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Engineering Structures 23 (2001) 407424

www.elsevier.com/locate/engstruct

Static pushover versus dynamic collapse analysis of RC buildings

A.M. Mwafy, A.S. Elnashai *

Department of Civil and Environmental Engineering, Imperial College, Imperial College Road, London SW7 2BU, UK

Received 2 February 2000; received in revised form 23 May 2000; accepted 26 May 2000

Abstract

Owing to the simplicity of inelastic static pushover analysis compared to inelastic dynamic analysis, the study of this technique

has been the subject of many investigations in recent years. In this paper, the validity and the applicability of this technique areassessed by comparison with dynamic pushover idealised envelopes obtained from incremental dynamic collapse analysis. Thisis undertaken using natural and artificial earthquake records imposed on 12 RC buildings of different characteristics. This involvessuccessive scaling and application of each accelerogram followed by assessment of the maximum response, up to the achievementof the structural collapse. The results of over one hundred inelastic dynamic analyses using a detailed 2D modelling approach foreach of the twelve RC buildings have been utilised to develop the dynamic pushover envelopes and compare these with the staticpushover results with different load patterns. Good correlation is obtained between the calculated idealised envelopes of the dynamicanalyses and static pushover results for a defined class of structure. Where discrepancies were observed, extensive investigationsbased on Fourier amplitude analysis of the response were undertaken and conservative assumptions were recommended. 2001Elsevier Science Ltd. All rights reserved.

Keywords: Pushover analysis; Timehistory collapse analysis; RC buildings; Fourier amplitude analysis

1. Introduction

Inelastic timehistory analysis is a powerful tool forthe study of structural seismic response. A set of care-fully selected ground motion records can give an accur-ate evaluation of the anticipated seismic performance ofstructures. Despite the fact that the accuracy andefficiency of the computational tools have increased sub-stantially, there are still some reservations about thedynamic inelastic analysis, which are mainly related toits complexity and suitability for practical design appli-cations. Moreover, the calculated inelastic dynamicresponse is quite sensitive to the characteristics of theinput motions, thus the selection of a suite of representa-tive acceleration timehistories is mandatory. Thisincreases the computational effort significantly. Theinelastic static pushover analysis is a simple option forestimating the strength capacity in the post-elastic range.The technique may be also used to highlight potentialweak areas in the structure. This procedure involves

* Corresponding author. Fax: +44 207 594 6053.

E-mail address:[email protected] (A.S. Elnashai).

0141-0296/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.

PII: S0 1 4 1 - 0 2 9 6 ( 0 0 ) 0 0 0 6 8 - 7

applying a predefined lateral load pattern which is dis-tributed along the building height. The lateral forces arethen monotonically increased in constant proportion witha displacement control at the top of the building until acertain level of deformation is reached. The target topdisplacement may be the deformation expected in thedesign earthquake in case of designing a new structure,or the drift corresponding to structural collapse forassessment purposes. The method allows tracing thesequence of yielding and failure on the member and thestructure levels as well as the progress of the overallcapacity curve of the structure.

The static pushover procedure has been presented anddeveloped over the past twenty years by Saiidi andSozen [1], Fajfar and Gaspersic [2] and Bracci et al. [3],among others. The method is also described and rec-ommended as a tool for design and assessment purposesby the National Earthquake Hazard Reduction ProgramNEHRP (FEMA 273) [4] guidelines for the seismicrehabilitation of existing buildings. Moreover, the tech-nique is accepted by the Structural Engineers Associ-ation of California SEAOC (Vision 2000) [5] amongother analysis procedures with various level of com-plexity. This analysis procedure is selected for its

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408 A.M. Mwafy, A.S. Elnashai / Engineering Structures 23 (2001) 407424

applicablity to performance-based seismic designapproaches and can be used at different design levels toverify the performance targets. Finally, it is clear fromrecent discussions in code-drafting committees in Eur-ope that this approach is likely to be recommended infuture codes.

The technique has been evaluated in several previousstudies [610], to some extent, with different emphasis.In most of the previous work, only comparative studiesbetween dynamic and static pushover analysis have beenassessed at certain loading levels, i.e. design level, or atequal top displacement (roof displacement from push-over equal to the maximum dynamic roof displacement).The results have been presented mainly in terms of glo-bal quantities, i.e. deformations, calculated hystereticenergy and structural damage indices. The main aim ofthis paper is to develop complete pushover-like loaddisplacement curves from incremental dynamic analysisup to collapse for a range of structural configurationsrepresenting the most common types of RC building,including different structural systems, building heights,design acceleration and level of ductility. The dynamicpushover envelopes are then compared with the forcedeformation curves obtained from inelastic static push-over analysis considering different lateral loading pat-terns. The procedure offers an opportunity for full com-parisons between the two methods of analysis up toultimate collapse.

2. Description of the buildings

In order to achieve the aforementioned objectives,twelve RC buildings are considered, split into threegroups: sets of four 8-storey irregular frame, four 12-storey regular frame and four 8-storey dual frame-wallstructures. Within each group, combination of twodesign ground accelerations (0.15 and 0.30 g) and threedesign ductility classes (High, Medium and Low) leadto the four cases mentioned above. The selection of fourcases for each configuration is motivated by the desireto compare the performance of structures design accord-ing to a ductility class set of rules but for differentground acceleration and for the same ground acceler-ation but different ductility class rules. The value of theforce reduction factor (behaviour factor q in EC8 andresponse modification factor R in UBC) increases andrigorous standards on member detailing requirements areimposed for higher ductility classes. Table 1 shows thedefinition of the set of structures under considerationwhere the elastic force reduction factors used in thedesign as well as the observed elastic fundamental per-iod, obtained from elastic free vibration analyses, arealso given.

Each building has been designed and detailed inaccordance with Eurocode 8 [11], Parts 1-1 to 1-3, as a

representative of a seismic design code applicable tomore than one country. While the second and thirdgroups are regular in plan and in elevation, the firstgroup exhibits two sources of irregularity in elevation.The first storey has a greater height than the remainingones and severance at the first storey of some intermedi-

ate columns, which are supported by long span beams.The geometric characteristics of the structures are illus-trated in Fig. 1.

