THREE PROPOSALS FOR CHARACTERIZING MDOF NONLINEAR

SEISMIC RESPONSE

By Paolo Bazzurro; C. Allin Cornell,:Z Member, ASCE, Nilesh Shome/and Jorge E. Carballo4

ABSTRACT: This paper reviews, compares, and contrasts the three methodologies proposed by C. A. Cornell,by Y. K. Wen, and by FEMA 273 (as proposed by H. Krawinkler) for predicting the postelastic response ofbuildings subject to seismic excitation. All three methods have the potential to contribute effectively to theproblem of improving the practical prediction of seismic response of structures. Applications include preliminarydesign, routine (code) use, calibration of codes, and specific structure assessment. These methods appear differentbecause their authors did not share the same objectives or information bases. The methods by Cornell and Wenare decisively prediction-oriented in explicitly probabilistic frameworks. The approach suggested by Krawinklerhas instead the goal of understanding in some average sense the seismic response of structures at variousperformance levels. This last method has been adopted, however, for prediction purposes by the FEMA 273Guidelines for rehabilitation of existing buildings. Despite the differences, the work proposed by these research­ers is profitably brought under the same perspective here.

INTRODUCTION

As the engineering profession begins to depend more andmore on the results of nonlinear time-history analyses, severaldifferent approaches have been proposed by researchers to ad­dress the challenging problem of practical estimation and rep­resentation of the response beyond maximum elastic capacityof realistic MOOF (multiple degrees of freedom) models ofstructures under severe ground motions (see Takada et al.1988, 1989; Uang 1991; Bertero and Bertero 1992; Mirandaand Bertero 1994; Nader and Astaneh 1994; Fischinger andFajfar 1994; Miranda 1997; and many others in the referencessection). In this note we shall compare and contrast only thethree methods put forward in the past few years by Cornell(CAC) and by Wen (YKW) and Foutch, as well as the oneadopted by the FEMA 273 document "NEHRP Guidelines forSeismic Rehabilitation of Buildings" (ATC 1996a, 1996b),which is based primarily on the work by Krawinkler (HK).The methods do not necessarily share the same objectives orinformation bases, hence they may at first appear quite differ­ent.

This paper has been prompted by a recent paper by Hanand Wen (1997) who, when motivating their methodology,made a per force brief comparison of the works done by thethree aforementioned research groups. Although important dis­similarities do exist in these procedures and should be pointedout, we realized that the work done and approaches proposedby these three groups are in fact quite similar. From this ob­servation follows the present attempt to bring these methodsunder the same perspective. They all hold potential for prac­tical application in the near term. Workers in the rapidly evolv­ing performance-based design field, which depends on thisprediction capability, should find of value this evaluation ofthe three methods.

A brief summary of these methodologies as originally pro-

'PhD, Dept. of Civ. and Envir. Engrg., Stanford Univ., Stanford, CA94305.

2Prof., Dept. of Civ. and Envir. Engrg., Stanford Univ., Stanford, CA.'PhD Candidate, Dept. of Civ. and Envir. Engrg., Stanford Univ., Stan­

ford, CA.'PhD Candidate, Dept. of Civ. and Envir. Engrg., Stanford Univ., Stan­

ford, CA.Note. Associate Editor: Chia-Ming Uang. Discussion open until April

1, 1999. To extend the closing date one month, a written request mustbe filed with the ASCE Manager of Journals. The manuscript for thispaper was submitted for review and possible publication on November19, 1997. This paper is part of the Journal of Structural Engineering,Vol. 124, No. II, November, 1998. @ASCE, ISSN 0733-9445/98/0011­1281-1289/$8.00 + $.50 per page. Paper No. 17035.

posed by their authors follows in the next section. In the lastsection these three procedures are first compared critically andthen applied to the problem of predicting the (nonlinear) max­imum lateral roof displacement of a building associated witha specified mean return period. The simple example is con­ducive in making the similarities among these methods moreapparent.

SUMMARIES OF THREE METHODS

Method Proposed by FEMA 273/HK

The goal of these guidelines is to establish an easy-to-usemethodology for conceptual performance-based seismic designand rehabilitation of buildings. In modern multi-level designprocedures, structural performance is gauged in terms of de­formation demands being limited within defined levels (e.g.,Bertero and Bertero 1994; SEAOC 1996; Bertero 1996; Cor­nell 1996b; Hamburger 1996; Krawinkler 1996; Moehle 1996;ATC 1996a, 1996b). Here, "deformation" generically refersto both global (e.g., roof displacement) and local response pa­rameters (e.g., interstory drifts and plastic hinge rotations).

Although the nonlinear dynamic procedure provides themost accurate tool for computing internal forces and defor­mations of a structure subject to severe ground motions, it isseldom-due to its inherent complexity and relative unfa­miliarity-carried out in everyday practice. Simplified anal­yses such as the linear static procedure, the linear dynamicprocedure (LOP), and the nonlinear static procedure (NSP) areroutinely used instead (ATC 1996a, 1996b; Hamburger 1997).

Following the idea of Newmark and Hall (1973), in orderto estimate the nonlinear response of MOOF structures bymeans of these simplified procedures, HK and his co-workers(see Osteraas and Krawinkler 1990; Nassar and Krawinkler1991; Rahnama and Krawinkler 1993, 1994; and Seneviratnaand Krawinkler 1994, 1996, 1997, for details) have developedperiod-dependent correction factors to be applied as modifierseither (1) of the applied shear force (as in the linear staticprocedure), or (2) of the computed actions (Le., forces andmoments) and deformations (as in the LDP), or (3) of thenonlinear deformations (in the NSP). Such factors have beenadopted by the FEMA 273 Guidelines. When linear analysesare performed, acceptability of the "adjusted" response maybe checked in terms of either internal forces or deformations;when nonlinear procedures are applied the checks must be interms of deformations. For brevity, in the following we shallconsider only the displacement-based case.

