Amplitude-Scaled versus Spectrum-Matched GroundMotions for Seismic Performance Assessment

YeongAe Heo1; Sashi K. Kunnath, F.ASCE2; and Norman Abrahamson3

Abstract: The need to consider only a small number of ground motions combined with the complexities of response sensitivity toboth modeling choices and ground motion variability calls for an assessment of current ground motion selection and modification methodsused in seismic performance evaluation of structures. Since the largest source of uncertainty and variability arises from groundmotion selection, this study examines the suitability of two ground motion modification (GMM) schemes: magnitude scaling (whereinthe ground motion is uniformly scaled so that the resulting spectrum matches the amplitude of the design spectrum at the structural funda-mental period) and spectrum matching. Comprehensive nonlinear time-history (NTH) simulations of two reinforced concrete moment framebuildings are carried out to evaluate the GMM approaches in the context of seismic demand prediction. Findings from the investigationindicate that spectrum matching is generally more stable than scaling both in terms of the bias as well as the resulting dispersion inthe predicted demands. It is also concluded that seven ground motions are inadequate to establish median demands for taller frames wheremultiple modes influence structural response. Both methods are found to be sensitive to the choice of records for the cases investigated in thisstudy. DOI: 10.1061/(ASCE)ST.1943-541X.0000340. © 2011 American Society of Civil Engineers.

CE Database subject headings: Drift; Ground motion; Probability distribution; Reinforced concrete; Seismic effects.

Author keywords: Interstory drift; Ground motion modification; Probability distribution; Reinforced concrete; Seismic simulation.

Introduction

The selection of ground motions for use in nonlinear dynamicsimulations is becoming an increasingly critical component ofperformance-based seismic evaluation. Given the limited databaseof earthquake records that satisfy the required site parameters, it isoften necessary to select empirical recordings from other similarsites and alter them suitably to meet the needs of the evaluation.Current practice in nonlinear seismic evaluation is to select groundmotion records that best represent the characteristics of theexpected event (such as magnitude, distance, and fault type) atthe site. Structural engineers tend to favor the use of intensity scal-ing methods, but since the intensity (such as spectral acceleration)at critical periods can vary considerably from record to record, theselected motions need to be modified to achieve certain targetintensities.

According to code requirements (ASCE 2005), the scaling ofselected ground motions for two-dimensional analysis should becarried out in a manner such that the average value of the 5%damped response spectra for the suite of motions is not less thanthe design response spectrum for the site for periods ranging from

0.2 to 1.5 times the fundamental period, T , of the structure for thedirection of response being analyzed. Depending on whetherthree or seven records are used, the maximum or mean value of theresponse parameter of interest is to be considered in design orevaluation. In addition to code-based scaling, modifying groundmotions by scaling them to the spectral acceleration at single spec-tral periods, such as the first-mode elastic period SaðT1Þ have alsobeen used, for example, Kunnath et al. (2006). Shome and Cornell(1998) and Shome et al. (1998) demonstrated that seismic demandsare strongly correlated with the elastic single-degree-of-freedom(SDOF) oscillator response acceleration at the fundamental periodof the system. More recently, Baker and Cornell (2006) suggestedthat if records are properly selected based on spectral shape, thereduction in bias and variance of resulting structural responseestimates are comparable to the reductions achieved by using avector-valued measure of earthquake intensity. In contrast, certainscaling methods such as scaling to target peak ground acceleration(PGA) produce biased estimates with large scatter in response(Nau and Hall 1984; Vidic et al. 1994; Shome and Cornell1998). Further, because scaling methods do not explicitly considerthe inelastic behavior of the structure, they may not be appropriatefor near-fault sites where the inelastic deformation can be signifi-cantly larger than the deformation of the corresponding linear sys-tem. For such sites, scaling methods that are based on the inelasticdeformation spectrum or consider the response of the first-modeinelastic SDOF system are more appropriate. Kalkan and Chopra(2010) used these concepts to develop a modal-pushover-based-scaling (MPS) procedure for selecting and scaling earthquakeground motion records. Other approaches to scaling include groundmotion modification over a selected period band (Alavi andKrawinkler 2004; Kalkan and Kunnath 2006). A good review ofseveral other scaling methods including scaling to effective peakacceleration, Arias intensity based parameter, effective peak veloc-ity, and maximum incremental velocity is reported by Kurama andFarrow (2003), where it is shown that scaling methods work well