The overall plan dimensions of the configurations con-sidered are 15 m20 m. The total heights are 25.5, 36and 24 m for groups 1, 2 and 3, respectively, with equalstorey heights of 3 m except the first storey of group 1,which is 4.5 m high. While the lateral force resistingsystem for groups 1 and 2 is moment frames, group 3possesses both a central core extending over the fullheight and moment frames on the perimeter. The floorsystem is solid slab in groups 1 and 2, and a waffle slabin group 3. Live loads and loading from floor finishesand partitions are both assumed to be 2.0 kN/m2. Allbuildings are assumed to be founded on medium soiltype B of EC8 (firm). The cross section capacities havebeen computed by considering a characteristic cylinderstrength of 25 N/mm2 for concrete and a characteristicyield strength of 500 N/mm2 for both longitudinal andtransverse steel. More details regarding member crosssection sizes and reinforcements are given in Fardis [12].

3. Modelling approach and assumptions

The inelastic analyses have been performed using theadaptive static and dynamic structural analysis programADAPTIC, a program developed at Imperial College[13] for the nonlinear analysis of steel, reinforced con-crete and composite structures under static and dynamicloading. The program utilises the layered fibreapproach for inelastic RC frame analysis and has thecapability of predicting the large displacement responseof elastic and inelastic plane and space frames. It hasalso the feature of representing the spread of inelasticitywithin the member cross section and along the memberlength. It is widely accepted that this technique is moreaccurate than the pointhinge models mainly used inmany other programs, especially when large axial forcevariations exist. The program has been verified else-where [1416].

To accurately predict the inelastic seismic response ofthe structure with sufficient accuracy, due care has beengiven to create detailed and efficient models of the struc-tures, taking into account all necessary geometric andstrength characteristics of columns, beams and beamcolumn connections. Towards minimising the compu-tational requirements and the volume of input and outputdata to be handled, an effort was made to select powerfultwo-dimensional models that can provide, with appropri-

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Table 1

Definition of the structural systems under analysis

Reference No. of storeys and structural DuctilityGroup Design acc. (g) Force red. factor Elas. fund. period (s)

name system class

1 IF-H030 8-storey irregular frame High 0.30 4.00 0.674

IF-M030 Medium 3.00 0.654IF-M015 Medium 0.15 3.00 0.719

IF-L015 Low 2.00 0.723

2 RF-H030 12-storey regular frame High 0.30 5.00 0.857

RF-M030 Medium 3.75 0.893

RF-M015 Medium 0.15 3.75 0.920

RF-L015 Low 2.50 0.913

3 FW-H030 8-storey regular frame-wall High 0.30 3.50 0.538

FW-M030 Medium 2.625 0.533

FW-M015 Medium 0.15 2.625 0.592

FW-L015 Low 1.75 0.588

Fig. 1. Plane and cross sectional elevation of the buildings: (a) 8-storey irregular frame buildings; (b) 12-storey regular frame buildings; (c) 8-

storey regular frame-wall buildings.

ate selection of parameter values, acceptable represen-tation of the cyclic inelastic behaviour on member andstructure levels, while guaranteeing numerical stability.The choice of two-dimensional modelling may be also

justified in the light of satisfying basic code requirements

for such type of modelling. Two-dimensional analysesare undertaken in one direction only (global X-directionof frame structures and global Z-direction of frame-wallones). This is supported by the fact that conservativeresponse parameters will be obtained as a result of the

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domination of gravity loads in long beam spans in theframe structures. On the other hand, the critical strainand shear criteria are expected to occur in the couplingbeams in the dual structural systems. Taking advantageof symmetry, only the interaction of two distinct frames(one internal and another external) is considered. Both

lateral load-resisting frames are assembled using anoverlay approach, which is illustrated in Fig. 2 for theirregular frame structure. The two frames are coupledappropriately with regard to translational and rotationaldegrees of freedom by 2D joint elements to meet theassumption of infinite in-plane stiffness of the slab inthe normal direction.

The structural mesh utilises three elements for beamcolumn members, the lengths of which are determinedon the basis of the critical member lengths. Theselengths are determined according to EC8 provisions fordifferent ductility classes. The ends of horizontalelements within the beamcolumn joints are consideredrigid. Consequently, two elements are added to eachbeam at its extremities. Furthermore, shear spring con-nection elements are introduced to represent the shearstiffness of the beamcolumn connection. To simplifycalculations of the shear stiffness of the joint, the forcedeformation relationship for both concrete and steelreinforcement within the joint is assumed to be linearelastic. Despite the simplicity of the joint modelling, glo-bal structural response obtained have been extensivelycompared and checked with analyses performed by Sal-vitti and Elnashai [17] and Panagiotakos and Fardis [18].These show a good conceptual agreement with the cur-

rent modelling results since the drift values are on thewhole higher than the values by Salvitti and Elnashai[17] where no provision for beamcolumn connectionbehaviour was made. For the sake of brevity, only someresults of the comparison are shown in Fig. 3. The resultsof the current study, for two of the 12-storey frame

Fig. 2. The overlay technique considered and description of the beamcolumn joint modelling.

buildings, are between results of the rigid beamcolumnjoint modelling of the former and the flexible, one mem-ber lumped plasticity modelling, of the latter where barslip effects within the joints and member shear defor-mations are considered.

Modelling of the core is achieved by making use of

two flexural elements, for each wall at each storey, inorder to account for splicing of bars at mid-storey height.The elements are located at the centroid of the core U-shaped cross section and connected with beams at eachstorey level using two rigid links. In addition, fiveelements are used to represent each coupling beam, withbidiagonal reinforcement represented by vectorial resol-ution of the inclined reinforcement area along the longi-tudinal and transverse directions. The same method isutilised to represent the bidiagonal shear reinforcementin some other beams and in the lower two storeys of thecore of the FW-H030 and FW-M030 building.

