The system considering only the first three

The nonlinear behavior and responseof structures to earthquake loads have resulted in inadequacies over thereliance on linear analysis for seismic design. Non-linear dynamic analysiswhich is the ideal analysis for seismic analysis is complex and characterizedby lengthy repetition of structural response to each group of groundacceleration. Engineers relied on pushover analysis which is a method of nonlinearstatic analysis as it offers a simplistic approach of estimating inelastic seismicresponse to structures. However, pushover analysis does not provide the exactbehavior of structures under lateral loads due to its inabilities in factoringhigher modes effects.

More studies are being conducted to ascertain ways toeither perfect nonlinear static or to simplify nonlinear dynamics. This is areview of 10 papers in recent years on pushover analysis using moment resistingsteel frames.REVIEWAND DISCUSSIONSMahdi et al (2015) comparedstructural response of 3-story (low rise) and 9-story (medium rise) steelresisting frames using Modal Incremental Dynamic Analysis (MIDA) proposed byMofid et al in 2005 and Incremental Modified Pushover (IMP) proposed by AZIMIet al in 2009 with Nonlinear Time-History Analysis (NTHA) which gives exactstructural response. In the MIDA approach, each multi-degree of freedom (MDOF)frame was modelled as several equivalent single degree of freedom (SDOF)systems and analyzed using modal analysis up to the nth mode effects(ie 3 for 3-story and 9 for 9-story). The IMP method modelled the MDOF as oneequivalent SDOF system considering only the first three mode effects.

Theresulting nonlinear Incremental Dynamic Analysis (IDA) curves were plotted andcompared with that of NTHA. Both methods (MIDA and IMP) were found to beaccurate for low rise frames with accuracy levels reducing for medium riseframes. The MIDA method produced more accurate results for both low rise andmedium rise structures. The results for both methods were on the conservativeside with the IMP being more conservative and can be used in design due to itssimplicity and safety margins.

G Tarta et al (2012)compared interstory drift obtained from standard and advanced pushover analysiswith exact results from nonlinear time history analysis on two moment resistingsteel frames of 8 and 12 story. This frames were loaded in accordance with Eurocodesand seismic response spectra for three ground motion. Two standard pushoveranalysis (PA) namely: PA with uniform distribution of vertical loads and PAwith a vertical distribution after first mode vibration were considered. Forcebased adaptive pushover analysis, interstory drift based scaling pushoveranalysis and method of modal combination were the advanced pushover analysisconsidered. Interstory drift errors which is a measure of the differencebetween the exact method and particular method under consideration werecomputed. The least error was obtained for adaptive pushover analysis methods withinterstory drift based scaling adaptive pushover analysis being the most accurate.The errors obtained from method of modal combination rose significantly for 12story frames recording the highest error making this method unreliable for highbuildings.

Hariri et al (2012) studied structuralresponse obtained from Endurance Time Analysis (ETA), Time History Analysis(THA) and Incremental Dynamic Analysis (IDA) for four steel moment resistingframes (9, 11, 13 and 15 stories). ETA, a dynamic pushover analysis capable ofestimating both linear and nonlinear utilizes endurance time accelerationfunctions (ETAFs) to evaluate structural response. Incremental Dynamic Analysis(IDA) which utilizes nonlinear time history analysis at different levels ofintensity of a specified ground motion. The four frames were analyzed using threeset of second generation ETAFs for the ETA method and seven scaled groundmotions for THA and IDA methods. Results indicated that ETA method producedresults with acceptable accuracy compared with THA and IDA.The inconsistencies associated withnonlinear static analysis was the basis for Rofooei et al (2012) to propose twoseparate pushover analyses namely modal spectra combination (MSC) and firstmode load pattern of ASCE41-06 as a combination rule using three 2-D (one 8story and two 15 story) moment resisting as a case study. The MSC load patterncombines mode shapes with their weighting factors which are based on modalfrequencies and spectral values.

Sway, plastic moment capacity and inelasticresponse envelope were developed from both procedure and compared to method ofmodal combination (MMC), modal pushover analysis (MPA) and nonlinear dynamictime history analysis to ascertain the efficiency of the proposed methods. Swayresults obtained for the MSC and combination rule were acceptable compared withnonlinear time history analysis. They also concluded that the results for MPAproposed by Chopra, et al in 2003 were also acceptable.Structural response (ductilitybased reduction factor and response modification factors) were computed by Izadiniaet al (2011) for 3, 9 and 20 moment resisting steel frames analyzed by AdaptivePushover Analysis (APA) using eight different constant and adaptive lateralload pattern to estimate proportion earthquake loads imposed on structures.

