Engineering Structures 32 (2010) 174–183

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

A story shear-based adaptive pushover procedure for estimating seismicdemands of buildingsKazem Shakeri a,∗, Mohsen A. Shayanfar b, Toshimi Kabeyasawa ca Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iranb Department of Civil Engineering, Iran University of Science and Technology, Tehran 16846, Iranc Earthquake Research Institute, University of Tokyo, Yayoi 1-1-1, Bunkyo-ku, Tokyo 113-0032, Japan

a r t i c l e i n f o

Article history:Received 24 March 2008Received in revised form27 August 2009Accepted 1 September 2009Available online 4 October 2009

Keywords:Seismic demandsAdaptive pushoverStory shear-basedSign reversalHigher modes

a b s t r a c t

In recent years some adaptive pushover methods have been proposed to include the effects of the highermodes and the changes in the vibration characteristics during the inelastic response. However, becauseof using the quadratic combination rules to combine themodal forces, the changes in the sign of the storycomponents in the higher modes are removed. Consequently, the magnitudes of the applied loads in allstory levels are positive and these adaptive methods are not superior to their non-adaptive counterparts.Here, an innovative adaptive pushovermethod, called ‘‘SSAP’’, is proposed based on the story shearswhichtakes into account the reversal of sign in the highermodes. In each step, the applied load pattern is derivedfrom the instantaneous combined modal story shear profile. The sign of the applied loads in consecutivesteps are changed and the structure is simultaneously pushed and pulled in different story levels. Anotheraspect of the proposed method is that at each step an assumed fundamental mode shape is derived fromthe load profile. Based on this adaptive fundamentalmode shape and the energy concept, themulti degreeof freedom system is converted into a single degree of freedom system. The proposedmethod is applied totwo steelmoment-frame buildings. The results show an admirable accuracy in prediction of peak inelasticdrift response, especially where the effects of the higher modes are important. A combination of thismethod with the conventional pushover approach, called ‘‘SS–M1’’, results in more accurate estimationof peak inelastic drift in all story levels compared to the other pushover approaches.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

During the last decade, pushover analysis has beendeveloped asa practical tool for estimating seismic demands of inelastic struc-tures [1–4]. In the conventional pushover analysis, an invariant lat-eral force pattern is incrementally applied to the structural modeluntil a predetermined target displacement is achieved. This proce-dure accurately estimates the global seismic response of low-risebuildings,where the response is dominatedby the firstmode [5–7].However, in high-rise buildings where the effects of the highermodes are important, this procedure is flawed [8–11]. The majordrawback of the conventional pushover is that it cannot considerthe effects of the higher modes and the progressive changes in themodal properties due to structural yielding.In order to include the effects of the higher modes, some ad-

vanced multi-run modal pushover procedures based on the elas-tic modal decomposition concepts have been developed [12–16].

∗ Corresponding author. Tel.: +98 914 451 3150; fax: +98 451 2252096.E-mail addresses: shakeri@uma.ac.ir, k_shakeri@iust.ac.ir (K. Shakeri).

0141-0296/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.09.004

In the most well-knownmodal pushover analysis (MPA) proposedby Chopra and Goel [15], multiple pushover analyses with a lat-eral load corresponding to the considered elastic mode shapes areconducted separately, and then the total seismic response is esti-mated by combining the responses due to eachmodal load. Since inthe higher modes the increase of the roof displacement is not pro-portional to the other stories’ displacement, andmay even reverse,Hernandez-Montes et al. [17] have developed the MPA procedurebased on the energy concepts. Also, a modified version of the MPA(MMPA) based on elastic spectral response has been proposed byChopra et al. [18]. Furthermore, in order to consider the progres-sive changes in the modal properties, adaptive form of the multi-run pushover procedures are developed [19–21]. However, thesemulti-run methods whether in an adaptive or non-adaptive formare not able to reflect the yielding effect of one mode in the othermodes and the interaction between modes in the nonlinear range,except the method proposed by Aydinoglu [20]. Furthermore, themodal combination rules such as square-root-of-the-sum-of-the-squares (SRSS) are valid to combine the responses of independentmodes. Since in the inelastic domain, the structural system couldnot be decomposed into independentmodes, the application of themodal combination rule in the inelastic phase is no longer valid.

