An Energy Based Adaptive Pushover Analysis For Nonlinear .

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Civil Engineering Infrastructures Journal, 49(2): 289 – 310, December 2016Print ISSN: 2322-2093; Online ISSN: 2423-6691DOI: 10.7508/ceij.2016.02.007An Energy Based Adaptive Pushover Analysis for Nonlinear StaticProceduresShayanfar, M.A.1, Rakhshanimehr, M.2* and Zare Bidoki, R.31Associate Professor, Center of Excellence for Fundamental Studies in StructuralEngineering, School of Civil Engineering, Iran University of Science and Technology,Tehran, Iran.2Assistant Professor, Department of Civil Engineering, University of Alzahra, Tehran,Iran.3Ph.D. Candidate, Center of Excellence for Fundamental Studies in StructuralEngineering, School of Civil Engineering, Iran University of Science and Technology,Tehran, Iran.Received: 14 Feb. 2016;Revised: 14 Sep. 2016;Accepted: 19 Sep. 2016ABSTRACT: Nonlinear static procedure (NSP) is a common technique to predictseismic demands on various building structures by subjecting a monotonically increasinghorizontal loading (pushover) to the structure. Therefore, the pushover analysis is animportant part of each NSP. Accordingly, the current paper aims at investigating theefficiency of various algorithms of lateral load patterns applied to the structure in NSPs. Inrecent years, fundamental advances have been made in the NSPs to enhance the response ofNSPs toward nonlinear time history analysis (NTHA). Among the NSPs, the philosophy of“adaptive procedures” has been focused by many researchers. In the case of utilizingadaptive procedures, the use of incremental force vector considering the effects of highermodes of vibration and stiffness deteriorations is possible and seems that it can lead to agood prediction of seismic response of structures. In this study, a new adaptive procedurecalled energy-based adaptive pushover analysis (EAPA) is implemented based on the workdone by modal forces in each level of the structure during the analysis and is examined forsteel moment resisting frames (SMRFs). EAPA is inspired by force-based adaptivepushover (FAP) and story shear-based adaptive pushover (SSAP). FAP has applied modalforces directly into load patterns; SSAP, on the other hand, has implemented the energymethod in system s capacity curve for measuring the equivalent movement. EAPA hasenforced the concept of energy directly in load pattern; so that by using the modal forcesmovements an energy-based adaptive algorithm is obtained. Hence, the effects of highermodes, deterioration in stiffness and strength, and characteristics of a specific site areincorporated and reflected in applied forces on the structure. Results obtained from themethod proposed a desirable accordance with the extracted results from NTHA over theheight of the structure.Keywords: Adaptive Pushover, EAPA Procedure, Seismic Response, SMRF Structures,Stiffness Deterioration.* Corresponding author E-mail: m.rakhshanimehr@alzahra.ac.ir289

Shayanfar, M.A. et al.energy-based modal pushover analysis wasproposed which hereafter referred to asEMPA.Based on what mentioned above and theimportance of NSPs in engineering practiceto predict seismic demands on buildingstructures, future attempts are needed toimprove the pushover analysis procedures.This need is confirmed by the fact that thesimplified procedures based on invariantload patterns are partially inadequate topredict inelastic seismic demands inbuildings when the issues such as effects ofhigher modes, inelastic effects, andcumulative damages are significant (Shakeriet al., 2012; Abbasnia et al., 2013; Kunnathand Kalkan, 2004). In recent years, so as toovercome some of these drawbacks, anumber of enhanced procedures have beenproposed considering the effects ofinstantaneous state of the system related toequivalent seismic loads at each pushoverstep (Poursha et al., 2011, Malekzadeh,2013; Belejo and Bento, 2016; Shakeri et al.,2010; Izadinia et al., 2012).However, there are still issues ofcontroversy such as selecting a suitable loadvectors, variations in nonlinear responseassociated with record-to-record variability,difficulty in selecting appropriate loadvectors, and the convergence of the analysisrelated to sudden drops in componentstrength (NIST, 2010). For this reason,proposed method namely energy-basedadaptive pushover analysis (EAPA) aims touse the energy concepts to provide anappropriate adaptive load pattern toapproach the responses of the NTHA. Also,in this load pattern, it is possible toimplement an incremental process that maylead to the good responses compatible withdrop in system strength in inelasticdeformation ranges; in other words,removingtheconcernsabouttheconvergence of the analysis during theinelastic ranges of deformation. Also,INTRODUCTIONNonlinear time history analysis (NTHA) isknown as the most accurate method toevaluate the response of the structuressubjected to earthquake excitations; though,all nonlinear static procedures (NSPs) sufferinvariably from some limitations due to theirinherent static assumptions (Krawinkler andSeneviratna, 1998). Nevertheless, some ofthem are still popular for assessing theseismic capacity of structures due to theirsimplicity and application (Jiang et al. 2010;Amini and Poursha, 2016; Izadinia et al.,2012).Considering an invariant load patternduring the analysis is a drawback ofconventional pushover, leading to ignoringthe effects of higher modes of vibration. Inorder to overcome this drawback, multi-runmodal pushover procedures such as the wellknown modal pushover analysis (MPA)(Chopra and Goel, 2002) subject the systemto separate lateral loads corresponding to theconsideredelasticmodeshapes.Successively, the total seismic response ofthe system is estimated through thecombination of the responses due to eachmodal load (Chopra and Goel, 2002; Shakeriet al., 2010). Also, a modified modalpushover analysis (MMPA) based on elasticspectral responses has been proposed byChopra et al. (Chopra et al., 2004) wherebythe problem with potential es et al., 2004). Moreover,Hernández-Montes proposed an energybased capacity curve method in whichdisplacements of all floors were involved(Hernandez-Montes et al., 2004; Shakeri andGhorbani, 2015). Using the methodproposed by Hernández-Montes can possiblydefine the capacity curve of the systemcorresponding to higher modes than the firstmode, removing the concerns about reversaldisplacement of roof. In other words, an290

