Engineering Structures 32 (2010) 1272–1283

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

A modified approach of energy balance concept based multimode pushoveranalysis to estimate seismic demands for buildingsYi Jiang, Gang Li ∗, Dixiong YangDepartment of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

a r t i c l e i n f o

Article history:Received 20 October 2009Received in revised form27 December 2009Accepted 4 January 2010Available online 18 January 2010

Keywords:Building structureEarthquake engineeringPushover analysisSeismic demandsEnergy balance conceptNear-fault ground motions

a b s t r a c t

The procedure of energy balance concept based multimode pushover analysis incorporates concepts incapacity spectrum method and retains computational attractiveness with invariant force distribution,which includes the contributions of higher significant modes, and estimates the seismic demands ofbuildings quite accurately. Developed in this paper is a modified energy balance concept basedmethod inwhich the response contributions of higher vibrationmodes are computed by assuming the building to belinearly elastic, thus reducing the computational effort. Themodal energy capacity and demand diagramsof the buildings are constructed, and the dynamic target point is the intersection of the correspondingtwo diagrams. Moreover, the proposed procedure’s accuracy is evaluated for buildings subjected to avariety of ground motion ensembles, including near-fault ground motions with fling-step and forwarddirectivity pulses, and in the process of investigation some specific observations of near-fault groundmotions are obtained. The statistical predictions of seismic demands of building under selected groundmotions compared with the estimates of nonlinear response history analysis (NL-RHA) demonstrate thatthe proposed procedure is an attractive alternative for practical application.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

A great challenge to performance based seismic design of struc-tures is to develop an effective and feasible procedure for analyz-ing and evaluating the seismic demands. Due to the consumingcomputational effort of nonlinear history analysis, engineers aremore likely to apply nonlinear static methods before finally transi-tioning to nonlinear history analysis.While nonlinear response dy-namic procedure (NDP) is themost rigorous procedure to computeseismic demands of structures, nonlinear static procedure (NSP) orpushover analysis is becoming the current standard practice ac-cepted by more guideline documents such as FEMA-440 due to itssimplicity and efficiency.The capacity spectrum method (CSM), capable of predicting

the demands of forces and deformations of low to medium build-ings dominated by the fundamental mode well, has been docu-mented in the ATC and FEMAguidelines [1–5]. Briefly speaking, thekey idea of CSM is to construct the capacity and demand diagramof an equivalent single degree of freedom (ESDOF) inelastic sys-tem by using the pushover analysis and site-specific ground mo-tion response spectra analysis, respectively, and obtain the targetdisplacement of seismic performance point corresponding to thegraphical intersection of the capacity and demand diagram. Then

∗ Corresponding author. Tel.: +86 411 84707267; fax: +86 411 84707267.E-mail address: ligang@dlut.edu.cn (G. Li).

0141-0296/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2010.01.003

the target displacement of the ESDOF system is converted to thatof the corresponding multiple degree of freedom (MDOF) system.By comparing the target displacementwith the allowable displace-ment, the building structure is evaluated as towhether itmeets theperformance requirement of seismic design or not. Although CSMbased on the idea of Freeman et al. [6] has advantages such as thesimple conception and easy operation, the significant higher modeeffects, theoretical foundation, and computational aspect shouldbe investigated and improved further.In recent years, some researchers in earthquake engineering

havemade lots of efforts to develop the related theory and applica-tion for pushover analysis. For example, Fajfar [7] proposed acomprehensive, relatively simple, N-2 method for seismic dam-age analysis of reinforced concrete buildings. Fajfar [8], Chopra andGoel [9,10] established the demand diagram of an inelastic sys-tem according to the constant-ductility inelastic response spec-tra, and calculated the ductility factor of the system based onthe intersection point of capacity and demand diagrams. Genc-turk and Elnashai [11] developed an advanced CSM, incorporatingthe inelastic response history analysis of SDOF system, in whichthe updating bilinear idealization of structural system accordingto the selected trial performance point on the capacity diagramimproves the accuracy of CSM. However, the higher mode re-sponse needs to be considered if they are judged to be impor-tant [12–26]. Consequently, Chopra and Goel [14–18] proposedthe modal pushover analysis (MPA) and modified MPA proceduresto deal with higher mode effects, including the symmetric- and

Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283 1273

asymmetric-plan buildings. Besides, Jan et al. [19] suggested anupper-bound pushover analysis procedure, Kim et al. [20] im-proved the mass proportional pushover procedure, and Pourshaet al. [21] proposed a new consecutive modal pushover procedurefor tall buildings. Furthermore, to overcome the reversal of con-ventional pushover curves of highermodes effectively, Hernandez-Montes et al. [22,23] constructed an energy based pushover curve,and proposed the energy based multimode pushover procedure.It is noted that the research work mentioned above, except the

N-2 method [8], adopted the invariant lateral load distributions tosimulate the inertia forces under the seismic hazards. However,after the structure yields, the invariant force distributions cannotaccount for the redistribution of inertia forces owing to thestructural yielding and the associated changes of the dynamicproperties of structures. For overcoming these limitations, theadaptive force distributions that attempt to follow the time-variantdistributions of inertia forces have been proposed and investigatedin detail [8,24–31]. For instance, Gupta andKunnath [24] presentedthe adaptive spectra based pushover procedure, in which theadaptive lateral force distributions are scaled with the elasticspectral acceleration for site-specific ground motions. Kalkanand Kunnath [25] proposed the adaptive modal combination(AMC) procedure which integrates the inherent merits of thecapacity spectrum method, and the modal pushover procedureadopting the energy based pushover curve. Shakeri et al. [27]proposed a story shear based adaptive pushover procedure, whichpredicts the peak inelastic drift response well especially whenthe higher mode effects are important. Antoniou and Pinho [28]investigated the advantages and limitations of the adaptive andnon-adaptive force and displacement based pushover procedures,and concluded that the adaptive force based pushover proceduredespite its conceptual superiority features a relatively minoradvantage over the traditional non-adaptive force based approach.However, the adaptive displacement based pushover procedure,which can even deal with structures with ground soft story, canpredict the capacity curve much more efficiently and effectivelythan the non-adaptive displacement based approach [29]. Andan adaptive capacity spectrum method (ACSM) was utilizedfor seismic assessment of continuous span bridges [30] andirregular frames [31] in detail. While adopting the adaptive forcedistributionsmay provide better estimates of the seismic demandsof structures [24,25], it may be conceptually complicated andcomputationally expensive to update the force distributions ateach step for application in structural engineering practice.Other attempts were also made to evaluate the seismic dema-

nds of buildings. Park et al. [26] suggested a factored modal com-bination approach, predicting the actual story shear profiles forthe prototype buildings with and without vertical irregularity.Leelataviwat et al. [32,33] established a new seismic evaluationprocedure based on an energy balance concept focusing on thefundamental mode effects. Aydinoglu [34] presented an incremen-tal response spectrum analysis procedure based on the inelasticspectral displacements for the evaluation of the multimode seis-mic performance. On the other hand, similar to the static pushoveranalysis, or CSM, the incremental dynamic analysis (IDA) [35–38],depicting the structural seismic behavior from elasticity to yield-ing and finally collapse, has caused a lot of interest. Vamvatsikosand Cornell [37] investigated the direct estimation of the seismicdemand and capacity ofmulti-degree-of-freedomsystems throughthe incremental dynamic analysis of single degree of freedom ap-proximation. Nevertheless, at present, a fully computer-intensiveincremental dynamic analysis especially for tall buildingsmight betoo much time consuming.Hence, the need to evaluate the existing static schemes and im-

prove their accuracy and efficiency is still a hot issue in performan-ce based earthquake engineering. The present work is motivated

by the framework of energy demand and capacity diagrams andhighmode effects in all mid and high rise buildings under the near-fault groundmotionswith forward directivity and fling-step pulsesand without pulses. The principal objective of this investigationis to develop a modified multimode pushover analysis procedurebased on the framework of modal energy capacity and demanddiagrams of structures, retaining the conceptual simplicity andcomputational attractiveness with invariant force distributionsimultaneously. Meanwhile, the response contributions of thehigher vibration modes of building structures are computed by as-suming the building to be linearly elastic. Finally, the effectivenessand accuracy of the proposed approach is validated by comparingthe response predictions with the estimates by the nonlinear re-sponse history analysis (NL-RHA) and modified MPA.

2. Development of the proposed procedure

This section describes themodified procedure of energy balanceconcept based multimode pushover analysis proposed in thispaper and its differences from MPA [14,15] and the modifiedMPA [16].