In this framework, the maximum lateral inelastic displace-

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regular buildings. Correction factors for MOOF buildings withvertical irregularities of mass, stiffness, and strength are cur­rently under study by HK and his co-workers.

The relationships for this correction factor, C, as a function ofthe level of nonlinear response (e.g., ductility), are establishedby regressing the values of the 8MDOP/8ENS ratio obtained frommultiple response analyses performed for a group of (e.g., 88)ground motions of different magnitude, M, recorded at differ­ent source-to-site distances, R.

The dependency of C on M and R is also studied. Thisdependency would exist if the MOOF response were depen­dent on characteristics of the ground motions (e.g., frequencycontent and duration, which are in turn potentially affected byM and R values) to a degree not already captured by the 8ENS '

However, the correction factor, which shows a limited record­to-record scatter compared to the variability of the ground mo­tion, is observed to be virtually independent of M and R. Thedependency of C on other characteristics of the ground motion,such as level of spectral acceleration or duration, is impliedby the writers but not explicitly investigated. This lack of de­pendency on M and R is exploited by adopting this structure­specific response correction factor, C, regardless of the build­ing location in the region and, hence, regardless of theparticular spectrum of potential magnitudes and distanceswhich threaten a site.

Therefore, when the MOOF building is thought to be lo­cated at a given site, the researchers suggest the followingstrategy to estimate the MDOF response of the structure. A

Method Proposed by Wen and His Co-Workers

The objective here is somewhat different. It is the estimationof the probabilities of exceeding various limit state conditionsof MOOF structures; these are to be used for calibration ofdesign parameters in fUlly reliability-based seismic codes (Wen1995; Han and Wen 1997). Unlike current NEHRP develop­ments, such code bases consider the range of possible groundmotion levels and the variability in responses and capacities.Therefore, the responses to multiple likelihood levels ofground motion of a suite of representative structures withineach building class (e.g., special moment resisting frames) lo­cated at several generic sites have to be systematically eval­uated. This process without any simplification obviouslywould require an unmanageable computational burden. Theresearchers are seeking ways to reduce this cost without sig­nificant loss of accuracy.

The generic-site inelastic response of each single structurebelonging to a certain building category is computed by per­forming comparative dynamic analyses of a nonlinear MOOFmodel and of an "equivalent" simplified nonlinear system(ENS) model of the building subject to a suite of unscaled realground motions. For the ENS models, Wen and Foutch haveemployed both SOOF systems (Collins et al. 1995, 1996a,1996b) and 200F systems (Han and Wen 1994, 1997) whosecharacteristics, such as initial frequency of vibration and modeshape(s), global yield displacement and post-yielding behavior,match those of the original nonlinear MOOF structure. Em­pirical correction factors applied to global (e.g., roof displace­ment) and local (e.g., interstory drift) response parameters­the latter available only when the 200F system is used-aredeveloped to obtain estimates of the MOOF (elastic and ine­lastic) responses directly from the computationally "cheaper"ENS responses.

In the same framework and notation as before, the elasticand post-elastic displacement of the MOOF building, 8 MDOF,

can be obtained from the response of the ENS, 8 ENS' multipliedby the either global or local correction factor, C, as follows:

(3)8MDOF "" CBENS

(1)

The three frequency- and damping-dependent coefficients arebased on comparative numerical analyses carried out on bothlinear Single-Oegree-of-Freedom (SOOF) systems, whosefirst-mode period equals that of the MOOF structure, and onnonlinear SOOF systems with the same initial period and re­storing force characteristics as described above. An additionalnonlinear response modification coefficient not included abovehas also been developed to incorporate the effect of soft soilat the site. The question of record selection and the record-to­record variability of responses is considered by using the av­erage value for C;'s obtained from a suite of unscaled records(typically 10-15) whose average acceleration response spec­trum shape somewhat matches the ATC design spectrum forsoil type Sl (Nassar and Krawinkler 1991).

When NSP (commonly known as a static "pushover anal­ysis' ') is performed instead, the force-displacement relation­ship extending from linear response to failure of the buildingis determined. This relationship is also used to compute the"effective" stiffness of the structure (corresponding to the ex­pected displacement range), which is in turn used to assess the"effective" first-mode structural frequency of vibration, j.(ATC 1996a). The design value of 8 MDOFNL (the roof displace­ment in this case) is then estimated by modifying the linearelastic spectral displacement, Sd' or the spectral acceleration,Sa, at the "effective" frequency, j., and damping of the linearSOOF system:

• C1 accounts for the expected ratio of maximum nonlinearinelastic to maximum linear elastic displacement ampli­tudes in structures with stable and full hysteretic loops(e.g., a bilinear model).

• C2 deals with the expected larger maximum nonlinear in­elastic displacement due to stiffness degradation andstrength deterioration.

• C3 represents increased nonlinear displacements due todynamic P - a effects.

~ ~UMDOFNL "" COC.C2C3Sd "" COC.C2C3 (2-rrJ.)2 (2)

where the additional factor, Co, is the ratio of 8MDOFL and Sd'Co values are made available for both moment resisting framesand structural (shear or bending) walls. To verify the struc­ture's adequacy, the local forces and deformations (e.g., mem­ber forces and deformations and interstory drifts) are evaluatedfrom the pushover analysis results at the level correspondingto the roof displacement value, 8 MDOFNL predicted by (2).

