Engineering Structures 31 (2009) 591–599

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Engineering Structures

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A consecutive modal pushover procedure for estimating the seismic demands oftall buildingsMehdi Poursha a, Faramarz Khoshnoudian a,∗, A.S. Moghadam ba Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iranb International Earthquake and Seismology Research Centre, Tehran, Iran

a r t i c l e i n f o

Article history:Received 6 August 2007Received in revised form19 October 2008Accepted 20 October 2008Available online 10 December 2008

Keywords:Consecutive modal pushover (CMP)procedureSeismic demandsHigher-mode effectsTall buildings

a b s t r a c t

The nonlinear static procedure (NSP), based on pushover analysis, has become a favourite tool for usein practical applications for building evaluation and design verification. The NSP is, however, restrictedto single-mode response. It is therefore valid for low-rise buildings where the behaviour is dominatedby the fundamental vibration mode. It is well recognized that the seismic demands derived from theconventional NSP are greatly underestimated in the upper storeys of tall buildings, in which higher-mode contributions to the response are important. This paper presents a new pushover procedurewhich can take into account higher-mode effects. The procedure, which has been named the consecutivemodal pushover (CMP) procedure, utilizes multi-stage and single-stage pushover analyses. The finalstructural responses are determined by enveloping the results of multi-stage and single-stage pushoveranalyses. The procedure is applied to four special steel moment-resisting frames with different heights.A comparison between estimates from the CMP procedure and the exact values obtained by nonlinearresponse history analysis (NL-RHA), as well as predictions frommodal pushover analysis (MPA), has beencarried out. It is demonstrated that the CMP procedure is able to effectively overcome the limitations oftraditional pushover analysis, and to accurately predict the seismic demands of tall buildings.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Nonlinear response history analysis (NL-RHA) is a robust toolfor calculating seismic demands, as well as for identifying plastichinge mechanisms in structures. However, the response of NL-RHA is strongly affected by the modelling parameters and by thecharacteristics of the earthquake input such as frequency content,intensity, and duration [20,24]. It is therefore necessary to carefullychoose a set of representative ground motion records. This causesadditional computational effort [24]. On the other hand, thenonlinear static procedure (NSP) or pushover analysis, unlikedynamic analysis, can easily provide valuable information aboutthe locations of structural weaknesses and failure mechanismsin the inelastic range [22]. Also, pushover analysis is capableof using a code’s response spectrum as a demand diagram andcomputing the earthquake-induced demands [7]. Thus, in recentyears theNSP has gained importance as a standard tool for buildingassessment and design verification. However, pushover analysissuffers from several inherent deficiencies [21,22]. Among them,invariant load distribution in the traditional NSP is one of the most

∗ Corresponding author.E-mail address: khoshnud@aut.ac.ir (F. Khoshnoudian).

0141-0296/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.10.009

important limitations, and it cannot take higher-mode effects intoconsideration [10,16,22]. Recently, attempts have been made todevelop enhanced pushover procedures and to consider higher-mode effects.About 10 years ago, the ‘‘multi-mode pushover’’ (MMP)

method [26] was proposed, but the seismic demands were notquantified. More recently, the ‘‘pushover results combination’’(PRC) method [23] was proposed, in which the maximum seismicresponse was derived from combining the results of severalpushover analyses and utilizing a mode-shape as a load pattern ineach analysis. The final response was determined as a weighted(using modal participation factors) summation of the resultsfrom each pushover analysis. About the same time, ‘‘modalpushover analysis’’ (MPA) [10] was developed, in which theseismic demands were separately determined for each of themodal pushover analyses and combined using the appropriatemodal combination rule. It was concluded that the MPA procedurewould be accurate enough in practical applications for buildingevaluation and design [10]. Nevertheless, the plastic rotationsof the hinges were considerably underestimated [10,11,17,25],even if a large number of modes were included. A modifiedversion of the MPA (MMPA) [12] was then proposed, in whichthe seismic demands of the structure are obtained by combiningthe inelastic response of first-mode pushover analysis with theelastic response of higher modes. In another investigation, an