2.1. Basic elements of the proposed procedure

The development of the proposed procedure is motivated byCSM [1–10,23], energy balance concept [32,33], and modifiedMPA [16] to adopt the concept of the nth modal energy capacityand demand of the ESDOF system assuming higher modes ofbuilding in linearly elastic phase. The primary concepts integratedand enhanced in the proposed methodology are listed as follows.

(1) Establish the modal energy capacity diagram of the ESDOFsystem by the lateral force distribution vs. energy baseddisplacement, using an energy based procedure [22,23,39].

(2) Construct the modal energy demand diagram of the ESDOFsystem under the selected ground motion vs. displacement.Meanwhile, an important numerical problem is met and dealtsuccessfully.

(3) Intersect the dynamic target point, which is similar tothe performance point in CSM, however, derived from theframework of energy capacity and demand of the ESDOFsystem.

(4) Apply the modal combination. The seismic demand estimateof each modal lateral force distribution is combined by certaincombination rule.

The details of conceptual elements of the process are described inthe following sections.

2.2. Energy based approach of nonlinear static procedure

The conventional pushover curve in the form of ‘‘base shear vs.roof displacement’’ may reverse in higher modes [22,23]. Actually,the inelastic response of structure may dissipate energy ratherthan generate energy. Therefore, it is inappropriate to use theforce–displacement relationship to depict the ESDOF system.Whatis more, it may cause the difficulty of idealizing the conventionalpushover curve as a bilinear curve of MPA and the fundamentalmode of the modified MPA.A general energy based approach of pushover analysis [39]

was proposed to calculate the work done by the external forcedistribution, whether invariant or adaptive. The N-story momentframe with floor diagrams in Fig. 1(a) is subjected to lateral loadprofiles s = [f1, f2, . . . , fN ], where fi is the lateral load acting on thefloor i andN is the total number of floors. In the process of pushoveranalysis, the external force distribution s gradually increases at themth step and cause horizontal displacementsu = [u1, u2, . . . , uN ],

1274 Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283

a b

c

Fig. 1. (a) N-story moment frame with diagrams at floor levels subjected to lateral force s resulting in horizontal displacement u at themth step. (b) Pushover curve of ithfloor. (c) Energy capacity diagram under the lateral force s of ESDOF system.

where ui is the horizontal displacement of the ith floor. At themth step of the analysis, the applied total force sm and horizontaldisplacement um are available from the analyzed results, and theincremental work can be expressed in the matrix form:

1Wm =12(sm−1 + sm) · (um − um−1). (1)

Then the energy based incremental displacement due to thelateral force s at themth step can be calculated,

1ume =1Wm

(Vm−1b + Vmb )/2. (2)

In which, the base shear at themth step equals Vmb = sm · 1.Finally, the total work Wm done by the lateral force s and the

total energy based displacement ume can be formulated as:

Wm = Wm−1 +1Wm

ume = um−1e +1ume .

(3)

Fig. 1(b) shows the pushover curve of a single ith floor, and Fig. 1(c)demonstrates the energy capacity diagram under the lateral forcedistribution s. Herein, the invariant lateralmodal force distributionis selected as

sn = mφn (4)

where m denotes the mass matrix, and φn is the nth naturalvibration mode of the structure.

2.3. Energy demand diagram and evaluation of dynamic target point

Determining the energy demand diagram is a key element inthe proposed procedure. Firstly, according to Fig. 2, the energyfactor [32,33] of an ESDOF system is defined as

γ =

12 fyuy + fy(um − uy)

12mS

2v

=

12 fyuy + fy(um − uy)

12 fouo

=2µ− 1R2y

(5)

where, fy, uy are the yield strength or yield base shear andyield displacement, respectively; fo, uo are the peak values

Fig. 2. Elastoplastic system and its corresponding linear system.

