brbf1 current nonlinear

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Journal of Constructional Steel Research 66 (2010) 11181127

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Journal of Constructional Steel Research

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Assessment of current nonlinear static procedures for seismic evaluation of BRBFbuildings

An Hong Nguyen a, Chatpan Chintanapakdee a,, Toshiro Hayashikawa b

a Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok, Thailandb Graduate School of Engineering, Hokkaido University, Sapporo, Japan

a r t i c l e i n f o

Article history:

Received 25 June 2009

Accepted 4 March 2010

Keywords:

Buckling-restrained braced framesEarthquake resistant system

Nonlinear static procedure

Modal pushover analysis

Seismic demands

Non-linear response history analysis

a b s t r a c t

Nonlinearstatic procedures (NSPs) are now standard in engineering practice to estimate seismic demandsin the design and evaluation of buildings. This paper aims to investigate comparatively the bias and ac-

curacy of modal, improved modal pushover analysis (MPA, IMPA) andmass proportional pushover (MPP)procedures when they are applied to buckling-restrained braced frame (BRBF) buildings which have be-

come a favorable lateral-force resisting system for earthquake resistantbuildings. Three-, 6-, 10-, and14-storey concentrically BRBF buildings were analyzed due to two sets of strong ground motions having 2%

and10%probabilityof beingexceededin 50years. Theassessmentis basedon comparingseismic displace-ment demands such as target roof displacements, peak floor/roof displacements and inter-storey drifts.The NSP estimates are compared to results from nonlinear response history analysis (NL-RHA). The re-

sponsestatistics presentedshow that theMPP procedure tends to inaccuratelyestimate seismic demandsof lower storiesof tall buildingsconsidered in this study while MPAand IMPA procedures provide reason-

ably accurate results in estimating maximum inter-storey drift over all stories of studied BRBF systems.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

To estimate seismic demands in the design and evaluation of

buildings, the nonlinear static procedures (NSPs) using the lateral

force distributions recommended in ATC-40 [1] and the FEMA-

356 [2]documents are now standard in engineering practice. The

nonlinear static procedure in these documents is based on the ca-

pacity spectrum method (ATC-40) and displacement coefficient

method (FEMA-356), and assumes that the lateral force distribu-

tion for the pushover analysis and the conversion of the results to

the capacity diagram arebased on the fundamental vibrationmode

of the elastic structure. Consequently, these NSPs based on invari-

ant load patterns provide accurate seismic demand estimates onlyforlow- andmedium-risemoment-framebuildings where thecon-

tributions of higher modes response are not significant and in-

adequate to predict inelastic seismic demands in buildings when

the higher modes contribute to the response[38]. To overcome

these drawbacks, an improved pushover procedure, called modal

pushover analysis (MPA), was proposed by Chopra and Goel [4] to

include the contributions of higher modes. The MPA procedure

Corresponding author. Tel.: +66 8 1612 3045; fax: +66 2 251 7304.E-mail addresses: [email protected](A.H. Nguyen),

[email protected] (C. Chintanapakdee).

has been demonstrated to increase the accuracy of seismic de-mand estimation in taller moment-frame buildings, e.g., 9- and 12-stories tall, compared to the conventionalpushover analysis [9,10].In spite of including the contribution of higher modes, MPAis conceptually no more difficult than standard procedures be-cause higher modes pushover analyses are similar to the firstmode pushover analysis. Moreover, MPA procedure consideringfor the first few (two or three) modes contribution are typicallysufficient [3,9].

Another pushover method is the adaptive pushover procedures,where the load pattern distributions are updated to consider thechange in structure during the inelastic phase [1113]. In thistype of procedure, equivalent seismic loads are calculated at each

pushover step using the immediate mode shape. Recently, a newadaptive pushover method,called the adaptive modal combination(AMC) procedure, has been developed by Kalkan and Kunnath [14]where a set of adaptive mode-shape based inertia force patterns isapplied to the structure. This procedure has been validated for reg-ular moment frame buildings[14,15]. However, it is conceptuallycomplicated and computationally demanding for routine applica-tion in structural engineering practice while the MPA method isgenerallysimpler,and thus, more practicalthan adaptive pushoverprocedures for seismic design.

