evaluationofsimplifiedanalysisproceduresforahigh-rise reinforcedconcretecorewallstructure · 2019....

Research ArticleEvaluation of Simplified Analysis Procedures for a High-RiseReinforced Concrete Core Wall Structure

Abdur Rahman , Qaiser Uz Zaman Khan, and Muhammad Irshad Qureshi

Department of Civil Engineering, University of Engineering and Technology (UET), Taxila 47080, Pakistan

Correspondence should be addressed to Abdur Rahman; [email protected]

Received 6 September 2018; Accepted 15 January 2019; Published 12 February 2019

Academic Editor: Rosario Montuori

Copyright © 2019 Abdur Rahman et al. +is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Nonlinear response history analysis (NLRHA) procedure is one of the most precise and accurate numerical method to compute theseismic demands of high-rise structures but is complex, rigorous, and time-consuming and requires a lot of expertise for nonlinearmodelling and results interpretation. +erefore, practicing engineers in developing countries like Pakistan still use the simplifiedanalysis procedures to compute the seismic demands. Among the simplified analysis procedures, equivalent lateral force andresponse spectrum analysis procedures are widely used for the design purpose. However, other procedures have also been proposedin the recent past to accurately capture the higher mode effects in mid-to-high-rise structures. In the current study, results of a forty-story core wall building are used to check the relative accuracy and ease of application of different simplified analysis procedures.Furthermore, a modal decomposition technique is used to separate the modal responses from the NLRHA results, and a mode wisecomparison of different demand parameters for different simplified procedures is performed. +e current study has been used toclearly identify the reasons of inaccuracies in different simplified procedures. Furthermore, a simplified analysis procedure isproposed to accurately estimate the seismic demands of high-rise buildings and the possible solutions to improve their predictions.

1. Introduction

Recent survey by the United Nations estimated that morethan half the population of the world lives in the urbanregions [1]. +e United Nations also predicted that, by theend of 2050, about 64% of the developing countries and 86%of the developed countries population will be living in thecities. To accommodate the increase in population in theurban areas, either the urban areas (cities) should be ex-panded or the high-rise buildings should be constructed.High-rise buildings are generally more vulnerable to seismichazard and wind actions as compared to low-rise buildings.+erefore, it is essential to use efficient lateral force-resistingsystems to reduce lateral demands. Various lateral force-resisting systems have been developed in the past for high-rise buildings. High-rise reinforced concrete (RC) buildingswith an RC core wall as a lateral force-resisting systempresent one such solution and provide the advantages offaster construction, flexible architecture, and the availabilityof more open spaces [2]. As the core wall is stiffer comparedto the RC columns, most of the lateral forces are resisted by

core wall. +us, the columns remain flexible to carry thegravity loads.

Nonlinear response history analysis (NLRHA) pro-cedure is one of the most precise and accurate numericalmethods to compute the dynamic responses of high-risestructures; however, it is time-consuming, computationallyexpensive, and requires expertise in nonlinear modelling. Arecent study of Mehmood et al. [3] shows that the com-putation time required to perform the NLRHA procedure topredict the seismic responses of a high-rise building underone groundmotion is about 30 hours and the postprocessingtakes another 5 hours. +erefore, several simplified analysisprocedures were developed in the past to avoid the afore-mentioned issues associated with the NLRHA procedure.Some of the simplified procedures require nonlinearmodelling, while for others, the linear elastic modellingoption suffices. But these methods are less accurate ascompared to the NLRHA procedure. Most of these sim-plified procedures were adopted in various codes such asUBC-97 [4], FEMA-356 [5], and ATC-40 [6]. Simplifiedanalysis procedures include response spectrum analysis

HindawiAdvances in Civil EngineeringVolume 2019, Article ID 1035015, 21 pageshttps://doi.org/10.1155/2019/1035015

mailto:[email protected]://orcid.org/0000-0002-3549-3955https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/1035015

(RSA) procedure, equivalent lateral force (ELF) procedure,modified modal superposition procedure (MMSP), modalpushover analysis (MPA) procedure, and weighted capacitydesign (WCD) method.

A number of simplified pushover analysis procedureshave also been developed recently for various specificstructural types such as bidirectional energy-based pushover(BEP) procedure [7], cyclic pushover analysis procedure [8],extended consecutive modal pushover procedure [9], im-proved modal pushover analysis procedure [10], extendedenergy-based pushover analysis procedure [11], simplifiedpushover analysis procedure for high-rise rocking buildings[12], envelope-based pushover analysis procedure [13], andadaptive modal pushover analysis procedure [14]. However,the current study is focused on the applications of moreestablished simplified analysis procedures.

+e simplest and straightforward design method is theELF procedure which was adopted by UBC-97. +is methodmainly considers the seismic responses of a fundamentalmode and also an extra force at top of a structure for highermodes. It has been found to be reasonably accurate inpredicting the seismic forces of low-rise buildings; however,it underestimates the seismic responses of high-rise build-ings in which higher modes contribute significantly.

Response spectrum analysis (RSA) procedure, on theother hand, explicitly takes into account the higher modeactions and is the most commonly adopted procedure tocompute the seismic demands of buildings in many coun-tries. In this method, first, the elastic seismic forces of eachmode are computed independently from the elastic responsespectrum, and then, modal seismic forces are added by usinga modal combination rule to obtain the total elastic re-sponses. +ese elastic demands are then divided by the forcereduction factor (R) to obtain the inelastic seismic demands.Research work of Eibl and Keintzel, Priestley and Amaris,Sullivan et al., and Munir and Warnitchai [15–18] show thatRSA procedure underestimates the inelastic seismic de-mands of high-rise buildings.

Priestley et al. [19] proposed the MMSP in which onlythe 1st mode structural responses are divided by the forcereduction factor (R) and then combined with elasticstructural responses of higher modes to obtain the maxi-mum response. +e results of this method have beenchecked with the results from NLRHA procedure for can-tilever walls having 2 to 20 stories, and it is found to slightlyunderestimate the seismic demands of shorter structures andoverestimate the seismic demands of taller structures.

A simplified lateral force analysis procedure called thenonlinear static pushover (NSP) analysis procedure isadopted by various codes such as FEMA-356, ASCE-41-06[20], and ATC-40. +is method is based on the structuralresponses of the fundamental mode only. In this method, thestructure is pushed laterally to a target displacement after thegravity loads are applied. +ese pushover lateral forces areproportional to fundamental modal inertial forces. +eseismic responses of the building are then determined at thetarget displacement.