of earthquake-induced resisting force and deformation in thecorresponding linear system; um is the maximum displacementof the bilinear system; Ry is the yield strength reduction factor,Ry = fo/fy; µ is the ductility factor, µ = um/uy; m is the mass ofthe ESDOF system; Sv is the pseudo-spectral velocity of the selectedground motion.In the proposed procedure, the underlying assumptions, which

could influence the accuracy of the proposed procedure, need tobe clarified: (1) the coupling effects between modal co-ordinatesarising from the yielding of the system are neglected [14,15]; (2)when the relationship of Ry-µ-Tn is used, the natural periods ofthe structure are set unchanged, unlike MPA updating the periodsof the ESDOF system according to the conventional pushovercurve; (3) to reduce computational effort, the energy factor isset to unit directly for higher modes, which means higher modesare assumed linearly elastic [15,16]; (4) the modal responses(such as floor displacement, interstory drift, etc.) are combined bySRSS (square-root of sum-of-squares) rule. Through validation inthe later section, the seismic demands of building predicted bymodified pushover analysis under these assumptions are feasible,which reach a reasonable accuracy.

Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283 1275

Fig. 3. Identification of the ductility factor of the dynamic target point.

Generally, the energy demand of the building system subjectedto a specific ground motion can be expressed below:

Edemand =12γM∗n S

2v (6)

where, M∗n represents the modal effective mass, Sv denotes thepseudo-spectral velocity, and γ is the energy factor.In this procedure, for the fundamental mode, the energy factor

can be calculated through Eq. (5). For the highermodes, the energyfactor equals unit corresponding to the third underlying assump-tion, which also means the ductility factor µ = 1, and the yieldstrength reduction factor Ry = 1. A key aspect of the proposedprocedure is to calculate the dynamic target ductility in Fig. 3. Fi-nally, the intersection of the energy capacity and demand diagramsis called the dynamic target point, as displayed in Fig. 4. The en-ergy based dynamic target displacement (u(p)e,t arg et ) can be trans-formed from the ESDOF system to the MDOF system using Eq. (7)expressed as

u(p)r,n = φr,nΓnu(p)e,t arg et (7)

where ur,n stands for the roof displacement obtained from nth-mode pushover analysis, φr,n and Γn are the roof component of thenth modal shape and the modal participation factor, respectively.The evaluation procedure of the dynamic target point is illustratedin Fig. 5 only considering a single mode response. Then, extract thevalues of desired response parameters (r (p)n , e.g., displacements, in-terstory drifts, member rotations, etc.) at the pth step of the nthmodal pushover analysis.Furthermore, compute the modal seismic responses as many

modes as deemed essential for the system under consideration.The fundamental modal response is enough for most low to med-ium rise buildings. However, for tall buildings the contributions ofthe first several modes cannot be neglected. The total response is

a b

Fig. 5. First three modes of the 9-story and 20-story buildings.

determined by some appropriate combination rule. Practically, theSRSS combination rule is typically valid if the dominant modal pe-riods are well separated, whereas the complete quadratic combi-nation (CQC) may be more appropriate for the systems with closermodes. The total response given in the following expression is ob-tained through SRSS combination of modal response quantities:

r =

(∑n

(r ipn )

)0.5. (8)

If the system remains elastic in any mode considered, thecomputation of the response parameters will be equivalent to theconventional response spectrum analysis (RSA). Finally, it is notedthat in the energy capacity and demand framework, there arethree features different from MPA: (a) the period of the ESDOFsystem is set to the corresponding structural period directly, whileMPA updates the period of the ESDOF system according to thecorresponding pushover curve; (b) the energy based displacementis adopted rather than the roof displacement used by MPA; (c) themodal energy demand and capacity based diagrams are adopted.

3. Evaluation of the proposed procedure

The proposed procedure is verified for different structuralbenchmark models and a range of near-fault ground motionrecords herein, and some observations are obtained. Validationstudies are presented for two SAC steel moment resisting frame(SMRF) buildings located at the Los Angeles, and results of the

Fig. 4. Dynamic target point evaluation using energy capacity and demand diagrams.

1276 Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283

Table 1Properties of prototype frame structures.

Structure Total height (m) Seismic mass per frame (×106 kg) Floor seismic mass (×105 kg) Periods (s)1st mode 2nd mode 3rd mode

9-story 37.18 4.00 4.82 (ground) 2.325 0.873 0.503(above ground level) 5.04 (1st)

4.94 (2nd–8th)5.34 (roof)

20-story 80.77 5.40 2.63 (ground) 3.837 1.340 0.779(above ground level) 2.81 (1st)

2.68 (2nd–19th)2.89 (roof)

Table 2Property parameters of three groups of ground motions.