ADAintroduced by Antoniou et al (2004) utilizes multi-mode effects and considersinelastic range of plastic hinges unlike the Conventional Pushover Analysis(CPA) which normally deals with the first elastic mode. ADA gave higherstructural response recording 16% for response modification factor and 17% forductility ratio above that obtained from CPA (Izadinia et al 2011).In attempt to reducecomputational demand of nonlinear plastic design, Liu et al (2009) proposed apushover analysis which utilizes a single element per member for seismic designconsidering initial imperfection, P-? and P-? effects.Two buildings consisting of 3 and 9 story steel moment resisting frames wereanalyzed in accordance with FEMA 440 and Nonlinear Integrated Design andAnalysis (NIDA). Base shear/weight were plotted against roofdisplacement/height for FEMA 440, NIDA without initial imperfection and NIDAwith initial imperfection for both frames. NIDA without initial imperfectiongave more accurate results (very close to FEMA 440) than NIDA with initialimperfection. NIDA procedure were less time consuming as compared to FEMA 440 (Liuet al.

2009). Azim et al (2009) developed theIncremental Modified Pushover (IMP) which considers high mode vibration bymodelling each 4, 8, 12 and 16 story moment resistant frames as SDOF system andone pushover analysis performed per ground motion from seventeen differentscaled earthquake as against Incremental Dynamic Analysis (IDA) which utilizesnonlinear time history analysis at different levels of intensity of a specifiedground motion. The resulting responses (maximum roof displacement, inter-storydrifts and plastic hinge rotation for each ground motion) were compared withthat obtained from IDA.

The results indicated that IMP method gave higher structuralresponse than the IDA.Progressive collapse occurs whenfailure/failures of any element of a structure results in partial or totalcollapse of the structure either by serviceability or ultimate limit state.Jinkoo et al (2007) studied progressive collapse capacity of steel resistingframes in accordance with General Services Administration (GSA) and Departmentof Defense guidelines using both linear static and nonlinear dynamic analysisand concluded that the nonlinear dynamic procedure may be used as more preciseprocedure whiles the linear procedure gave conservative decision forprogressive collapse potential of modeled structures. Rofooei et al (2006) analyzed structuralresponse of dynamic nonlinear time history analysis on five steel resistingmoment frames (2, 5, 10, 15 and 20 stories) modelled as a single degree offreedom (SDOF) system using five different load patterns for each of the fiveearthquake records considered. Pushover curves were constructed from nonlinearstatic analysis using (FEMA 360) as well as nonlinear dynamic SDOF analysisusing both bilinear and trilinear approximation to assess maximum roofdisplacements (target displacement).

Percentage error which measures the ratioof the differences between studied procedures analyzed as SDOF and THA on MDOFsystem, increases from low rise to high rise structures. Nonlinear static inthe first mode shape gave the least error for low rise buildings (2 and 5 stories)with trilinear approximation having least error for high rise buildings (15 and20 stories). First mode shape analysis is therefore the preferred nonlinearstatic procedure for low rise structures compared with the complicated staticprocedure in codes as both results are structurally close (Rofooei et al. 2006).R. Hasan et al (2002), simplifiedperformance based design pushover analysis of buildings subjected to increasinglateral loads using both linear and nonlinear analysis.

Two buildings consistingof three story and nine story steel moment frames representing low and highrise buildings respectively were used for the study. Loading was done by usingconstant gravity loads and lateral loads applied increasingly based on the differentspectral ground acceleration. The buildings were analyzed with the nodaldisplacement, member displacement and forces measured. This analysis wasrepeated with incremental lateral loads until the horizontal displacement atthe roof reached the collapse prevention displacement. They concluded that theabove described procedure and method offer adequate structural response (bothelastic and inelastic), information and results for performance based ofseismic design for new moment resisting frame structures as well asrehabilitation of existing steel frame structures.CONCLUSIONSThe studied methods provided variedlimits for structure height classifications. The upper limit for high risestructures of most of the reviewed papers was 12 story which is not a goodrepresentation of high rise building although Izadinia et al (2011) and Rofooeiet al (2006) pegged it at 20 story.

Structural response of buildings to lateralloads are more severe for taller buildings, therefore there is a probabilitythat the method being proposed may not be application to buildings with storiesabove 20. All the methods proposed andreviewed in the above mentioned publications produced similar structuralresponse with errors increasing from low rise buildings to high rise buildingswhen compared with the exact time history nonlinear dynamic analysis. Most ofthe errors for low rise buildings were on the conservative side and hence canbe acceptable in design due to high factor of safety, however relying of thesemethods will results in higher cross-sectional elements and uneconomicaldesigns. Since high errors were recorded for medium and high rise building, itis not acceptable to consider the use of the above methods for design althoughmost of the above mentioned methods were on conservative side. Also, there isthe risk of the proposed methods not being on conservative side for structuresabove 20 story which are yet to be reviewed using the above mentioned methods.

Morestudies need to be done for structural response to the proposed methods ofmedium to high rise building (including buildings above 20 story).Almost all the reviewed papersexcept for Liu et al (2009) were silent on the actual time involved in usingthe proposed methods to analyze the various structures compared with thenonlinear dynamic analysis. Whiles argument were made that nonlinear dynamicanalysis is time consuming and therefore not practical for everyday design,little was said about the associated time used in the proposed methods thusmaking it difficult to ascertain if the proposed methods are less timeconsuming.