Reinforced concrete column-section and T-section areutilised for modelling of columns and beams, respect-ively. Both sections, taken from ADAPTIC library,allow the geometrical definition of the section as wellas that of the confined concrete region within it. Takinginto account the available cross sections in ADAPTIClibrary, a reasonable approximation is made to replacethe original U-shaped section of the core of the frame-wall structures by a T-section, with the same stiffnessproperties. The approximation may be justified in thelight of the two-dimensional modelling which neglectstorsion and the regularity of the structure both in termsof stiffness and strength. Reinforcement patterns are

varied for each section as a function of stirrup spacingin accordance with those specified in the design. Con-finement factors are evaluated as described in Eurocode8, and varied along the member length according to thearrangement of transverse reinforcements. The effectiveslab width participating in beam deformation is taken as

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Fig. 3. Verification of modelling assumptions (a) RF-H030 building and (b) RF-L015 building (average for four artificial accelerograms).

the beam width plus 7% of the clear span of the struc-tural member on either side of the web. Horizontal andvertical structural members including core walls aremodelled using two-dimensional cubic elasto-plasticbeamcolumn elements, where a cubic shape function isused for the transverse displacement [13]. This formu-lation is intended to represent short lengths of RCelements, consequently, axial strain is assumed to beconstant along the element length. The numerical inte-gration of the governing equations for this element isperformed over two Gauss sections, which have a fixedposition within the element length. The inelastic

response of the cross section is assembled from contri-butions of individual layers for which inelastic cyclicmaterial constitutive relationships are applied. The cubicelasto-plastic elements are combined with material mod-els for concrete, which account for active confinementand reinforcing steel with nonlinear hardening. On theconcrete side, the uniaxial constant confinement concretemodel, Martinez-Rueda and Elnashai [19], has beenchosen. For steel, the advanced multisurface steel modelfor cyclic plasticity, which defines the stressstrainresponse of steel in terms of a series of cubic poly-nomials, Elnashai and Izzuddin [20], is utilised. Theparameters used in the material models are the meanvalues.

4. Load pattern and seismic action

According to the data used in the design [12], a liveload Q=2.0 kN/m2 is considered to calculate the totalgravity loads on the frames, which is applied as pointloads at nodes. Using the appropriate coefficients fromEC8, the vertical loads are combined with seismicactions in a combination of 1.0G+0.15Q+EL for all sto-ries except the top floor, where it was taken equal to

1.0G+0.30Q+EL. To account for inertia effects duringdynamic analysis, masses are calculated in a mannerconsistent with the gravity loading combinations and arerepresented by lumped 2D mass elements.

Due to the fact that the lateral force profiles in staticpushover analyses will influence the structural response,three different load patterns have been utilised to rep-resent the distribution of inertia forces imposed on thebuilding. The first shape is calculated as SRSS combi-nations (for the first three modes) of the load distri-butions obtained from modal analyses of the buildings.The choice of this load shape is made to take into con-

sideration the anticipated effect of higher modes ofvibrations for moderate long period and irregular struc-tures (the 12-storey and the 8-storey frame buildings),as well as for buildings with hybrid lateral resistancesystems (the 8-storey frame-core structures). The designcode lateral load pattern and a uniform load distributionshape have been also utilised. The latter represents thelateral forces that are proportional to the vertical distri-bution of the mass at various levels. On the other hand,the code lateral load shape represents the forces obtainedfrom the predominant mode of vibration. The use of theuniform load shape may be justified in the light of apossible soft storey mechanism of the 8-storey irregularbuildings. If this mechanism occurs the response will becontrolled by a large drift in the first storey. Therefore,this load distribution may give better predictions of theoverall response. The inverted triangular (code) and therectangular (uniform) load shapes also represent theextreme cases from the linear distribution point of view.

The shape of the lateral load should be selected onthe light of anticipated changes in inertia forces as thestructure moves from the elastic to the plastic phases.Ideally, this shape should be modified with the changesin inertia forces during the actual earthquake. Thesechanges mainly depend on the characteristics of both the

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record and the structure. Several trials [2,3] have beenmade to permit of changes in inertia forces with the levelof inelasticity through the use of adaptive load patterns.The underlying approach of this technique is to redistrib-ute the lateral load shape with the extent of inelasticdeformations. The load shape is suggested to be redis-

tributed according to the global displacement shape, thelevel of storey shear demands or a combination of modeshapes obtained from secant stiffnesses. This redistri-bution is performed at each time step, which leads to asubstantial increase in the computational effort. More-over, the pushover analysis has not been widely estab-lished as yet in the design office environment. Therefore,for common types of building the need for more taxingapproaches is by no mean fully established. The variableload distribution option may be appropriate for specialand long period structures, despite that eminence of thistechnique has not been confirmed yet [9,10]. On thisbasis, the aforementioned fixed load distribution shapeshave been utilised for the current study. It is also worthmentioning that the NEHRP (FEMA 273) guidelines rec-ommend utilising fixed load patterns with at least twoload profiles. The first shape should be the uniform loaddistribution and the other is the code profile or the loadshape obtained from multimodal analyses. The code lat-eral load distribution is allowed if more than 75% of thetotal mass participates in the predominant mode.

Timehistory analyses employ four artificially gener-ated 10-s duration acceleration records, referred to asArt-rec1 to Art-rec4, as well as two natural records. Theartificial accelerograms were generated to fit the Euroc-

ode 8 elastic response spectrum for medium soil classas shown in Fig. 4 for a PGA=0.3 g. The use of theartificial accelerograms is in order to allow effectivecomparisons and calibrations with the design code.Moreover, the effect of the vertical component of theseismic excitation is worthy of consideration [21], parti-cularly for the irregular frame structures where the

Fig. 4. Acceleration spectra for the artificial accelerograms (5%

damping).

planted columns are supported by long span beams.Towards this end, two natural ground motions have beenselected in terms of the V/H ratio (peak vertical-to-hori-zontal acceleration). The Kobe (Hyogo-ken Nanbu atKobe University, Japan, 1995) and the Loma Prieta(Northern California at Saratoga Aloha Ave, USA,

1989) earthquakes are employed and applied with andwithout the vertical components, giving two analyses foreach record. However, for the sake of brevity, results ofthe effect of vertical ground motion on the seismicresponse are not presented herein. Comprehensiveresults of this study are given elsewhere [22]. Character-istics of the records that have been used are given inTable 2, while their acceleration response spectra for 5%damping are shown in Fig. 5.