More recently, an improved modal pushover analysis (IMPA)procedure was proposed by Jianmeng et al. [16] to considerthe re-distribution of inertia forces after the structure yields. The struc-tural stiffness changes after it yields, so the displacement shape

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A.H. Nguyen et al. / Journal of Constructional Steel Research 66 (2010) 11181127 1119

vector also changes. The IMPA procedure uses the product of thetime variant floor displacement vector (as the displacement shapevector) and the structural mass matrix as the lateral force dis-tribution at each applied-load step beyond the yield point of thestructure. However, to avoid a large computation, only two phaselateral load distribution was recommended. In the first phase, thepushover analysis is performedby using thefirst fewelastic naturalmodes of structure, i.e., similar to the MPA. In the second phase,

only for the first mode the lateral load distribution is based on as-sumption that the floor displacement vector at the initial yieldingpoint is the displacement shape vector.

An alternative pushover analysis method to estimate the seis-mic displacement demands, referred to as the mass proportionalpushover (MPP) procedure,was proposed by Kimand Kurama[17].The main advantage of the MPP is that the effectsof higher modeson the lateral displacement demands are lumped into a single in-variant lateral force distribution that is proportional to the totalseismic masses at the floor and roof levels. However, the accuracyof both IMPA and MPP procedures has been verified for a limitednumber of cases.

With the increase in the number of alternative pushover anal-ysis procedure proposed in recent years, it is useful to assess the

accuracy and classify the potential limitations of these methods.An assessment on accuracy of MPA and FEMA pushover analysesfor moment resisting frame buildings was investigated by Chopraand Chintanapakdee[9]. Then, an investigation on the accuracy ofimproved nonlinearstatic proceduresin FEMA-440 wascarried outby Akkar and Metin [18]. Meanwhile, the ability of FEMA-356, MPAand AMC in estimating seismic demands of a set of existing steeland reinforced concrete buildings was examined by Kalkan andKunnath [15]. More recently, an investigation into the effects ofnonlinear static analysis procedures which are the DisplacementCoefficient Method (DCM) recommended in FEMA 356 and theCapacity Spectrum Method (CSM) recommended in ATC 40 to per-formance evaluation on low-rise RC buildings was carried out byIrtem and Hasgul [19].

To assess the ability of current procedures, this paper aimsto investigate comparatively the bias and accuracy of MPA, IMPAand MPP procedures when applied to buckling-restrained bracedframes (BRBFs), which have become a favorable lateral-force re-sisting system for earthquake resistant buildings as its hystereticbehavior is non-degrading and much hysteretic energy can bedissipated. BRBF is an innovative structural system that preventsbuckling of the braces by using a steel core and an outer casingfilled with mortar for the brace. Brace axial forces are resistedonly by the steel core and not by the surrounding mortar andsteel encasement. The steel core is restrained from buckling by theouter shell and the infill mortar. The BRBF system is consideredto have favorable seismic performance over conventional bracedframesin that the bracesare capable of yielding in both tension andcompression instead of buckling, making it an attractive option to

structural engineers. BRBF systems are currently used as primarylateral force resisting elements both in new construction and seis-mic retrofit projects. A more comprehensive background on thissystem can be found in [20,21].

2. Review of selected nonlinear static procedures

This section briefly introduces the modal, improved modalpushover analysis (MPA, IMPA) and mass proportional pushover(MPP) procedures in estimating seismic demands for buildingdesign.

2.1. Modal pushover analysis (MPA)

The modal pushover analysis (MPA), which has been proposedby Chopra and Goel[4], is an extension of conventional pushover

Fig.1. (a)Pushovercurve and(b) forceand deformationrelationshipof SDFsystem.

analysis to include contribution of higher modes. A step-by-step

summary of the MPA procedure to estimate the seismic demands

for building is presented as a sequence of steps:

(1) Compute the natural frequencies,n, and mode shape vectors,

n, for linearly elastic vibration modes of the building.

(2) Forthe nth-mode, develop the base-shearroof-displacement

(Vbnurn) pushover curve by nonlinear static analysis of the

building using the force distribution sn = mn where m isthe mass matrix.

