.

The proposed framework for deriving analytically-basedfragility curves for RC high-rise buildings is presentedin Fig. 1 which illustrates the key features such asselection of representative building structures, appropriateanalytical modelling, uncertainty consideration, limit statedefinitions and numerical simulations. Ideally, these analyticalassessments would be conducted by running DRHA usingthree-dimensional inelastic and geometrically non-linear finiteelement models. This is not possible as these models are notsufficiently mature to provide reliable predictions and the timerequired to build and run the analyses is prohibitive by ordersof magnitude. Hence, the success of an analysis framework is

J. Ji et al. / Engineering Structures 29 (2007) 3197–3209 3199

dependent on the efficiency, appropriateness, and reliability ofthe selected simplified model of the RC high-rise structure.

The primary technical contribution presented in this paperis the suggested process for developing a simple lumped-parameter model that can capture key aspects of the dynamicbehaviour of RC high-rise structures including RC materialinelasticity, second-order geometrical non-linearity, and frame-wall interactions. The selection of appropriate parameterswas made possible using genetic algorithms. Following thepresentation of this model, its effectiveness and computationalefficiency is demonstrated in a case study for an existingcomplex high-rise building. Sample fragility relationships arethen presented for three performance limit states, which aredefined as “Serviceability”, “Damage control” and “Collapseprevention” in a companion paper by Ji et al. [13]. Whereasthis paper presents the framework and a reference application,more extensive presentation of fragility relationships is givenin the companion paper and in which the sensitivity analysesare presented that are essential for developing final fragilityrelationships.

2. Analytical structural modelling

For typically complex wall-frame systems in RC high-risebuildings, the behaviour of the beams and columns can usuallybe adequately captured by fibre or multi-layer beam elementsin which only a strength check is made for shear (Spaconeet al. [14], Elnashai et al. [15]). For the walls, either continuumanalysis is required or the effects of shear must be handledseparately. It is essential to consider the contribution of theframe and the frame-wall interactions to obtain sufficientlyaccurate results from dynamic response history analyses.

A two-stage lumped modelling approach was used inthis study to provide an efficient computational model forconducting DRHA from which fragility relationships could bederived. In Stage I, the outer frame was modelled as equivalentnon-linear springs at the wall, while in Stage II the wall wasmodelled using lumped elements.

2.1. Analysis software platforms

The ZEUS–NL environment, developed by Elnashaiet al. [15], was adopted for running the analyses as: (i) itcan be used to model all elements of the frame FE modeland utilizes non-linear axial and rotational springs as well asBernoulli–Euler beam elements that employ a fibred sectionalapproach in which realistic constitutive relationships are usedfor the response of structural concrete to cyclic loadings, and(ii) it is capable of conducting conventional pushover, adaptivepushover, Eigen-value analysis and dynamic response historyanalyses.

It is inadequate to simply predict the behaviour of RCwalls using Bernoulli–Euler beam models as the behavior ofwalls is strongly influenced by their shear behaviour. Thenon-linear 2D continuum analysis tool VecTor2 (developed atUniversity of Toronto, Vecchio [16], Wong and Vecchio [17])was used to predict the detailed behaviour of the walls.

VecTor2 employs a rotating-angle smeared crack modelingapproach and implements both the Modified Compression FieldTheory (MCFT) by Vecchio and Collins [18] and DisturbedStress Field Model (DSFM) by Vecchio [19]. VecTor2 canmodel concrete expansion and confinement, cyclic loading andhysteretic response, construction and loading chronology forrepair applications, bond slip, crack shear slip deformations,reinforcement dowel action, reinforcement buckling and crackallocation processes.

2.2. Lumped modelling approach

The proposed two-stage optimization procedure is nowdescribed for the selection and calibration of the lumpedZEUS–NL model that was selected for running the non-linearDRHA from which example fragility relationships can then bederived.

2.2.1. Stage I — Outer frame elimination and global systemsimplification

The walls and core systems in RC high-rise buildings areusually dominant in resisting seismic loads especially at lowerbuilding levels, while the frame components principally supportgravity loads and contribute to the lateral resistance with frame-wall interactions (Ali [1]). It is therefore acceptable to replacethe frames in the dynamic analyses with equivalent non-linearboundary springs at the connection point of the wall and frameat each floor. This approach, as illustrated in Fig. 2, employsa joint element containing three springs which simulate thereaction forces [Fx , Fy , Mz] for the outer frame (as within aplane) based on the displacements [ux , u y , θz] at the wall joints.