592 M. Poursha et al. / Engineering Structures 31 (2009) 591–599

Notations

c damping matrix of the structureh height of the structurek lateral stiffness-matrix of the structurem diagonal mass-matrix of the structuremi lumped mass of the ith floorM∗n effective modal mass of the nth modeM∗ total mass of the structureN number of storeysNs number of stages in the multi-stage pushover

analysisPeff (t) the effective earthquake forcesqn(t) modal co-ordinater the peak response of the structure in the CMP

procedures∗n incremental lateral force distribution for the nth

stage of multi-stage pushover analysisT predominant period of the structureug(t) acceleration of the ground motionu(t) displacement of the NDOF systemφn nth mode-shape of the structureζn damping ratio of the nth modeωn natural frequency of the nth modeΓn modal participating factor of the nth modeαn effective modal participating mass ratio of the nth

modeδt target displacement of the roof

upper-bound pushover analysis [18] was developed in which thecontributions of the first two elastic modes to the invariant loadpattern were combined through the absolute sum rule. A singlepushover analysis was then implemented with the resulting loadpattern. Other researchers proposed a two-phase load pattern [19],and found it suitable for approximating the seismic capacity curveonly for low- tomid-rise buildings. This procedurewas not verifiedwith respect to the estimation of the seismic demands of buildings.In another study, an ‘‘incremental response spectrum analysis’’(IRSA) [3] was developed. In this procedure, whenever a newplastic hinge occurs, elasticmodal spectrum analysis is executed inorder to take into account the changes in the dynamic propertiesof the structure. More recently, an adaptive modal combination(AMC) [20] procedure was proposed, in which the applied lateralforces are updated in accordance with the changes in the dynamiccharacteristics during inelastic analysis, for each mode. The totalresponse is obtained by combining the peak modal responses,using an appropriate combination scheme at the end. The accuracyof this procedure was not demonstrated for the estimation ofplastic rotations of the hinges.Themain objective of the present paper is therefore to propose a

new procedurewhich can take into account higher-mode effects inthe pushover analysis of tall buildings, and can improve estimatesof seismic demands, especially of the plastic rotations of the hinges.The new procedure, called the consecutive modal pushover (CMP)procedure, benefits from the concepts of structural dynamicsand uses multi-stage and single-stage pushover analyses that areelaborated in detail in this paper. To demonstrate its applicabilityand effectiveness, the procedure has been applied to four specialsteel moment-resisting frames. Predictions based on the MPAprocedure are also presented for the sake of comparison tothose obtained by the CMP procedure. Seismic demands obtainedby approximate pushover procedures are compared with exactsolutions derived from nonlinear response history analysis. Theresults of the comprehensive analyses, which are summarized

in the body of this paper, indicate the effectiveness of the CMPprocedure in estimating the seismic demands of tall buildings.

2. Modal response analysis and the required concepts

The differential equation governing the response of a multi-degree-of-freedom (MDOF) system to earthquake-induced groundmotion is as follows [8]:

mu+ cu+ ku = −miug(t) (1)

where m, c , and k are the diagonal mass, damping, and lateralstiffness matrices of the structure, respectively, and i is the unitvector. The right-hand side of Eq. (1) represents the effectiveearthquake forces, Peff(t), and can be written as

Peff(t) = −miug(t) = −sug (t) (2)

where s defines the spatial distribution of the effective earthquakeforces over the height of the building, and can be expanded as asummation of the modal inertia force distributions, sn, as follows:

s = mi =N∑n=1

sn =N∑n=1

Γnmφn (3)

in which Γn is the modal participating factor of the nth mode andφn is the corresponding mode-shape. The displacement of an Ndegree-of-freedom (NDOF) system, u, can be defined by the sumof the modal contributions:

u(t) =N∑n=1

φnqn(t) (4)

where the modal co-ordinate, qn(t), is governed by

qn + 2ζnωnqn + ω2nqn = −Γnug(t) (5)

in which ωn and ζn are the natural vibration frequency anddamping ratio of the nth mode, respectively. Γn is obtained asfollows:

Γn =φTnmiφTnmφn

. (6)

The solution to Eq. (5) is

qn(t) = ΓnDn(t) (7)

where Dn(t) is governed by the equation of motion for a single-degree-of-freedom (SDOF) system subjected to ug (t):

Dn + 2ζnωnDn + ω2nDn = −ug(t). (8)

The floor displacements, u(t), can be expressed by substituting Eq.(7) into Eq. (4), so that

u(t) =N∑n=1

ΓnφnDn (t) . (9)

Making use of Eq. (6), the effective modal mass, M∗n , and theeffective modal participating mass ratio for the nth mode, αn, canbe defined as

M∗n = LnΓn (10)

αn =M∗nM∗

(11)

in which

Ln = φTnmi (12)

M∗ =N∑j=1

mj (13)

M. Poursha et al. / Engineering Structures 31 (2009) 591–599 593

whereM∗ is the totalmass of the structure obtained by summationof the lumped masses,mj, over all floor levels. An important resultcan be derived by pre-multiplying Eq. (3) by iT :

N∑n=1

M∗n =N∑j=1

mj. (14)

This result shows that the sum of the effective modal masses overall modes (N) is equal to the total mass of the building [8]. As aresult from Eqs. (11), (13) and (14), the summation of effectivemodal participating mass ratios over all modes is equal to unity.

3. Consecutive modal pushover procedure

The consecutivemodal pushover (CMP) procedure is presented,which can be used to estimate the peak response of inelastic struc-tures subjected to earthquake excitation. Some pushover analysesare employed in the CMP procedure since it is possible to use dif-ferent pushover analyses and to envelope the results [16]. The pro-cedure uses multi-stage and single-stage pushover analyses. Themulti-stage pushover analysis benefits from consecutive imple-mentation ofmodal pushover analyses, including a limited numberof modes, such that when one stage (onemodal pushover analysis)has been completely performed, the next stage (the next modalpushover analysis) begins with an initial structural state (stressand deformation) which is the same as the condition at the endof the previous stage. Consecutive modal pushover analyses arecarried out with force distributions using mode-shapes obtainedfrom eigen-analysis of the linearly elastic structure. Changes inthe modal properties of the structure are ignored when the struc-ture experiences nonlinear yielding under increasing lateral loadsduring pushover analysis. The number of modes in the consec-utive modal pushover analyses (i.e. the number of stages in themulti-stage pushover analysis) depends on the fundamental pe-riod, T , of the building structure. When the fundamental period ofthe building (with a moment-resisting frame system) is less than2.2 s, themulti-stage pushover analysis is carried out in two stages.For buildings with fundamental periods of 2.2 s or more, both two-and three-stage pushover analyses are used. The displacementincrement at the roof is obtained, in each stage ofmulti-stage anal-ysis, as the product of a factor and the total target displacement ofthe roof. This factor is determined from the initial modal proper-ties of the structure. The displacement increment, uri, at the roof inthe ith stage of multi-stage pushover analysis, is therefore calcu-lated as

uri = βiδt (15)

in which

βi = αi for the stages before the last stage (16)

and

βi = 1−Ns−1∑j=1

αj for the last stage (17)

where δt is the total target displacement at the roof, and Ns is thenumber of stages included in the multi-stage pushover analysis.Also, αi is the effective modal mass ratio for the ith mode, whichis derived from Eq. (11). Several different approaches can be usedto establish the total target displacement at the roof level. Thisdisplacement can be determined by using the capacity spectrummethod [2], the displacement coefficient approach [4,5], the N2method [15,16], or dynamic analysis of the structure [14,23,24,28].As demonstrated previously, in addition to multi-stage pushoveranalysis, single-stage pushover analysis is used in the procedure.The single-stage pushover analysis is performed separately with

a triangular or uniform load distribution. Finally, the seismicdemands are obtained by enveloping the peak responses derivedfrom the multi-stage and single-stage pushover analyses.The details of the CMP procedure are expressed as a sequence

of the following steps:

1. Calculate the natural frequencies, ωn, and the mode-shapes,φn. These properties are determined by eigen-analysis of thelinearly elastic structure for the first three modes. The mode-shapes are normalized so that the roof component of φn equalsunity (φrn = 1).