(3) Idealize the pushover curve as a bilinear curve(Fig. 1(a)).

(4) Convert the idealized pushover curve to the forcedeformation

(Fsn/LnDn) relation of the nth-mode inelastic SDF system and

determine the elastic modal frequencyn, and yield deforma-

tionDny. Thenth-mode inelastic SDF system is defined by the

forcedeformation curve ofFig. 1(b) (with post-yield stiffness

ratio n) and damping ratio n specified for the nth mode.

WhereMn = nLn is the effective modal mass, Ln = Tn m,

n = Tn m

Tn mn, andeach element of the influencevector is equal

to unity.

(5) Computethe peak deformation, Dn maxt|Dn(t)|,ofthe nth-mode inelastic SDF system with the forcedeformation rela-

tion ofFig. 1(b) due to ground excitation ug(t)by solving:

Dn+ 2nnDn+Fsn

Dn, Dn

Ln= ug(t) . (1)

(6) Calculate the peak roof displacementurno associated with the

nth-mode inelastic SDF system from

urno =nrnDn. (2)

(7) Extract other desired responses, rno , from the pushover

database when roof displacement equal tourno .

(8) Repeat Steps 27 for as many modes as required for sufficientaccuracy; usually the first two or three modes will suffice for

buildings shorter than 10 stories.

(9) Determine the total response rMPA by combining the peak

modal responses using appropriate modal combination rule,

e.g., Square-Root-of-Sum-of-Squares (SRSS) as shown by

Eq.(3)or Complete Quadratic Combination (CQC) rule:

rMPA =

jn=1

r2no (3)

wherejis the number of modes included.

The MPA procedure summarized in this paper is developed for

symmetric buildings[4].


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1120 A.H. Nguyen et al. / Journal of Constructional Steel Research 66 (2010) 11181127

Fig. 2. Frame elevations of 3-, 6-, 10-, and 14-storey BRBF buildings.

2.2. Improved modal pushover analysis (IMPA)

Unlike the MPAprocedure where the response is obtained frominvariant multi-mode lateral load pattern vectors, the improvedmodal pushover analysis (IMPA) proposed by Jianmeng et al. [16]considering the redistribution of inertia forces after the structureyields. The principal improvement of the IMPA is to use deflectionshape of structure after yielding as an invariant later load pattern.However, to avoid a large computation, a two-phase lateral loaddistribution is suggested for the first mode while the force pat-terns forhighermodes are similar to the MPAapproach. The IMPAprocedure is summarized by following steps:

(1) Implement the Steps 13 of the MPA procedure described inprevious section for first mode. The lateral force distribution

s1 = m1is considered as the first-phase load pattern.(2) Determine the displacements vector of structure, 1y, at the

yielding point with the pushover analysis obtained from Step1.(3) Continue pushover analysis from the structure yielding point

by applying the load distribution s1y =m1y, which is consid-ered as the second-phase lateral load pattern to obtain a newpushover curve. Then, this new pushover curve is used for de-termining the response of the structure by Steps 47 of MPAprocedure described in Section2.1.

(4) Determine the total response rIMPA with SRSS or CQC combi-nation rules by combining the response for the first modeobtained from Step 3 and the responses due to other highermodes obtained from MPA procedure.

2.3. Mass proportional pushover (MPP) procedure

An alternative pushover analysis procedure, called the massproportional pushover (MPP), was proposed by Kim and Kurama[17] to estimate the peak seismic lateral displacement demands forbuildings. The main advantage of the MPP procedure over otherapproximate procedures is the use of a single pushover analysisfor the structure with no need to conduct a modal analysis to cap-ture the effect of higher modes. A summary of the mass propor-tional pushover procedure, whose details can be found in Kim andKurama[17], is as follows:

(1) Determine the multi-degree-of-freedom (MDOF) base shearforce versus the roof displacement (Vbur) relationship usingthe force distribution given by mg= w where m is the massmatrix andwis weight matrix.

Table 1

Natural periods of BRBF building models in this study.