2.2.2. Stage II — Simplify the structural wall into lumpedelements

It is possible to simplify the model for the wall into lumpedelements that capture the non-linear longitudinal behaviouracross the width of the wall as well as its shear behaviour. Twodifferent simplified lumped models were used and comparedin this Stage II optimization procedure. The simple-vertical-line-element model (SVLEM) proposed by Vulcano [20] isshown in Fig. 3(a) where single linear elastic beam elementsare connected with nonlinear axial and rotational springs.Ghobarah and Youssef [21] proposed a multiple-vertical-line-element model (MVLEM) as shown in Fig. 3(b). Theymodelled wall boundary elements as elastic truss elementswhich are connected with horizontal rigid beams supported byfour non-linear springs simulating the potential plastic hingeregion. An elastic beam element is used to represent wall webregion which is connected with the rigid bars at the center, andthen cut and linked with a spring kH for simulating the shearbehaviour.

2.2.3. Genetic algorithm application for parametric studyAlthough lumped-modelling approaches are conceptually

simple, the selection of a suitable replacement structure thatproperly considers the influence of the dominant parameterson the non-linear response is not trivial. There is no explicit

3200 J. Ji et al. / Engineering Structures 29 (2007) 3197–3209

Fig. 2. Equivalent non-linear springs at wall joint.

(a) Single-vertical-line-element(SVLEM) (After Vulcano [9]).

(b) Multiple-vertical-line-element (MVLEM) (After Ghobarahand Youssef [10]).

Fig. 3. Macroscopic structural wall models.

Table 1Standard versus genetic algorithms (adapted from Goldberg [12])

Requirement Standard algorithm Genetic algorithm

General process Generates a single point at each iteration Generates a population of points at each iterationDeal with parameters themselves Deal with coding of parameter setsThe sequence of points approaches an optimal solution The population approaches an optimal solution

Seeking algorithm Use functional derivatives Use objective function (payoff information)Selects the next point in the sequence by a deterministiccomputation

Selects the next population by computations thatinvolve random probabilistic choices

analytical approach to derive this model from inelastic FEAand direct optimization techniques for parametric studiesare not appropriate since they all depend on explicitfunctions. To develop a suitable lumped model, a reliableand effective computational approach is needed in StagesI and II to implicitly optimize the essential parametersthrough consecutive FEA runs and progressively lead tothe convergence of typical structural response to standarddata for calibrations. Genetic Algorithm (GA), a well-knownimplicit goal-seeking technique, was employed as a primarymethodology in this analytical approach. A brief introductionto GA and its use in this study is now presented.

The Genetic Algorithm, developed by John Holland(Holland [22]) and his colleagues (Goldberg [23]), is a goal-seeking technique used for solving optimization problems,based on natural selection. Since the appearance as aninnovative subject, GA application has been expanded fromsocial science, biology and computer science to the fields ofengineering topology and optimizations. For example, Adeliand Sarma [24] have developed computational approaches for

cost-based structural optimizations of tall buildings using GAand Fuzzy Logic. However the GA applications in structuralengineering are still in their early stages and this work by theauthors is the first application of GA for the tuning of FEmodel parameters with these software platforms and for usein modelling both global and local behaviour of RC high-risebuildings with a complex dual core wall-frame system.

Conceptually different from standard algorithms, GArepeatedly modifies a population of individual solutions called“generation”, as described in Table 1. In each generation, GAevaluates all individuals and selects those better performing orother specific individuals randomly as parents and reproducechildren for the next generation using three methods: elite,cross-over and mutation. Over successive generations, thepopulation “evolves” toward an optimal solution as shown inFig. 4.

The two-stage structural optimization procedure using GA isillustrated in Fig. 5 in which the left column shows the overalltwo-stage process, the middle column presents the structure ofthe two lumped modelling steps, and in the right column is

J. Ji et al. / Engineering Structures 29 (2007) 3197–3209 3201

Fig. 4. Structure of genetic algorithm (adapted from Goldberg [12]).

Fig. 5. Global lumped model derivations using genetic algorithm.

given a flow chart of the generic algorithm toolbox used in thisparametric study.

MATLAB codes were developed and used to realize thefunctionalities presented in Fig. 5. The described procedure isillustrated in the following case study.