In order to apply the outlined procedure for the evalu-ation of dynamic collapse envelopes, scaling of the rec-ords utilised is frequently required. The technique of sca-

ling earthquake records to possess equal values ofspectrum intensity was based on a proposal by Housner[23]. The spectrum intensity is defined as the area underthe pseudo-velocity spectrum between certain periodlimits. It is suggested in the current study to modify thelimits employed in the original method (between 0.1 and2.5 s) to be between 0.8 Ty and 1.2 T2D, where Ty andT2D are the inelastic periods of the structure at globalyielding and at twice the design ground acceleration,respectively. This follows the proposal of Martinez-Rueda [24,25], modified for the range used here. There-fore, the normalisation factor for an accelerogram (n) isequal to the ratio SI

c/SI

n. Where, SI

c and SI

n are the

areas under the code velocity spectrum and the velocityspectrum of the scaled accelerogram, respectively. SIcand SInare calculated between periods of 0.8 Ty and 1.2T2D, as explained above.

It is also worth mentioning that there is no need touse the aforementioned scaling method with the artificialaccelerograms since they are already spectrum-compat-ible. Hence, the four artificial records are scaled accord-ing to their PGA. The buildings are analysed first underthe artificial records at different PGA levels and therecorded top response time history is utilised to obtainthe inelastic periods Ty and T2D of each building from

Fourier analyses (average for four artificial records). Thescaling factors are then calculated for the longitudinalcomponent of natural records and used for scaling the

accelerograms up to collapse. The factors used to scalethe longitudinal earthquake component are also used toscale the vertical component of the motions, whenemployed, to keep the V/H ratio constant. Table 3 showsthe average normalisation factors to ground acceleration0.30 g for each of the three groups of buildings. Finally,it should be noted that the quoted values of PGA are not

of the natural or scaled records but rather multiples ofthe design ground acceleration.

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Table 2

Characteristics of records used in analysis

Earthquake Date Ms Station PGA (g) V/H No. of input runs

Horiz. Vert.

Kobe (Japan) 17/01/95 7.20 KBU 0.276 0.431 1.56 2Loma Prieta (USA) 18/10/89 7.17 SAR 0.319 0.349 1.09 2

Artificial Records Art-rec1, Art-rec2, Art-rec3,4

and Art-rec4

Fig. 5. Elastic spectra for the long. component of the natural records

(5% damping).

Table 3

Normalisation factors for ground acceleration 0.30 g

Earthquake IF-buildings RF-buildings FW-buildings Average

Artificial 0.30 0.30 0.30 0.30

Kobe (KBU) 0.54 0.61 0.56 0.57

Loma Prieta1.15 1.25 1.32 1.24

(SAR)

5. Collapse criteria and incremental dynamic

collapse dynamic pushover results

Three types of analyses have been performed usingthe structural models described earlier. Eigenvalueanalyses are conducted to determine the elastic periodsand the mode shapes of the buildings needed for calcu-lating the first lateral load profile of the static pushoveranalysis (combination of loads from modal shapes).Inelastic static pushover and dynamic analyses are thenperformed using the calculated lateral load shapes andthe seismic actions with increasing severity. The analy-ses are progressed until all the predefined collapse limitsare exceeded. In both static and dynamic analyses, per-manent loads are first applied and iteration to equilib-rium is performed. This is followed by applying the hori-zontal action (loads or ground acceleration). The analysis

is both inelastic and geometrically nonlinear. The largedisplacement formulation is an updated Lagrangianform, where convected member axes are used to derivemember deformations.

The criteria used for defining collapse are classifiedinto two groups; local and global criteria. Two failurecriteria on the member-level are applied: the ultimate

curvature, which is normally controlled by the maximumcompression strain at the extreme fibre of the confinedconcrete and shear failure in any structural member. Anempirical axial load-sensitive shear model capable ofproviding an experimentally verifiable estimate of shearsupply in RC members was proposed by Priestley et al.[26] and has been utilised in this study [27]. The codeshear supply model has also been employed after elimin-ating the design safety factors. On the structure level,three collapse criteria are chosen: a limit corresponds toa maximum inter-storey drift of 3% of the storey height,formation of a sidesway mechanism and reduction in lat-

eral resistance by considering the loaddisplacementcurve of the structure. Additionally, the criterion used todefine global yield threshold, which is essential for theproposed scaling method of the records, is selected asthe yield displacement of the equivalent elasto-plasticsystem with reduced stiffness evaluated as the secantstiffness at 75% of the ultimate load of the real system.The utilised shear models are implemented with othercollapse and yield criteria in a post-processing programconnected to ADAPTIC [28]. This post-processsor tracesthe shear supplydemand situation at each time step atboth ends of all members. It also performs the appropri-ate calculations to evaluate the local and global response

parameters of the structure and directly apply the selec-ted criteria.

The results of more than 1300 inelastic timehistoryanalyses were employed to perform regression analysesto obtain the dynamic pushover (ideal) envelope for eachof the twelve examined buildings. Figs. 68 depictdynamic response points and the fitted regression equa-tions of the response of the buildings subjected to theeight seismic actions considered for all limit states. Thefitted envelopes for the upper and lower response points,the number of analyses carried out, the design base shearand the correlation coefficient for each case are also

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Fig. 6. Dynamic collapse analysis results for the irregular frame structures.

shown. The actual response of the 8-storey irregularframe structures illustrated in Fig. 6 show how theresults of the eight seismic actions follow the same trendand shape the pushover envelopes without the need toapply curve fitting. This is clear from the correlationcoefficient values, which are almost invariably above0.9. It is worth mentioning that the number of timehis-tory analyses shown on each graph varies according tothe number of trials needed to identify the collapse andthe yield limits, as discussed above.