Mode Modal natural periodsTn (s)

3-story 6-story 10-story 14-story

1 0.504 0.797 0.982 1.274

2 0.197 0.296 0.338 0.423

3 0.120 0.174 0.187 0.230

(2) Idealize the pushover curve as a bilinear curve.(3) Convert the idealized pushover curve to the pseudo-acc-

eleration versus the displacement (AD) relationship of anequivalent SDF system using:

A= Vb

M; D=

ur

(4)

where M is the total mass and is the participation factor

calculated as: = uTe m

uTe mue;ue is the lateral floor displacement

vector (normalized with respect to the roof) obtained from thelinear-elastic response rangeof the pushover analysis using themg = w force distribution which is the same as uniformdistribution of FEMA-356.

(4) Determine the maximum SDF displacement, Dmax by solvingEq.(1)withFs/L= A.

(5) Calculate the maximum MDOF roof and floor displacements ofstructure as:umax = Dmaxue.

3. Structural systems and analytical models

Three-, 6-, 10-, and 14-storey BRBF buildings, which weredesigned to meet seismic code criteria, were analyzed to evaluatethe bias and accuracy of MPA, IMPA and MPP procedures. Buildingdesignsfor the BRBF system in boththe 3-storey and 6-storey casesadhered to the criteria for the 3vb2 and 6vb2 model cases studiedby Sabelli et al. [22] while the characteristics of the 10- and 14-

storey buildings were adopted from Asgarian and Shokrgozar [23].

The elevation view of all BRBF systemsis shown in Fig. 2. Analyticalmodelswere created to analyze these BRBF buildings whose detailscan be found in Chintanapakdee et al. [24]. A Rayleigh dampingmodel was used with 5% critical damping ratios for the first twomodes, according to common practice for code designed steel

structures [22]. P effect was also considered for this study.Nonlinear static and dynamic analyses were carried out using thecomputer program DRAIN-2DX [25]. The natural periods of allmodels are shown inTable 1.

4. Ground motions and response statistics

Two sets of ground motions, referred as LA2/50 and LA10/50,corresponding to 2% and 10% probabilities of exceedence in a

50-year period are used in this study. These ground motionswere compiled by the SAC Phase II Steel Project for a site in Los

Angeles, California [26]. These acceleration time histories were de-rived from historical recordings or from simulations of physicalfault rupture processes. Each set of ground motions consists of 20records which are the fault-normal and fault-parallel componentsof 10 recordings. The records in these suites include near-faultand far-fault records.Table 2provides the information of LA2/50

set of records including: recording station, earthquake magnitude,distance, scaling factor, and peak ground acceleration (PGA).

To determine the seismic demands of a building due to a setof ground motions, each record was scaled such that the spectralacceleration at the fundamental natural period of the building isequal to the median spectral acceleration for that period (Table 3).

This method of scaling helps reduce the dispersion of results [27].More details of these scaling ground motions can be found in [24].


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Table 2

Set of ground motions having 2% probability of being exceeded in 50 years (LA2/50).

Record Earthquake/recording station Earthquake magnitude Distance (km) Scaling factor PGA (cm/s2)

LA21 1995 Kobe 6.9 3.4 1.15 1258

LA22 1995 Kobe 6.9 3.4 1.15 903

LA23 1989 Loma Prieta 7.0 3.5 0.82 410

LA24 1989 Loma Prieta 7.0 3.5 0.82 464

LA25 1994 Northridge 6.7 7.5 1.29 854

LA26 1994 Northridge 6.7 7.5 1.29 925LA27 1994 Northridge 6.7 6.4 1.61 909

LA28 1994 Northridge 6.7 6.4 1.61 1304

LA29 1974 Tabas 7.4 1.2 1.08 793

LA30 1974 Tabas 7.4 1.2 1.08 973

LA31 Elysian Park (simulated) 7.1 17.5 1.43 1271





LA36 Elysian Park (simulated) 7.1 11.2 1.10 1079LA37 Palos Verdes (simulated) 7.1 1.5 0.90 698

LA38 Palos Verdes (simulated) 7.1 1.5 0.90 761



Table 3

Median spectral acceleration at the fundamental periodA(T1)/gof each building.