2.3. Case study: Development of lumped-parameter model forsample building

The framework was implemented for the development ofa simple lumped-parameter model for a single frame (F4) ofan existing RC high-rise structure, the newly constructed high-rise Tower C03 in the Jumerirah Beach development, Dubai,United Arab Emirates as illustrated in Fig. 6. The primarycharacteristics of this tower are given in Table 2.

2.3.1. Simplification Stage I using genetic algorithmThe objective of the Stage I simplification is to replace

the outer frame with non-linear springs at the point of the

Table 2Main features of sample building

Features Description

Height (m) 184.000Total stories 54Regular storey height (m) 3.400Irregular storey height (m) 4.488Core walls (exterior and interior size) 9.43×3.25 (8.48×2.55) (m)

9.33×3.15 (8.48×2.55) (m)

9.18×3.05 (8.48×2.55) (m)

Concrete f ′c (MPa) 60 (wall); 40 (slab)Reinforcing bars f y (MPa) 421 (Grade 60)

frame-wall connection at each storey level. GA will be usedfor the selection of model parameters. The assessment ofthe suitability of the simplified model will be made by acomparison of natural modes as well as the results frompushover and DRHA. The complete frame model is shown inFig. 7(a) and typical cross-sections in Fig. 7(b)–(d).

3202 J. Ji et al. / Engineering Structures 29 (2007) 3197–3209

Fig. 6. Sample building structure and half plane view.

(a) Whole model. (b) Core wall (400 fibres). (c) Normal column (100 fibres). (d) Slab beam (250 fibres).

Fig. 7. Structural models in ZEUS–NL typical component cross-sections.

Table 3Modal periods and mass participation factors

Mode 1 2 3 4 (vertical) 5

Period (s) 3.05323 0.81950 0.36427 0.32787 0.22872Mass participation factor 0.5610 0.2637 0.0729 Neglected 0.0433Sum of mass participation factor of listed 5 modes 0.9409

From the results of the ZEUS–NL complete model of theouter frame, the modal mass participation factors (MPF) can bedetermined by Eq. (2):

MPF: Γi ={φi }

T [M] {1}

{φi }T [M] {φi }

(2)

where, for i-th mode φi — normalized i-th mode shape vector,M represents the mass.

The first five modal periods and related MPFs are listed inTable 3 in which the sum of the modal MPFs of four modesamong the first five is 94%.

Because of this, it is reasonable to evaluate the lateraldistributed loads for a pushover analysis from the 4 horizontalmodes as illustrated in Eqs. (3) and (4) and this leads to the finallateral load shape that is shown in Fig. 8. The vertical mode 4is not considered as it has little effect on the lateral response

J. Ji et al. / Engineering Structures 29 (2007) 3197–3209 3203

Fig. 8. Distributed lateral loads following mixed modal shape.

o

which is the focus of this study.

Proportional load vector with single modal shape:

Fi j =M jφ j i∑

jM jφ j i

. (3)

Proportional load vector with mixed modal shape:

F∗

j =

4∑i = 1

Fi jΓi (4)

where, i — selected mode, j — floor level.Using this force distribution, a pushover analysis was

undertaken in ZEUS–NL from which the overall nodaldisplacements and wall-frame interface forces were evaluated.In order to obtain a reasonable starting point for all the springconstants in GA parametric study, a non-linear least squaremethod was used to simulate the inelastic behaviour of the outerframe resistant forces corresponding to wall deformations ateach floor as shown in Fig. 2. Thereby, the outer frames canbe replaced by springs at joints that connect the wall nodes tofixed supports as shown in Fig. 9.

As discussed previously, the first step is to define thepopulation of individuals and fitness functions used in GA.In ZEUS–NL, the main properties of each single tri-linearspring, as shown in Fig. 9(b), are now defined where the initialand two-stage yielding stiffness and corresponding limit statedisplacements at interested DOFs (x, y, r z) are:[K1x K1y K1r z K2x K2y K2r z K3x K3y K3r z

]j

and[u1x u1y θ1z u2x u2y θ2z

]j

where, j refers to the floor level. The strain hardening andstrength degradation behaviours can be simulated by the tri-linear stiffness properties defined using joint elements inZEUS–NL.

(a) Simplified model protocol.

(b) Equivalent tri-linear joint spring.

Fig. 9. Main features in simplified model stage I.