Concerning the 12-storey regular frame structures,Fig. 7 shows a higher scatter in the dynamic analysisresults of different ground motions than the results ofthe 8-storey irregular frame buildings. Moreover, thescatter for the two buildings designed for the higherdesign ground acceleration gives the impression of beinghigher than the other pair of results. The low correlationof the former and the high correlation of the latter arereflected in the correlation values which are equal to 0.69and 0.66 for the first pair and 0.93 and 0.88 for thesecond one. It should be noted that the main differencebetween the two pairs is in the longitudinal and trans-verse reinforcement of the structural members, while the

dimensions of the cross-sections for this group are thesame except a slight changes, mainly in the beams cross-section width (from 0.35 m for the first pair to 0.30 mfor the second one). In spite of the aforementionedobservations, the difference in scatter between the higherand the lower design ground acceleration pair diminisheswhen calculating the difference between the lower andthe upper response envelope for each case (quotient ofminimum and maximum strength for the eight records).This value is equal to 0.69 and 0.72 for the 0.30 g designground motion pair and 0.76 and 0.74 for the other pair.This is more consistent since the calculated inelastic per-iods, which are the main cause of the different response,are very close for the four buildings, as subsequentlydiscussed. Finally, the difference in the correlationvalues between the two pair of buildings can only be

justified in the light of the lower number of runs (orresponse points) needed to achieve yield and collapsefor each pair. This number is equal to 97 and 105 forthe first pair and 60 and 73 for the second one.

The high sensitivity to changes in the input motionobserved in the 12-storey frame buildings are alsoreflected, to a lesser extent, in the 8-storey frame-wall

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Fig. 7. Dynamic collapse analysis results for the regular frame structures.

group. Fig. 8 illustrates the results of the timehistoryanalyses for this group. At collapse limit state, a scatteris observed for values of VMin/VMax shown in Table 4.A higher hardening stiffness is also observed for the 0.30g design ground acceleration pair compared to the otherpair. This is not observed in the other two groups ofbuilding. Previous analytical investigations [29,30] haveindicated that base shear demands of wall structures aresensitive to higher mode effects. Once a plastic hingehas formed at the base of the wall, higher mode effectscan considerably amplify the base shear as well as theshear at each storey level. The results shown in Fig. 8confirm that such amplification may occur and could belarge. It is also worth mentioning that the thickness ofthe core-walls for the higher design ground accelerationpair is 0.35 m, compared to 0.25 m for the other pair.This causes an increase in the mass at each storey levelfor the former, hence higher amplification of base sheardemand. Alongside the high initial stiffness of this pair,the difference in response between the two pairs ofbuilding shown in Fig. 8 can be explained.

6. Contribution of the elongated period to the

seismic response

The scatter observed for some buildings is mainly inthe post-elastic range, and is associated with the spreadof yielding and member failure throughout the structure.Subsequently, the stiffness of the structure decreases, thefundamental period elongates and the distribution of theinertia forces along the building undergoes continuouschange. To provide insight into the response of theinvestigated buildings, extensive analyses in the fre-quency domain (Fourier analyses) of the accelerationresponse at the top have been conducted to identify thepredominant inelastic period of each building under con-sideration. Fig. 9 illustrates the calculated periods(average for the eight seismic actions) at the design andtwice the design ground acceleration, along with theelastic period for each building calculated from eigen-value analyses.

It is clear to what extent the fundamental periods ofthe buildings are elongated as a result of the spread of

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Fig. 8. Dynamic collapse analysis results for the frame-wall structures.

Table 4

Observed response at global collapse for the eight records

Group Reference name Roof disp. (mm) Min/Max Base shear (kN) VMin/VMax Storeya

Max Min Mean VMax VMin Mean

1 IF-H030 613 503 542 0.82 11,614 9918 10,567 0.85 1, 2, 3, 5

IF-M030 635 500 570 0.79 13,930 12,713 13,146 0.91 1, 2, 4, 5

IF-M015 492 381 449 0.77 7699 6663 7123 0.87 1, 4, 5

IF-L015 590 380 465 0.64 9229 8102 8685 0.88 1, 2, 4, 5

2 RF-H030 690 580 625 0.84 15,647 11,568 13,689 0.74 2, 4, 8, 9

RF-M030 796 611 684 0.77 16,278 12,076 13,990 0.74 4, 5, 8, 9

RF-M015 735 630 681 0.86 9743 9234 9453 0.95 5, 6, 8

RF-L015 785 607 694 0.77 12,735 11,009 11,972 0.86 4, 5, 9

3 FW-H030 643 599 631 0.93 20,821 15,520 17,849 0.75 2, 3, 6, 7

FW-M030 660 576 625 0.87 23,300 18,123 20,738 0.78 2, 5, 7

FW-M015 643 590 621 0.92 12,724 8769 10,642 0.69 2, 3

FW-L015 652 598 626 0.92 16,153 11,604 13,425 0.72 2, 3, 7

a Location where interstorey drift collapse criterion is observed for the eight ground motions.

cracks and yielding. The average elastic periods for thethree groups of building are 0.69, 0.90, and 0.56 s,respectively. On the other hand, the calculated inelasticperiods at the design and twice the design ground accel-eration are (1.301.46), (1.651.80), and (0.811.00) s,

respectively. It is observed that the average percentageof elongation in the period is (100%), (90%), and (60%).The percentage increase is clearly related to the overallstiffness of the structural system of the building. Themaximum calculated elongation is recorded in the most

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Fig. 9. Elastic and inelastic (at the design and twice the design ground acceleration) predominant response periods of the buildings average

for the eight seismic actions.

flexible system, where the first storey can be consideredas a soft storey; whereas the minimum elongation isobserved in the stiff frame-wall structural system. Theresults point towards an important conclusion, employ-ment of elastic periods of vibration in estimating designforces leads to high levels of overstrength (ratio ofactual-to-required strength). Moreover, they lead to non-uniform safety margins for different structural systems.