Set of records Spectral accelerationA(T1)/g


LA10/50 1.187 0.896 0.781 0.619

LA2/50 1.775 1.783 1.390 1.205

The response of each building to each set of the ground mo-tions was determined by nonlinear response history analysis(NL-RHA), and a nonlinear static procedure (NSP), e.g., MPA, IMPAandMPP. Thepeak value of inter-storeydrift,, determined by NL-RHA is denoted by NL-RHA, and from NSP byNSP. From these datafor each ground motion, a response ratio was determined from thefollowing equation:NSP = NSP/NL-RHA. The median values, x,defined as the geometric mean, ofn observed values (xi) ofNSP,NL-RHAand NSP; andthe dispersion measures of

NSPdefined as

the standard deviation of logarithm of the nobserved values werecalculated:

x= exp

ni=1

lnxi

n

(5)

=

n

i=1

lnxi lnx

2

n1. (6)

An advantage of using the geometric mean as the estimator ofmedian is that the ratio of the median of

NSP to the median of

NL-RHAis equal to the median of the ratio NSP, i.e., the biasof NSP

in estimating themedian response is equal to themedian of bias inestimating response to individual excitation.

5. Evaluation of nonlinear static procedures

The bias and accuracy of the MPA, IMPA and MPP proceduresapplied to BRBF buildings are evaluated by comparing the targetroof displacements, peak floor (or roof) displacements and inter-storey drifts compared to more accurate results from nonlinearresponse history analysis (NL-RHA).

5.1. Target roof displacements

Pushover curves, which show therelationshipbetween the baseshear force and the roof displacement, for the 3-, 6-, 10- and

Table 4

Median ductility factors for building models calculated from a NL-RHA estimate ofthe peak roof displacement.

Set of records Building model


LA10/50 6.02 3.25 2.20 1.94

LA2/50 14.85 7.29 3.52 2.93

14-storey BRBF buildings due to the first mode load pattern

(MPA), variable lateral force distribution (IMPA) and seismic mass

(or weight) distribution (MPP) are plotted in Fig. 3.The pushover

curves for these frames are approximately tri-linear in nature

whose details were discussed by Chintanapakdee et al. [24]. The

variable lateral force distribution of IMPA procedure in this study

is taken as a three-phase load pattern, which changes at the firstand second yielding points of the pushover curve. Fig. 3 shows

that the pushover curve of IMPA is similar to MPA. This results

in nearly identical estimates of target roof displacements of both

procedures. It implies that the changes of lateral load distribution

of IMPA procedure are not significant whereas the mg= w forcedistribution of MPP leads to different results. Pushover curves of

MPP are always higher and stiffer than both MPAs and IMPAs for

all cases.

On each pushover curve, diamond (MPA), star (IMPA) and circle

(MPP) markers show the peak roof displacements of buildings de-

termined by NL-RHA of the equivalent single-degree-of-freedom

(SDF) system due to 20 records in each set of ground motions. The

ductility factors of the first mode, defined here as the ratio be-

tween median of peak roof displacements determined by NL-RHAand yield roof displacement estimated by first mode load pattern,

are about 1.94 to 6.02 for LA10/50 and 2.93 to 14.85 for LA2/50

ground motions, respectively.Table 4shows the median ductility

factors for these BRBF buildings calculated from NL-RHA estimate.

Themedianductility factornoticeably decreaseswhen the building

height increases.

The accuracy of target maximum roof displacements predicted

by displacement of the equivalent SDF systems: (ur)SDF = 1r1D1for MPA and IMPA or (ur)SDF = Dmax for MPP are exam-ined by calculating the ratio between the SDF systems esti-

mate and roof displacement determined from NL-RHA: (ur)SDF =(ur)SDF/(ur)NL-RHA. The ratio (u

r)SDFbeing close to 1 indicates good

accuracy. The histograms of these ratios are shown in Fig. 4.The

median and dispersion of the peak roof displacements are alsonoted.Fig. 4shows that the SDF systems of these nonlinear static


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Fig. 3. First mode pushover curves of 3-, 6-, 10-, and 14-storey BRBF buildings due to (a) LA10/50 and (b) LA2/50 ground motions.

procedures slightly over-estimate the maximum roof displace-ments but the bias of MPA and IMPA is no larger than 15% for

set of LA10/50 ground motions and 19% for stronger ground mo-tions LA2/50 while the bias of MPP is 14% and 28% for LA10/50 and

LA2/50 records, respectively. The IMPA tends to predict the me-dian and dispersion of target roof displacements better than MPA;however, the difference is not significant while the MPP tends to

estimate the maximum roof displacements slightly more accuratethan both MPA and IMPA for set of LA10/50 but less accurate forstronger records LA2/50.