K. Shakeri et al. / Engineering Structures 32 (2010) 174–183 175

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Fig. 1. The process for defining the applied load pattern at one step of the proposed SSAP and FAP methods. (a) Modal story forces. (b) Story shears profile for each mode.(c) Combined modal story shears profile. (d) Load pattern in the SSAP method. (d′) Load pattern in the FAP method.

On the other hand some researchers have developed enhancedsingle-runmodalmethods inwhich the structures are pushedwithcombined modal forces. The modal combination concept is usedto define the load pattern [22–24] rather than to combine thenonlinear responses due to each mode. Adaptive forms of thesemethods have also been proposed [25–28]. At each step, the loadpattern is defined by combining the instantaneous modal loadsand is applied through a single pushover analysis (single-run). Theapplied load vector reflects the contributions of the instantaneoushigher modes, and the interaction between the modes in theinelastic phase. This study, however, has focused on the single-run adaptive pushover procedures and a new single-run adaptivepushover method has been proposed.

2. Single-run adaptive pushover procedures

Single-run adaptive pushover analysis was first proposed byReinhorn [29] and Bracci et al. [30] where at each step thelateral load pattern is updated based on the inelastic story forcesequilibrated in the previous step. Satyarno et al. [31] have proposedan adaptive method based on the variation of the fundamentalmode in consecutive steps. At each step, the fundamental modeis estimated through a modified Rayleigh method based on theincrement of the displacement in the previous step. Requena andAyala [25] have established an adaptive pushover method basedon the contribution of the instantaneous higher modes. On theother hand, an advanced adaptive pushover procedure has beenproposed by Elnashai [26], which has been subsequently furtherdeveloped by Antoniou and Pinho [27] through a fiber finiteelement model, called force-based adaptive pushover (FAP). Inthis method, at each step according to the instantaneous stiffnessmatrix and the corresponding elastic spectral accelerations, themodal story forces are obtained for the interestedmodes. Then thelateral load pattern is calculated by combining the story forces ofeach vibration mode (see Fig. 1a and d′). This method conceptuallyhas a strong background compared to the conventional pushoverand considers the effects of the higher modes, interaction betweenmodes, progressive changes in the modal properties, stiffness

degradation and the frequency content of a design or particularresponse spectra. Furthermore, the behavior of the structure andthe plastic hinges formed at the consecutive steps are tracedeasily through a single-run pushover. However, generally the FAPprocedure cannot provide a major advantage compared to its non-adaptive counterparts [27,32].As described by Pinho and his co-workers [27,33] as well as

by Papanikolaou et al. [32], such an unsuccessful performancecan be the consequence of the quadratic modal combination rules(e.g. SRSS) used to combine the modal loads which, always leadsto a positive value for all the story levels in the incremental loadpattern (see Figs. 1d′ and 2a). Consequently, the effects of the signreversal in the higher modes forces are not reflected in the appliedload pattern and thus only the amount of the modal forces arereflected.Antoniou and Pinho [28] have also developed an interesting

displacement-based adaptive pushover method (DAP), in whichat each step a displacement loading is applied rather than forceactions. In this method the story forces are not directly appliedto the structure but rather calculated as a response of the applieddisplacement loads. The story forces are the result from structuralequilibrium and could take reversal of sign in the story forceprofile. This reversal of sign can be interpreted as an effect of thesign reversal of the modal forces in the higher modes, which areimplicitly considered in the DAP procedure. The performance ofthe DAP procedure is better than the FAP [28,33] and has beenfurther investigated through multi-ground motion incrementaldynamic analyses [34]. This paper, however, mainly focuses on theforce-based single-run procedures and the displacement-basedprocedures are not within its scope.In order to remedy the drawbacks of the FAP procedure through

a force-based procedure, this paper proposes a story shear-basedadaptive pushover method, where at each step the load pattern isderived from themodal story shears profile. The proposedmethodexplicitly takes into account the changes in the sign of the storycomponents in the higher modes. Also a solution is provided todetermine the target displacement in the adaptive nonlinear staticanalysis.