To facilitate the comparison between the input acce-lerograms utilised in this study in terms of the frequencycontent, the records are scaled to ground accelerationequal to 0.30 gand used to obtain the Fourier spectrumfor each record. The normalisation factors used are theaverage of the scaling factors utilised to perform the col-lapse analysis for the three groups of buildings, as shown

in Table 3. The Fourier amplitude spectra for the acceler-ation records (one of the utilised artificial records, Art-rec1, and the horizontal component of the two naturalrecords) are shown in Fig. 10. The average inelastic per-iod for each group of building is also shown on thegraphs. It is clear that the input motions, with the excep-tion of the Kobe (KBU) record, have high amplitude thatmay amplify the effect of the second mode of vibrationfor structures with period between 0.35 and 0.50 s. Theamplitude in this period range is higher than the ampli-tude corresponding to the fundamental period of thethree groups of structures. This is one of the reasons forthe scatter in the results of the 12-storey regular frameand the 8-storey frame-wall structures. It is verified thatthe high response points in Figs. 7 and 8 are for theartificial and Loma Prieta (SAR) records, while the lowresponse is for the Kobe (KBU) record. This may also

justify obtaining a higher maximum base shear corre-sponding to almost an identical top deflection whenapplying the same ground motion with higher PGA. Fur-thermore, the Fourier spectral ordinate corresponding tothe inelastic fundamental period of the buildings can alsobe utilised to justify the scatter in the results of thesecond and the third group of buildings. For the 12-sto-rey buildings, the average inelastic fundamental period

is about 1.75 s, which corresponds to high amplificationin the Loma Prieta (SAR) record only. This alsoaccounts for the high response of the 12-storey framestructures when subjected to the latter record. The sameapplies, to a lesser extent, to the artificial and Loma Pri-eta (SAR) records when imposed on the frame-wallstructures (inelastic period 0.91 s), compared to the Kobe(KBU) record. For this reason the observed scatter forthis group is less than the 12-storey buildings. On theother hand, the ordinates of the spectra correspond to theinelastic period of the 8-storey irregular buildings areequivalent, hence the high correlation for this group.

7. Inelastic static-to-collapse static pushover

analyses

Following the success in obtaining the incrementaldynamic response envelopes for the twelve buildingsunder investigation, inelastic static pushover analyses areperformed to assess the applicability of the technique(for different load distributions) in predicting the overalldynamic response of structures. Figs. 1113 illustrate thebase shear vs top displacement plots for the three lateralload profiles utilised along with the incremental dynamicenvelopes for the twelve buildings. The dynamic push-over curve for each case is shown in the form of theupper and lower response envelope as well as the bestfit of the timehistory analysis results. The global yieldand collapse thresholds are also shown. It should bepointed out that it was decided to choose only one globalyield limit from the limits obtained from the four push-over envelopes (the three static and the dynamic one).This is due to the need to unify and simplify obtainingthis limit, which is necessary for the suggested methodof scaling the input seismic actions explained earlier.The yield limit state obtained from the static pushoveranalysis using the code lateral load shape is selected forthis purpose. For collapse, the observed upper and lowerglobal collapse limits from the eight earthquake records

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Fig. 10. Fourier amplitude spectra for the input accelerograms (scaled to 0.30 g) and the average inelastic period of the buildings.

Fig. 11. Static and dynamic pushover analysis results for the irregular frame structures.

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Fig. 12. Static and dynamic pushover analysis results for the regular frame structures.

as well as from the three pushover analyses arepresented. It is important, however, to note that the staticpushover and the incremental dynamic collapse analysesare continued beyond all the predefined collapse thresh-olds. This is to ensure that all collapse states are boundedby the dynamic analysis.

The static pushover method is rarely used to predictseismic demands when a particular ground motion isimposed on a structure [9,10]. If this is needed, the toptarget displacement expected when this ground motionis imposed on the building should be estimated. It isbeyond the scope of this study to address the approachesof estimating the target displacement. A review of thesemethods was given in the latter two references. Sincethe main application of the static pushover analysis is toestimate the seismic capacity of structures, the followingobservations are driven by this requirement. In thisapplication of the procedure, the analysis is usually con-tinued until any of the predefined collapse criteria isexceeded.

In general, it is clear in all cases that the response ofthe buildings is sensitive to the shape of the lateral loaddistribution. This is particularly true when moving from

the code and the multimodal load patterns to the uniformload shape. It is also noticeable that the differencebetween load shape A (the design code load pattern,which is almost an inverted triangle) and load shape B(load shape from multimodal analysis) is very small.Although higher mode effects are confirmed in theresponse of the second and the third group of buildings,as explained earlier, the multimodal analysis load patterndid not show an enhanced capability to predict theseeffects. This is due to the fact that this load shape rep-resents the distribution of inertia forces in the elasticrange only, while the amplification of higher modeeffects are observed in the post-elastic phase. Table 5presents the results at global collapse limit state for thethree load shapes. In terms of the predicted ultimatestrength and drift at collapse, the differences betweenload A and B are less than 4%, for the twelve buildings.

As a general trend, the collapse is observed earlierwhen applying the uniform load than the triangular load.Collapse is observed slightly earlier than the triangulardistribution when imposing the multimodal load. In Figs.1113 the lower collapse limits from static analyses arealways from the uniform load and the upper limits are

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Fig. 13. Static and dynamic pushover analysis results for the frame-wall structures.