Chopra and Goel [21] suggested the modal pushoveranalysis (MPA) procedure to overcome shortcomings of

NSP procedure based on the fundamental mode only. In thistechnique, firstly, all significant modes in which the buildingwill vibrate are determined. An equivalent single degree offreedom (SDOF) system for each mode is subjected to thespecified ground motion to determine the target displace-ment for each mode.+e SDOF system is controlled by a fewmodal properties which are determined from the hystereticbehavior of the structure. +e maximum structural re-sponses are then determined at the target displacement foreach mode. +ese individual responses are then added by amodal combination rule to obtain the overall structuralresponses. Goel and Chopra [22] also modified the MPAprocedure by suggesting that the higher modes remain in theelastic region which is called the modified modal pushoveranalysis (MMPA) procedure.

+e developing countries like Pakistan still use the sim-plified analysis procedures for the design of buildings againstlateral forces due to limited expertise. +ese procedures aresimple and easy to apply as compared to the NLRHA pro-cedure to design the buildings for lateral forces, but theiraccuracy is still in question. +e above discussed simplifiedanalysis procedures have been compared individually withNLRHAprocedure in the past to check their accuracy, but stillthere is a lack of literature in which all these simplifiedprocedures have been checked especially for high-rise corewall buildings. +is paper aims at comparing all simplifiedprocedures for their relative accuracy, computational efforts,and computation time for high-rise core wall structures and todetail merits and demerits of various simplified analysisprocedures. Furthermore, based on an in-depth analysis ofmodal responses using modal decomposition techniques, amodified simple analysis procedure is proposed. For thispurpose, a high-rise core wall building is chosen as a casestudy building as explained in the next section.

2. Case Study Building

To check the relative accuracy of various simplified analysisprocedures, a high-rise reinforced concrete (RC) core wallbuilding is chosen as a case study building. According toEmporis Standards Committee, a multistory building with12 to 39 stories having unknown height or a multistorybuilding having a total height of 43 to 122m (140 to 400 ft) isconsidered as a high-rise building [23]. +e chosen casestudy building is a residential building having a total heightof 126.50m (415 ft) and 40 stories as shown in theFigures 1(a) and 1(b). It has a lobby level height of 6.0m(20 ft), while above the lobby level, typical story height is3.0m (10 ft). It has three basement-level parking floors with3.0m (10 ft) story height.+e lateral force-resisting system inthe building comprises of fourteen peripheral columns andcentral reinforced concrete core wall, while the vertical loadresisting system comprises of a 20.20 cm (8 inch) thickposttensioned concrete slab laying on the central core walland the peripheral columns.

Klemencic Magnusson Associates and Arup used the LosAngeles tall building structural design council’s alternativedesign procedure for tall buildings to design this building[24, 25]. +is building is in a high seismic zone, which is

2 Advances in Civil Engineering

correspondent to seismic zone 4 in the UBC-97 code. e soilbelow the foundation of building and in the surrounding re-gion is considered as a sti� soil corresponding to soil type SD inUBC-97. e seismic demands for the design of building were

obtained from the design-based response spectrum in theUBC-97. As the seismic demands decrease with height, thethickness of the core wall and reinforcement were reducedalong the height. e thickness of the reinforced concrete

Couplingbeams

3′–0″ 3′–0″30′–0″30′–0″30′–0″

6′–0

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24″ from level20-roof

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14″-thick basement

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Figure 1: (a) Plan view of 40-story building [24, 25]; (b) elevation of 40-story building [24, 25].

Advances in Civil Engineering 3

central wall is kept 76.20 cm (30 inches) up to the story level 20and 60.90 cm (24 inches) from story level 20 to the roof. +ecross sections of the columns were also reduced along theheight from 91.0 cm× 91.0 cm (36 inches× 36 inches) at thelower level to 60.90 cm× 60.90 cm (24 inches× 24 inches),71.0 cm× 71.0 cm (28 inches× 28 inches) to 60.90 cm×60.90 cm (24 inches× 24 inches) at the top level. +e com-pressive strength of concrete was also reduced from 55MPa(8000psi) up to level 20 to 40MPa (6000psi) from story level20 to the roof.+e steel bars having a nominal yield strength of415MPa (60 ksi) were used. +e core wall in the center of thebuilding is a group of cantilever RC walls, having open spaceswhich are connected by coupling beams. +e coupling beamdimensions were also reduced with the height of the corewall. +e coupling beams have the dimensions of 76.0 cm×152.0 cm (30 inches× 60 inches) above the lobby level opening,76.20 cm× 81.28 cm (30 inches× 32 inches) at other floorlevel up to story level 20, and 60.90 cm× 81.20 cm (24 inches×32 inches) from story level 20 to the roof.

3. Numerical Modelling of Case Study Building

As explained earlier, the current study focuses on the com-parison of results from different simplified analysis pro-cedures with the results from the time history analysis. Toperform the nonlinear response history analysis (NLRHA), anonlinear numerical model of the forty-story building hasbeen developed in the Perform-3D version 4 [26] as pre-viously created by Munir and Warnitchai [18]. +e nonlinearmodel is a conventional fiber model, which is usually used forthe modelling of high-rise RC core wall buildings. In thismodel, only the flexural behavior is considered nonlinear,whereas shear response is considered as linear elasticaccording to the principles of capacity design for the shearstrength design. +e core wall up to story level 5 has beenmodelled by nonlinear shear wall element in which each wallelement consists of eight concrete and eight steel vertical fibersegments in each layer. +e hysteretic curve used for steelfibers is a nondegrading bilinear type as shown in theFigure 2(a). +e postyield stiffness of steel is assumed to beabout 1.2 percent of initial stiffness. As the expected strengthof the material is greater than the strength specified by thecodes, the expected yield strength of steel rebar of grade 60 issupposed to be 485.0MPa (70.20 ksi) which is greater than thenominal yield strength. Similarly, the compressive strength(fc′) of concrete is assumed 1.3 times of the specified com-pressive strength. +e concrete tensile strength is assumed tobe (7.5

��

fc′

) psi as in UBC-97. +e core wall above storylevel 5 is modelled by the elastic shear wall element segmentsas this portion is expected to remain in the elastic region. +estress-strain curve of Mander’s model [27] for the unconfinedand confined concrete is used for modelling the concretefibers and is approximated by the trilinear envelope as shownin Figure 2(b). +is is to account the confinement effect ofpostpeak strain ductility capacity and ultimate compressivestrength by the transverse reinforcement in the core wall. InPerform-3D, the unloading stiffness of concrete in com-pression in the hysteretic curve is assumed to be approxi-mately equal to the initial stiffness. However, the reloading

stiffness can be modified, and it is set such that the stiffnessdegradation is directly proportional to the plastic strain.Columns and slabs are also modelled by elastic column andshell elements, respectively. As the plastic hinges are permittedat the end of coupling beams, the coupling beams aremodelledby the elastic beam-column elements with the plastic hinges atboth ends.+e geometric nonlinear effects (P-delta effects) arealso considered in the model. According to the Tall BuildingInitiatives (TBI) recommendations, a modal damping ratio of2% is more reasonable for the high-rise building as comparedto the traditionally assumed 5% damping ratio [28].+erefore,the 2% damping ratio has been assigned to each translationalmode. A small Rayleigh damping is also considered for sta-bilization of other higher modes. Further details about non-linear modelling of the case study building can be found in thestudy of Munir and Warnitchai [18].