Earthquake Station, Component d (km) S PGA (g) PGV (cm/s) PGD (cm) Scale factors9-story building 20-story building

Forwarddirectivity pulses

Northridge SCE018 6.1 C 0.828 117.5 34.22 1.0 1.0Northridge RRS228 7.1 C 0.838 166.1 28.78 0.6 0.78Northridge SYL360 6.4 C 0.843 129.6 32.68 1.0 1.0Northridge SCS052 6.2 C 0.612 117.4 53.47 0.50 0.76Northridge JEN022 6.2 C 0.424 106.2 43.06 0.63 1.08Imperial valley H−E06230 1.0 C 0.439 109.8 65.89 0.58 0.40Imperial valley H−E07230 0.6 C 0.463 109.3 44.74 0.70 0.75

Fling-step pulses

Chi–Chi TCU052NS 1.84 D 0.448 220.7 723.1 (676.9) 0.52 0.62Chi–Chi TCU068NS 3.01 D 0.365 292.2 867.7 (619.3) 0.68 0.40Chi–Chi TCU068EW 3.01 D 0.505 280.5 709.5 (−567.3) 0.60 0.53Chi–Chi TCU075EW 3.38 D 0.332 116.2 171.0 (120.0) 0.60 0.54Chi–Chi TCU087NS 3.42 C 0.113 45.3 93.0 (−81.2) 1.2 1.2Chi–Chi TCU102NS 1.19 D 0.171 71.5 107.2 (−77.1) 0.6 1.0Chi–Chi TCU076EW 3.17 D 0.343 69.3 108.0 (87.6) 1.2 0.9

Without pulse

Northridge WST270 29.0 B 0.361 20.9 4.27 2.8 3.0Northridge CEN155 30.9 C 0.465 19.3 3.48 3.3 3.3Loma prieta PAE000 36.1 D 0.204 22.0 11.67 3.0 4.5Loma prieta WAH090 16.9 – 0.638 38.0 5.85 1.7 1.5Imperial valley El Centro-NS – – 0.319 33.0 21.33 3.0 3.0Kobe NIS000 11.1 D 0.509 37.3 9.52 2.0 2.0

Note: d denotes the closest distance to the rupture surface. PGA, PGV and PGD represent the peak ground acceleration, velocity and displacement respectively. The numberin bracket stands for the permanent displacement of near-fault ground motions.

suggested approach are compared with those using Modified MPA(considering 3 modes) and the first-mode lateral load pattern.The simulation results of different pushover analyses are thencompared with those of nonlinear history analysis under a set ofground motion records in the near-fault region with and withoutvelocity pulses, which were carefully selected so as to signifythe higher mode contributions. Moreover, the modified pushoveranalysis is implemented using the open source finite elementplatform, OpenSees (2005) in conjunction with MATLAB (TheMathWorks, Inc. 2005) routines [40–42].

3.1. Prototype structures

The seismic demands of a set of benchmarked 9-story and 20-story prototype SMRF structures are evaluated in the investigation.The detailed information of the two SMRF including the informa-tion of bay width and story height and the sizes of beams andcolumns can be referred to the paper [43]. Table 1 and Fig. 5 pro-vide the basicmodal properties of the two typical SMRF structures.A nonlinear beam–column element utilizing a layered fiber sectionis utilized to model all the components of building structures, andthe constitutive law of steel is assumed elastoplastic. Moreover, allthe numerical results of steel frames by nonlinear history analy-sis and the proposed methodology consider the gravity load ef-fects. Additionally, the Rayleigh damping model is adopted for thepushover analysis of buildings, in which for the fundamental modethe damping ratio is 2%, and for the highermode the damping ratio1% [16].

3.2. Ground motion ensemble

A set of ground motion records from near-fault region areselected to enable the dynamic responses of frame structuresinto the inelastic range, thus the higher mode responses ofbuildings are triggered. The selected impulsive near-fault groundmotion records contain rupture forward directivity and fling-stepeffects with rich coherent low frequency velocity pulses (see inFig. 6) [44]. Because of the large velocity pulses, these recordsare scaled accordingly so that the maximum value of story driftsof buildings via nonlinear history analysis is larger than 1/50.Furthermore, the non-pulse records are amplified by a scale factorto approximate to 1 g (980 gal, 1 gal = 1 cm/s2) to induce inelasticresponses. The property parameters of these ground motions aresummarized in Table 2. The groundmotions without pulses with dless than 40 km can be viewed as the near-fault records. Moreover,elastic pseudo-velocity spectra of the four representative scaledrecords (SCE018, RRS228, TCU075EW and WST270) with threemodal periods corresponding to 9-story and 20-story buildings arepresented in Fig. 7.