Table 5

Results at global collapse limit state for the three load patterns

Group Reference name Roof disp. (mm) Base shear (kN) Storeyd

Aa Bb Cc Aa Bb Cc Aa Bb Cc

1 IF-H030 534 528 508 10,091 10,446 11,592 4th 3rd 3rd

IF-M030 552 534 480 12,690 13,056 14,219 3rd 3rd 2nd

IF-M015 474 462 432 6652 6914 7620 3rd 3rd 2ndIF-L015 516 498 450 8253 8508 9147 3rd 2nd 2nd

2 RF-H030 648 624 552 12,135 12,499 14,650 5th 4th 3rd

RF-M030 712 688 568 13,083 13,444 15,748 5th 5th 3rd

RF-M015 656 640 600 7332 7554 9235 5th 5th 4th

RF-L015 688 664 592 9817 10,136 12,175 5th 5th 4th

3 FW-H030 570 560 535 13,243 13,796 16,425 3rd 3rd 3rd

FW-M030 580 570 545 16,671 17,241 20,754 3rd 3rd 3rd

FW-M015 575 565 530 7880 7988 9843 3rd 3rd 3rd

FW-L015 590 580 545 10,001 10,119 12,490 3rd 3rd 2nd

a Triangular load.b Multimodal load.c Uniform load.d Storey at which collapse is observed.

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from the triangular load. Moreover, the maximum inter-storey drift collapse limit when employing the uniformload pattern is observed in lower storeys than therecorded collapse from the triangular or the multimodalloads. This observation is also recorded in dynamicanalyses, where collapse is observed in lower storeys for

records that impose higher base shear. Fig. 14 depictsthe relationship between collapse limit states of the threeload shapes for one of the investigated buildings(building RF-L015). It should be emphasised that theloaddeformation envelope is for global response, whichis a function of the point of application of resultant force.The uniformly distributed load gives the lowest point;hence the maximum strength and earlier global yield andcollapse limit states. On the other hand, resultant in thetriangular load case is applied at a higher point; conse-quently lower strength and delayed global yield and col-lapse are observed in all cases.

Despite the fact that all load shapes do not representthe actual distribution of relative inertia forces duringthe dynamic analysis, almost an identical response isobserved in the first group of buildings between thedynamic analysis best-fit envelopes and the staticresponse obtained from the triangular and the multimo-dal distributions. On the other hand, the uniform loadoverestimates the initial stiffness and the maximum baseshear in the four buildings. Table 6 illustrates graphicallythe differences between the results of the static pushoveranalysis for the triangular and the uniform load patternson one side, and the incremental dynamic analysis(average for eight records) on the other, at global col-

lapse limit state. Since the triangular load shape is simpleand show very close results with the multimodal loadpattern, it was decided to exclude the latter from thiscomparison. It is clear that the uniformly distributed loadis unconservative in predicting collapse limit states(underestimates the drift and overestimates the strength).The overall prediction of collapse using the triangularload is significantly better. Although it slightly under-

Fig. 14. Differences between the three lateral load patterns.

estimates the average drift of dynamic analysis in twobuildings, it is between the upper and lower drift limitsobtained from the eight records, as shown in Fig. 11.

Higher modes effect and concentration of inelasticdeformations are expected to be significant in the firstgroup of structures where the buildings exhibit two

sources of irregularity and weak storey. Notwithstand-ing, four static pushover analyses using the simple tri-angular load pattern have succeeded in predicting theaverage results of more than 600 inelastic timehistoryanalyses. It is also important to note that the good designof these buildings and the high overstrength associatedwith structural elements, particularly the columns, pre-vented any undesirable mode of failure. The results showthat utilising the triangular load shape only to predict theglobal response of low rise frames as well as welldesigned irregular frame structures is adequate.

In contrast to the first group of buildings, the resultsof the static pushover of the 12-storey group, illustratedin Fig. 12, show discrepancies with the dynamicresponse envelope in the post-elastic range. While thestatic pushover results of the triangular and the multimo-dal load pattern show a good agreement with thedynamic results best fit in the elastic range, both give aconservative prediction of the maximum lateral strength,as also shown in Table 6 for the triangular load. How-ever, in the four buildings the triangular load responseis higher than the lower limit envelopes obtained fromdynamic analyses employing natural and artificial rec-ords. On the other hand, the capacity curve obtainedfrom the uniformly distributed load overestimates the

response in the elastic range. However, it gives betterprediction of the ultimate strength. It is also clear fromFig. 12 that the triangular load shape gives good predic-tion of the deformation at collapse, while the uniformload underestimates the collapse limit state in the fourbuildings. It is concluded for this group of buildings thatthe triangular distribution is again the most suitable loadpattern given that the uniform load, which is rec-

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Table 6

Differences between static and dynamic pushover analysis at global collapse limit state

Group Difference in drift Difference in strength

Triangular Uniform Triangular Uniform

1

2

3

ommended by the NEHRP guidelines (FEMA 273), isunconservative in predicting the response and the driftat collapse.

Concerning the frame-wall group of buildings, the dif-ferences between the static pushover and the dynamicanalysis results are larger than for the other two groups.In terms of the predicted elastic response and initial stiff-ness, the triangular and the multimodal load shapes showgood correlation with the dynamic analysis best fit forthree buildings and a conservative prediction for the

fourth (FW-M030). In the post-elastic range, the twoload shapes underestimate the lateral strength obtainedfrom the timehistory analyses. Table 6 shows that thetriangular load prediction of strength at global collapseis between 20 and 26% less than the average results ofdynamic analyses. Similar to the 12-storey buildings, theuniform load overestimates the elastic response but givesbetter prediction of the lateral strength at collapse forthis group of buildings. Moreover, none of the investi-gated load patterns give reasonable prediction of the highhardening stiffness obtained from dynamic analysis forthe higher design ground motion pair. With regard topredicting the drift at collapse, both the triangular andthe uniform load patterns are unconservative. This isclear from Fig. 13 and Table 6.

The comparison between static and dynamic pushoveranalysis for this group of buildings shows more discrep-ancies than the second group, especially for the 0.30 gdesign ground motion pair. As explained earlier, thesedifferences are mainly due to higher mode effects, whichamplify the base shear following formation of first plas-tic hinge at the base of the wall. In pushover analysis,once the wall attained its ultimate lateral strength, it willdeform by plastic hinging at the base [31]. Clearly, forthis type of structure (shear frame response plus cantil-

ever wall response) the amplification of the base shearduring the dynamic analysis is difficult to predict bypushover analysis. However, the triangular load profileshows good correlation in the elastic range, conservativepredictions of the ultimate strength and reasonable esti-mations of the collapse limit state (underestimates thedrift by about 8%). Hence, it may be employed for esti-mating the seismic capacity and collapse limit state.