Civil Engineering Infrastructures Journal, 49(2): 289 – 310, December 2016equivalent forces of a given earthquake canbe computed and practically consider thevariations related to record-to-recordvariability. Moreover, it depends onengineering design decisions to simply usethe design earthquake spectral responserecommended in seismic codes such asASCE 7-05 (S.E. Institute, 2006) for aparticular site. This study shows that byusing the concept of energy to illustrate thelateral load vector in nonlinear staticmethods is closer to reality and gives betteranswers.Pushover analysis as an important part ofeach NSP, is a static technique that directlyinvolves the nonlinear properties ofmaterials (Mazza, 2014; Poursha et al. 2014)investigated by many researchers for variousstructures (e.g. Nguyen et al., 2010;KhoshnoudianandKashani,2012;Malekzadeh, 2013; Panyakapo, 2014).Conventional pushover methods apply anincreasingly single direction predeterminedload pattern which is kept constantthroughout the analysis (EN, 2004; FEMA,2005; Camara and Astiz, 2012; Manoukas etal., 2012; Giorgi and Scotta, 2013; BeheshtiAval and Keshani, 2014). Althoughchoosing a constant load pattern is simple, itmay lead to uncertain predictions ofresponses in high-raise structures; since, forexample, the modal characteristics of thestructure can be varied during the analysis.One of the most important assumptions ofmulti-run modal pushover procedures(Chopra, 2001) can be related to theindependent analysis of system in each modeby pushing the structure with thecorresponding modal load patterns. In orderto define the system's overall responses inmost well-known modal pushover analysis(MPA), the obtained results of each modeare combined using an appropriate modalcombination rule such as square root of sumof squares (SRSS) or complete quadraticcombination (CQC) (Chopra and Goel,2002, 2007). Also, a consecutive modalpushover (CMP) procedure is proposed byPoursha et al. (2009) considering the effectsof higher modes. In the same way, amodified version of CMP is proposed andexamined in braced frames (Khoshnoudianand Kashani, 2012). Even though modalnon-adaptiveapproachesofferimprovements over conventional methods,the limitations about ignoring changes instructural properties of the system during theanalysis are existent yet (Krawinkler andSeneviratna, 1998; Tarta and Pintea, 2012).Using these patterns, some of seismicbehaviors of system such as materialaccumulated strain, reducing stiffness, andsubsequent increase in the structure periodof vibration are not taken into account.Therefore, it seems that non-adaptive NSPswhich do not consider these changes presentunreliable responses.Based on the above-mentioned issues andthe influence of inelastic deformations onseismic behavior of system, many studieshave been carried out to consider thechanges of modal characteristics in theiranalysis which are known as adaptivepushover analysis (Araújo et al., 2014;Tarbali and Shakeri, 2014; Beheshti-Avaland Jahanfekr, 2015). In an adaptiveprocedure, the load pattern as shown inFigure 1 is updated at each step of theanalysis and the progress of structuralstiffness deterioration is reflected during theinelastic deformation stages.In recent years, many researchers haveproposed different lateral load patterns toimprove the compatibility of ics of the system. Reinhorn(1997), and Bracci et al. (1997) utilized theconcepts of adaptive pushover to improvethe NSPs. In these procedures the analysis isstarted with the assumption of an initiallateral load distribution, and continued withthe changing instantaneous patterns in291