1Senior Researcher, Marine Research Institute of Samsung HeavyIndustries, Korea; formerly, Graduate Student Researcher, Dept. of Civiland Environmental Engineering, Univ. of California, Davis, CA 95616.E-mail: yaheo@ucdavis.edu

2Professor, Dept. of Civil and Environmental Engineering, Univ. ofCalifornia, Davis, CA 95616. E-mail: skkunnath@ucdavis.edu

3Seismologist, Pacific Gas and Electric, San Francisco, CA 94105;Adjunct Professor, Dept. of Civil and Environmental Engineering, Univ.of California, Davis, CA 95616.

Note. This manuscript was submitted on April 20, 2010; approved onOctober 27, 2010; published online on October 29, 2010. Discussion periodopen until August 1, 2011; separate discussions must be submitted for in-dividual papers. This paper is part of the Journal of Structural Engineer-ing, Vol. 137, No. 3, March 1, 2011. ©ASCE, ISSN 0733-9445/2011/3-278–288/$25.00.

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for ground motions representative of stiff soil and far-fieldconditions, but lose their effectiveness for soft soil and near-fieldconditions.

As an alternative to scaling by a constant factor, the method ofspectral matching (in time or frequency domain) is gaining atten-tion among researchers and practitioners. With growing demand inusing nonlinear time-history (NTH) analysis for seismic assess-ment of structures, this method is attractive for engineers becauseit minimizes the number of records needed to obtain reasonableaccuracy of the performance estimate; thereby the computationalcost is significantly reduced. In this method, the frequency contentand phasing of actual recordings are manipulated to match asmooth target spectrum (e.g., Silva and Lee 1987; Lilhahandand Tseng 1989; Bolt and Gregor 1993; Carballo and Cornell2000; Hancock et al. 2006). While scaling methods keep the fre-quency content of ground motions intact, spectral matching tech-niques may alter the physical characteristics of the accelerogramsthough more recent methods also try to minimize the alteration ofthe overall nonstationary characteristics of the motions.

In the case of scaling approaches, the central questions that needto be answered include the following: How many ground motionsshould be used? What number should form the basis for decidingwhether the average or maximum response measure should be con-sidered? Should the scaling be limited to a single period or a periodrange? If two alternative sets of records produce different perfor-mance estimates, how should the findings be reconciled? For spec-trum-matching methods, one of the chief concerns is themodification of the frequency content that can distort the nonsta-tionary characteristics of the time series. Other concerns include theuncertain effects of leveling or flattening all the peaks and troughsof the spectrum on the computed structural response.

Both scaling andmatching methods are considered in the presentstudy to evaluate not only the effectiveness of eachmethodology butalso to assess the reliability of structural performance estimates for agiven site hazard. A study by Iervolino and Cornell (2005) showedthat there is little evidence to support the need for a careful site-specific process of record selection based onmagnitude and distanceand that concern over scenario-to-scenario record scalingmay not bejustified.Hence, this study focuses primarily on groundmotion char-acteristics such as spectral acceleration that affect structural responsecharacteristics rather than site-specific sources.

Buildings Considered in Evaluation

The two ground motion modification methods are evaluated byconsidering the bias and dispersion of seismic demand predictions

of a four-story and 12-story RC frame building. The structuresare assumed to be standard office buildings located in SanFrancisco (37.46N, 122.25W). The lowest level of the buildingis 4.57 m (15 ft) high while the remaining floor heights are3.66 m (12 ft). The plan dimensions of the building are 36:6 ×36:6 m (120 × 120 ft) with five equal bays in each direction.The choice of a symmetric floor plan allows a single typical framein either direction to be analyzed as a two-dimensional frame. Thetypical plan and elevation of the buildings is shown in Fig. 1.

The buildings are designed to meet the provisions of SeismicDesign Category D as specified in ASCE 7-05 (2005). The follow-ing parameters were used in the design:

SS ¼ 1:50 g; S1 ¼ 0:67 g; I ¼ 1:0;

Cd ¼ 5:5; R ¼ 8:0

Using the preceeding values, the design base shears are2,325 kips and 1,386 kips for the 12-story and four-story buildings,respectively. The approximate code-based fundamental periods are1.43 s and 0.55 s while the true first mode periods based on inputmaterial and element properties were determined to be 2.1 s and0.88 s. The final design is based on using normal weight concrete(150 pcf) with a compressive strength f 0c ¼ 4;000 psi and reinforc-ing steel with a nominal yield strength f y ¼ 60;000 psi. Sectionsizes and details of the required flexural and shear reinforcement,all of which conform to the requirements of ACI-318 (2009), arepresented in Table 1.