2. Compute s∗n = mφn, where s∗n shows the distribution ofincremental lateral forces over the height of the structure forthe nth stage of multi-stage pushover analysis.

3. Compute the total target displacement of the structure at theroof, δt .

4. The CMP procedure consists of single-stage and multi-stagepushover analyses. First apply the gravity loads and then per-form these displacement-control pushover analyses accordingto the following sub-steps:4.1 Perform the single-stage pushover analysis using aninverted triangular load pattern for medium-rise buildingsand a uniform force distribution for high-rise buildings,until the control node at the roof sways to the predefinedtotal target displacement, δt .

4.2 The second pushover analysis is a two-stage pushoveranalysis. In the first stage, perform the nonlinear staticanalysis, using the incremental lateral forces s∗1 = mφ1,until the displacement increment at the roof reaches ur1 =β1δt (Eq. (15); i = 1), where β1 = α1 (Eq. (16); i = 1).Then implement the second stage of analysis using theincremental lateral forces s∗2 = mφ2 until the displacementincrement at the roof equals ur2 = β2δt (Eq. (15); i = 2),where β2 = 1 − α1 (Eq. (17); i = 2). It is noted thatthe initial condition in the second stage of the two-stagepushover analysis is the same as the state at the last step ofanalysis in the first stage.

4.3 The third analysis is a three-stage pushover analysis. It isonly performed for buildings having a fundamental periodof 2.2 s or more. The first stage is exactly the same as thefirst stage of the two-stage pushover analysis describedabove. After the first stage, continue the nonlinear staticanalysis using the incremental lateral forces s∗2 = mφ2until the displacement increment at the roof reaches ur2 =β2δt (Eq. (15); i = 2) where β2 = α2 (Eq. (16); i = 2).Thereafter, perform the third (last) stage of the three-stagepushover analysis using the incremental lateral forces s∗3 =mφ3. The displacement increment at the roof at this stageis equal to ur3 = β3δt (Eq. (15); i = 3) where β3 =1−α1−α2(Eq. (17); i = 3). It is again noted that the initialcondition at each stage of the analysis is the same as thestate at the end of the analysis in the previous stage.

5. Calculate the peak values of the desired responses, such asdisplacements, storey drifts, and hinge plastic rotations, for thepushover analyses described above. The peak values resultingfrom the one-, two-, and three-stage pushover analyses aredenoted by r1, r2, and r3, respectively.

6. Calculate the envelope, r , of the peak responses as follows:

r = Max {r1, r2} T < 2.2 s (18)r = Max {r1, r2, r3} T ≥ 2.2 s (19)

showing that the seismic demand of the inelastic structurein the CMP procedure is obtained by enveloping the peakresponses resulting from the single- and multi-stage pushoveranalyses.

594 M. Poursha et al. / Engineering Structures 31 (2009) 591–599

Table 1Characteristics of the analyzed frames.

No. No. of storeys h (m) b (m) Seismic mass of floors (kg s2/m) PeriodsT1 (s) T2 (s) T3 (s)

S1 10 32 15 5440 1.697 0.605 0.347S2 15 48 15 5546 2.338 0.854 0.493S3 20 64 15 5600 3.092 1.135 0.670S4 30 96 15 5650 3.866 1.381 0.798

Fig. 1. Configuration of two-dimensional frames.

It is notable that the nonlinear behaviour of the structure de-pends on the loading path, and separation between the loading in-put and the structural response is not possible [21]. Consequently,modal pushover analyses must, in the CMP procedure, be carriedout consecutively in the order ofmodes, from the first to the higherones, as has been demonstrated in detail.