5.2. Peak floor/roof displacements

The responses of the BRBF buildings studied to the two sets ofground motions were determined by MPA, IMPA, MPP nonlinear

static procedures and by nonlinear response history analysis

(NL-RHA). The MPA and IMPA were considered as many modes as

to include participating mass at least 95% of the total mass. For the

structures in thisstudy, the contribution of the first two modesfor

a 3-storey building, three modes for 6- and 10-storey buildings,

and four modes for a 14-storey building were considered to esti-

mate the seismic demands. The combined values of floor displace-

ments and storey drifts were computed by using the SRSS modal

combination rule.

The peak floor/roof displacement demands from the four meth-

ods are compared inFig. 5;the results from modal pushover anal-

ysis (MPA) including only the fundamental mode are also shown

by the dashed line. These results lead to the following observa-

tions for the BRBF system. The contributions of higher modesof MPA and IMPA procedures to floor displacements are not


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Fig. 4. Histograms of ratio(ur)SDFfor 3-, 6-, 10-, and 14-storey BRBF buildings due to (a) LA10/50 and (b) LA2/50 ground motions.

significant. One mode pushover analysis, MPA, and IMPA can esti-mate thepeak floor displacementsreasonablywell witha tendencyto slightly overestimate the floor/roof displacement compared toNL-RHA while the MPP tends to significantly overestimate peakfloor displacements of lower stories(Fig. 5).

Fig. 6 shows the median floor displacement ratio, uNSP =uNSP/uNL-RHA, due to the two sets of ground motions. It can beseen that the MPA procedure can accurately estimate floor dis-placements of the 3-, 6-, 10-, and 14-storey BRBF buildings; thebias is generally less than 20% and 30% for LA10/50 and LA2/50ground motions, respectively. The IMPA tends to overlap the MPAwith slight difference whereas the MPP tends to much overesti-mate peak floor displacements of lower stories with increasingbias when the building height increases. The bias of MPP is verylarge for BRBF buildings taller than 6 stories considered in thisstudy.

5.3. Story drift demands

Unlike the floor/roof displacements, the contributions of highermodes in estimating the storey drifts of MPA and IMPA proce-

dures are more significant, especially in upper stories of tall BRBFbuildings.Fig. 7shows that the storey drift demands of 10-, and

14-story BRBF buildings predicted by MPA are able to follow theNL-RHA results whereas the first mode alone is inadequate. With

three or four modes included, the storey drifts estimated byMPA are generally similar to the results from the nonlinear RHA.However, the MPA storey drift results including two modes for 3-

storey and three modes for 6-storey BRBF buildings are close tothe one mode results indicating that the contributions of higher

modes are not significant for these buildings. Both one mode

pushover analysis andMPA canestimate the response of structuresreasonably well, although their results differ from the NL-RHA


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Fig. 5. Median floor displacements of 3-, 6-, 10- and 14-storey BRBF buildings determined by one mode pushover analysis, MPA, IMPA, MPP and NL-RHA due to LA10/50

(first row), and LA2/50 (second row) ground motions.

Fig. 6. Floor displacement ratio of 3-, 6-, 10- and 14-story BRBF buildings due to LA10/50 (first row) and LA2/50 (second row) ground motions.


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Fig. 7. Median storey drifts of 3-, 6-, 10- and 14-storey BRBF buildings determined by one mode pushover analysis, MPA, IMPA, MPP and NL-RHA due to LA10/50 (firstrow), and LA2/50 (second row) ground motions.

Table 5

Median and dispersion of maximum storey drift ratios over all stories determine by MPA, IMPA and MPP,NSP, versus exact values,NL-RHA.