Ground Motion Selection and Modification

A study by Watson-Lamprey (2007) introduced the notion of a“point of comparison” wherein the “true” (statistically most likely)response of a system is established. It is proposed that the “true”structural response measure is best obtained by a high-end predic-tion, wherein NTH simulations are carried out using an extremelylarge number of scaled and unscaled ground motions. A responsemodel is then derived that relates relevant structural demand mea-sures to critical ground motion parameters. Such a model not onlyprovides the best estimate of structural demand for any site but alsoprovides a convenient baseline against which the validity of differ-ent ground motion modification methodologies can be assessed.The notion proposed by Watson-Lamprey (2007) is adopted inthe present study with a change in that only unscaled groundmotions are used. Watson-Lamprey (2007) used scaled groundmotions to drive the structure well into the nonlinear range. In this

Fig. 1. Building plan and typical frame elevation of buildings

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paper, ground motions that cause nonlinear response is also of pri-mary interest, therefore, a subset of 200 ground motions from thePacific Earthquake Engineering Research (PEER) Next GenerationAttenuation (NGA) database (PEER 2005), whose PGA exceeded0.2 g, was used in the simulations. The computed inelastic responsemodels to these 200 records are used to determine the dependenceof the structural response on the spectral acceleration at a suite ofspectral periods. This model of the structural response is then usedto compute the expected response and the variability of theresponse given the design spectral values. This distribution ofthe predicted response, given the design spectrum, is termed the“true” response—from the perspective that record-to-record vari-ability and other inherent uncertainties in the ground motions havebeen more comprehensively considered in this data set than in asmaller subset of records. A methodology that uses a limited dataset (ground motions) can be classified as being reliable if theresulting performance model comes reasonably close to the “true”prediction (in terms of bias and variance). Note that the term pre-diction model refers to a model that provides a demand or perfor-mance measure (interstory drift demand or damage index, forexample) as a function of ground motion parameters. Fig. 2 shows

the 5% damped response spectra for all records in the database.Also shown alongside is the mean spectrum of the 200 recordssuperimposed on the ASCE 7-05 (2005) design spectrum for thesite. The mean spectrum is observed to be slightly lower thanthe design spectrum. Complete details of the ground motion char-acteristics are described in Heo (2009).

Scaling Methods

Numerous approaches to ground-motion scaling exist in current lit-erature. The simplest and most commonly used amplitude-scalingapproach is selected in the present study—namely, scaling of themotion such that the ordinate of the spectral acceleration matchesthe design spectral acceleration at the fundamental period of thestructure in the direction of loading. As pointed out in the introduc-tion, this method is also known to typically have scatter less thanother scaling approaches.

A limited subset of 17 records was randomly selected from thelarger bin of 200 records. Three bins of seven records each werethen created from this set of 17 records and are referred to in thispaper as Set 1, Set 2, and Set 3. The accelerograms in each setwere selected to represent three possible scenarios: records whosemean spectral value at the fundamental period was lower thanthe design spectrum, records with a mean spectral value higher thanthe design spectrum (note that this outcome was not achieved forthe four-story structure), and a final set that comprised recordswhose mean spectral acceleration was relatively closer to the designspectrum at the fundamental period. Each of the records in the binwas scaled to match the spectral acceleration of the design spectraat the fundamental period of the structure. The 5% dampedresponse spectrum of the records in each set as well as the meanspectrum for both the original set and the scaled set is displayed inFig. 3. Also shown in these plots are the first-mode periods of thetwo buildings. Tables 2–4 list the scale factors applied to eachrecord.