There are two ways to optimize the parameters using GA;ne is to directly define the individual population as a group

3204 J. Ji et al. / Engineering Structures 29 (2007) 3197–3209

containing all parameters and the other is to simulate the springconstants as functions of other properties. For the first method,the protocol parameter vector of the population is:

C j =[K1x , K1y, K1r z, K2x , K2y, K2r z, K3x ,

K3y, K3r z, u1x u1y θ1z u2x u2y θ2z]

j (5)

j = 1, 2, . . . , 54.

The initial values Ci are obtained through the post-processing of the original ZEUS–NL analysis results in theprevious step. In the population, the independent variablevector size is determined by the total number of joint springparameters which are 810 for the 54-storey tower. Eachindividual vector is a set of 810 generated random numbers ofuniform distribution as follows:

X = [Rand(L B, UB)]1×810 . (6)

(L B, UB) are lower and upper bounds, set as (0.5, 2.0) here.So for each trial in GA, the individual joint parameter vector

K is computed as

K =[C1 C2 · · · C54,

]Kn = Kn · Xn (7)

where, n = 1, 2, . . . , 810. K is of dimension 810 by 1, and ineach generation the population consists of N such individualvectors where N is the population size.

For the second method, only selected parameters aresubjected to optimization. This is encouraged by theobservation that the stiffness values are decreasing withincreasing height. Thereby, the spring parameters C j can beassumed to follow certain functional trends, approximately asbelow:

(C j

)k =

(C1)k · jak , k = 1, 2, . . . , 15 for j ≤ 25

(C1)k · 25ak ·

(j

25

)bk

, k = 1, 2, . . . , 15

for j ≥ 25.

(8)

Since the building configuration changes at the 25th flooras shown in Fig. 6, the stiffness variations are different forthe structure below and above 25th floor, thus the functionalexpressions for C j need to change accordingly. Post-processingof the ZEUS–NL analysis is again applied to obtain the initialvalues. Only C1 is needed here, but additional estimations ofa and b for each parameter in C1 are also desired for theoptimization using genetic algorithms. The final number ofindependent variables in the vector is 45. Each individual is aset of 45 generated random numbers with a uniform distributionas follows:

X = [Rand(L B, UB)]1×45 (9)

where, (L B, UB) are lower and upper bounds, set as (0.5, 2.0)here.

So for each trial in GA, the individual joint parameter vectorK is computed as:

K =[C1 a b

], Kn = Kn · Xn (10)

where, n = 1, 2, . . . , 45. K is of dimension 45 by 1, and ineach generation the population consists of 50 such individualvectors.

The primary comparison between the lumped and morecomplete model was of the deformed shapes under thesame loadings. On account of different influences on wholebuilding behavior from boundary conditions at differentheights, weighting factors are introduced in the deformation-based fitness function. The fitness function is defined as:

R =

∣∣∣∣∣∣

(Dt

j − Doj

)j × Do

j

∣∣∣∣∣∣ , (11)

where, j—Floor level = 1, 2, . . . , 54.Dt

j — Wall nodal displacement along X at i th storey, fromlumped model.Do

j — Wall nodal displacement along X at i th storey, fromoriginal model.

Both approaches set the maximum generations at 100.The second approach was finally selected due to its greatercomputational efficiency. For the evolution, all three techniquesincluding Elite (survival selection), Crossover and Mutation,were coded in the MATLAB toolbox and employed to enhanceperformance of the genetic algorithm searching.

For Elite procedure, 20% of all parents with lower fitnessvalues were selected directly for the next generation:

Xm+1 = Xm, m — Generation number. (12)

For Crossover procedure, two methods were used: one wasthe partial crossover method as expressed in Eq. (13) and theother one was the complete crossover method as describedin Eq. (14). Crossover was the major reproduction methodgenerating 70% of the population of each generation

Xm+1 =

[X(1)

1 , X(1)2 , . . . , αlX

(1)l

+ (1 − αl)X(2)l , . . . , X(2)

n−1, X(2)n

](13)

where, αl — Preset or random number, ∈ [0 1] , 1 < l < n, nis variable vector length, m — Generation number

Xm+1 = αX(1)m + (1 − α)X(2)

m (14)

where, α — Preset value or random number, ∈ [0 1]; m —Generation number.

For the Mutation procedure, random scaling or shrinkingwas imposed on selected individuals according to fitnessperformance level as shown in Eq. (15). The mutation rate is10%.