4. Seismic Demands by NLRHA and SimplifiedAnalysis Procedures

+e NLRHA procedure is one of the most precise methods tocompute the seismic demands of buildings. +erefore, it willserve as a benchmark to check the accuracy of all simplifiedanalysis procedures. Two levels of the earthquake are definedfor the present case study, one is design basis earthquake(DBE) and the other is maximum considered earthquake(MCE). DBE has 10% probability of exceedance in 50 years.MCE can be described as the earthquake having a 2%probability of exceedance in 50 years. DBE is considered two-third of the MCE for the present study. A set of three groundmotions is chosen for this study as shown in Table 1. +esegroundmotions were chosen from PEERNGA and COSMOSdatabase [29, 30], whose response spectra are similar with thetarget spectrum. Further information about ground motionsselection, scaling, and matching can be found in the study ofMunir and Warnitchai [18].

With nonlinear models of the building as discussedabove, nonlinear response history analyses are executed foreach of the ground motion for DBE and MCE levels appliedin the X direction. +e maximum shear force and themaximum bending moment at each story level of thebuilding for each ground motion are determined and thenplotted along the height of the building, but here the resultsare plotted only for HM ground motion as shown in theFigures 3(a) and 3(b). +e plastic hinges are formed at theend portions of the coupling beams in the X direction and atthe bottom region of the core wall as expected. +e MCEshear demand at the base of the building is about 1.5 of theDBE level shear demand, while at the midheight, it is about1.3 times the DBE level shear demand. Similarly, the MCElevel bending moment at the base and midheight of thebuilding is about 1.25 times of DBE-level earthquake. In thenext section, seismic demands through simplified analysesprocedures will be computed and will be compared with trueseismic demands from the NLRHA procedure.

4.1. Equivalent Lateral Force Procedure. +e ELF procedurein UBC-97 is still used in many developing countries for the


fu

ξy ξsh ξu

fy

Strain

Stre

ssEo

Park envelopePerform-3DCyclic

fy = rebar yield stress, fu = rebar ultimate stress capacityξy = rebar yield strain, ξsh = strain in rebar at onset of strainhardening,ξu = rebar ultimate strain cpacity, Eo = modulus of elasticity

(a)

f ′cc

Strain

Stre

ss

Mander envelopePerform-3D

fccξ′c = concrete strain at fc′ξ′cc = concrete strain at fccξcu = ultimate strain capacity for confined concrete

ft = tensile strength of concrete =

Eo = tangent modulus of elasticity, dc = energy dissipation factor for thereloading stiffness

UnloadingReloading

Eo

Eo

ft

dcEo

ξ′c ξ′cc ξcu

7.5 √fc′ (fc′ is in psi)

Unloading and reloadingare the same in the case oftension

fc′ = compressive strength of unconfined concretefcc = comprssive strength of confined concrete′

′

(b)

Figure 2: (a) Stress-strain curve of steel rebar [18]; (b) Mander’s stress-strain model for concrete [18].

Table 1: Detail record of the three ground motions for the present study.

S. No. Earthquake event Years Abbreviation Mw R (km) PGA (g) Duration (seconds)1 Superstition Hills 1987 SH 6.5 11 0.30 22.302 Hector Mine 1999 HM 7.1 26 0.27 45.313 Loma Prieta 1989 LP 6.9 48 0.37 59.95

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Figure 3: Building maximum (a) shear and (b) bending moment demands by the NLRHA procedure at DBE and MCE levels for groundmotion HM.


design of building structures against lateral forces. As thismethod is developed on the fundamental mode structuralresponses of the building and on considering an extra forceat top of the building for higher modes, it is therefore moresuitable to be used for low-rise regular building in highseismic zones, all regular and irregular buildings in low-seismic zone, and irregular buildings less than five stories or20m (65 ft) in height. In accordance with above limitationsof UBC-97, this procedure cannot be used for the presentcase study building which is a high-rise building located inthe high seismicity zone. However, seismic demands by thismethod are computed only to show its underestimation forhigh-rise buildings because it is still used in some developingcountries such as Pakistan. +e ELF procedure in UBC-97 isadopted to compute the seismic demands for the presentcase study building as shown in Table 2.

+e computed shear force and moment at each floorlevel are then compared with true shear force and truemoment computed by the NLRHA procedure, which isconsidered the most accurate procedure for determining theseismic demands as shown in Figures 4(a) and 4(b). +ecomparison clearly shows that the true seismic demands aremuch higher than those computed by the ELF method. +etrue base shear is about 3 to 6 times higher than the baseshear computed by the ELF procedure, while the midheightmoment is about 1.85 to 3.5 times higher than the midheightmoment of the ELF procedure. +ese results clearly showthat the higher vibration modes significantly contribute tothe total seismic demands.

4.2. SeismicDemands by RSAProcedure. +e RSA procedureis a dynamic analysis procedure in which the elastic re-sponses of the significant vibration modes are obtained fromthe DBE elastic response spectrum and then combined toobtain the total elastic responses of the structure either bySRSS or CQC modal combination rule. Total elastic re-sponses are then reduced by response modification factor(R) to get the inelastic responses for the design purposes.

Traditionally, buildings are designed for the DBE butevaluated for MCE-level seismic intensity. +erefore, we willinvestigate the seismic demands for both levels of seismicintensity. Firstly, the DBE elastic response spectrum is de-fined by using the values of Ca and Cv, and damping ratio of5%. +is spectrum is then multiplied by the factor of 1.5 toobtain the MCE level spectrum for the 5% damping ratio. Asa constant modal damping ratio of 2% is used in the NLRHAprocedure, the same will be used here. 2% DBE and MCEspectrum are constructed by multiplying 5% DBE and 5%MCE spectrum with factor Rζ . +e value of Rζ is computedby the equation Rζ � (0.07/0.02 + ζ)

0.5, where ζ is thedamping ratio.