An examination of the spectra in Fig. 3 indicates that the meanof the original records in Set 1 was significantly lower than thecorresponding design spectral value at the fundamental periodwhile the records in Set 3 produce a mean spectral demand(at T1) that is marginally higher than the design spectrum. Themean of the Set 2 motions were also lower but closer to the design

Table 1. Component Sizes and Reinforcement Details for Both Buildings

Story levels

1-2 3 4

Column Size 711 × 711 610 × 610 559 × 559

Long reinf 16#29 16#25 16#25

Trans reinf #9.5@152 #9.5@152 #9.5@127

4-story Beam Size 711 × 558 610 × 508 559 × 457

Long reinfa 16#29 14#29 14#29

Trans reinf #12.7@140 #12.7@127 #12.7@114

Story levels

1-3 4-6 7-8 9-10 11-12

Column Size 864 × 864 813 × 813 762 × 762 711 × 711 559 × 559

Long reinf 20#36 16#32 16#29 16#29 16#25

Trans reinf #12.7@127 #9.5@152 #9.5@152 #9.5@152 #9.5@140

12-story Beam Size 864 × 762 813 × 711 762 × 660 711 × 559 559 × 457

Long reinfa 14#32 14#32 16#29 16#29 14#29

Trans reinf #9.5@190 #9.5@178 #9.5@165 #9.5@140 #9.5@114

Note: n#d ¼ number of bars “n” and bar # with diameter “d”; #d@s ¼ bar#with diameter “d” at spacing “s”; all dimensions in mm.aTotal reinforcement (equal top and bottom reinforcement).

Fig. 2. 5% damped response spectra of selected 200 ground motionsand comparison of mean spectrum with ASCE 7-05 design spectrum(note: T1, T2, and T3 are the first, second, and third mode periods;subscripts indicate whether the modes are for the 4- or 12-story frame)

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Fig. 3. Spectra of amplitude-scaled records: (a) Set 1; (b) Set 2; (c) Set 3 (Notation ¼ orgAvg: average spectrum of unscaled records; Design:ASCE 7-05 design spectra; sclAvg04: average spectrum of scaled records for 4-story frame; sclAvg12: average spectrum of scaled records for12-story frame)

Table 2. Intensity and Scale Factors of Scaled Record Set 1

4 Story 12 Story

GM # PGA SaðT1Þ SF SaðT1Þ SF

1 0.386 1.482 1.51 0.128 2.49

3 0.597 1.681 0.80 0.129 2.47

5 0.447 1.416 1.45 0.135 2.36

6 0.400 1.350 0.78 0.102 3.12

26 0.402 1.594 0.69 0.268 1.19

71 0.241 1.338 2.05 0.093 3.42

145 0.208 1.749 2.96 0.155 2.05

Table 3. Intensity and Scale Factors of Scaled Record Set 2

4 Story 12 Story

GM # PGA SaðT1Þ SF SaðT1Þ SF

1 0.386 1.482 1.51 0.128 2.49

3 0.597 1.681 0.80 0.129 2.47

5 0.447 1.416 1.45 0.135 2.36

16 0.464 1.378 1.27 0.520 0.61

81 0.349 1.566 0.95 0.216 1.47

82 0.536 1.756 0.75 0.421 0.76

145 0.208 1.749 2.96 0.155 2.05

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spectrum at the fundamental period that Set 1 motions. Therefore,the scaling process results in a mean spectrum that is generallyhigher than the design spectrum (particularly at smaller periods,which influence higher modes) for the first two sets and a meanspectrum that is somewhat lower than the design spectrum forSet 3. These facts are also reflected in the scale factors shownin Tables 1–3. For the four-story frame, since the scaling is anch-ored at a lower period, the resulting mean spectra for all sets are notas significantly altered as in the case of the 12-story building. How-ever, the mean spectral demands at periods higher than the firstmode are lower than the design spectral values for Set 1, almostsimilar to the design spectra for Set 2, and higher than the designspectral values for Set 3. These observations provide valuable in-sights into examining and evaluating the features of the buildingresponses to the three record sets.