4. Description of the analytical models and assumptions

The structures considered were three-bay frames with fourdifferent heights of 10, 15, 20, and 30 storeys, covering a widerange of fundamental periods. All the frames had 5mbays. A storeyheight of 3.2 m was assumed throughout. The configuration of theframes is shown in Fig. 1. The dead and live loadswere equal to 650and 200 kg/m2 on the floor area, assuming the loadingwidth of theframes to be 5 m. The concentrated seismic masses were assumedto be equal at all floors of each structure and to consist of the deadload plus 20% of the live load. The lateral load-resisting system ofthe structures was a special steel moment-resisting frame (SMRF).More characteristics of the frames and their first three natural-vibration periods are listed in Table 1. All of the frames wereassumed to be founded on type ‘II’ firm soil of the Iranian seismiccode [27] (class C of NEHRP [6]), and located in the region of highestseismicity. The seismic effectswere determined in accordancewiththe requirements of the Iranian code of practice for the seismic-resistant design of buildings [27]. The frames were designedaccording to the allowable stress design method [1]. All buildingswere also designed to satisfy the drift criterion and the strong-column/weak-beam philosophy. P–∆ (second-order) effects wereincluded, but the panel zone size, strength, and deformation wereneglected. The sections of the beams and columns were assumedto be of the plate girder and box type, respectively. More detailsabout the members can be found in [25].The nonlinear behaviour of the structures occurred in discrete

hinges in the nonlinear static and dynamic analyses. Hinges weredefined at the ends of the frame members. Hinges based on theinteraction of the axial forces and bending moments were definedfor column members, whereas hinges based on bending momentswere assigned to beams. Fig. 2 shows the load–deformation curveof the hinges. The hinge properties and modelling parameters a, b,and c (Fig. 2) were specified according to FEMA-273 [4]. The post-yield slope was assumed to be 3% of the elastic slope. It is notedthat Q and Qy, in Fig. 2, are the generalized and yield componentloads, respectively.

Fig. 2. Generalized load–deformation curve for hinges (BSSC 1997).

5. Ground motion characteristics

Seven ground motions were selected from the strong groundmotion database of the Pacific Earthquake Engineering Research(PEER) Centre (http://peer.berkeley.edu). The selected groundmotions were far-field records, and corresponded to locationswhich were at least 12 km from a rupturing fault. Also, the soilat the site corresponds to NEHRP site class C. To ensure thatthe structures respond into the inelastic range when subjectedto ground motions, the records were scaled up to 0.7g . Morecharacteristics of the ground motion records used are given inTable 2. The elastic pseudo-acceleration and displacement spectra,together with the corresponding the mean spectra, are presented,for a 5% damping ratio, in Fig. 3. The mean spectra are shown by athicker line.

6. Types of analysis

To verify the proposed CMP procedure, elaborated in detailearlier, the procedure was performed, together with NL-RHA, forframes S1, S2, S3, and S4. The predicted results from the MPAprocedure are also presented for the purpose of comparison. TheMPA procedure was carried out for each ground motion. Theseismic responses were calculated for the medium-rise (i.e. 10 and15 storey) frames, including three modes, and for the high-rise(i.e. 20 and 30 storey) frames, including five modes. The meanvalue of the responses was then determined over the set of usedground motions. P–∆ effects were included in the CMP and MPAfor all modes. In the CMP procedure, the target displacement atthe roof was obtained as the mean of the maximum top floordisplacements resulting from the NL-RHA for the selected groundmotions. The target displacements were equal to 26.92, 33.05,38.27, and 61.26 cm for frames S1, S2, S3, and S4, respectively.The nonlinear response history analyses were performed using thenumerical implicit Wilson–θ time integration method, in whichthe stability and accuracy characteristics are determined by theparameter θ . This parameter was assumed to have a value of 1.4. Adamping ratio of 5% was considered for the first and third modesof vibration, in order to define the Rayleigh damping matrix. Theresponse resulting from each pushover procedure was comparedwith the mean value of the maximum seismic demands computedby rigorous nonlinear response history analyses. It is noted that thenonlinear version of the computer programSAP2000 [13]was usedto perform these analyses.

M. Poursha et al. / Engineering Structures 31 (2009) 591–599 595

Fig. 3. (a) Pseudo-acceleration spectra and (b) displacement spectra of the set of far-field records of ground motions, damping ratio = 5%.