+e eigen value analysis method is performed to find thenatural time periods, modal participating factors, massparticipation factors, and mode shape factors for all thesignificant modes, and they are shown in Table 3.+e sum ofmass participating factors of the first five translationalmodes is 97.72% which is greater than 90% as recommendedby the UBC-97 code. +e elastic responses of the first five

translational modes are determined from response spectrumfor both DBE and MCE levels. +e elastic modal responsesfor each case are combined by SRSS rule to obtain the totalelastic responses and then divided by R factor to obtain thetotal inelastic responses. +e total inelastic responses arethen compared with the results of the NLRHA procedure asshown in Figures 5(a)–5(d). Both the shear force andbending moment at each story level for both DBE and MCEcomputed by the NLRHA procedure are much higher thanthose determined by the RSA procedure. +e base shear ofthe NLRHA procedure is about 3 to 5 times higher than baseshear of the RSA procedure. Similarly, the midheightbending moment determined by the NLRHA procedure isalso about 3 to 5 times higher than that determined by theRSA procedure. +ese results evidently illustrate that theRSA procedure undervalues the seismic demands for high-rise core wall structures.

4.3. SeismicDemandsbyMMSP. Priestley et al. [19] modifiedthe modal response spectrum analysis procedure by as-suming that ductility primarily acts in the 1st mode to limitthe 1st mode responses while assuming the higher modes tobe in the elastic range. +is means that 1st mode responsesare independent of the groundmotion intensity, whereas thehigher modes responses are proportional to intensity. +etotal response obtained is an approximation to the trueresponse. +e responses of the higher modes will be mod-ified to little extent due to 1st mode ductility, but they cannotincrease the base moment which will be anchored by thebase plastic hinge moment capacity.

+e seismic demands by MMSP are also computed forthe present case study building and then evaluated with thetrue seismic demands of the NLRHA procedure for groundmotion HM as shown in Figures 5(a)–5(d). +e MMSPunderestimates the shear force at the midheight of thebuilding while overestimate at the base shear. However, theoverall shear force at each floor level matches well with thetrue shear force computed by the NLRHA procedure ascompared to the RSA procedure. Similarly, the bending

Table 2: Equivalent lateral force procedure.

Parameters ValueSeismic zone factor (zone 4) Z � 0.4Type of soil SDSeismic acceleration coefficient Ca � 0.44Seismic velocity coefficient Cv � 0.64Response modification factor R � 5.5Importance factor I � 1.0Natural time period of the firstmode (by method A) T � 2.91 sec

Total seismic dead load W � 4.0 × 105 kN (89750 kips)Equivalent static lateral loadprocedureDesign base shearV � (CvΙW)/(RT)

V � 0.039 W � 1.60 ×104 kN (3600 kips)

Minimum design base shearV � 0.11CaΙW

V � 0.046 W � 1.92 ×104 kN (4300 kips)

Minimum design base shearV � 0.8ZNvΙ/R

V � 0.06 W � 2.31 ×104 kN (5200 kips) (governed)


moment at each floor level computed by the MMSP pro-cedure as compared to the RSA procedure is in goodagreement with bending moment of the NLRHA procedurefor the upper half height of the building, while it over-estimates the bending moment at the lower half height. +eoverestimation for the lower half height is noticeable in thecase of MCE.

4.4. Seismic Demands by MPA Procedure. +e modalpushover analysis (MPA) procedure was developed byChopra and Goel. In this procedure, the target displacements ofeach significant mode are first computed from the equivalentSDOF system which is subjected to the specified ground mo-tion, and then the structure is pushed laterally to each targetdisplacement. +e various seismic responses are then de-termined for each mode at the target displacement fromstandard pushover analysis and then combined by a modalcombination rule such as SRSS or CQC rule to obtain the totalseismic responses of the structure. Using the MPA procedure,seismic demands are determined for both DBE andMCE levelsand then compared with seismic demands from the NLRHAprocedure. Figures 6(a)–6(d) shows the comparison of shearand moment demands at each story level. +e comparisonshows that the shear demands computed by theMPAprocedurefor both DBE and MCE levels match approximately with thoseobtained by the NLRHA procedure, except from story level 26

to story level 33 where the discrepancy is noticeable. However, itunderestimates the bending moment at the middle and over-estimates at the base of the core wall. Despite relatively accurateresults, it is important to mention here that, as opposed topreviously shown simplified procedures, the MPA procedurerequires nonlinear modelling of the structure.

4.5. Seismic Demands by MMPA Procedure. Goel andChopra [22] modified the MPA procedure by consideringthat higher vibration modes of the building having usuallyvery limited inelastic action to remain in the elastic region.+is is called modified modal pushover analysis (MMPA)procedure. +e need of pushover analysis for higher modes isalso eliminated. +is procedure is limited only to systemshaving a moderate damping ratio.+is is because the increasein seismic demands of higher modes are unacceptably verylarge for lightly damped systems, while for modest dampedsystems, the increase in the seismic demand is small andacceptable. For the present case study building, the highermode target displacements are determined from the equiv-alent elastic SDOF systems and modal seismic demands arethen obtained from these target displacements and combinedby the modal combination rule to determine the total seismicdemands. +ese seismic demands are compared with trueseismic demands of the NLRHA procedure. Figures 6(a)–6(d)show the comparison of shear force and bending moment ofthe MMPA procedure with the results of the NLRHA pro-cedure at each story level of the building. +e comparisonshows that shear forces at each floor level computed by theMMPA procedure for both DBE- and MCE-level groundmotion match well over the upper half of the building withtrue shear forces of the NLRHA, while it overestimates fromthe base to midheight of the building.

Various simplified procedures presented in this sectionshows that the ELF procedure, RSA procedure, and MMSPare relatively less accurate in predicting the seismic demands

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Figure 4: Comparison of (a) shear and (b) bending moment demands of ELF and NLRHA procedures for ground motion HM.

Table 3: Modal properties of the case study building.

Modeshape

Naturaltime period

(sec)

Massparticipating

factors

Modalparticipating

factors1 3.84 66.08 1.60482 0.745 20.29 −0.92523 0.3026 06.75 0.53554 0.1724 03.06 −0.3653


of the case study building. Although MPA and MMPAprocedures show better results, these procedures requirenonlinear modelling. To further explore the results fromdifferent procedures, a modal decomposition technique isused in the current study to separate the total true seismicdemands from NLRHA to their respective modal demands.+is allows the mode by mode comparison of true andpredicted seismic demands of various procedures and hencecan give better understanding of the shortcomings in dif-ferent procedures.

For this purpose, a modal decomposition method calledthe uncoupled modal response history analysis (UMRHA)procedure has been adopted in the next section to explore

and understand the discrepancies in different simplifiedanalysis procedures.