(Xm+1)n = (Xm)n ·(βm

)n , n = 1, 2, . . . , 45 (15)

where, βm — Random vector generated from normaldistribution Norm

[1,

(1 − rnp

)], rnp — Ranking of parent

fitness performance level in precentage, m—Generationnumber, np—Parent number within [1, N ].

Lumped model Stage I was obtained as expected through theGA parametric study. Series of analyses with derived lumped

J. Ji et al. / Engineering Structures 29 (2007) 3197–3209 3205

Fig. 10. Pushover response comparisons between original and lumped models.

Table 4Modal analyses comparisons between original and simplified models

Mode Modal period (s)Original model Lumped model Stage I

Value Error (%)

1 3.05323 3.08966 1.22 0.81950 0.80381 −1.93 0.36427 0.34249 −6.04 0.32787 0.28195 −14.05 0.22872 0.19274 −15.7

models were performed for both modal and pushover analysesand comparisons with original model results were made formodal analyses (Table 4) and typical pushover curves (Fig. 10).

By these comparisons, it was observed that the relativeerror for first three significant modal periods, for which thesum of MPF at 90%, was less than 10% between thesetwo models. It was also observed that the load-deformationpushover responses from the lumped model matched well thosefrom the original model. These comparisons thereby indicatethat the derived lumped model was able to provide accuratepredictions of natural modes and static pushover behaviours.There were some expected errors including shorter periods forhigher modes and larger resistant forces in the highly inelasticrange. These errors are probably due to the computational jointmodels used in ZEUS–NL that require non-negative tangentstiffness values and thereby make it difficult to detect actualstrain softening of the substitute outer frame. Hence, thelumped model is a little stiffer than the complete frame model insome cases, especially for higher modes and large deformationranges where the RC outer frame members are more likely toget damaged. Such errors may be taken into consideration by

slight adjustments on epistemic errors in uncertainty modellingfor fragility assessment.

Similar comparisons were also made using DRHA as shownin Fig. 11(b) and (c); the sample ground motion record fromKocaeli Earthquake (1999, Turkey, Duzce station) is given inFig. 11(a).

At intermediate and top height levels, relative displacementtime histories from the lumped model are quite close to thosefrom the original model. That means that the lumped modelreplicates well the predicted seismic behaviour of the selectedbuilding.

2.3.2. Simplification Stage II using genetic algorithmThe objective of this simplification is to produce a simple

lumped-parameter model that will provide a similar predictionof the pushover response as that which would be predicted froma 2D continuum analysis for the core wall panels. The programVecTor2 is used to provide the continuum analysis predictions.Both the SVLEM and MVLEM models, as were describedin Fig. 3, were considered for developing the lumped modelin ZEUS–NL. The GA process was employed for parametervalue selection. In addition, the “hsv” type joint was applied toconsider the axial load-shear interaction effects in ZEUS–NL(Lee and Elnashai [25]). For all wall panels of the samesize, only one parametric study for lumped modelling wasrequired. The core wall panel prototypes used two typicalwall dimensions at lower and higher storey levels as shown inFig. 12(a); total storey forces imposed on the wall panel arediscretized into equivalent nodal forces at the top as shown inFig. 12(b) and (c) which also present the FE mesh used in theVecTor2 analyses.

The axial loads at different levels, as shown in Table 5and used in the VecTor2 pushover analyses, were imposed

3206 J. Ji et al. / Engineering Structures 29 (2007) 3197–3209

Table 5Designed capacities and applied load information

Design No (kN) 255 658Compressive capacity Cmax (kN) 620 000Tensile capacity Tmax (kN) 80 000

Applied loads 1 2 3 4 5 6Axial load 0 0.5No 1.0No 1.5No 0.075Tmax 0.225TmaxHorizontal displacement (mm) 20

(a) Strong motion.

(b) 25th floor.

(c) Roof.

Fig. 11. DRHA comparisons between original and lumped models.

as the initial load for the different analyses as required bythe non-linear hsv joints in ZEUS–NL. Horizontal loads wereincrementally applied in displacement control. All loads wereequivalent to uniformly distributed nodal loads along the floorat the top of the wall panel.

In the VecTor2 model, the concrete compressive stress–straincurve by Popovics [26] was used for normal strength concreteand Modified Popovics curve by Collins and Porasz [27] forhigh-strength concrete was used for both pre-peak and post-peak concrete behaviour. The effect of confinement stresses fol-lows the suggestions by Kupfer et al. [28].