It should be noted, however, that when 8 MDOFNL represents alocal response measure (e.g., interstory drift or plastic hingerotation), or when significant higher mode effects are expectedin the structure, the response cannot be adequately capturedby nonlinear SOOF response analyses or by static pushoveranalyses. In such cases, an additional factor, CMDOF (Nassarand Krawinkler 1991; Seneviratna and Krawinkler 1996,1997), has been obtained by comparing structural responsesfrom both linear and nonlinear MOOF analyses of the build­ing. This factor is not reported above because it is not includedin the FEMA 273 document. Finally, it is worth remarkingthat all these correction factors were derived for structurally

where

ment of interest, 8 MDO!'NL' to be expected in the nonlinearMOOF structure subject to the design ground motion can beestimated by modifying the maximum elastic displacement,8MDOFL' computed by applying the LOP to a MOOF linearmodel of the structure, as follows:

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large number of artificial ground motion records is employedto compute the ENS responses, 8ENS' which in turn are mod­ified by means of the global or local correction factor, C, pre­viously established (from the generic suite of real ground mo­tions). The synthetic records are generated, via the MonteCarlo method, based on the seismotectonic features of the re­gion around the site (most likely recurrence models, magnitudeand distance values, appropriate mean and variance of groundmotion attenuation characteristics, etc.), in order to be statis­tically representative of all possible future events (Le., all Mand R pairs) which may occur (e.g., Wen et al. 1995). Thissimulation-based method, unless some kind of "smart" sam­pling techniques are adopted, may require the generation ofhundreds to thousands of ground motions (e.g., 1,292 recordsin Wen 1995).

In general, for any given fault i, the conditional probabilityof exceeding a limit state described by a given displacementx, P[8MDOF ~ xlevent]/ is evaluated by fitting an extreme-valuemodel through the MDOF response data, 8MDOF, obtained bymodifying the ENS response values (to the synthetic recordsgenerated for this pair), 8ENS' via the correction factor, C. Themagnitude and distance values of each simulated event onsource i are drawn according to the joint M - R probabilitydensity function, !M,R(m, r)/, for fault i. Finally, for the MDOFbuilding the annual (unconditional) limit-state probability,P[8MDOF ~ x], is computed by summing the contributions ofall faults as follows:

P[8MDOF 2: xl "'" P[C8ENS 2: xl "'" 2: v I P[C8ENS 2: xlevent]1I (4)

where VI represents the mean annual rate of activity of eventsgenerated by fault i above a specified magnitude level, mo (say,mo = 5).

The dispersion on the correction factor C can be incorpo­rated in the previous limit-state probability calculations bymodeling the factor as a random variable (Han and Wen 1997).The effect of the comparatively small scatter on C on the over­all probability estimation is predicted by YKW to be generallyquite limited. An explicit formulation, however, is not carriedout.

Method Proposed by Cornell and His Co-Workers

The methodology devised by CAC has the objective of es­timating the annual probability of exceeding a given targetlevel of post-elastic response (or demand) in a specific MDOFstructure located at a given site due to potential seismic loads.As before, the response is intended here to represent both localdemands (e.g., interstory drift or plastic hinge rotation) andglobal demands (e.g., lateral roof displacement) which can begauged in terms of any desired damage measure (e.g., ductilityor dissipated hysteretic energy).

CAC and his co-workers developed a dual procedure whichcouples conventional probabilistic seismic hazard analysis(PSHA) with nonlinear dynamic structural response analyses.The damage assessment problem is addressed from two alter­native yet equivalent standpoints: the displacement-based andthe force-based perspectives, each of which has its own ad­vantages and disadvantages. For consistency, in the followingthe emphasis shall be put on the more direct, displacement­based approach (Shome et al. 1998). The interested reader canfind the details of the force-based method in Cornell and Sew­ell (1987), Sewell (1989), Inoue (1990), Inoue and Cornell(1991), Bazzurro and Cornell (1994), and Cornell (1994 and1996a).

The main idea is that the post-elastic response of a nonlinearMDOF structure subject to a set of ground motions shows astrong dependence on a record-specific quantity: typically the

linear elastic spectral acceleration, Sa(f, ~), at the fundamentalfrequency,f, and damping level, ~, of the undamaged structure.Sa(f, ~) (in short, Sa) is a parameter very well studied by earth­quake engineers and strong-motion seismologists, who in thelast 30 years have put a great deal of effort into improving itsprediction, given the occurrence of an event of magnitude, M,at a distance, R, from the site. A large collection of ground­motion predictive equations developed for this purpose are infact available in the literature. These equations are routinelyused in a probabilistic context (Le., PSHA) to provide quan­titative estimates of annual probabilities of exceeding any levelof Sa at a specific site. CAC's methodology builds directly onthis readily available knowledge.

The second step involves the estimation of the probabilityof exceeding a limit-state response in the nonlinear MDOFstructure, given that the M- and R-dependent quantity Sa equalsa specified level x at the site. It is important to outline that theconditioning is only on the Sa value; any additional potentialdependency on M and R is typically dropped. This importantsimplifying assumption must be confirmed by empirical re­search. It has indeed been consistently observed for a largevariety of SDOF systems (Sewell 1989) and for several MDOFsystems (e.g., Inoue 1990; Bazzurro and Cornell 1994) thatlinear and nonlinear structural responses are, for all practicalpurposes, not statistically dependent on M and R beyond theirdependence through the first-mode spectral acceleration level.Exceptions include responses measured in terms of dissipatedhysteretic energy (e.g., Shome et al. 1998).

The strong trend with Sa and this empirically observed lackof (conditional) dependency on M and R have been confirmed,for example, by computing the post-elastic structural responsesof a MDOF building for two different sets of 20 real ground­motion records belonging to two nonoverlapping, narrowranges of M and R (Shome et al. 1998). The two pairs of Mand R were selected in order that an attenuation law wouldpredict at the site approximately the same median spectral ac­celeration, Sa' The structural strength was designed in orderthat virtually all the ground motions would induce a responsein excess of yielding.

The response values showed a clear dependence on Sa butalso a great dispersion, since the Sa values of different recordswithin the same M and R bin spanned over one order of mag­nitude. The median magnitudes of the two bins differed byone and a half magnitude units. Nonetheless, the two bin me­dian responses differed only to the (limited) degree that thetwo median Sa's differed. The three regression lines of theresponse measure (e.g., maximum interstory drift measured interms of story ductility) on Sa computed by using each bindata set separately and then by combining data from both sets,were statistically indistinguishable. This confirms that condi­tional on Sa this nonlinear response measure shows no signif­icant dependency on M and R.