176 K. Shakeri et al. / Engineering Structures 32 (2010) 174–183

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3. Story shear-based adaptive procedure for nonlinear staticanalysis

The proposed procedure consists of the following three keyelements: (1) Updating the load pattern at each step based on themodal story shear profile. (2) Deriving the assumed fundamentalmode shape at each step from the existing load profile. (3)Converting the capacity curve coordinates of theMDOF system intoan SDOF system based on the assumed fundamental mode shapeand the energy concepts. These key elements are explained in thefollowing sections.

3.1. Load pattern

As mentioned before, in the FAP method which uses the SRSSrule, the amount of the applied load at each story is definedindependently and regardless of the sign and amount of the loadin the other story levels. However, this may not be appropriatesince the inter-story drift (a crucial index in damage assessment) isaffected by the amount of story shear which in turn is affected bythe sign and amount of the applied forces at the upper story levels.Hence, a story shear-based adaptive pushover (SSAP)method is

herein proposed. The proposed procedure considers the contribu-tion of the instantaneous higher modes and the effect of the signreversal of the modal forces in the higher modes. At each analy-sis step, based on the instantaneous dynamic properties, the storyshears associated with each considered mode are calculated byEqs. (1) and (2) (see Fig. 1a and b). The story shears associatedwith each mode are combined using the SRSS rule (Eq. (3)); this isdefined as the combined modal story shear (see Fig. 1c). In thecalculation of the story shears for each mode using Eq. (2), thesign reversal effects of the modal forces in the upper stories areconsidered.Fij = Γj ∅ijmiSaj (1)

SSij =n∑k=i

Fkj (2)

SSi =

√√√√ m∑j=1

SS2ij (3)

where, i is the story number, j: themode number, ∅ij: the i-th com-ponent of the j-th eigenvector (mode shape), mi: the mass of thei-th story, Saj: the spectral acceleration corresponding to the j-thmode, Γj: modal participation factor for the j-th mode, SSij: thestory shear in level i associated with mode j, SSi: the modal storyshear in level i associated with all the considered modes.The lateral forces required to generate the combined modal

story shears profile are assumed as the lateral load pattern. Therequired story forces are calculated by subtracting the combinedmodal shear of consecutive stories using Eq. (4) (see Fig. 1d). Thelateral load pattern is normalized with respect to its total value byEq. (5). The incremental applied load profile at each step is thencomputed by Eq. (6).

Fi = SSi − SSi+1 i = 1, 2, . . . , (n− 1)Fn = SSn i = n (4)

Fi =Fi∑Fi

(5)

1Fi = 1Vb × Fi (6)

where,1Vb is the incremental base shear,1Fi: the ith componentof the incremental applied load at each step.Hence, if even the SRSS combination rule is used, the calculated

applied force at each story can be negative whenever the amountof the modal shear in one story is less than the amount of the up-per story. Therefore, this method is able to simulate the force dis-tribution profile with story force reversal as observed in dynamicanalysis (see Fig. 1d). The variation of the incremental applied loadpattern at different steps during the SSAP and FAP procedures fora sample model are shown in Fig. 2.

3.2. Target displacement

In the conventional nonlinear static procedure (NSP), the tar-get displacement is approximated through the maximum inelasticdisplacement of the equivalent SDOF system. The lateral load pat-tern is defined based on the assumed constant fundamental modeshape using Eq. (7). The base shear versus the top displacementcurve of the MDOF system is converted to a spectral acceleration

K. Shakeri et al. / Engineering Structures 32 (2010) 174–183 177

versus spectral displacement curve by Eqs. (8) and (9). In otherwords, the capacity curve of the MDOF is converted to a force ver-sus displacement curve of the equivalent SDOF system with unitmass. The mode shape used in the Eqs. (8) and (9) is the sameas fundamental mode shape used to define the load pattern inEq. (7) [1,3].