Such joints in ZEUS–NL have a similar tri-linear nature asprevious except the shear stiffness represented by ‘hsv’model,which is defined as “Hysteretic Shear model under axial forcevariation”, requests series of tri-linear shear response curveswith stiffness values and related displacements under differentaxial loads including four compressions and two tensions(in reference to ZEUS–NL version 1.7 manual by Elnashaiet al. [29]).

For the SVLEM model in ZEUS–NL, only one joint elementis needed for each wall, with parameters at x , y and rotationabout z, and there are only three types of wall sections.Therefore, even considering direct GA optimizations on alljoint element parameters, the population size of each generationis small enough not to significantly increase the computationalcost. Initial elastic and the two-stage yielding stiffness valuesand corresponding limit state displacements are defined as

KWx =[[K 1

1x K 12x K 1

3x

],[K 2

1x K 22x K 2

3x

], . . . ,

[K 6

1x K 62x K 6

3x

]]1×18

UWx =[[u1

1x u12x u1

3x

],[u2

1x u22x u2

3x

], . . . ,

[u6

1x u62x u6

3x

]]1×18

KWy =

[[K1yT K2yT K3yT

],[K1yC K2yC K3yC

]]1×6

UWy =

[[u1yT u2yT

],[u1yC u2yC

]]1×4

KWRz =

[K1r z K2r z K3r z

]1×3

UWRz =

[θ1r z θ2r z

]1×2

(16)

where, values with T and C for the y direction indicate differentresponses under tension and compression. The six rows for thex direction present the data sets for the six axial load levels.

The protocol parameter vector of the population is

CW=

[KW

x , KWy , KW

Rz, UWx , UW

y , UWRz

]. (17)

The initial values for CW are obtained from the VecTor2analysis results in the previous step. In this situation, the sizeof the independent variable vector within each population is51. As done for Stage I, each individual vector is a set of 51generated random numbers which have a uniform distributionas follows:

XW= [Rand (L B, UB)]1×51 (18)

(L B, UB) are lower and upper bounds, set as (0.5, 2.0) here. Sofor each trial in GA, the individual joint parameter vector K iscomputed as:

Kn = CWn · XW

n n = 1, 2, . . . , 51. (19)

A similar procedure is applied for MVLEM model, exceptnow for a total of five joint elements including four vertical

J. Ji et al. / Engineering Structures 29 (2007) 3197–3209 3207

(a) Wall prototype. (b) Equivalent nodal loads. (c) FEM model in VecTor2.

Fig. 12. Discrete FEM model of core wall panel.

(a) SVLEM. (b) MVLEM.

Fig. 13. Pushover comparisons for SVLEM and MVLEM with VecTor2 model.

springs at the bottom and one horizontal spring at the lower partof the wall panel. As in Stage I, the three techniques of Elite(survival selection), Crossover and Mutation, were employedfor generating populations using the genetic algorithm toolbox,as described by Eqs. (12)–(15). In the ZEUS analysis forfragility assessment, nodal displacements at the centre of thewall panel at each floor level were required. Therefore, the maincomparison between the optimized lumped model and the RCcontinuum model were control point displacements. Becausethe floor slab was treated as rigid in the ZEUS–NL model, whilenot in VecTor2, the nodal rotation at control points is excludedfrom the comparisons. It follows that the fitness function isdefined as

R =

∣∣Dt− Do

∣∣|Do|

(20)

where, Dt — Wall nodal displacement at X and Y directioncomputed from lumped model, Do — Wall nodal displacementat X and Y direction obtained from VecTor2 results.

The GA optimizations for the parametric study successfullyhelp to finalize the lumped model for Stage II. A comparisonof the pushover prediction under designed dead loads for wallsection types is presented in Fig. 13.

The static pushover response from both the SVLEM andMVLEM are close to the response from the VecTor2 detailed

continuum FEM analysis. The SVLEM model does not providequite as close a fit particularly around the ultimate storeyshear but even this error is acceptably small. The comparisonsillustrate that using GA optimization in parametric studies, bothSVLEM and MVLEM can provide lumped models for wallpanels that are reliable and sufficiently accurate for capturingthe inelastic behaviours including cracking, steel yielding andconcrete crushing.