Spectrum Matching Methods

Although the objective of the scaling approach is to match only asingle spectral value at a target period, the concept of spectralmatching is to modify the original acceleration time series to matchthe entire range of the target spectrum with minimal alteration ofthe velocity and displacement history of the record. The approachadopted in this study is based on the time domain spectral matchingprocedure proposed by Hancock et al. (2006) and assumes that thetime of the peak response does not change because of wavelet ad-justment. Given N target spectral points to match, at the ith targetperiod, the spectral misfit to be altered can be computed by thedifference between the target spectral value (Qi) and the initial timeseries spectral value (Ri)

ΔRi ¼ ðQi � RiÞPi ð1Þ

where Pi = the polarity of the peak response of the oscillator.Hancock et al. (2006) shows that the response of an adjustmenttime series should be equal to ΔRi

ΔRi ¼XN

j¼1

bjf jðtÞ ð2Þ

where f jðtÞ = a set of adjustment functions and bj = the set ofamplitudes of the adjustment functions. The modified amplitudeof the responses to the wavelet (which can be considered as a scalefactor) is determined by not only the misfit at each spectral pointbut also neighboring spectral points

b ¼ C�1δR ð3Þ

Each component of a square matrix C = the amplitude of thewavelet response for the jth spectral point at the peak oscillatortime (ti) of the initial time series response for the ith spectral point.A sample matched spectrum using this procedure is shown inFig. 4(a). The mean spectrum for an ensemble of seven matchedrecords in illustrated in Fig. 4(b). Since the matching processresults in a spectrum that closely matches the target spectrum, themean spectrum of multiple records is almost indistinguishable fromthe design spectrum; hence, only a sample mean spectrum is shown.

Development of Seismic Demand Prediction Model

The assessment of GMM approaches begins with the developmentof a statistically reliable prediction model of the selected seismicdemand parameter. In the present study, the prediction model isessentially a regression model based on the analysis of large datasets that consist of empirical data (such as ground motion param-eters from recorded motions) and simulated data from detailednumerical procedures (such as nonlinear structural simulations).The purpose of a prediction model for structural response istwofold: It provides a simple means to predict a reliable probabi-listic structural response quantity with considerable reduction incomputational effort, and it offers a means to compare different ap-proaches in seismic structural assessment. In the present study, it isused to evaluate ground motion selection and modificationmethods.

The typical procedure for developing a regression model of asample model response parameter Y in terms of a set of predictive

Table 4. Intensity and Scale Factors of Scaled Record Set 3

4 Story 12 Story

GM # PGA SaðT1Þ SF SaðT1Þ SF

16 0.464 1.378 1.27 0.520 0.61

77 0.352 1.323 1.53 0.358 0.89

78 0.283 1.923 1.12 0.202 1.58

82 0.536 1.756 0.75 0.421 0.76

86 0.498 1.309 1.18 0.360 0.88

91 0.333 1.214 1.96 0.226 1.41

98 0.296 2.096 1.83 0.507 0.63

Fig. 4. Spectrum matching: (a) sample spectral matched record; (b) mean spectrum (Notation ¼ Design: ASCE 7-05 design spectra; orgAvg: averagespectrum of unscaled records; matAvg: average spectrum of 7 spectrum-matched records for Set 1)

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parameters Xi consists of the following steps. An expression of thefollowing form can be generated from available (simulated) data

lnðYÞ ¼ c0 þ c1 lnðX1Þ þ � � � � � � þ cn lnðXnÞ ð4ÞIn the preceeding expression, n predictive parameters are

selected. The constants are determined by data analysis. Theimportance of the selected variables can be established by examin-ing the magnitude of the constants, the standard deviation of theresiduals, and the correlation among the predictive parameters.The correlation can typically be assessed by examining thefollowing parameter:

εδln Y ¼ δln Yσδln Y

ð5Þ

where εln Y = the residual normalized by the standard deviation ofthe residual. The residual of the model Y is determined from

δln Y ¼ ln Y � Eðln YÞ ð6ÞEðln YÞ = the expected value of the parameter Y as established

by the regression model. Once it is confirmed that a correlationexists between some of the predictive parameters, these variablesshould be linked by correlation functions. If, for example, εlnX1,εlnX2, and εlnX3 are correlated, then the following correlation func-tions can be developed:

εlnX2 ¼ c0 þ c1εlnX1 ð7Þ

εlnX3 ¼ c00 þ c01εlnX1 þ c02εlnX2 ð8ÞNote that in the preceeding expressions, each of the predictive

parameters can be formulated as a separate set of regression mod-els. Hence, they can be applied with generality to the developmentof regression models for both the structural response in terms ofground motion parameters and ground motion parameters in termsof earthquake parameters. Ideally, the objective of the probabilisticassessment is to establish the probability of exceeding a certaindamage threshold given an earthquake scenario. To accomplish thisobjective, it is also essential to develop a ground motion predictionmodel in terms of earthquake parameters (such as moment magni-tude of the earthquake and closest distance to the rupture zone)

ln SaT1 ¼ c0 þ c1 · lnM þ c2 · lnR ð9ÞIn the preceding example, only a single ground motion param-

eter and two earthquake parameters are considered. Additionalforms of the preceding model are obviously possible, and thechoice of parameters depends on numerous factors. The develop-ment of a ground motion prediction model based on earthquake siteand source characteristics is beyond the scope of the present studybecause a large body of literature currently exists on this topic in thefield of seismology. In the present study, spectral ordinates are gen-erated by using seismic design provisions in ASCE 7-05 (2005).