{f } = [m]× {∅} (7)

F∗ = Sa =VbM∗

(8)

D∗ = Sd =urΓ φr

(9)

where, {∅} is the assumed fundamental mode shape, [m]:the mass matrix, Vb: the base shear, ur : the roof displace-ment, ∅r : the component of the {∅} in the roof level, Γ =({∅}

T . [m] . {1})/({∅}

T . [m] . {∅}): the participation factor, M∗ =

Γ · L: the effective mass and L = {∅}T . [m] . {1}.Since in the single-run adaptive modal pushover methods, at

each step the load pattern reflects the contribution of the instan-taneous higher modes, the load pattern is not compatible with anydistinct dynamic modal shape. Consequently, using each constantmodal shape as a fundamental mode shape in Eqs. (8) and (9) is notmeaningful. In this regard, Casarotti and Pinho have developed anadaptive capacity spectrummethod (ACSM) [35,36]. In this paper asimilar concept is used, where at each step (k) an assumed equiva-lent fundamental mode shape (EFMS) is derived from the existingload profile using Eq. (10) (inverted form of the Eq. (7)).

{∅}(k)= [m]−1 × {f }(k) (10)

where, {f }(k) is the vector of the total forces in the structure at stepk, {∅}(k): the equivalent fundamental mode shape at step k, [m]:the mass matrix.This assumed equivalent fundamental mode shape is compati-

ble with the load pattern at each step and also reflects the highermodes contribution. Based on the assumed equivalent fundamen-tal mode shape, {∅}(k), the base shear of the MDOF system at eachstep is converted to the equivalent force of the SDOF system withunit mass, F∗(k), using Eq. (8).In the assumed equivalent fundamental mode shape as well as

in the higher modes, the increase of the roof displacement is notproportional to the other stories and may even reverse. Therefore,using the roof displacement as a conversion parameter from theMDOF system to the SDOF system cannot bemeaningful except forthe first mode. In this regard, an energy-based method is used todefine the value of the displacement in the equivalent SDOF systemwhile considering the amount and sign of the displacements in allstories [17]. At each step (k) the increment of the displacement inthe SDOF system is defined based on the sum of the work done atthe different story levels through each incremental force, dF (k)i . Ateach step, the total work done in all stories is assumed to be equalto the work done by the base shear (Eq. (11)), and the equivalentdisplacement is calculated using Eqs. (12) and (13)n∑i=1

((F (k−1)i +

12dF (k)i

)×1d(k)i

)

=

(n∑i=1

(F (k−1)i +

12dF (k)i

))×1D(k) (11)

1D(k) =n∑i=1

((F (k−1)i +

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)1d(k)i

)/n∑i=1

(F (k−1)i +

12dF (k)i

)(12)

D(k) = D(k−1) +1D(k) (13)

where, F (k−1)i : the existing force in the story i at the end of step k−1

dF (k)i : incremental applied force in the story i at step k1d(k)i : incremental displacement in the story i due to increasedapplied load at step k1D(k): incremental displacement of the equivalent SDOFsystem at step kD(k) : displacement of the equivalent SDOF system at step k.

The derived capacity curve for the equivalent SDOF system(F∗–D curve) is idealized as a bilinear curve and is plotted againstthe inelastic acceleration–displacement response spectra (demandspectra). The intersection of the capacity curve and the demandspectra is defined as the target displacement (or performance pointfor the system). In this study, in order to verify the proposedmethod against the nonlinear time history (NTH) analysis, thepeak displacement of the bilinear inelastic SDOF system (targetdisplacement) is thus computed through theNTHanalysis. For eachanalysis step carried out within the SSAP procedure, the structuraldemands can be treated as a pseudo seismic demand.The proposed procedure is summarized in a sequence of basic

stages as follow:

1. Create a structural model incorporating nonlinear materialcharacteristics.

2. Perform an eigenvalue analysis to compute the instantaneousnatural frequencies, ω j and mode shapes, ∅j.

3. For a selected number of modes that are considered, computethe modal story forces at each level by Eq. (1).

4. Calculate modal story shears (SSij) for the considered modesusing Eq. (2) and combine them using SRSS to define thecombined modal shear (SSi) at each story (Eq. (3)).