2.3.3. Evaluation and commentsThrough the implementation of the above two-stage model

optimization procedure, a comparatively simple lumped-parameter model is created that consists of beam elements, rigidbars and non-linear springs. The simplified model is sufficientlyaccurate for evaluating nodal displacements, global internalforces and at the same time accounting for shear deformationswithin the structural walls. The required computational timefor completing a DRHA is reduced to a small fraction ofwhat it would have been for an analysis of the originalmodel. Table 6 compares the details and computational timesbetween the original whole building frame model and thefinal lumped model in ZEUS–NL. Though it took around oneweek to actually complete the lumped modelling process forthis specific building, the computational saving of next stepDRHA runtime is much more significant compared to this

3208 J. Ji et al. / Engineering Structures 29 (2007) 3197–3209

Table 6Variation from original model to final lumped model

Variation From original to lumped models

Model geometry

Node number 876 → 472Element number 1910 → 632DRHA runtime 10–15 → 1

cost. The total estimated runtime of DRHA and post-processfor the reference structure was about 3 h per 1000 time stepson a Pentium 4 CPU running at 2.66 GHz and with 1 GBof RAM. This would have required 1800 h (2.5 months) toconsider 600 ground motion records with an average of 1000time steps. Using the same computer, the derived lumped modelapproach resulted in substantial time savings which createsgreater flexibility for making fragility assessments.

3. Fragility assessment

The analytical framework described in this paper is used todevelop sample fragility curves for Tower C03 of the JumeirahBeach development in Dubai. The fragility assessment is morefully described in another paper by Ji et al. [13], includingevaluation and discussion of the effect of uncertainties inmaterial properties, seismic action, and numerical modelling.Herein, a sample fragility curve assessment is briefly introducedto illustrate the basic ideas and main features.

Three categories of natural records were used in thisinvestigation to account for variation in magnitude as well asthe distance from fault and soil conditions. Hence the effectsof uncertainties in ground excitation are taken into accountwhen performing numerical simulations with seismic load inputconsisting of records varying in all three categories.

Considering their unique structural configurations andseismic behaviours new performance limit states are proposedfor RC high-rise buildings under earthquake strikes, correlatingglobal system response and local resisting componentbehaviour together, as defined by: (1) Serviceability; (2)Damage Control; (3) Collapse Prevention. The quantitativecriteria corresponding to the definitions were derived from theresults of the pushover analyses based on the knowledge of bothglobal structural response and local damage patterns.

Numerical simulations were conducted using the derivedlumped-parameter structural model and 30 ground motionrecords for three defined limit states. This involved the useof a developed code combining pre- and post-processors for

Fig. 14. Sample fragility curves based on SA (T = 1 s).

ZEUS–NL. The probability cumulative function of Log-normaldistribution was employed to obtain a comprehensive functionalfragility relationship. Fig. 14 presents fragility examples ofthe reference building based on the intensity measure (IM) ofspectral acceleration (SA) at a natural period of T = 1.0 s.

4. Conclusions

In this study, an analytical framework was presented formaking fragility assessments of RC high-rise buildings and itsuse was demonstrated in a reference application. The principalcontribution presented in this paper is the developmentof simplified lumped-parameter analytical models that werecreated to run in the ZEUS–NL environment and that consistedof beam elements, rigid bars and non-linear springs. This modelwas derived through a two-stage optimization procedure thatemployed Genetic Algorithms to capture the predicted inelasticnon-linear response from more detailed and refined analyticalmodels. This simplified model is capable of representing theinteractions between core wall and external frame membersby providing sets of non-linear interface springs at X , Y andRz directions at each storey over the height of the wall. Thecomplex behaviour of the wall, including its flexural, shear andaxial response and their interactions, were also accounted for.The ZEUS–NL program was also used in the determination ofthe spring stiffness values to represent the outer frame (Stage Ioptimization) while the VecTor2 non-linear continuum analysistool was used in the selection of the lumped-wall model (StageII optimization). The use of the simplified model resultedin very significant reductions in analysis times that render itpractical to develop probabilistic fragility curves for complexhigh-rise structures, where none have existed in the literature.

By the demonstrated functionality of the proposed frame-work, it is possible to extend the approach to the assessment ofthe seismic response of other types of RC high-rise buildings.Uncertainties of this proposed approach could be reduced byfurther model validation with field experiences and experimen-tal test data.

Acknowledgements

This study is a product of project EE-1 ‘VulnerabilityFunctions’ of the Mid-America Earthquake Center. The MAE

J. Ji et al. / Engineering Structures 29 (2007) 3197–3209 3209

Center is an Engineering Research Centre funded by theNational Science Foundation under cooperative agreementreference EEC 97-01785.

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