Modeling of True Response

The procedure described in the previous section is applied todevelop predictive models of the response of the two RC buildingsselected for this study. As previously indicated, the true responseprediction model is generated through a complete set of 200 NTHsimulations of each building. All nonlinear simulations werecarried out using the open-source software OpenSEES (2009).Frame elements were modeled using nonlinear beam-with-hinges

elements wherein the hinge lengths were set to 10% of the elementlength. This value was based on preliminary numerical validationstudies using observed experimental response of cyclically loadedcolumns under constant axial load. Concrete 01 and Steel 02 wereused as the material models: the former ignores the tensile resis-tance of concrete whereas the latter is a complete cyclic descriptionof the behavior of reinforcing steel based on the uniaxial Giuffre-Menegotto-Pinto model with isotropic strain hardening. A 5%mass-proportional damping in the first mode was specified in allthe dynamic analyses.

The following ground motion intensity measures are consideredin the development of the response model: spectral acceleration atthe fundamental period, SaT1, the spectral acceleration at the sec-ond-mode period SaT2, the spectral acceleration at the third-modeperiod SaT3, the spectral acceleration corresponding to 1.5 timesthe fundamental period SaT4, and the spectral acceleration at 2.0times the fundamental period SaT5. The parameters SaT4 andSaT5 were selected after preliminary studies indicated that the soft-ened fundamental period (following some structural damage) canbe a reliable parameter, particularly for ground motions that pushthe system well into the inelastic response region. The primaryresponse variable considered in this study is the maximum inter-story drift (IDR). Before determining a probable form of theresponse model, the correlation between peak IDRs and the groundmotion intensity measures are examined independently. The corre-lations of IDR with respect to the spectral accelerations at the firstthree elastic modal periods are plotted in Fig. 5 in natural logarith-mic scale. For the four-story frame it is evident that the response iswell correlated only with respect to the first-mode spectral accel-eration while all three intensity measures are reasonably wellcorrelated with IDR for the 12-story frame. This outcome mightsuggest that the inclusion of SaT2 and SaT3 in the response modelmay not be necessary for the four-story structure. However, thesedirect correlations by themselves only provide a measure of thelinear dependence of the demand parameter on each of the selectedvariables. Among the features of the response that are of particularinterest in seismic evaluation is the variation of the responsemeasure as a function of some intensity measure at higher demandlevels (in the inelastic range). Expressing the demand as a linear ornonlinear combination of one or more intensity measures andexamining the resulting residuals [Eq. (6)] is a simple way toassess the effectiveness of a predictive demand model. In this study,the following regression models, where the peak (IDR) expressedin percentage values) is selected as the primary response parameter,were considered:

Model A: Eðln IDRÞ ¼ c0 þ c1 ln SaT1 þ c2 ln SaT2 þ c3 ln SaT3

þ c4 ln SaT4 þ c5 ln SaT5 ð10Þ

Model B: Eðln IDRÞ ¼ c0 þ c1 ln SaT1 þ c2 ln SaT2

þ c3 ln SaT3 þ c4 ln SaT4 ð11Þ

Model C: Eðln IDRÞ ¼ c0 þ c1 ln SaT1 þ c2 ln SaT2 þ c3 ln SaT3ð12Þ

Model D: Eðln IDRÞ ¼ c0 þ c1 ln SaT1 þ c2 ln SaT2 ð13Þ

Model E: Eðln IDRÞ ¼ c0 þ c1 ln SaT1 ð14Þ

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Results of the data analysis using the response prediction mod-els expressed by Eqs. (10)–(14) are presented in Tables 5 and 6.Note that σ ln(IDR) and μ(IDR) refer to the standard deviationand median of the IDR. The regression coefficients show thatc1 and c4 have a significant effect on the response of the four-storybuilding; whereas c1, c2, and c3 have a significant effect on theresponse of the 12-story building. In general, the introduction ofadditional variables improves the reliability of the prediction mod-els, as shown by the reduction in the standard deviation for modelswith more terms; however, it is also evident that the spectral accel-eration at first mode SaT1 is the most significant parameter control-ling the response of the 4-story frame, but the spectral accelerationat the second mode (SaT2) is most significant for the 12-storyframe. For the 12-story structure, Model C is adequate to representthe response because the additional reduction in the standarddeviation for Models A and B is negligible. For the 4-story struc-ture, model B is adequate to represent the response.

Figs. 6 and 7 show the IDR residuals for the 4- and 12-storyframes, respectively, based on Models A and C for the 200

simulations as a function of selected ground motion intensityparameters. The regression model used has a linear scaling ofthe response with spectral acceleration. The residual plots canbe used to determine if nonlinear scaling effects are observed.Nonlinear effects would be seen as a curvature in the residualsat the higher intensity ground motions. These residual plots donot show a curvature except for the T4 dependence for the 12-storybuilding for model A—there is a trend to increase the response forhigher SaT4 values, but this trend is not well constrained. Part of thereason that nonlinear effects are not seen in the residuals is thatmost of the unscaled ground motions used in the study did notdrive the systems well into the inelastic range of response. Analternative approach would have been to scale the motions, as inthe study by Watson-Lamprey (2007), and observe trends at higherintensity levels.

The probability density functions of Models A, C, and E forboth structures, shown in Fig. 8, confirm that though Model Ahas the least dispersion in both cases, Models C and E in the caseof the 4-story frame, or Models A and C in the case of the 12-story

Table 5. Regression Coefficients for Different Response Models Using4-Story Building

Model A Model B Model C Model D Model E

σlnðIDRÞ 0.2593 0.2601 0.2846 0.2856 0.2883

μðIDRÞ (%) 1.4955 1.5078 1.4036 1.4272 1.4168

c0 0.7151 0.7414 0.5492 0.5676 0.5667

c1 0.4921 0.4871 0.7685 0.7746 0.7981

c2 0.1340 0.1336 0.1297 0.0882

c3 �0:0564 �0:0527 �0:0690

c4 0.3547 0.2906

c5 �0:0653

Table 6. Regression Coefficients for Different Response Models Using12-Story Building

Model A Model B Model C Model D Model E

σlnðIDRÞ 0.3548 0.3563 0.3563 0.3736 0.4469

μðIDRÞ (%) 1.5119 1.4916 1.4892 1.4425 1.4534

c0 0.8611 0.8235 0.8189 0.8200 1.0904

c1 0.2759 0.2758 0.2888 0.2813 0.6264

c2 0.3800 0.3793 0.3783 0.5527

c3 0.2889 0.2973 0.2971

c4 �0:0897 0.0114

c5 0.0983

Fig. 5. Correlation of structural demand measures with primary ground motion parameters [Note: (log) refers to natural logarithm]

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frame, both provide similar measures of the median response andvariability of the response.

Spectrum Matching versus Scaling

The two GMM methods are evaluated against the so-called truesolution represented by the predictive regression model generatedthrough 200 high-end numerical simulations presented in theprevious section. The regression model considered in the evaluationcorresponds to the function representing Model C [Eq. (12)] but thefindings can be generalized to the other models as well. Recall fromthe preceding discussion that a subset of 17 records was randomlyselected from the 200 ground motions used to establish the trueresponse models. Three records of 7 motions each were thenselected from this bin to create three possible scenarios.

To compare the stability and bias of the results obtained withsets of seven scaled and matched ground motions, the peakIDRs for each simulation and arithmetic mean for each set aresuperimposed on the probability distribution of the correspondingtrue solution for each frame. The comparisons are presented inFigs. 9(a)–9(f) for each of the three ground motion sets and for bothframe structures. Figs. 9(a) and 9(b) compare computed demandsfor both frames subjected to the ground motions comprising Set 1.In the case of the four-story frame, the mean of the maximumIDRs using scaled motions is 1.52%, whereas that of the matched

motions is 1.42%. Although the dispersion in the computed IDRs iscomparable for both methods, the mean estimate using spectrum-matched records is closer to the true mean estimate of 1.40% (basedon Model C). In the case of the 12-story frame, the results usingspectrum-matched records are clearly more consistent than scaledrecords in terms of both the mean and dispersion of the computedmaximum IDRs. The true mean of the expected peak IDR isapproximately 1.49%, whereas the mean estimate is 1.46% usingmatched motions and 2.06% using scaled motions.