5. Define the amount and sign of the incremental load patternprofile at each story from subtracting the combined modalshear of consecutive stories by Eq. (4)

6. Normalize the incremental load pattern with respect to thetotal value by Eq. (5).

7. Scale the normal incremental load profile by the predefinedincremental base shear,1Vb at each step using Eq. (6).

8. Apply the scaled incremental load profile to the structuralmodel; compute displacements, inter-story drifts, and elementforces, etc.

9. Compute the assumed equivalent fundamental mode shape(EFMS) by Eq. (10) and convert the base shear of the MDOFsystem to the equivalent force of the SDOF system with unitmass, based on the assumed EFMS, Eq. (8).

10. Compute the equivalent displacement in the SDOF systememploying Eqs. (12) and (13).

11. Go back to step 2 and repeat the process until the supposedmaximum base shear (an extreme value) is achieved.

12. Develop the Force–Displacement curve, (F∗–D curve) of theequivalent inelastic SDOF system with unit mass (spectralacceleration versus spectral displacement, Sa–Sd) based on thecomputed value in steps 8 and 9 in the previous cycles andidealize it as a bilinear curve.

13. Plot the capacity curve, (i.e. F∗–D curve) against the inelasticacceleration–displacement response spectra and define thetarget displacement. In this study, in order to verify themethod against the NTH analysis, the peak displacement of thebilinear inelastic SDOF system (target displacement) is directlycomputed through the NTH analysis.

14. Determine the corresponding step to the target displacementin the pushover procedure and obtain the interested pseudoseismic demands (e.g. story drift, element force, etc.)

178 K. Shakeri et al. / Engineering Structures 32 (2010) 174–183

Table 1Ground motion properties.

Earthquake Year M Ms Station Component Closest distance to fault (km) PGA (g) PGV (cm/s) Site conditionCWB USGS

Northridge 1994 6.7 6.7 77 Rinaldi receiving 228 7.1 0.838 166.1 C CLoma Prieta 1989 6.9 7.1 16 LGPC 0 6.1 0.563 94.8 A –Landers 1992 7.3 7.4 24 Lucerne 275 1.1 0.721 97.6 A AKobe 1995 6.9 – 0 KJMA 0 0.6 0.821 81.3 B BErzican 1992 6.9 – 95 Erzican NS 2 0.515 83.9 D CTabas 1978 7.4 7.4 9101 Tabas LN – 0.836 97.8 C –

FAP

M1

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SS-M1

MMPA

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20

Loma Prieta Landers Kobe Erzican

SAC 20

TabasNorthridge

Err

or

on

dri

ft (

%)

0

25

Fig. 3. Observed errors in the different NSPs for the SAC-20 building under the different records.

4. Validation of the proposed method

The proposed algorithm in the SSAP procedure is implementedthrough a computer code incorporated the DRAIN-2DX [37].The SSAP procedure is verified for two levels of the typicalmedium- and high-rise steel buildings denoted as SAC-9 and SAC-20 buildings. The responses resulted from the SSAP procedure arecompared to those resulted from the NTH analysis. Six earthquakerecords are considered in this study. Three of these records havebeen recorded in California and the other earthquakes have beenrecorded in Japan, Turkey and Iran.