3.3. Comparison of seismic demands of buildings

This section applies the energy balance concept based multi-mode pushover procedure to estimate the seismic demands forprototype structures. The demand estimates using the proposedapproach are compared with the results employing nonlinear his-tory analysis as well as the modified MPA (MMPA) suggested by

Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283 1277

RRS (228)

-50

0

50

100

150

-100

200

Vel

ocity

(cm

/s)

RRS (228)

2 4 6 8 10 12 14

Time (s)

0 16Time (s)

-20

-10

0

10

20

30

Dis

plac

emen

t (cm

)

-30

2 4 6 8 10 12 140 16

a b

(a) RRS-228 with forward directivity pulse.

-50

0

50

100

150

200

Vel

ocity

(cm

)

-100

250

10 20 30 40 50 60 70 80

Time (s)

0 90

TCU052 (NS)

0

100

200

300

400

500

600

700

-100

800

Dis

plac

emen

t (cm

)10 20 30 40 50 60 70 80

Time (s)

0 90

TCU052 (NS)

a b

(b) TCU052NS with fling-step pulse.

Fig. 6. (a) Velocity and displacement time histories of near-fault ground motion RRS-228 with forward directivity pulse. (b) Velocity and displacement time histories ofnear-fault ground motion TCU052NS with fling-step pulse.

a b

Fig. 7. Elastic pseudo-velocity spectra (2% damped) with modal periods corresponding to 9-story (a) and 20-story (b) buildings.

Fig. 8. Predicted floor displacements and interstory drift ratios of 9-story building by the proposed procedure compared to MMPA and NL-RHA under RRS228 and SCE018with forward directivity pulses.

1278 Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283

Fig. 9. Predicted floor displacements and interstory drift ratios of 9-story building by the proposed procedure compared to MMPA and NL-RHA under TCU075EW withfling-step pulses and WST270 without velocity pulse.

Fig. 10. Error distributions of predicted floor displacements and interstory drift ratios of 9-story building by the proposed procedure compared toMMPA and NL-RHA underRRS228 and SCE018 with forward directivity pulses.

Fig. 11. Error distributions of predicted floor displacements and interstory drift ratios of 9-story building by the proposed procedure compared toMMPA and NL-RHA underTCU075EWwith fling-step pulses and WST270 without velocity pulse.

Chopra et al. [14–16]. The dynamic responses of buildings by non-linear history analysis are regarded as the exact solutions. For thebenchmark SMRF in this study, only the first three modes areconsidered for the proposed procedure and MMPA. Furthermore,in the MMPA, only the first-mode target estimates are obtainedbased on the inelastic dynamic analysis on the ESDOF systems.Moreover, the target responses for the proposed procedure just

considering the first-mode responses are also plotted for evalua-tion of the higher mode effects.In Figs. 8 and 9, the floor displacement and interstory drift

ratio profiles are presented for 9-story buildings subjected tothe four typical ground motions. For the cases of the recordsRRS228 and SCE018with forward directivity pulses and the recordWST270without velocity pulse, higher mode effects result in large

Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283 1279

Fig. 12. Predicted floor displacements and interstory drift ratios of 20-story building by the proposed procedure compared toMMPA andNL-RHAunder SCE018with forwarddirectivity pulses and WST270 without velocity pulse.

Fig. 13. Predicted floor displacements and interstory drift ratios of 20-story building by the proposed procedure compared to MMPA and NL-RHA under TCU068NS andTCU052NS with fling-step pulses.

Fig. 14. Error distributions of predicted floor displacements and interstory drift ratios of 20-story building by the proposed procedure compared to MMPA and NL-RHAunder SCE018 with forward directivity pulses and WST270 without velocity pulse.

demand at the upper story (story 8, 9) level. However, the onlydifference between the two cases is that, although the peak groundacceleration (PGA) for records without velocity pulse is over 1 g,the fundamental mode of building still responds elastic. For thecase of record TCU075EW with fling-step pulses, higher mode

effects are insignificant enough. Figs. 10 and 11 show the height-wise error distributions for 9-story building subjected to the fourtypical ground motions, in which the error of floor displacementsof the proposed procedure is less than 22.8% and the error ofinterstory drift ratios of the proposed procedure is less than 26.1%.