It is also observed that when the records in the two binswere normalized (scaled) to a common value, like Sa, prior totheir use in the structural analyses, the (statistically) same me­dian nonlinear response, 8 MDOI'NL' obtained from the unscaledsets is recovered but with greatly reduced variability (the COYis about half). Further, despite the systematic differences incertain ground motion characteristics, the same conclusions(i.e., same 8MDOFNL and reduced variability) were obtainedwhen records from much "stronger" and "weaker" M and Rbins (inter-bin Sa's different by a factor of 3) were successivelyscaled to the same Sa value of each bin. Subsequent efforts(Shome and Cornell 1998) have shown that if sample sizesare increased substantially, one can in some cases discern asystematic, albeit weak (no more than 10-15% per magnitudeunit), dependence of (median) ductility on magnitude. It canapparently be ignored for most practical purposes.

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These results imply that:

1. A statistically unbiased estimate of 8MDOFNt., given a spe­cific value of So, namely 8MDOFNt.1 SO, can be estimatedwith the same accuracy from fewer records if they arescaled beforehand to the same level of So' Given the re­duced dispersion, the median nonlinear MDOF responseconditional on the So level can be estimated within 15%by only 5 appropriately scaled records. This small num­ber of nonlinear dynamic analyses can be affordable evenfor a large MDOF structure.

2, A careful search of the records that closely match thehypothesized M and R scenarios for the site with the aimof computing the structural response of the building isnot crucial. Beyond the spectral So level, whose depen­dency on M and R is already captured in the site-specificPSHA, M and R play little systematic role in typical non­linear response measures. A selection of records accord­ing to M and R values may of course be important, how­ever, in those cases where near-source directivity effectsare expected to be significant (e.g., Somerville et al.1997).

Since the dependency on M and R of the response, given So,is weak and its variability, given So, is much less than that ofSo, given M and R, the (unconditional) probability of exceed­ing a given level x of nonlinear response, 8MDOFNt.' in a MDOFbuilding located at a specific site can be computed as follows:

P&", =P[8MDOFNL ~ x]

"" 2: P[8MDOFNL ~ xls.(f, ~) =s.,JP[Sa(f, ~) =s.,.t]k (5)

where the first factor in the right-hand side can be computedby fitting a regression model to nonlinear MDOF responses ofrecords scaled to two or more different levels of spectral ac­celeration. The model can be assumed to be of the form8MDOFNL = 13oS~l, where 130 and 131 are regression parameters.The second factor can be easily extracted from a conventionalsite hazard analysis.

If the variability of the structural response for the same Sa,tlevel is neglected, (5) considerably simplifies to the following(first-order) relationship:

(P&",)1 "" peS. ~ 05.",] =H(S.",) (6)

where H(Sa",) is the value of the site's seismic hazard curvefor Sa(f, ~) corresponding to the median level of So, (i,e., So",)capable of inducing the specified response value x as predictedby the regression model (i.e., So", = (.x1130) 11131).

An estimate, (P3.x)2, of P(8MDOPNL ~ x] more accurate than(P3",)1 is obtained by considering the response variability forthe same Sa,t level. Under the mild assumption supported bythe data (see Shome et al. 1998) that the response 8MDOPNLgiven So = Sa,t follows a lognormal distribution (e.g., Cornell1994, 1996a):

where

• k, is the slope (in log - log scale) of the straight linetangent to the hazard curve at the point So",, This impliesthat in the vicinity of So", the hazard curve is replaced byits tangent of equation H(Sa) = koS;/I. Such a lineariza­tion is usually a very good approximation over an orderof magnitude probability range around the tangency point.To quantify this term note that k, = l/log alQ, where alQis the factor by which one must increase So in order to

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increase the hazard (or reduce the mean return period) byone order of magnitude.

• 0' is the standard deviation of the (natural log) response~alues, In(8MDOFNt.), given the spectral acceleration valueSo",, An estimate of 0' can be acquired from the responseanalysis results (see Shome et al. 1998).

• 13, is a parameter of the fitted regression model.

Hence, loosely speaking, the annual probability of exceedinga specified level x of 8MDOPNL is simply the value of the hazardcurve at the level So", capable of generating a response in thestructure exceeding x by 50% of all relevant records, multi­plied by an exponential correction factor. Given the typicalvalues of 0', in the order of 0.3, and of the slopes of the hazardcurves for most sites, commonly varying from 2.5 to 6, and131 equal to 1 (a value commonly observed), such a correctionfactor in most practical cases is not larger than 2.0.

For the discussion which will follow in the last section, itis advantageous to consider (7) from a performance-based de­sign viewpoint. In this context, it is desirable to estimate the(unknown) median response 8MDOPNL = (8MDOFNL)T associatedwith a specified mean return period, T (e.g., 1,000 years or,alternatively, with a mean annual probability of exceedance p&= 0.001).

If the variability in the response is disregarded, it followsf~om (6) that a first order estimate of (8MDOPNL)T, i.e.,(8MDOFNL,I)T' can be easily obtai~ed by reading off from theregression model the value of 8MDOFNt. corresponding to thevalue Sa.T such that H(Sa,T) = P& = liT. From (7), however, itis clear that the explicit consideration of such variability in­creases the probability of exceedance of a specified level ofr~sponse. Hepce, a second order estimate, (8MDOFNL ,2)T' of(8MDOFNL)T is (8MDOFNL)T is associated with a spectral accelerationlevel Sa,2 somewhat larger than Sa,T' The value of Sa,2 can beestimated from (7) by replacing the seismic hazard curveH(Sa) by its linear approximation H(Sa) = koS;~ at the un­known value Sa,2' and by solving for it:

(8)

Finally, it is worth noting that, for historical reasons, thisprocedure uses spectral acceleration, So, as the "intensity"measure of ground motion, In this context, the procedure sum­marized above could alternatively make use of the spectraldisplacement, Sd' quantity which, of course, is only a constantapart (i,e., (21T f)2) from So' This simple change brings thismethod even closer to the previous two. For example, (5)could be interpreted as a relationship for estimating the prob­ability of exceeding a specified level of inelastic displacementin a MDOF structure, given that the maximum linear response,Sd' of the first-mode counterpart SDOF model of the structureis equal to each of a series of increasing values, Sd,.t.