Shayanfar, M.A. et al.accordance with floor shear strength in thenext steps. Lefort (2000) extended this workby applying a scaled additional force thatwas associated with considering theparticipation of higher modes.Albanesi et al. (2002) proposed anotherpushover method where the analysis isdefined based on the concept of energy. Theproposed adaptive energy-based pushoveranalysis (AEPOA) not only include theinternal structural properties in lateral forcethat is applied at each step, but also theexpected kinetic energy of the motion of thesystem under earthquake loading is takeninto account. Antoniou and Pinho (2004)proposed a force-based adaptive pushover(FAP) algorithm in which the load vector isupdated based on modal forces. They alsooffered an adaptive load vector schemebased on drift and displacement and verifiedthrough multi-ground motion incrementaldynamic analyses (Antoniou and Pinho,2004; Ferracuti et al., 2009).Recently, a pushover method based onthe developed story shear force in each levelof structure is proposed by Shakeri et al.(2010), namely story shear-based adaptivepushover (SSAP) that uses the energyconcepts to define the capacity curve ofstructure. In several cases, this method leadsto underestimation of responses especially inlower stories. Therefore, the author (Shakeriet al., 2010) proposed that the maximumresponse of SSAP and the conventionalpushover approach be selected to obtainbetter prediction of responses. The combinedprocedure was called “SSM1”.To attain an adaptive approach thatincorporates the effect of specific-sitespectrums, contribution of higher modes ofvibration, change in local resistances andstructural modal characteristics due tocumulative damages, and predictingmaximum seismic response of system by areliable accuracy, the current paper willfocus on EAPA where lateral load pattern isupdated in each step based on concepts ofenergy.The other feature of this load pattern canbe noted as involving the effects of modalforces and displacements, simultaneously.Thus, this adaptive load pattern not onlyconsiders the instantaneous state of thesystem under deterioration in stiffness andstrength, but also incorporates the movementof structure in updating the applied loadpattern. In other words, the novelty of EAPAmethod is related to entering the concept ofenergy to define the incremental adaptiveload pattern to achieve better prediction ofresponses. As a result, in this load pattern,the sign and amount of modal forces, as wellas the story displacements are directlyinvolved to define and update the loadpattern. In addition, the effects of highermodes and stiffness deteriorations aredirectly reflected to update the load pattern.For assessing the accuracy of EAPA withrespect to NTHA, a parametric study onseismic response prediction of 3, 9, and 20story steel moment resistant buildings whichdesigned by consulting structural engineersfor the Phase II of the SAC project ispresented under a range of non-elasticresponses. These models are exposed to asuitable collection of natural earthquakeswith 10% probability of being exceed in 50years. In addition, the performance of somecommon NSPs along with EAPA isevaluated to predict the peak inter-storydrift, peak story shear, and peak flooroverturning moment (OTM) profiles as thebasic engineering design parameters.292

Base Shear (KN)Civil Engineering Infrastructures Journal, 49(2): 289 – 310, December 2016Total DriftFig. 1. The capacity curve obtained from an adaptive pushover analysis along with the applied load vectors thatare updated during the analysis (Elnashai and Di Sarno, 2008)[given by Eq. (2)] in each floorcorresponding to each mode (Antoniou andPinho, 2004). Where i indicates the floorlevel, and j shows the number of consideredmodes. Spectral amplification factors aredefined for adoption of a weighted storyforce and inter-story drift using the value ofresponse spectrums corresponding to jthvibrational mode period, i.e. Sij and Sdj. It isthought that this idea leads to considerableimprovement in prediction of both capacitycurve and drift profile (Antoniou and Pinho,2004a,b).MATERIALS AND METHODSTo conduct the analysis, a mathematicalmodel of the building is made up of alloriginal laterally resistance members whichis subjected to an incremental lateral forceloading up to reaching a predetermineddisplacement value or structural collapsethreshold.Thisalgorithmcanbesummarized in four main steps as follows.The Initial VectorThe fundamental mode shape vector isused in the first step of analysis to determinethe initial load distribution. This vector isthen automatically updated with the progressof the analysis algorithm. S ijdj j i , ji 1, j F m Saijj i, j i jCalculate the Scaled Load VectorScaled load vector, Ē, determines theshape of increasing lateral load vector takinga real stiffness state of structure in each stepof the analysis. For this aim, the eigenvalueanalysis is performed in each step of thepushover analysis and the obtained results atthe end of previous time step areimplemented to determine the modalcharacteristics of the system. The results areused to determine the inter-story drifts, ij,[given by Eq. (1)] and modal forces, Fij,(1)(2)where Sij and Sdj are spectral accelerationand spectral displacement, respectively,corresponding to period of system in jthmode, ϕi,j is the component of mode shapefor the ith story and the jth mode, and mi isthe lumped mass in level i. Γj is the modalparticipation factor of mode j which isobtained from Eq. (3). In this equation, [m]is the mass matrix of structure, and {Фj} isreferred to the component of structural modeshape matrix corresponding to jth vibrationmode.293