1280 Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283

Fig. 15. Error distributions of predicted floor displacements and interstory drift ratios of 20-story building by the proposed procedure compared to MMPA and NL-RHAunder TCU068NS and TCU052NS with fling-step pulses.

Fig. 16. Mean, 16, and 84 percentile predictions of floor displacements and interstory drift demands for 9-story and 20-story buildings (note: 16 and 84 percentile predictionsare shown by unfilled markers).

The comparison of the predicted results demonstrates that both ofthe results of the proposed procedure and MMPA approximate tothe predictions of nonlinear history analysis.In Figs. 12 and 13, the floor displacement and interstory drift

ratio profiles are presented for 20-story buildings subjected to thefour representative ground motions. For the cases of the recordsSCE018 with forward directivity pulses and WST270 withoutvelocity pulse, higher mode effects can be captured at the upperstory (story 16–20) generally. For the case of the record TCU068NSwith fling-step pulse, the fundamental responses are dominated;but for the case of the record TCU052NS with fling-step pulses,the higher mode effects are fairly remarkable. Figs. 14 and 15illustrate the height-wise error distributions for 20-story buildingssubjected to the four ground motions, which the error of floordisplacements of the proposed procedure is less than 31.8% and themaximumerror of interstory drift ratios of the proposed procedureis up to 49.0%. However, the predicted demands of the proposedprocedure at most story levels approximate to the results vianonlinear history analysis and MMPA.Although MMPA captures the overall responses in many cases

(see Figs. 8–15), the proposed procedure predicts the demandswhich are generally similar to the results of nonlinear history anal-ysis at most story levels. Nevertheless, in some cases neither theproposed procedure nor MMPA are able to reproduce the dynamic

response at some story level. The results of nonlinear history analy-sis shown in Figs. 8–11 highlight the underestimation of the struc-tural response using the first-mode pushover analysis, especiallyfor interstory drift ratios. This implies that the approximate com-putation of seismic estimates using the first-mode behavior maynot be conservative and may vary from record to record.It is worth pointing out that, the computational results in

Figs. 8, 9, 12 and 13 consider the response of typical framessubjected to the selected records to emphasize the distinctivefeatures of structural responses, especially the highermode effectsand applicability of different ground motion sets. These resultsrepresent the particular cases from the entire set of simulationswherein the largest discrepancy between the proposed procedureand nonlinear history analysis scheme is observed. Furthermore,the effectiveness of the proposedprocedure to estimate the seismicdemands is studied statistically. In Fig. 16, the mean and mean± standard deviations (16, 84 percentiles) for both the proposedprocedure and NL-RHA predictions of floor displacements andinterstory drift demands are presented for the two prototypebuildings subjected to all ground motion sets. Generally speaking,the mean estimates of the proposed procedure are overestimatedslightly. However, the mean interstory drift demands at theintermediate stories are underestimated slightly (i.e. from 9thto 14th story for 20-story SMRF), which may be caused by thenumbers of the higher modes. More numbers of higher mode

Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283 1281

Fig. 17. Evaluation of dynamic target point of mode 1 in the proposed procedure for 20-story building.

considered, the resultsmay be improved accordingly. These resultsprovide a measure of confidence in the estimated ability of theproposed pushover scheme.

3.4. Some discussions on energy demand and assumptions

The computation of the energy factor γ is the key of the struc-tural energy demand under the groundmotions, inwhich themod-ified detailed relationship between the yield strength reductionfactor Ry and ductility factor µ is adopted. For most of the cases,the energy factor γ is less than unit when the structure respondsinto inelastic range. However, for some of near-fault ground mo-tions (i.e. TCU068NS etc.), when the structure responds into in-elastic range, the energy factor γ is more than unit displayed inFig. 17,which can be explained easily using Fig. 18 and Eq. (6). It isa unique characteristic of the near-fault groundmotion with fling-step pulses, which can cause larger energy demand for structuresso that input energy cannot be dissipated timely which could leadto much more severe damage of structures.It should bepointed out that, for the specific period T such as the