COMPARISON OF THREE METHODOLOGIES

The estimation of the seismic response (or demand) ofMDOF systems beyond their maximum elastic capacity is thecommon denominator of all three procedures. These methods,however, are substantially different in several aspects, such as

1. Original goal2. Probabilistic versus deterministic perspective3. Ground motion interface and use of time histories4. Solutions adopted for achieving efficiency and making

the method of practical use

Despite the dissimilarities, which will be discussed now in thesubsection, these methods can be brought under the same lightwith the aid of a simple example.

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Main Similarities and Differences

HK's work focuses mainly on developing a simplified toolfor use in conceptual design of classes of buildings subject toseismic loads. These classes may be, for example, special mo­ment resisting frames, whose seismic design is controlled byinterstory shear forces, or structural shear and bending walls.This method, however, has also been adopted for rehabilitationof existing buildings in the FEMA 273 Guidelines (ATC1996a, 1996b). Unlike in the methods proposed by YKW andCAC, no particular attention is given by HK to quantitativelyestimating the likelihoods of the structure (being designed/re­habilitated) experiencing specified states of damage.

In HK's procedure it is assumed that the structure design isacceptable when it sustains without excessive deformations thedemands imposed on it by ground motions consistent with thelevel prescribed in the building code. The consistency is de­fined as matching in some "average" sense the response spec­tra of the selected ground motions to the design response spec­trum shape and level corresponding to a predefined meanprobability of exceedance for a given exposure time (e.g., 10%change in 50 years). Times, locations, and sizes of future earth­quakes are not explicitly considered in HK's method since theyhave been already "accounted for" during the definition ofthe seismic spectral input. Hence, possible dependency ofstructural response on ground motion parameters, such as Mand R, poses a very limited concern and, therefore, is onlypartially investigated (Lawson 1996). We shall return to thismatter further below.

On the contrary, CAC's and YKW's methodologies are aftera more explicit quantification of the mean frequency of ex­ceedance of specified nonlinear structural limit states. How­ever, while CAC's approach focuses on a single, particularstructure located at a given site, the procedure by YKW ismuch more ambitious. Its scope is the calibration of the designparameters of the seismic code based on the performance as­sessment of a large ensemble of structures designed accordingto the code itself. Although HK's approach is not explicitlyprobability-based, in the next subsection we make an attemptto present it in a probabilistic format which facilitates the com­parison with these other two proposals.

In the methods of both CAC and YKW, the evaluation ofthe structural performance is done with explicit considerationof the uncertainties in the ground motion loads. Since the seis­mic threat is usually posed by a collection of faults around thesite, these two methodologies consider the whole spectrum ofpotential magnitudes and source distances. Allowance for theuncertainty in the ground motion interface of the problem ismade in CAC's method by coupling the nonlinear structuralanalyses with the conventional site-specific PSHA approach.For the same purpose, YKW adopts instead a Monte-Carlo­based simulation technique.

Because of the significant randomness of the ground motionintensity, given M and R, and of the generally large numberof combinations of M and R pairs potentially hazardous forthe site, a main concern of the latter two procedures is thelimitation of the computational cost required to evaluate theMDOF structural performance. Although the method is notprobabilistically centered, the efficiency requirement is presentin HK's method as well. In fact, to be of practical use a con­ceptual design tool must limit the required effort. Therefore,both FEMA 273/HK's and YKW's methodologies make useof various simplified mechanical models, including SDOFmodels, of either representative "generic" structures or of theMDOF structure at hand, as will be discussed more fully be­low.

In contrast, the method proposed by CAC does not makeuse of any linear (or nonlinear) SDOF model to gain effi­ciency. (Incidentally, the paper by Han and Wen (1997) in

discussing the CAC method states the contrary.) It is well rec­ognized that the use of linear SDOF responses modified by acorrection factor to infer the local nonlinear MOOF responses(e.g., interstory drifts) may not yield satisfactory results. InCAC's method the post-elastic response of the MDOF struc­ture is always computed via nonlinear dynamic analyses of theMDOF model; no more simplified models are ever used. Theonly linear SDOF-based quantity in this method is the con­ventional response spectral parameter (i.e., Sd or Sd) selectedto represent the intensity of a ground motion record (e.g.,Shome and Cornell 1998).

In CAC's method, efficiency is not achieved by making"cheaper" the computation of the MDOF response (Le., byconsidering a simple substitute structure whose response needsto be calculated hundreds to thousands of times), but by lim­iting the number of response calculations of the full nonlinearMDOF building model. What makes this procedure practicalis that, given the resulting, comparatively small record-to-rec­ord variability and the weak dependency on M and R of theMDOF response, its median value can be estimated (withoutbias) with a sufficient level of accuracy by using as input alimited number of properly scaled real ground motion records(e.g., to the same Sd)'

Unlike CAC's procedure, both YKW's and HK's methods(with the exception of the early work of Nassar and Krawink­ler 1991) do not use any explicit scaling strategy for the inputground motions. There is, in fact, widespread concern in theengineering community regarding the practice of scalingrecords. For example in Han and Wen (1994) it is stated that"scaling an earthquake to attain a target damage level of dif­ferent intensity is questionable since scaling a ground motiondoes not account for variations in ground motion characteris­tics (e.g., frequency content) which change with intensity."Many researchers have stated that scaling procedures based ona single parameter (frequently peak ground acceleration) donot work well across the entire spectrum of structural oscillatorfrequencies.