Shayanfar, M.A. et al. j T . m . 1 jT j . m . j Now, scaled load vector, Ē, can bedefined via dividing the work done in eachfloor by the maximum of floordisplacements as shown by followingexpression:(3)Then, the modal drifts and modal forcesare combined with a suitable combinationrule to achieve the inter-story drift andinertia force at each level by means of Eqs.(4) and (5), respectively. In these equations,m is the number of considered vibrationmodes. Generally, estimated responses inany modal method are practically affectedby assumed modal combination rule.However, it is assumed that frequencies ofvarious vibration modes are enough far fromeach other to simply use SRSS combinationrule. Therefore, SRSS combination rule is inthe current paper used in all of implementedmodal analyses.Displacement value of ith level, Di iscomputed through summation of combinedmodal drifts from lower levels up to ith level[given by Eq. (6)] (Antoniou and Pinho,2004). Consequently, the work done in leveli, Ei can be easily calculated from theproduct of the force by the correspondingdisplacement as denoted by Eq. (7) where fiis effective force in level i defined by Eq.(8), and Di is incremental displacement inlevel i. In Eq. (8), dFi(t) is the incrementalapplied force in the level i at step t, and Fi(t-1)is the existing force in the level i at the endof step t-1 of the analysis (Shakeri et al.,2010). ij2 j 1Fi Fij2 j 1(5)iD ikk 1(6)E f . Di ii(7)2i (9)(10)Updating Force-Based Load VectorWhen both of the scaled load vector, Ē,and incremental base shear, Vb, aredetermined, the force vector, Vt, at each stepof the analysis can be updated by theevolution pattern shown in Figure 2. Eq. (11)(Antoniou and Pinho, 2004) demonstratesthe new load vector applied at step t, Vt,where Vt-1 is the previous load vector, Vb,tis the current incremental base shear, and Ētis the current scaled load vector.V t V t 1 V b ,t Et(11)EAPA in SequenceThe main stages of the nonlinear staticprocedure can be as follows: Stage 1: Perform a pushover analysisbased on the concepts of energy. Stage 2: Convert the capacity curve ofmulti-degree of freedom (MDOF)system to an equivalent single degreeof freedom (SDOF) system. Stage3:Estimatethetotaldisplacement demand of the equivalentSDOF system and obtain ly, the sequence of NSP basedon EAPA is as follows:m i max D V V .Eb(4) t F t 1 1 dF t iEiNew Increment of LoadsNew increment of loads, V, at each stepis obtained from product of scaled loadvector, Ē, and incremental base shear, Vb,as introduced in Eq. (10).m i fEi (8)294

Civil Engineering Infrastructures Journal, 49(2): 289 – 310, December 2016Incremental baseshearScaled loadvectorat step tNew increment ofloadsΔVtBalanced loads atprevious stepNew increment ofloadsUpdated loadvector at step tVt Vt-1 ΔVtFig. 2. Updating the load vector10. Calculating the new increment of loadsusing the predetermined incremental baseshear, Vb, via Eq. (10).11. Calculating the updated load schemefrom Eq. (11) (as shown in Figure 2) andapplying it to structural model.12. Returning to step 2 and repeating theprocess until a predefined assumedcontrol point value is achieved.Stage 11. Establishing the structural model inwhich the non-linear properties of thematerial have been considered.2. Performing eigenvalue analysis in orderto calculate the natural instantaneousfrequencies, {ω}, and the mode shapes,[Ф], of the system.3. Calculating the modal drifts at each levelfor considered modes, ij, via Eq. (1).4. Calculating the modal forces at eachlevel for considered modes, Fij, using Eq.(2).5. Combining the obtained modal drifts ofstep 3 using a suitable combination ruleto define total drift, i, [Eq. (4)] at eachlevel.6. Combining the obtained modal forces ofstep 4 to define the total force, Fi, [Eq.(5)] at each level.7. Determining the displacement of the ithlevel, Di, by summation of story driftsfrom base up to level i as shown in Eq.(5).8. Calculating the work done in each levelby multiplying the modal force and thecorresponding displacement given by Eq.(7).9. Divi

An Energy Based Adaptive Pushover Analysis for Nonlinear Static Procedures Shayanfar, M.A.1, . called energy-based adaptive pushover analysis (EAPA) is implemented based on the work done by modal forces in each level of the structure during the analysis and is examined for steel moment resisting frames (SMRFs). EAPA is inspired by force-based adaptive pushover (FAP) and story shear-based .

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