Table 2List of the ground motions used.

No. Earthquake name Date Magnitude Station name Station number Component (deg) PGA (g)

1 Duzce, Turkey 1999/11/12 Ms (7.3) Lamont 1061 E 0.1342 Northridge 1994/01/17 Ms (6.7) LA - Baldwin Hills 24157 90 0.2393 Trinidad, California 1980/11/08 Ms (7.2) Rio Dell Overpass, FF 1498 270 0.1474 Victoria, Mexico 1980/06/09 Ms (6.4) Cerro Prieto 6604 45 0.6215 Hollister 1986/01/26 Ml (5.5) SAGO South - Surface 47189 295 0.096 Imperial Valley 1979/10/15 Ms (6.9) Parachute Test Site 5051 315 .2047 Morgan Hill 1984/04/24 Ms (6.1) Corralitos 57007 310 0.109

Fig. 4. Peak values of storey drift ratios derived from pushover analyses used in theCMP procedure and from NL-RHA for the 10-storey frame.

For a yielding structure, the occurrence of structural damage isclosely related to storey drift [9]. The reduction of drift protectsthe structural components and elements, as well as non-structuralcomponents which are sensitive to drift-induced damage. Toevaluate the structural performance, the hinge plastic rotationsare usually compared to the acceptability criteria specified inFEMA-273/356 [4,5]. For this reason, storey drift ratios (storeydrifts/height of the storey) and the hinge plastic rotations of theinternal beams at all floor levels are presented in the current paper.The errors in the seismic demands obtained from the approximatepushover procedures are also shown. Because floor displacementsare incapable of indicating structural damage [9], they are notdisplayed.

7. Discussion of the results

First, the implication of using multi-stage and single-stagepushover analyses in the CMP procedure is discussed in detail.As explained earlier, the seismic demands in the CMP procedureare obtained by enveloping the peak responses from the multi-stage and single-stage pushover analyses. Figs. 4 and 5 show the

Fig. 5. Peak values of storey drift ratios derived from pushover analyses used in theCMP procedure and from NL-RHA for the 20-storey frame.

peak storey drift ratios resulting from the multi-stage and single-stage pushover analyses in the CMP procedure, as well as fromNL-RHA, for the 10- and 20-storey frames. As could be expected,the single-stage pushover analysis with triangular or uniformload distribution controls the seismic demands only at the lowerstoreys. Fig. 4 provides evidence that the storey drifts derived fromthe two-stage pushover analysis for the 10-storey frame are closeto the results of NL-RHA for the five upper storeys. As can be seenfrom Fig. 5, the seismic demands at the mid and upper storeys ofthe 20-storey frame are controlled by two- and three-stage modalpushover analyses, respectively, because the higher modes in theconsecutivemodal pushover analyses strongly affect the responsesat the mid and upper storeys, contrary to the lower storeys. Thesame trend can be demonstrated for the other seismic demands.Shown in Fig. 6 are the storey drift ratios obtained by the MPA

and CMP procedures, together with NL-RHA, for the medium-rise(10- and 15-storey) frames. The errors in the storey drifts from thepushover procedures are displayed in Fig. 7. The figures illustratethat the MPA and CMP procedures produce satisfactory estimatesof storey drifts. The CMP provides better estimates than the MPAat some upper storeys, whereas the errors from the MPA are less

596 M. Poursha et al. / Engineering Structures 31 (2009) 591–599

(a) Storey drift ratios of the 10-storey frame. (b) Storey drift ratios of the 15-storey frame.

Fig. 6. Height-wise variation of the storey drifts for the 10- and 15-storey frames.

(a) Storey drift ratios of the 10-storey frame. (b) Storey drift ratios of the 15-storey frame.