The dependency of ground motion spectral characteristicson M and R (and therefore "intensity") has been recognizedby seismologists for many years. For example, a M = 5 recordscaled to match the PGA of a M = 7 record will certainly bedeficient in the frequency content below 1 Hz, and its usewould therefore underestimate the response of a long periodbuilding. Unlike the single-parameter all-structures scalingprocedures proposed in the past, the one used in the CACmethod is structure-frequency-specijic. If the seismic responseof two structures with same damping, ~, but different funda­mental frequencies, f, and f2, were to be analyzed, the samerecord would be scaled differently to match the different val­ues of the target SoCf,., ~) and SoCh, ~). As discussed above, ithas been shown (e.g., Shome et al. 1998; Shome and Cornell1998) that within reasonable limits, scaling records using thisfrequency-specific approach not only does not (significantly)alter the median structural displacements but also considerablyreduces the (conditional) variability in the response itself. Thislast property enables one to run many fewer analyses to attainthe same desired level of accuracy in the response estimation.

In the methods proposed by HK and YKW, however, thestructural response is evaluated using ground motion recordscarefully "searched" for in order to fulfill specific criteria: thematching of the code spectrum (in HK's method) or the se­lection during simulation of the magnitude and distance valuesproper to each hazardous fault (in YKW's approach). Thesecriteria alone do not at all ensure small variability in the com­puted response and, thus, a large number of runs is generallyneeded to achieve sufficient accuracy. In fact, when unscaledreal records belonging to the same narrow M and R bin (say,O.5M-units and 20 km wide) are run through a MDOF struc-

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ture, it is not uncommon to observe response measures (e.g.,maximum interstory ductility) differing by an order of mag­nitude (Shome et al. 1998; Shome and Cornell 1998). As dis­cussed earlier, this large response scatter can be attributed pri­marily to the large variation of intensity Sa among groundmotions from the same bin (variation which, for the same Mand R, is captured in the error term of each attenuation equa­tion, and, hence, in the PSHA). At least when the performanceof the structure is monitored in terms of damage measures suchas ductility or the Park-Ang damage index, these careful strat­egies for selection of the records seem to be unnecessary whenan appropriate scaling procedure is adopted (unless near­source directivity effects are expected).

Regarding again the use of simplified models of the struc­ture, it should be recognized, however, that the reason behindtheir adoption is markedly different in the methods by FEMA273/HK and by YKW. In the latter procedure, the ENS is themost accurate, as well as simplest, model that is able to retainthe main characteristics of the nonlinear MDOF response. Inorder to ensure this, the pre- and post-yielding structural prop­erties of the "equivalent" system are derived from a staticpushover analysis of the original MDOF structure. lt is be­cause of this "equivalence" that YKW's procedure requiresonly one carefully tuned, empirical response modification fac­tor, C [see (3)].

On the other hand, the method by FEMA 273/HK calls fora series of correction factors, C/s, based on SDOF systemswith increasingly more complex inelastic load-deformationcharacteristics; such factors are to be used, when appropriatefor the structure under consideration, for MDOF response pre­diction [e.g., (2)]. This is a direct consequence of the fact thatat a preliminary design stage structural member details are notknown with sufficient accuracy to enable the use of a singlecorrection factor based on only one "dynamically equivalent"SDOF system.

It should also be noted that the correction factors employedby FEMA 273/HK and by YKW are intrinsically different.With the exception of Co (and CMDOF if used), the C/s factorsused by FEMA 273 are tailored to capture the difference inthe maximum post-elastic response of a nonlinear SDOF sys­tem and the maximum linear elastic response of its linearcounterpart. On the other hand, values of the C factor proposedby YKW are obtained by performing analyses of structuralmodels with different number of degrees of freedom (SDOFor 2DOF versus MDOF). Furthermore, C is obtained by re­gressing response values from both the nonlinear ENS andMDOF models of the structure which contain both pre- andpost-yielding data. Actually, most of the values in the samplesize (75% or more of the 88 runs in Han and Wen 1997)represent linear elastic analyses (from which, incidentally, therelatively low regression error is derived) and the rest do notinduce a significant damage (global ductility less than 4 andlocal ductility less than 3). In this respect, the YKW's C factorplays mainly the role of FEMA 273/HK's Co.

lt is important to recognize that all three methodologies al­low the prediction of any single response measure of interest,both at the global and at the local level. The prediction of aresponse measure at the local level (e.g., member forces anddeformations) is direct both in CAC and YKW methods (al­though the latter would require performing a regression anal­ysis for each and every such measure selected) and in FEMA273/HK method when LDP is used. When NSP is adoptedinstead, the FEMA 273/HK method enables the direct esti­mation only of global measures, 8 MDOFm.' such as the roof dis­placement, and the prediction of local response measures canbe extracted from the pushover analysis results when the lat­eral displacement 8MDOFNL is imposed at the roof level.

Finally, the three methods deal differently with the issue of

1286/ JOURNAL OF STRUCTURAL ENGINEERING / NOVEMBER 1998

possible dependency of the structural response on ground mo­tion parameters, such as M and R. The method by FEMA 273/HK does not explicitly consider this dependence and impliesthat this issue has generally been resolved at the code levelwhen the design spectra are obtained. CAC's procedure con­cludes that such dependency is very weak provided that theground motion is previously scaled to the specified level ofSa' The latter procedure, at least in the simpler version de­scribed here, keeps the dependency of the structural responseon ground motion parameters only via the dependency onspectral acceleration. The method proposed by YKW accountsimplicitly, although approximately, for any such dependencyby means of the ENS response quantity, 8ENS '

Example

In this subsection we apply the three methods just discussedto a same simple example. Let us consider a single regularbuilding of fundamental frequency f and damping ~, whoseresponse does not show significant high-order modal effectsduring seismically induced motion. The structure is foundedon firm soil and is located close to a fault which is capable ofgenerating only characteristic earthquakes of a narrow mag­nitude interval, m ::!: am. Let the mean annual activity rate ofevents of the characteristic type generated by this fault be v(e.g., 0.01).