4.6. UMRHA Procedure. +e UMRHA procedure has beendeveloped by Chopra and Goel. Initially, this procedure wasused to calculate the target drift of the SDOF system which isthen used in the MPA procedure to determine the modalseismic responses. However, this procedure is used in therecent past by many researchers to comprehend the complexseismic responses of buildings by disintegrating the totalseismic response into modal responses. +e UMRHA pro-cedure has also been recently used by Tahir et al. as a

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Figure 5: Evaluation of seismic demands by the RSA procedure andMMSPwith the NLRHA procedure for groundmotion HM: (a) shear atDBE level; (b) shear at MCE level; (c) bending moment at DBE level; (d) bending moment at MCE level.


simplified analysis procedure to accurately determine theseismic demands of high-rise buildings [3]. UMRHA takesless time as compared to the NLRHA procedure.

+e equation of motion for a multidegree of freedom(MDOF) system subjected to a horizontal ground motion inthe nonlinear range is

M€u(t) + C _u(t) + fs(u, _u) � −Mι€ug(t), (1)

where C and M are the damping constants and story lumpedmass matrices, u(t) denotes the displacement vector consistsof N lateral floor displacements, fs denotes the lateralresisting force vector, and ι denotes the influence vector ofthe MDOF system.

+e floor displacement for Equation (1) can be shown assum of the modal responses:

u(t) � N

i�1ϕiqi(t), (2)

where qi(t) is the modal coordinate of the ith mode and ϕi isthe mode shape vector of the ith natural vibration modewithin the elastic range. +e UMRHA procedure assumesthat Equation (2) is approximately valid for buildings in thenonlinear range.

Mι€ug(t) is the effective earthquake force and its spatialdistribution is defined as s � Mι and can be described as asum of modal inertial forces distribution:

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Figure 6: Evaluation seismic demands by MPA andMMPA with the NLRHA procedure for ground motion HM: (a) shear at DBE level; (b)shear at MCE level; (c) bending moment at DBE level; (d) bending moment at MCE level.


−Mι€ug(t) � −s€ug(t) � N

i�1− si €ug(t) �

N

i�1− ΓiMϕi €ug(t),

(3)

where Γi � ϕTi Mι/ϕTi Mϕ.

+e seismic response of the structure against ith modalinertial force vector is expressed by

M€u(t) + C _u(t) + fs(u, _u) � −si €ug(t). (4)

+e response vector u(t) in Equation (4) can besubstituted by ϕiqi(t). Premultiplying Equation (4) by ϕiT,we obtain:

Mi €qi + 2ζMi _qi + Fs qi, _qi( � −ΓiMi €ug(t), (5)

where M � ϕTi Mϕi and ωi and ζ i denote the angular fre-quency of vibration and damping ratio of the ith mode, re-spectively. +e lateral resisting force Fsi � ϕ

Ti fs(u � ϕiqi, _u �

ϕi _qi), and it is therefore a nonlinear function of qi and _qi.A newmodal coordinate Di(t) is introduced and is given

by_qi(t) � ΓiDi(t). (6)

Equation (5) can be written as

Di + 2ζ iωi _Di +Fsi Di,

_Di

Li� €ug(t), (7)

where Li � MiΓi.Equation (7) is the governing equation of motion for

inelastic SDOF systems in the standard form. +e inelasticresponse time history of Di can be determined from Equation(7) if the nonlinear function Fsi(Di, _Di) is known. Chopraand Goel showed that Fsi(Di, _Di) can be accurately definedby a simple bilinear hysteretic curve for steel moment-resisting frame building. +e bilinear hysteretic curve wasobtained by standard pushover analysis using the ith modalinertial force distribution si � Mϕi. Due to strength andstiffness degradation under cyclic loading in reinforcedconcrete (RC) special moment-resisting frame buildings, thebilinear hysteric curve does not accurately describe the truenonlinear force-deformation relationship of the structures.+erefore, Bobadilla and Chopra [31] proposed a cyclicpushover analysis to obtain the nonlinear curve of structureswhere their modal hysteretic behavior is not known.

+e ith mode’s cyclic pushover analysis can be performedby applying a force vector with the ith modal inertial forcepattern s∗i to the structure together with gravity loads. +elateral force vector is altered in magnitude and then re-versed, while the gravity loads remain in its position, tocreate cyclic responses with gradually increasing amplitude.With above force distribution, the lateral displacement re-sponse and the other seismic responses are considered to bedominated by those of the ith mode, and therefore, the roofdisplacement (uri ) and Di are approximately given by

Di �uriΓiϕri

, (8)

where ϕri is the value of mode shape at the roof level.

+e base shear Vbi and resisting force Fsi with the ith

mode inertial force distribution pattern is given by thefollowing relationship:

FsiLi

�VbiΓiLi

. (9)

+e cyclic pushover results are first expressed in the formof the cyclic base shear (Vbi)-roof displacement (uri ) re-lationship and then presented in the Fsi −Di relationshipusing Equations (8) and (9) and is shown in Figures 7(a) and7(b). Against a low-displacement demand, the structurebehaves in a linear elastic manner and the gradient of theforce-deformation curve represents the initial linear elasticstiffness. As soon as the roof displacement increases againstan increasing load, the building goes into inelastic range andthe tension cracks appear in RC walls and columns resultingin softening of the overall structural stiffness as shown in theFigure 7(a). During the unloading phase, the cracks in theRC walls and columns tend to close under the action of highgravity force and low reinforcement ratio. As the cracks fullyclose, the structure almost completely restores its initialstiffness, resulting in a minimal residual drift.+is causes thestructure to exhibit a bilinear elastic or flag-shaped hysteresisbehavior. Similar behavior has been observed by Adebaret al. [32] during experimental testing of a large-scale high-rise wall model simulating the first mode lateral load pattern.To model this flag-shaped behavior obtained from pushoveranalysis in Perform-3D, Ruaumoko-2D program [33] is usedwhich has a readily available flag-shaped hysteretic model.+e comparison of actual and idealized hysteretic behaviorsis shown in Figure 7(b).

Equations (8) and (9) are then used to compute thedeformation and force related modal responses of theMDOF system.

+e symbol Ri(t) is generally used to represent each ofdeformation-related or force-related responses of the ithvibration mode. +ese individual responses are then com-bined in the time domain to obtain the total response R(t) ofthe structure as follows:

R(t) � n

i�1Ri(t), (10)

where n is the number of vibration modes.

4.6.1. UMRHA Results and Discussion. +e UMRHA pro-cedure is used to determine the seismic demands of the 40-story building to each of the three ground motions, both forthe DBE and MCE levels. +e modal seismic response timehistories are first determined and then combined in the timedomain for each of the floor level using Equation (10) toobtain the total seismic response time history of the building.+e modal response time histories are determined for thefirst four translational vibration modes since the contri-bution from other higher modes is very low. +e totalseismic response time history at each floor obtained from theUMRHA procedure for each of the ground motion is


expected to match with those determined from the NLRHAprocedure. +e maximum responses throughout the totaltime history for each floor is obtained and then comparedwith maximum responses from the NLRHA for HM groundmotion as shown in Figures 8(a)–8(d). +e comparisonsshow that the shear forces and the bending moments ob-tained from the UMRHA procedure for each ground motionmatches well with those of the NLRHA procedure.