Set of records 3-story 6-story 10-story 14-story

LA10/50

MPA = 0.982; = 0.119 MPA = 0.949; = 0.205

MPA = 1.058; = 0.214

MPA = 0.986; = 0.248

IMPA = 0.983; = 0.117 IMPA = 0.952; = 0.206 IMPA = 1.101; = 0.220 IMPA = 0.987; = 0.249MPP = 1.353; = 0.163

MPP = 1.244; = 0.209

MPP = 2.154; = 0.284

MPP = 1.831; = 0.317

LA2/50

MPA = 0.926; = 0.132 MPA = 1.013; = 0.203

MPA = 1.143; = 0.226

MPA = 1.046; = 0.298

IMPA = 0.922; = 0.128 IMPA = 1.015; = 0.202

IMPA = 1.161; = 0.227

IMPA = 1.048; = 0.297

MPP = 1.225; = 0.149 MPP = 1.422; = 0.212

MPP = 1.839; = 0.364

MPP = 2.256; = 0.287

results at some stories. Similar to investigations of peak floor/roofdisplacements, IMPA estimates tend to overlap the MPA estimatesin estimating storey drift demands. The MPP excessively overes-timates storey drifts in the lower stories but underestimates thestoreydrift in the upper stories in these cases.Moreover, the storeydrifts predicted by the MPP procedure seem to be uniform in theupper stories, especially for 10- and 14-storey BRBF buildings.

Fig.8 shows themedianstorey drift ratio,NSP =NSP/NL-RHA,due to the two sets of ground motions. The bias of MPA, IMPA andMPP nonlinear static procedures in estimating seismic demandstends to increase for stronger excitations and the variation of theNSP bias in estimating seismic demands along building height pri-marily depends on the building height rather than the intensityof ground motions. The bias of MPA and IMPA in estimating peakstorey drifts at an individual storey can be as large as 50% and60% at certain locations for LA10/50 and LA2/50 ground motions,respectively. Meanwhile, the bias of MPP in peak storey drifts es-timation at an individual storey can be as large as 80% for two setsof ground motions; however, it overestimates storey drifts by asmuch as 200%for the lower stories of 14-storey building.This is be-cause the MPP significantly overestimates floor displacements forlower stories due to the total seismic mass (or weight) load pattern(Fig. 5).

To verify a building design or to evaluate an existing structure,building codes usually require the maximum storey drift in any

stories to be less than its allowable value. Fig. 9plots the maxi-

mum storey drifts over all stories determined by NL-RHA and NSPas abscissa and ordinate, respectively, with diamond markers for

MPA, star markers for IMPA and circle markers for MPP. The MPA

and IMPA data points are clustered along the diagonal line indicat-ing that the maximum storey drifts over all stories estimated by

MPA and IMPA are close to the value from NL-RHA. The median

and dispersion of storey-drift ratio NSP considering maximum

storey drift over all stories are also shown inTable 5.The medianstorey-drift ratios of MPA,MPA, range from 0.93 to 1.14 while the

median storey-drift ratios of IMPA, IMPA, from 0.92 to 1.16 in-

dicating that both MPA and IMPA procedures predict maximum

storey drifts over all stories with a bias less than 14% and 16% forthese BRBF buildings, respectively. On the contrary, the bias in es-

timating the maximum storey drifts over all stories of MPP can be

considerable in the range from 1.22 to 2.26. This implies that

MPP significantly overestimates the maximum storey drift over allstories.

The dispersion of storey-drift ratios of MPA and IMPA range

from 0.117 for a 3-storey building to 0.298 for 14-storey buildingwith a tendency to increase as the building becomes taller or

ground motions become stronger. Meantime, the dispersion of

storey-drift ratios of MPP range from 0.149 to 0.364 for these BRBFbuildings. This implies that the accuracy of NSPs in predicting the


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Fig. 8. Storey drift ratio of 3-, 6-, 10- and 14-storey BRBF buildings due to LA10/50 (first row) and LA2/50 (second row) ground motions.