In this case, it is reasonable to postulate that the only eventsdominating the hazard have magnitude M = m. When occur­ring, such events tend to break a portion of the fault longenough that the distance considered in ground motion predic­tive equations can be assumed to be the shortest from the faultto the site, Le., R = r. Further, assume a site-specific PSHA isavailable and the ~-damped uniform hazard spectrum (UHS)associated to a mean return period of Tyears (say, 1,000 years)is to be used for seismic rehabilitation of the building. Usingthe same notation as before, define the ordinate of the UHS atfrequency f to be Sa.T' Suppose also that for such m and r, theselected attenuation equation predicts a median Sit. ~) value,Sa, which exceeds the maximum elastic capacity, Sa.y/dg, of thestructure. Under these conditions, the potential ground motionsat the site are likely to induce post-elastic structural responses.The performance of the building is checked in terms of themaximum lateral displacement of the roof, 8 MDOFNL• More pre­cisely, the structure's adequacy is verified if it can sustainwithout

Asignificant damage the T-year median roof displace­

ment, (8MDOFNL)T'

Although we shall not use it directly in what follows, it isinteresting to recognize that when, as in this example, onlyone pair of M and R contributes to the hazard, the ordinate ofthe T-year UHS at the frequency f can be expressed analyti­cally as follows:

(9)

where

• (J'.n s. is the standard deviation of the (natural log) Sa val­ues, given m and r. This standard error term is providedby every attenuation relationship.

• <1>;1(.) is the inverse of the standardized Gaussian com­plementary cumulative distribution function. This func­tion, which is widely tabulated in any probability textbook(e.g., Benjamin and Cornell 1970), returns the valuewhich is exceeded by a standard Gaussian variable U withprobability equal to the argument (e.g., it would returnapproximately 1.0 when the argument is 0.16). Of course,T must be greater than lIv.

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The spectral displacement associated with the same return pe­riod T is obviously Sd,T = Sa,T/(27Tfi.

FEMA 273/HK's Method

According to the methodology review~d in the pre.vious sec­tion, the first step consists in the selectIOn of a sUIte of re~ground motions whose average response spectrum matches, mthis case, the UHS. In this particular case, if available, theearthquake records (e.g., 15) might be p~ferably (~ut not n~c­essarily) chosen from a magnitude and dIstance ~m ~atchmg

the m and r values dominating the hazard for thIS SIte. Onlytime histories recorded on stiff soil conditions consistent withthe soil at the site are selected. This suite of (unscaled) recordsis used to compute the responses of a series of nonlinear SDOFsystems with restoring force behaviors, appropriate ~or thestructure under consideration. To help thIS task, a statIc non­linear pushover analysis of the MDOF building would mostlikely be performed. .

Assume that in this structure stiffness degradatIOn andstrength deterioration along with significant P - a effects areexpected during the ground shaking. Hence, all the Clo C2, andC3 factors from (2) are needed to assess the MDOF response.The average value of Clo C2 , and C3 are computed from theresults of the dynamic analyses of the linear and nonlinearSDOF representations of the structure. Comparative analysesof these records passed through both linear first-mode SDOFmodel and linear MDOF model of the building are also carriedout to capture the effects of the higher modes in the linearresponse. The latter analyses yield the average value of Co.Given the assumptions on the building behavior stated in theprevious subsection, no other correction factor is needed toestimate the T-year return period roof displacement of theMDOF building, (BMDOfNL)T' From (2) it follows:

Sa,T(BMDOFNJT"'" COC\C2C3Sd ,T "'" COC1C2C3 (2Trf)2 (10)

where Sa,T is given in (9). The previous equation implicitlyassumes that

1P[BMDOFNL ~ (BMDOFNJT] "'" P[Sd ~ Sd,T] "'" P[Sa ~ Sa,T] =T

(11)

In other words, since the variability in the correction factorsis not accounted for, the likelihoods of exceeding the desiredresponse and the spectral acceleration value (at the first-modefrequency) which on average induces such response are con­sidered to be approximately equal.

It should be recognized that when the method proposed byFEMA 273/HK is expressed in probability terms by means of(10) and (11) it becomes very similar to CAC's method. Moreprecisely, if, as in this example, the ground motion intensitymeasure (here Sa,T), were taken from a PSHA-based UHS, thenthe method by FEMA 273/HK could be interpreted as the firstorder approach by CAC described by (6). However, when theSa,T (and thus Sd,T) is instead adopted from a building codespectrum, a certain (but unknown) degree of conservatism(usually included in the definition of the spectrum itself) isimplicitly introduced. This interpretation of FEMA 273/HKmethod also supports his use of mean values of the C/s cor­rection factors. From the discussion surrounding (6)-(8), it isclear that one would like these correction factors to producean unbiased (i.e., mean or perhaps better median) estimate ofl'iMDOFNL, given Sa,T.

YKW's Method

A preliminary pushover analysis of the MDOF building isperformed to assess its behavior beyond the elastic range, and

the most "accurate" ENS model is developed accordingly. Anensemble of real ground motion records are selected from theearthquake database. In this particular case the selecti~n wouldmost likely be carried out to match, as close as pOSSIble, boththe magnitude m and the distance r o.f interest, and the ~ocal

soil conditions. A time-history analySIS of both the nonlInearMDOF and ENS models is performed for each of the real(unsealed) records. The results are used to find a relations~ip

for the (in this case) global correction factor C as a functIonof the severity of the response in the ENS model, BENS '

Using a Monte Carlo simulation technique, a large nu~ber

of synthetic records is generated and for each one a non1tn~ar

dynamic analysis of the ENS is performed. For each analySiS,the displacement of the ENS, BENS ' is modified by the ~alue

of C, corresponding to the computed BENS value, to ob~~ anestimate of the roof displacement in the MDOF bUlldmg,

BMDOfNL• "The estimate of the T-year return penod roof dIsplacement

of the MDOF structure, (BMDOfNLh, is based on the followingequation:

1P[BMDol'NL ~ x] "'" P[ CBENS ~ x] "'" vP[ CBENS ~ xlm, r] =T

(12)

The displacement (BMDOfNL)T is the particular x value.whichmakes the conditional probability P[CBENS ;=: xlm, r] times vequal to liT. For any given value of x, t1;Ie conditional ~rob­

ability above is estimated simply by countmg how many tImesout of the total number of runs the MDOF displacement,BMDOfNL, exceeds x. For example, if T = 1,000 years and v =0.01, x would be the displacement value which is exceeded,on average, for lout of 10 simulated synthetic records.