+e modal shear forces and bending moments forthe ground motion HM at MCE level is shown in theFigures 9(a)–9(d). Figures 9(a) and 9(b) illustrate that theshear demand is dominated by the 2nd and 3rd modes, whileFigures 9(c) and 9(d) show that the moment demand iscontrolled by the first three vibration modes. +e 1st modeattains its highest shear and moment demand at the basedecreasing along the height, while the higher modes attainmaximum and minimum values of shear force and bendingmoment at certain heights. +e modal decomposition resultsof the shear demand show that, from the base to level 6, shearforce is dominated by the second and third vibration modes,while, at level 13, it is dominated by the first twomodes and thefourthmode.+emodal decomposition results of themomentdemand show that, at level 7, moment demand is controlled by1st mode while higher modes moment demand at this level isapproximately equal to zero. On the contrary, the momentdemand at story level 20 is dominated by the second mode.

+e UMRHA procedure has been confirmed in theabove section to be reasonably accurate, and hence themodal results can be analyzed to draw conclusions about thesimplified analysis procedures. Reason behind the un-derestimation of seismic demands by simplified analysisprocedure can be comprehended from the modal de-composition results of the UMRHA procedure. +e modewise seismic demands comparison of all simplified pro-cedures with that of the UMRHA procedure have beenpresented in the Figures 10(a)–10(d) and 11(a)–11(d) forground motion HM at MCE level. +e comparison inFigure 10(a) shows that the inelastic shear forces at each

story level of the 1st mode computed by the RSA procedurematches well with 1st mode shear forces from the UMRHAprocedure, while Figures 10(b)–10(d) shows that the in-elastic shear forces at each floor of the higher modescomputed by the RSA procedure are much lower than that ofthe UMRHA procedure. Similarly, the bending moment ateach story level of the 1st mode determined by the RSAprocedure is in good agreement with that of the UMRHAprocedure. +e bending moment at each story level of thehigher modes computed by the UMRHA procedure is muchhigher than that of the RSA procedure.

Figures 10(a)–10(d) and 11(a)–11(d) also show thecomparison of shear force and bending moment computedby the MPA procedure with the UMRHA procedure. +eMPA procedure’s first mode shear forces at each story levelmatches well with that of the UMRHA procedure. +e 2ndmode shear forces of the MPA procedure are also in goodagreement. +e other higher mode shear forces are also ingood correspondence with the UMRHA procedure. Simi-larly, the bending moment at each story level computed bythe MPA procedure is close with that computed by theUMRHA procedure. +e higher mode seismic demands arein good agreement due to the use of nonlinear hystericbehavior in seismic demand computation.+e overall resultsof the MPA procedure are lower due to modal combinationrule. Also, it is important to note that the UMRHA pro-cedure also showed some underestimation.

+e shear force and bending moment at each floorcomputed by the MMPA procedure for first, third, andfourth mode is the same with those of the MPA procedureand match well with those of the UMRHA procedure, butthe second mode shear force and bending moment are muchhigher than those of the UMRHA procedure. +is is due tothe fact that the MMPA procedure considered the secondmode to be elastic, but UMRHA results are showing yieldingin the second mode.

+e MMSP procedure’s shear force and bending mo-ment at each story level for the 1st mode is the same with

Pushover forcef = α M ϕi

Vbi

Uir

(a)

Di

Fbi

Cyclic pushover analysisFlag-shaped hysteretic model (Ruaumoko 2D)

(b)

Figure 7: Hysteretic behavior of the ith mode inelastic SDOF system: (a) cyclic pushover curve; (b) Fsi-Di relationship.


those of the RSA procedure and is in good correspondencewith those of the UMRHA procedure. +e 2nd mode shearforce and bending moment of the MMSP procedure is muchhigher than those of the UMRHA procedure. +e 3rd and 4th

mode shear force and bendingmoment at each floor level of theMMSP procedure matches well with those of the UMRHAprocedure as it remains in the elastic region in both procedures.From the above discussion, it is concluded that themain sourceof error in determining the seismic demand through simplifiedanalysis procedures lies in the 2nd mode seismic demand.

To further explore the reason of inaccuracies in sim-plified analysis procedures, the force-deformation relationsof first four modes for ground motion HM at MCE level, asillustrated in the Figures 12(a) and 12(d), are considered.+ehysteretic behavior for the first and second modes is flagshaped, while it is straight line for other higher modes whichshows that, for the first twomodes, the building goes into thenonlinear range as the base moment demand exceeds theflexural yield capacity. When the yielding occurs at the baseand the deformation continues, then the base moment

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Figure 8: Comparison of peak shear and peak moment demands between UMRHA and NLRHA procedures for both DBE and MCE levelsfor groundmotion HM: (a) shear at DBE level; (b) shear at MCE level; (c) bending moment at DBE level; (d) bending moment at MCE level.


increases but at much lower rate due to low postyieldstiffness as compared to the initial stiffness. +erefore, thebase moment will saturate around the flexural yieldingstrength and consequently the shear force and moment ofeach floor level will also saturate for that mode. For the first

mode, the seismic responses from the UMRHA procedureare therefore equal to the 1st mode seismic demands ob-tained by the RSA procedure. +e slight difference betweenthe 1st mode responses is due to the actual material strengthused in the nonlinear model. +e 1st mode seismic demands

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Figure 9: Modal peak shear and moment demands from the UMRHA procedure for MCE level ground motion. Modal peak shear of (a) 1st

and 2nd modes; (b) 3rd and 4th modes. Modal peak moment of (c) 1st and 2nd modes; (d) 3rd and 4th modes.


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Figure 10: Comparison of modal shear MCE level demands of RSA,MMSP, MPA, andMMPAwith that of the UMRHA procedure: (a) firstmode; (b) second mode; (c) third mode; (d) fourth mode.