Fig. 7. Errors in the storey drifts for the 10- and 15-storey frames.

than those from the CMP at some lower storeys. For example, inthe 15-storey frame, the storey drift ratios are underestimated byup to 11.7% and 25% at the upper eight storeys by the CMP andMPAprocedures, respectively. In the lower seven storeys, the errorsfrom theMPA and CMP procedures reach 5% and 14%, respectively.More detailed discussion about the results of the MPA procedureis available in [25].The height-wise variation of hinge plastic rotations and the

errors from pushover procedures for the medium-rise frames areshown in Figs. 8 and 9, respectively. The figures show that theMPA procedure fails to accurately predict hinge plastic rotationsat the upper floor levels of the 10- and 15-storey frames, whereasnoticeable improvement was achieved in the estimates of theserotations by the CMP procedure. The figures illustrate that the CMPprocedure is in closer agreement with the NL-RHA. As a result, thehinge plastic rotations obtained by the CMP procedure are moreaccurate than those obtained by the MPA procedure, especially atthe mid and upper floors. It is noted that the 100% errors resultingfrom the CMPprocedure at the last floor of the 10-storey frame andat the first floor of the 15-storey frame are ignored since the plasticrotations predicted by the CMP procedure are zero, whereas therotations obtained by NL-RHA are very small. At some lower floorlevels, the CMPprocedure occasionally provides better estimates ofplastic rotations than the MPA procedure, and vice versa. Also, theCMP procedure tends to slightly overestimate the plastic rotationof the hinges at some lower floor levels.From Figs. 10–13, consequences similar to those for the

medium-rise frames can be almost deduced for the high-rise (20-and 30-storey) frames. Figs. 10 and 11 provide evidence that theMPA and CMP procedures are accurate enough in predicting storeydrift ratios. As can be seen from Figs. 12 and 13, the CMPprocedure,

in general, provides substantially better estimates of hinge plasticrotations than the MPA procedure for high-rise frames. It isnoteworthy that a key aspect of the CMP procedure is the factthat modal pushover analyses are carried out continuously. Theconsecutive implementation of modal pushover analyses meansthat rotations of the plastic hinges are continuously accumulatedat the mid and upper floor levels during the modes of interest inthe multi-stage pushover analysis, whereas the MPA procedureattempts to estimate the total response quantities by combiningthe individual peak responses obtained separately for eachmode. The trend of the higher-mode influences and incrementalapplication of lateral forces in the multi-stage pushover analysisof the CMP procedure indicate a significant improvement incomparison with the MPA procedure in predicting hinge plasticrotations at the mid and upper floor levels. However, the pushoverprocedure suffers from the limitation that it is unable to take intoaccount the cumulative rotation of hinges due to cyclic hystereticbehaviour [21]. It is noted that the results derived from the CMPprocedure may be occasionally inaccurate at some mid or lowerfloors of high-rise frames inwhich the plastic rotations obtained byNL-RHAare small. Some large errors in hingeplastic rotations (suchas in the case of the last floor of frame S3) are also not importantsince the rotations computed by NL-RHA are very small.As can be seen from Figs. 6, 8, 10 and 12, the height-wise

distribution of storey drifts and hinge plastic rotations derivedfrom the CMP procedure is more similar to the distributionobtained by the benchmark solution (NL-RHA) than to thatobtained by the MPA procedure. This achievement by the CMPprocedure ismore remarkable for the plastic rotations of the hingesin comparison with the MPA procedure.

M. Poursha et al. / Engineering Structures 31 (2009) 591–599 597

(a) Hinge plastic rotations of the 10-storey frame. (b) Hinge plastic rotations of the 15-storey frame.

Fig. 8. Height-wise variation of the hinge plastic rotations for the 10- and 15-storey frames.

(a) Hinge plastic rotations of the 10-storey frame. (b) Hinge plastic rotations of the 15-storey frame

Fig. 9. Errors in the hinge plastic rotations for the 10- and 15-storey frames.

(a) Storey drift ratios of the 20-storey frame. (b) Storey drift ratios of the 30-storey frame.

Fig. 10. Height-wise variation of the storey drifts for the 20- and 30-storey frames.