The previous equation implicitly neglects the regression er­ror in the relationship between C and the response level of theENS. An explicit formulation which includes this variabilityin the Monte Carlo simulation framework would be straight­forward. We shall make no attempt, however, to include it heresince such a procedure was only alluded to in the works pub­lished by the author.

CAC's Method

Time-history analyses are performed using a nonlinearMDOF model of the building for a limited number (say,S to7) of real ground-motion records, The frequency of vibrationof the first mode, f, is computed, and the damping ratio, ~,

estimated. The records are chosen from the database with thesame criterion exposed in the previous subsections: matchingof m and r values, and soil conditions similar to those at thesite, However, since in this method a very small number oftime histories is needed, the selection is greatly simplified. Itis worth recalling that, since the severity of the structural re­sponse is not significantly affected by M and R values oncethe records are scaled to the same Sa value, an extremely care­ful search is not required.

The pool of time histories is scaled to at least two levels ofspectral accelerations, SaCf, ~), prior to doing structural anal­yses. Since the interest is on assessing the roof displacementassociated with a mean return period of T years, the lower ofthe two Sa values may be chosen equal to Sa,T [in general,available numerically from the UHS and, in this simple ex­ample, in analytical form from (9)]. The higher Sa value shouldbe selected large enough that the regression model would beable to capture any curvature present. A rule of thumb may beto pick the latter value, say, 50% higher than Sa,T'

A regression model of the form 8MDOFNL =~OS~1 (see pre­vious section) is then fitted through the structural responses

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0.75Sealed records •

Unsealed records r:J

0.6 • ••• ••

Sa,2=0.51 ---....-

- Sa,r=0.420) •-;u;:e

III(fJ

0.3 (8t.100FNL•1h/Syldg

0.2 (8t.1ooFNL,2h/Syldg

0.1Sa,yldg

00 1 2 3 4 5 6 7 8 9 10

J.l=8t.100PSyldg

FIG. 1. Application of CAC Procedure: Roof Displacements and Fitted Model

FIG. 2. Application of CAC Procedure: Seismic Hazard Curvefor S.

computed by nonlinear dynamic time-history analyses of theMDOF building subject to the ground motions scaled at thedifferent Sa levels. It should be recognized, however, that it isnot strictly required that all the records be scaled to each oneof the considered Sa level. A reasonably accurate estimate of8MDOFNL at a given Sa value can be obtained by performing atotal of 5 to 7 analyses spread among all the considered Salevels. With such a limited number of analyses, a reliable es­timate of the response variability, 0', cannot be obtained; lit­erature values may be used instead.

A simple example showing this procedure is displayed inFig. 1. In this particular case, however, a large suite of 20records all scaled to five different levels were used, with thepurpose of directly estimating from the data the scatter in theresponse even at severe damage levels. Although not necessaryfor this procedure, for visual comparison the responses of theoriginal 20 unsealed records are shown as well. The regressionline (not shown for clarity) fitted only to the unsealed data isalmost coincident with the one displayed in the figure. The

1288/ JOURNAL OF STRUCTURAL ENGINEERING / NOVEMBER 1998

regression parameters of the line fitted through the responsevalues from the scaled records are 130 =8.90 and 131 =0.83.

From (6), it is clear that if the variability in the responseobtained for a given level of Sa (i.e., the scatter in the stripesof data in Fig. 1) is neglected, then a first order estimate,(8MDoFNL)n of the T-year response is 13oS~}. Let us take T tobe equal to 1,000 years in this example. The value of Sa.T =0..:42 g is known from PSHA results (Fig. 2) and the value of(&MDOFNL)T can be read directly from the graph in Fig. 1 (orcomputed from the equation of the fitted model), i.e., 4.34&yldg.

If the variability in the response for a given Sa is accountedfor, a second order estimate of (~FNL)T is associated with aspectral acceleration value, Sa.2' larger than Sa.T = 0.42 g [see(8)]. For the case depicted in Figs. 1 and 2, 0' ..,. 0.26 (at the0.42 g level), 131 = 0.83, and k. assessed graphically fromPSHA results is 3.8. With such values the exponential term in(8) is equal to 1.20, giving a value of Sa.2 of 0.51. This spectralacceleration level corresponds to a mean return period of ap­proximately 2,000 years. Therefore, a more accurate estimate,(&MDOFNL,,)T, of the 1,000-year median response is 5.07&yldg'

A more complete recent example prepared for the SAC SteelProject is displayed in Luco and Cornell (1998).

CONCLUSIONS

We conclude that despite their apparent differences-whichare related in differences in objectives-these three proposalsfor characterizing the seismic behavior of realistically modeled(i.e., nonlinear and MDOF) structures can be usefully com­pared and contrasted with respect to their assumptions, theirefficiency, and their potential for application. Such proceduresare applicable to problems as disparate as routine (code) use,preliminary design, calibration of simpler codes, and specificstructure assessment. It is hoped that this paper will help im­prove communication across the various specialists involved.These range from ground motion experts through structuralanalysts to those concerned with the format of future codes.All the methods have the potential to contribute effectively tothe common problem of improving the practical prediction ofseismic response of buildings and other structures.

0.80.510.35 0.42S.(f.~ (g)

3000

2000

I1500

T.looo

IE 500

II 200

100

0.2

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ACKNOWLEDGMENTS

Funding for the preparation of this paper was provided by the U.S.Nuclear Regulatory Commission through the contract NRC-04-95-075,by the National Science Foundation under Grant CMS-94-23596, and bythe Mineral Management Services under Grant 1435-01-96-CT-30833-1.This support is gratefully acknowledged. Great appreciation is expressedto Helmut Krawinkler and Y. K. Wen for their constructive commentsand reviews.

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