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Figure 11: Comparison of modal moment MCE level demands of RSA, MMSP, MPA, and MMPA with that of the UMRHA procedure: (a)first mode; (b) second mode; (c) third mode; (d) fourth mode.


of the MMSP procedure are also the same as those of theRSA procedure as these are reduced by the same R factor.+e MPA and MMPA procedures use the nonlinear force-deformation relationship for computing the 1st mode seis-mic demand, and therefore, it matches well with 1st modeseismic demand of the UMRHA procedure. +e hystereticbehavior of the 2nd mode also shows yielding but it is lessextensive as compared to that of the 1st mode yielding. +eforce-deformation relationship of the 2nd mode is also flag-shaped self-centering as shown in the Figure 12(b). +eseismic demands of the 2nd mode are also limited at the levelwhen the base moment of the 2nd mode reaches flexuralyielding strength. However, the RSA procedure assumes thesame R factor for second and higher modes which UMRHAresults have shown to be inaccurate and the 2nd mode isshown to exhibit a much less nonlinear behavior producing asmaller R value for this mode.+eMMSP’s 2nd mode seismic

demands are the elastic demands of the RSA procedure andare much higher than the seismic demands from theUMRHA procedure. Only the MPA 2nd mode seismic de-mands match with the seismic demands of the UMRHAprocedure as it incorporates the nonlinear behavior intoseismic demand computation. +e 2nd mode demands fromMMPA are the elastic demands and are much greater thanthe seismic demands from the UMRHA procedure. +ehysteretic behavior of the third and fourth mode are illus-trated in Figures 12(c) and 12(d), and it shows that thesehigher modes remain in the elastic region and there is noyielding occurring in these modes to reduce or limit theirseismic demands. Consequently, the 3rd and 4th modeseismic demands from UMRHA are much higher than theseismic demands of the RSA procedure and equal to theelastic seismic demands of the RSA procedure. +e MMSP3rd and 4th mode demands match well with those of the

–2.5

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× 1

04 )


–3

–2

–1

0

1

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3

4

–0.01 –0.005 0 0.005 0.01

Mode 4

(d)

Figure 12: Hysteretic behavior of first four translational modes: (a) first mode; (b) second mode; (c) third mode; (d) fourth mode.


UMRHA procedure. +e MPA and MMPA procedure 3rdand 4th mode seismic demands are equal to the UMRHAprocedure. If these modes seismic demands were higher thanyield moment at the base, then the MMSP and MMPAprocedure seismic demands will be inaccurate, while MPAprocedure will accurately estimate the seismic demandsbecause it uses the actual nonlinear behavior of the building.+e above discussion suggests that the response modifica-tion factor (R) from the RSA procedure is suitable only forthe 1st mode, while a smaller value of force reduction factorshould be used for the 2nd mode, and the higher modesshould not be reduced by any force reduction factor as theseremain in the elastic region. Although the modal seismicdemands of theMPA procedure match well with those of theUMRHA procedure, the total seismic demands of the MPAprocedure differ. +is is due to the use of the modalcombination rule for determining the total seismic demandsfrom modal seismic demands in the MPA procedure incontrast to the addition of modal time histories to get thetotal response in the UMRHA procedure. +e above dis-cussion suggests that the current simplified procedurescannot be reliably used for the prediction of seismic de-mands of high-rise core wall structures. +erefore, an effortis made in the next section to further explore the reasons ofinaccuracies and to come up with a simplified yet effectiveprocedure with improved accuracy.

5. Proposed Modifications in SimplifiedAnalysis Procedure

Based on the reasons of inaccuracies in simplified analysisprocedures discussed in the previous section, we can suggestsome modifications in simplified analyses procedure to im-prove their accuracy. As the MPA and MMPA proceduresrequire nonlinear modelling, these are not used in the currentsection. +e RSA procedure, on the contrary, does not requirenonlinear modelling, is easy to use and most practicing en-gineers are aware of this method and hence is used in thissection.

+e RSA procedure assumes that inelastic demands of eachmode can be calculated by dividing their elastic seismic de-mands with a constant response reduction factor but as theaforementioned discussion shows only the 1st mode seismicdemands reduce by the code-based response reduction factorwhile higher modes do not reduce by the same factor.+erefore, each modal elastic demand computed by the RSAprocedure should be divided by separate responsemodificationfactor and then combined by SRSS combination rule to obtainthe total seismic demand of the high-rise building. To calculatethe responsemodification factor for eachmode, a rather simplemethod is adopted. +e elastic modal overturning moment atthe base for each mode is divided by the yield base momentcapacity/design base moment capacity of the building. A valueequal to or less than unity means the absence of yielding in thatmode, and hence, the elastic seismic demand for that particularmode is used. +e value of the force reduction factor obtainedfor the 1st mode of DBE level ground motion HM is 4.3 whichis smaller than the code-based value of 5.5, while it is 6.4 forMCE level groundmotionwhich is greater than the code-based

value. +e force reduction factor for the 2nd mode is 1.53 forDBE level, while it is 2.30 for MCE level ground motion.+esevalues for 2nd mode are much smaller than the code-basedvalues used in the RSA procedure. +e higher mode elasticmodal moments at the base are smaller than the yield momentcapacity at the base. +erefore, the force reduction factor forthese higher modes is unity. After obtaining the value of eachmodal force reduction factor, the elastic modal shear forces andbending moments at each floor computed by the RSA pro-cedure are divided by these force reduction factors and thencombined by the SRSS combination rule to get the total seismicdemands.

+e seismic demands obtained by this modified RSAprocedure are then compared with seismic demands of RSA,MPA, and NLRHA procedures for ground motion HM forboth DBE and MCE levels as shown in Figures 13(a)–13(d).+e results show that the seismic demands computed by thismodified method are much higher and accurate than thoseof the RSA procedure. It is important to note here that thismodified RSA procedure does not require nonlinear mod-elling, and the accuracy is comparable to that of the MPAprocedure which requires nonlinear modelling and push-over analysis. However, still this procedure is under-estimating the seismic demands and hence needs to beimproved. As discussed earlier in UMRHA results, themodal pushover behavior shows yielding at the base againstboth 1st and 2nd mode seismic demands, while higher modesremain in the linear elastic range. However, in a realstructure with the coupling of different modes, such a sit-uation seems unrealistic. As a structure yields at base againstthe 1st mode forces, it is physically not possible for the highermode seismic demands to cause another yielding at the samepoint. However, higher modes can cause yielding at themidheight where the seismic demands from the 2nd modecan exceed the midheight moment capacity, but as the casestudy building is modelled as elastic above the base, ayielding against higher modes above the base is not possible.+is implies that the MMPA procedure and MMSP shouldbe able to accurately estimate the seismic demands as theseprocedures consider yielding in only first mode. However, asdiscussed earlier, these procedures also did not accuratelyestimate the seismic demands of the case study building.+efirst reason behind this is that when the 1st mode yields andplastic hinge forms at the base, the stiffness at the basereduces, and due to the reduction in the stiffness, the highermode natural time period increases. When the natural timeperiod increases, the spectra show a decrease in the pseu-doacceleration due to which the 2nd mode seismic demandsdecrease. +e second reason behind the inaccuracy of theMMSP and MMPA procedure is that the when the 1st modeyields at the base, then the pattern of mode shapes alsochanges due to change in the boundary condition while theseprocedures are based on the elastic mode shape patterns.