Fig. 9. Maximum storey drifts over all stories determine by NSP,NSP, versus exact valuesNL-RHA, for 3-, 6-, 10- and 14-storey BRBF buildings due to LA10/50 (first row),and LA2/50 (second row) ground motions.

response due to an individual ground motion deteriorates when

applied to taller BRBF buildings or subjected to stronger ground

motions. Among these cases, the dispersion is still small, less than

0.298 for MPA and IMPA and 0.364 for MPP, when NSPs are used to

estimate the maximum storey drift over all stories. Moreover, the

dispersion of storey-drift ratios of MPP is always larger than both

MPAs and IMPAs. Thus, MPA and IMPA can be a useful analysis

tool to estimate the peak storey drift over all stories in evaluating

existing buildings or design of new buildings using BRBFs. Both

of these procedures provide practically the same results but MPA

is simpler and more practical than IMPA because it involves an

invariant load pattern. On the contrary, the MPP method is simplewith no need to conduct a modal analysis to capture the effects

of higher modes but it may be inaccurate in estimating seismicdemands for BRBF tall buildings due to strong ground motions.

6. Conclusions

The following conclusions are obtained from the accuracy as-sessment of nonlinear static procedures in estimating the seismicdemands of BRBF buildings using LA10/50 and LA2/50 sets of in-tense ground motions. These conclusions are based on a compar-ison of NSP estimates of seismic demands and the correspondingvalues determined by NL-RHA for 3-, 6-, 10-, and 14-storey BRBFbuildings which were designed to meet seismic code criteria.

(1) The equivalent bilinear SDF systems of nonlinear staticprocedures can estimate the peak roof displacement quite


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accurately with a bias no larger than 15% and 19% for LA10/50

and LA2/50 sets of ground motions, respectively. The IMPAtends to predict the median and dispersion of target roof dis-placements better than MPA; however, the difference is notsignificant while the MPPtends to estimate the maximum roof

displacements slightly more accurately than both MPA andIMPA for the set of LA10/50 but less accurate for the strongerrecords LA2/50.

(2) The storey drift demands predicted by MPA and IMPA are able

to followthe NL-RHAresults.However, the highermodes con-tributions of these procedures in the response of 3-, and 6-storey BRBF buildings are generally not significant, so the firstmode alone may be adequate.

(3) Despite considering the redistribution of inertia forces afterstructure yields, the pushover curve of IMPA is similar to MPA,resulting in nearly identical estimates of target roof displace-ments by both procedures. The IMPA tends to overlap the MPA

in estimating storey drifts with slight differences.(4) The MPP tends to significantly overestimate seismic demands

forlowerstories but underestimates storeydrifts foruppersto-ries with increasing bias when the building height increases.

Moreover, the storey drifts predicted by the MPP procedure

seem to be uniform in upper stories, especially for 10- and 14-storey BRBF buildings considered in this study.

(5) The bias and dispersion of nonlinear static procedures in es-

timating seismic demands tends to increase for taller BRBFbuildings and stronger excitations. The height-wise variationof bias primarily depends on the structural properties, e.g.,building height, rather than the intensity of ground motions.

(6) The bias of MPA and IMPA procedures in estimating the max-imum storey drift over all stories is generally small; however,the bias of these procedures in estimating peak storey drift atan individual storey can be considerable for certain cases. Both

of these procedures provide practically similar results whereasMPA is slightly simpler and more practical than IMPA as it in-volves an invariant load pattern. On the contrary, the bias in

estimating maximum storey drifts over all stories of MPP canbe large.

Acknowledgements

The authors would like to acknowledge the financial supportprovided by Japan International Cooperation Agency(JICA) through

the ASEAN University Network/Southeast Asia Engineering Edu-cation Development Network (AUN/SEED-Net) program and theCommission on Higher Education, Ministry of Education, Thailand.Our research has benefited from correspondence with Professor

Stephen A. Mahin of Universityof California, Berkeley and Dr. SutatLeelataviwat of King Mongkuts University of Technology. The au-thors also would like to thank Associate Professor Behrouz Asgar-

ian and Mr. H.R. Shokrgozar of K.N. Toosi University of Technologyfor providing the data of 10- and 14-story frames used this study.

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