Finally, if the static pushover analysis is utilised as atool for predicting seismic demands instead of estimating

capacities, the analysis is usually performed until theroof drift corresponding to the design ground acceler-ation is attained. Table 7 presents the average for eightground motions of the maximum top displacementobserved from timehistory analyses at the designground acceleration. Clearly, the target displacement isalmost always below the global yield limit state. Thecomparison between the static pushover and the dynamicanalysis discussed above show that the triangular loadgives better estimation of the response in the elasticrange. In few buildings, however, it underestimates theinitial stiffness. In contrast, in the same range the uni-form load shape overestimates the stiffness and the baseshear in all cases. From the design point of view, theuniformly distributed load is conservative for the twelvebuildings investigated. It is concluded that the use of twoload distributions is needed for estimating the seismicdemand. The simple triangular or the multimodal shape,which correlate well with dynamic analysis results andthe uniform load pattern which shows a conservativeprediction of demands in almost all cases considered.This conclusion is supported by the observationsobtained from the results of the frame-wall group ofbuildings. The uniform load pattern can provide a con-servative estimation of shear demand below collapse

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Table 7

Observed maximum roof displacement at the design ground acceleration (average for eight ground motions)

Group 1 Roof disp. (mm) Group 2 Roof disp. (mm) Group 3 Roof disp. (mm)

IF-H030 285 RF-H030 290 FW-H030 115

IF-M030 202 RF-M030 304 FW-M030 120

IF-M015 138 RF-M015 146 FW-M015 56IF-L015 136 RF-L015 163 FW-L015 67

limit states; hence it can support preventing undesirableshear failure. Moreover, it shows reasonable estimationof shear at global collapse limit state (at 3% interstoreydrift). It is worth mentioning that, according to theSEAOC (Vision 2000), complete collapse is consideredonce the interstorey drift exceeds 2.5%. Utilising thisdefinition of collapse leads to obtaining conservativeprediction of shear demand in all buildings investigatedwhen employing the uniformly distributed load.

8. Conclusions

The applicability and accuracy of inelastic static push-over analysis in predicting the seismic response of RCbuildings are investigated. Twelve RC buildings withvarious characteristics, incremental dynamic analysisemploying eight natural and artificial records, staticpushover analysis using three lateral load distributionsand local and global limit state criteria are utilised.Based on the large amount of information obtained,

which is nonetheless far from comprehensive, the fol-lowing conclusions are drawn:

Subject to adequate modelling of the structure, carefulselection of the lateral load distribution and articulateinterpretation of the results, pushover analysis canprovide insight into the elastic as well as the inelasticresponse of buildings when subjected to earthquakeground motions.

Static pushover analysis is more appropriate for lowrise and short period frame structures. For well-designed buildings but with structural irregularities,the results of the procedure also show good corre-lation with the dynamic analysis. In this study,response obtained for a group of four 8-storey irregu-lar frame buildings using an inverted triangular lateralload distribution is identical to inelastic timehis-tory analysis.

The experience gained from previous studies can helpto eliminate the discrepancies between static anddynamic analysis results for special and long periodbuildings. These differences are mainly due to thelimited capability of the fixed load distribution to pre-dict higher mode effects in the post-elastic range. Toovercome this problem, more than one load pattern

should be selected to guarantee providing an accurateor slightly conservative prediction of capacities anddemands.

The investigation carried out on two sets of four 12-storey frame buildings and four 8-storey frame-wallstructures show that a conservative prediction ofcapacity and a reasonable estimation of deformationis obtained using the simple triangular or the multi-modal load distribution. The same load patternsslightly underestimate the demand of some buildingsin the elastic range. On the other hand, the uniformload provides a conservative prediction of seismicdemands in the range before first collapse. It alsoyields an acceptable estimation of shear demands atthe collapse limit state.

Comparison between the triangular and the multimo-dal distribution results show differences less than 4%,for the twelve buildings, since the former captures thecharacteristics of the most important mode ofvibration. The load distribution from multimodalanalysis only represents the distribution of inertia

forces in the elastic range; hence higher mode effectsare not entirely accounted for in the post-elasticdomain.

The elongation in the fundamental period of structuresdue to extensive yielding and cracking during earth-quakes depends on the overall stiffness of the struc-tural system of the building. In the current study, theobserved elongation ranges between 100% for themost flexible irregular frame system and 60% for thestiff frame-wall structural system. Employment ofelastic periods in seismic code does not therefore pro-vide uniform levels of safety for different structuralsystems.

The results of the dynamic collapse analysis showclearly that each earthquake record exhibits its ownpeculiarities, dictated by frequency content, duration,sequence of peaks and their amplitude. The dispersionin the results of different ground motions depends onthe characteristics of both the structure and the record.The Fourier spectral analysis is an important tool toinvestigate the observed variability of the results andto identify the elongated inelastic periods of the struc-ture.

The importance of pushover analysis as an assessmentand design tool warrants much needed further devel-

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opments. These may be classified as tools andbehaviour. There is considerable scope for develop-ment of tools for more efficient and versatile pushoveranalysis techniques. One such development would bethe continuous assessment of the effect of inelasticityon the load distribution used, taking into account the

shape of the spectrum. This would enable the accurateand realistic analysis of highly irregular structures.With regard to behaviour, analysis of a larger sam-ple of buildings that includes high-rise structures andstructures with heavily irregular strength distributionis needed.

To close, it is emphasised that, notwithstanding the rangeof structures analysed, the number of records employedand the rigour of the limit state criteria monitored, theconclusions are, strictly speaking, applicable to the rangeinvestigated. However, some generality may be claimed

by noting that every effort has been made to select dis-tinct structural systems, comprehensive limit states andverified investigation tools.

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