The locations of the plastic hinges for the 15- and 20-storeyframes are shown in Figs. 14 and 15, respectively. They wereobtained by the MPA and CMP procedures, together with NL-RHA. The results of the MPA procedure, shown in the figures,were obtained by using the mean spectrum of the selected groundmotions. As shown in the figures, the MPA and CMP proceduresare almost able to identify the yielding of beams in the upperfloor levels. The CMP procedure is nearly able to recognize yieldingat most of the floors, but it fails to predict yielding at somelocations. It was shown in the previous investigation [25] thatFEMA force distributions [4] are unable to identify the yielding ofbeams at the upper floor levels of tall frames, in which higher-mode contributions to the response are important. It has also

been demonstrated that the seismic demands are considerablyunderestimated in the upper storeys of tall frames through theFEMA force distributions. It can therefore be concluded that, inthe case of tall building structures, the CMP procedure providessubstantially better estimates of seismic response than the FEMAforce distributions.

8. Conclusions

To take into account higher-mode effects in pushover analysisfor estimating the seismic demands of tall building structures, theconsecutive modal pushover (CMP) procedure has been proposed.The procedure employs multi-stage and single-stage pushover

598 M. Poursha et al. / Engineering Structures 31 (2009) 591–599

(a) Storey drift ratios of the 20-storey frame. (b) Storey drift ratios of the 30-storey frame.

Fig. 11. Errors in the storey drifts for the 20- and 30-storey frames.

(a) Hinge plastic rotations of the 20-storey frame. (b) Hinge plastic rotations of the 30-storey frame.

Fig. 12. Height-wise variation of the hinge plastic rotations for the 20- and 30-storey frames.

(a) Hinge plastic rotations of the 20-storey frame. (b) Hinge plastic rotations of the 30-storey frame.

Fig. 13. Errors in the hinge plastic rotations for the 20- and 30-storey frames.

analyses. In the multi-stage pushover analyses, modal pushoveranalyses are conducted consecutively with force distributions,using mode-shapes derived from the eigen-analysis of the linearlyelastic structure. The single-stage pushover analysis is performedwith a triangular or uniform load distribution. The seismicdemands are then determined by enveloping the peak responsesresulting from themulti-stage and single-stage pushover analyses.The former analyses control the seismic demands in the mid andupper storeys, whereas the latter analysis controls the responsesin the lower storeys of tall buildings.Using the CMP procedure, storey drifts can be estimated with

acceptable accuracy. At some (upper) storeys, the storey driftsare more accurately estimated by the CMP than by the MPA,whereas the MPA yields better estimates of storey drift than theCMP at some other (lower) storeys of the analyzed tall frames. A

significant improvement has been achieved in estimating the hingeplastic rotations through the CMP procedure. The plastic rotationsproduced by the CMP procedure are substantially better than thoseobtained by the MPA procedure, especially at the mid and upperfloor levels, when compared to NL-RHA. The improvement in theCMP procedure results from the consecutive implementation ofmodal pushover analyses, so that rotations of the plastic hingesare continuously accumulated at the mid and upper floor levelsduring the modes of interest in the multi-stage pushover analysis.Also, the height-wise distribution of the hinge plastic rotationsproduced by the CMP procedure is, in general, more similar to thatobtained by the benchmark solution (NL-RHA) than that producedby the MPA procedure. The CMP procedure is more accurate thanthe FEMA load distributions in predicting the seismic demands oftall buildings.

M. Poursha et al. / Engineering Structures 31 (2009) 591–599 599

Fig. 14. Locations of the plastic hinges yielded by several analyses for the 15-storeyframe.

Fig. 15. Locations of the plastic hinges yielded by several analyses for the 20-storeyframe.

In order to confirm these conclusions with greater certainty,the CMP procedure should be verified for different lateral force-resisting systems, reinforced concrete (RC) buildings, and a varietyof ground motion sets. Work in this area is underway.

Acknowledgements

The first author wishes to express his deep thanks to ProfessorPeter Fajfar for having given him the opportunity to stay anddo research at the University of Ljubljana. He has also benefitedfrom valuable discussions with Professor Fajfar. Comments fromanonymous reviewers helped to improve the manuscript.

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