Results for ground motion Set 2 and Set 3 are displayed inFigs. 9(c) and 9(d) and Figs. 9(e) and 9(f), respectively. In bothcases, the observations noted for Set 1 with respect to the meanand variability of the computed maximum IDR demands remainsvalid. The mean IDR for the four-story frame using matched Set 2ground motions is 1.45%, which is slightly higher than the trueestimate but significantly better than the estimated mean valueof 1.58% for scaled motions. In the 12-story frame, the meanestimate using spectrum-matched records is 1.33% (which is lowerthan the true mean of 1.49%), whereas the mean IDR demand usingscaled records is 1.83%. For the ground motions from Set 3, themean IDR estimates for the four-story frame using matched andscaled motions are 1.52% and 1.57%, respectively. For the 12-storyframe, the mean IDR using matched ground motions is 1.35%(which is lower than the true mean), but the mean estimate of1.15% using scaled motions is significantly lower than thetrue mean.

Fig. 6. Correlation of IDR residuals with ground motion intensity parameters for 4-story frame

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In addition to the previously noted general observations, it isobserved that the predicted mean for matched motions is closerto the true mean for the four-story frame than for the 12-story struc-ture. Of the two frames, there is considerably more variability in thepredictions for the for 12-story frame for both scaled and matchedmotions. This outcome indicates that additional parameters (such ashigher modes and modal changes attributable to inelastic effects)are influencing the response of the taller frame, hence more ground

motions need to be considered to reduce the bias in the computeddemands.

Concluding Remarks

Findings from a comprehensive set of NTH simulations of tworeinforced concrete moment frame structures indicate that peak

Fig. 7. Correlation of IDR residuals with ground motion intensity parameters for 12-story frame

Fig. 8. IDR distribution for three regression models: (a) 4-story frame; (b) 12-story frame

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IDR demands estimated from two different ground motion scalingmethods are generally a function both structural parameters andground motion characteristics. For the two case studies investi-gated, the estimates of the peak IDR demands resulting from arelatively small subset (seven ground motions were used in eachsubset in the present study) of spectrum-matched records wereconsistently closer to the median estimates of the true solution thanestimates from records of the same subset that were scaled to match

the target design spectral accelerations at the fundamental period ofthe structures. More importantly, the dispersion in the computedIDRs was generally smaller for spectrum-matched records thanfor scaled records. On the basis of the observed variability, itmay also be concluded that seven ground motions are inadequatefor the 12-story structure, indicating that taller or more complexstructures that are influenced by higher modes, inelastic effects,and other ground motion parameters will require a larger subset

Fig. 9. Stability and bias in predicted responses using scaled and matched motions: (a) 4-story frame (Set 1); (b) 12-story frame (Set 1); (c) 4-storyframe (Set 2); (d) 12-story frame (Set 2); (e) 4-story frame (Set 3); (f) 12-story frame (Set 3) (Notation ¼ Model C: probability distribution of truesolution based on Model C; Scaled: Peak IDR for each scaled ground motion; Matched: Peak IDR for each matched ground motion; AVGs: Averageof max IDR of all scaled motions; AVGM : Average of max IDR of all matched motions)

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of records to arrive at statistically reliable results. Since spectrum-matched records, like scaled records, produced mean demandestimates that were both lower and higher than the true mean val-ues, the relative accuracy or reliability of either approach cannot beassessed with certainty since all critical ground motion parametersassociated with structural demand are yet to be identified. It canalso be argued that the spectral accelerations of ground motionsat elastic modal periods of the system are not necessarily reliableground-motion intensity measures. Further studies using scaledmotions that produce larger inelastic drift demands, a significantlylarger set of ground motion bins, additional demand parameters(such as member plastic rotations and heightwise distribution ofdemands), enhanced performance measures (such as damage mod-els as opposed to IDRs), and demand models that utilize additionalground motion intensity measures are still needed to further exam-ine the advantages and shortcomings of the two ground motionmodification methods investigated in this study.

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