+e modified RSA procedure accurately estimates theseismic demands of the case study building compared to theRSA and matches well with MPA procedure, but still itunderestimates the seismic demand compared to UMRHAand NLRHA procedures. Qureshi [34] and Pennucci et al.[35] showed that when the 1st mode yields at the base, then


plastic hinge forms which changes the mode shape patternandmodal responses.+emodal periods also lengthen whenthe 1st mode yields.+erefore, the use of separate R factor forhigher modes is not a feasible solution in the current case. Infact, the lengthened modal periods should be used for de-termination of seismic demands for higher modes.

To calculate the reduced modal stiffness, Sullivan usedsecant stiffness to find out the lengthened time periods andeventually used them to calculate the nonlinear modal re-sponses of higher modes. Irshad [34] used postyield andsecant and tangent stiffness to estimate the reduced modalstiffnesses. Secant stiffness was found out to give relatively

accurate results. However, that study used the direct dis-placement based design (DDBD) for the design procedurewhich means that the total deformation demand was knownat the start, and hence, it can be used to calculate the secantstiffness by joining the origin with the maximum de-formation point on the force-deformation curve. In thecurrent study, deformation demand is not readily available;however, the force-deformation hysteresis behavior of thestructure against ground motions calculated with theUMRHA procedure can be used to calculate the maximumdeformation to be used for the secant stiffness. +is de-formation value can also be calculated by using the natural

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(d)

Figure 13: Evaluation of shear and moment demands of the modified RSA procedure with RSA, MPA, and NLRHA procedures for groundmotion HM for both DBE and MCE levels: (a) shear at DBE level; (b) shear at MCE level; (c) bending moment at DBE level; (d) bendingmoment at MCE level.


time period of the structure using design displacementspectra by assuming that the equal displacement assumptionholds which states that the linear and nonlinear deformationsin a structure are approximately equal. It is important tomention here that the yielding does not only lengthen thetime periods but also changes the shape of the inertial modalforce patterns. However, this change in the modal shapepattern is ignored in the current study for simplicity.

+e seismic demands of higher modes are obtainedby using the secant stiffness for both DBE and MCElevels and joining the 1st mode inelastic or design demandswith higher modes seismic demands with elongated period.Figures 14(a)–14(d) shows that both the shear force and

bending moment results from the abovementioned for-mulation (stiffness-modified RSA procedure) are moreaccurate as compared to the modified RSA and MPAprocedures. +e limitation of this new procedure is that themodal secant stiffness cannot be obtained if the elasticmodelling is considered. Hence, future work should focuson the calculation of deformation demand to obtain thesecant stiffness. One possible solution to be considered is toassume the postyield stiffness of the structure as a per-centage of initial stiffness which is known. +en calculatethe maximum deformation demand from the displacementspectrum by relying on the equal displacement assumption.Now the secant stiffness can easily be calculated by joining

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0 1 2 3 4

Leve

l no.

(d)

Figure 14: Evaluation of shear and moment demands of stiffness-modified RSA procedure with Modified RSA, MPA, and NLRHAprocedures for groundmotion HM for both DBE andMCE levels: (a) shear at DBE level; (b) shear atMCE level; (c) bendingmoment at DBElevel; (d) bending moment at MCE level.


origin with maximum deformation point, and all of this canbe done without any nonlinear modelling and pushoveranalysis.

6. Conclusions

+e current study is focused on the comparison of anumber of simplified analysis procedures based on theirrelative accuracy and ease of application to a real structure.For this purpose, a 40-story RC core wall building wasselected to compute the nonlinear seismic demands usingthe various simplified analysis procedures and NLRHAprocedure. Comparison of individual simplified analysisprocedures with the NLRHA procedure shows that ELFand RSA procedures highly underestimate the seismicdemands while the MMSP and MMPA procedures over-estimate the seismic demands. Only the MPA procedureseismic demands are comparable with the NLRHA pro-cedure. To further explore the seismic responses and un-derstand the reason of inaccuracies of simplified analysisprocedures, a modal decomposition method is employed todecompose the total responses into model responses. +emodal decomposition results show that actual modalseismic demands of the 1st mode reduced by the samecode-based R factor used in the RSA procedure and MMSPprocedure and matches well with the 1st mode seismicdemands of MPA and MMPA procedures. +e modalseismic demands of the 2nd mode of RSA procedure doesnot reduce by the code-based response reduction factor.MMSP and MMPA procedure seismic demands of the 2ndmode were higher than the actual seismic demands. Onlythe 2nd mode seismic demands of the MPA procedurematched with true seismic demands of the 2nd mode as theMPA procedure considers the actual nonlinear behavior inthe model. +e actual modal seismic demands of the 3rdand 4th mode are much higher than the modal seismicdemands of the RSA procedure due to the use of R factor forall modes and match well with modal seismic demands ofMMSP, MPA, and MMPA procedures as these highermodes were found to remain in linear elastic range. +ereason behind the inaccuracy in simplified analysis pro-cedures lie in the computation of the 2nd mode seismicdemands. However, the modal seismic demands of MPAprocedures for all modes match well with actual modalseismic demands but still the total seismic demands areinaccurate due to the use of modal combination rule anduse of elastic mode shapes in computing the modal de-mands. In the current study, a modified RSA procedure wassuggested to compute the seismic demands by using sep-arate force reduction factors for each mode to reduce theelastic demands into nonlinear seismic demands. +e re-sults of this procedure, based on linear elastic modellingand without any need for pushover analysis, are compa-rable with those of the MPA procedure for the present casestudy building, but still it underestimates the seismic de-mands. To further improve the results of this modified RSAprocedure, nonlinear modal properties are calculated byusing secant stiffness. +ese modified modal properties arethen used to calculate the higher mode seismic responses

and are then combined with 1st mode seismic demandsbased on R factor. +e proposed procedure is found to besimple to use with a higher level of accuracy.

Abbreviations

NLRHA: Nonlinear response history analysisRC: Reinforced concreteUMRHA: Uncoupled modal response history analysisELF: Equivalent lateral forceRSA: Response spectrum analysisMMSP: Modified modal superposition procedureWCD: Weighted capacity designMPA: Modal pushover analysisNSP: Nonlinear static procedureMMPA: Modified modal pushover analysisSDOF: Single degree of freedomMDOF: Multidegree of freedomMCE: Maximum considered earthquakeDBE: Design basis earthquakeTBI: Tall Building InitiativesSRSS: Square root of the sum of square methodCQC: Complete quadratic combinationR: Force reduction factor.

Data Availability

+e data of this research article are available from thecorresponding author upon request.

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper.

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