fundamental period of the structure, the yield strength reductionfactor Ry does not always increase monotonically as the ductilityfactor µ increases [45]. Thus, for a specific ductility this point mayresult in that the yield strength reduction factor Ry is not a singlevalue with respect to the energy factor γ in some case shownin Fig. 19, which possibly makes it quite difficult to capture thetarget point. In the proposed approach, the actual relationship ofRy-µ of the elastoplastic SDOF system is modified monotonicallydisplaying in Fig. 19 for the computational convenience, which isfeasible and effective, although this can cause someminor error fordetermining the target point, which is less than the error from theprocedure itself.

Fig. 18. The relationship of the yield strength reduction factor Ry and ductilityfactor µ of ESDOF system with period T = 3.84 under typical ground motions.

Fig. 19. Modified and original relationship of Ry-µ of the elastoplastic SDOF systemunder El Centro ground motion.

1282 Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283

(a) Mode 1 pushover curve. (b) Mode 2 pushover curve.

(c) Mode 3 pushover curve.

Fig. 20. Energy based pushover curves of 20-story building with displacements identified for 0.25, 0.5, 0.75, 1.0, 1.5, 2.0, and 3.0 × El Centro ground motion using theproposed procedure.

Fig. 21. Predicted floor displacements and interstory drift ratios of 20-story building by the proposed procedure compared to NL-RHA under JEN022 and El Centro.

In addition, for most of the cases, the higher modal responsesof 20-story building still remain elastic (see Fig. 20), which meansthe assumption is reasonable formost of the cases. In the proposedprocedure, the highermodes of building are assumedelastic,whichcan cause conservative results, but reduce the computation effort.However, such conservatism is unacceptably large for some of thecases with damping significantly less than 5%, thus damping ratioof 5% is recommended.Finally, a key aspect of the assumptions for the proposed proce-

dure is to neglect the coupling effects of nonlinear vibration equa-tion and adopt a particular combination rule. Chopra and Goel [14,15] investigated the uncoupled modal response history analysis

(UMRHA) in detail, and verified the errors caused by UMRHA. Itis shown that even for very intense excitation the errors causedby UMRHA are only a few percent. Thus neglecting the couplingeffects in the proposed procedure is reasonable confidently. Nev-ertheless, the predicted floor displacements and interstory drift ra-tios of 20-story building by the proposed procedure under JEN022and El Centro in Fig. 21 illustrate that, the intense coupling effectsshould be concerned in some cases.

4. Conclusions

In performance based earthquake engineering, identifying andassessing the performance capability of a building relies greatly on

Y. Jiang et al. / Engineering Structures 32 (2010) 1272–1283 1283

the advancement in analytical methods to estimate the inelasticseismic demand of building structures. This investigation developsan energy balance concept based multimode pushover analysisto estimate the seismic demand of new and retrofitted buildings,which incorporates the concept of modal energy demand andcapacity and retains computational attractiveness.Firstly, energy based energy capacity curve and modal demand

curve of buildings, in which higher modes are assumed elastic,are constructed. Then, the dynamic target displacement is de-termined based on the intersection of the two diagrams. More-over, the underlying assumptions and inherent shortcomings ofthe proposed procedure are clarified and investigated in detail. Bycomparing with nonlinear history analysis and other successfuladvanced pushover procedure (MMPA), the accuracy and highermode effects of the proposed procedure for the benchmark framebuildings subjected to near-fault groundmotionswith andwithoutvelocity pulses are verified. It is demonstrated that the proposedprocedure is an attractive alternative for practical application.Finally, it is noted that in the process of the investigation, some

specific observations of near-fault ground motions are obtained(i.e. TCU068NS, TCU052NS etc.). The energy factor is larger than1 and it demonstrates that these near-fault ground motions maycause larger energy demand for structures, so that the input energyto the building structure cannot be dissipated timely, probablyleading to much more severe damage of structures. However,the conclusions herein are based on the study of regular steelresisting moment frames, and further work is needed to explorethe applicability of the developed approach and investigate theintensive coupling effects in some scenarios.

Acknowledgement

The support of the National Natural Science Foundation ofChina (Grant nos. 90815023, 10721062 and 50978047) is muchappreciated.

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