4.1. Structural models (SAC-9 and SAC-20 buildings)

SAC-9 and SAC-20 buildings are respectively nine-story andtwenty-story perimeter steel moment resistant frame (SMRF)buildings designed by consulting structural engineers for the PhaseII of the SAC project [38]. These structures meet seismic coderequirement of the 1994 UBC for the Los Angeles, California region.The SAC-9 and SAC-20 have been used as benchmark structuresin SAC projects and also by numerous researchers in recent years,e.g. Ref. [11,39,40]. In this study, the two dimensional models areconsidered. Only one of the five-bay perimeter SMRFs representinghalf of the building in the N–S direction is modeled. The effectsof the gravity loads are neglected in the analysis. The structuralmodels are implemented in the DRAIN-2DX computer program.Further details of these buildings can be found in Ref. [40].

4.2. Ground motions

In order to investigate the accuracy of the SSAP method underdifferent ground motions, six near-fault motions are consideredfor the selected samples. The records are available in the PacificEarthquake Engineering Research (PEER) site, http://peer.berkeley.edu/smcat. The important objective in selection of the records isthat they induce an inelastic response in the sample buildings andexcite the effect of the higher modes. Among the two horizontalcomponents of a near-field motion, the considered component isthe one that posses the maximum ground velocity [21]. The mainproperties of the considered records are summarized in Table 1.

4.3. Comparative evaluation

The structural models of the considered buildings (SAC-20 andSAC-9) are subjected to the different ground motions listed inTable 1. The peak responses resulting from the NTH analyses areconsidered as benchmark responses. The resulting responses fromthe different NSPs are compared to these benchmark responses.The SSAP and FAP methods as well as the conventional proce-

dure based on the first mode (M1) are evaluated. Furthermore, theMMPA procedure as amodifiedmulti-run procedure and amethodcomposed of the SSAP andM1 procedures (SS–M1) explained later,are also evaluated. In the FAP, SSAP, SS–M1 and MMPA methods,the first threemodes are considered. The target displacement in allconsidered procedures except the FAP has been estimated throughthe maximum inelastic displacement resulted from the NTH anal-ysis of the equivalent SDOF system. Also in the SSAP procedure, theequivalent SDOF system is developed based on the energy concept.In the FAP procedure, the structural models are pushed until theroof displacement becomes equal to the response resulting fromthe NTH analysis.

4.3.1. Inter-story drift predictionTo compare the accuracy of the different NSPs, we use an error

index defined by Eq. (14) which has been presented by Lopez-Menjivar and Pinho [41].

Error1(%) = 100×1n

√√√√ n∑i=1

(1i-NTHA −1i-NSP

1i-NTHA

)2(14)

where, 1i-NTHA is the peak inter-story drift at a given level i,resulting from the NTH analysis,1i-NSP is the corresponding inter-story drift of the NSP, and n is the number of the stories. Wheneverthe error index is close to zero, the NSP response approaches theNTH analysis response.The error indices of the different NSPs are computed for the

SAC-20 and SAC-9 under the different records as summarized inFigs. 3 and 4. Furthermore, in Figs. 5 and 6, the inter-story drifts andthe displacements resulting from the different NSPs, respectively,

K. Shakeri et al. / Engineering Structures 32 (2010) 174–183 179

FAP

M1

SSAP

SS-M1

MMPA

5

10

15

20

25

Loma Prieta Landers Kobe Erzican

SAC 9

TabasNorthridge

Err

or

on

dri

ft (

%)

0

30

Fig. 4. Observed errors in the different NSPs for the SAC-9 building under the different records.

Fig. 5. Peak inter-story drift profiles resulting from the different NSPs and the NTH analysis for the SAC-20 and SAC-9 buildings under the sample ground motions.

are compared to the responses of theNTH analysis. In the followingsections, the observed errors and profiles are discussed.As presented in Figs. 3 and 4, the errors of the SSAP procedure

are less than the errors of the FAP procedure in all cases except theSAC-9 building under the Landers record. It is rather surprising thatthe errors of the FAP procedure for both buildings are more thanthe errors of the M1 method using constant load pattern based onthe first elastic mode. The reason for such underperformance wasdescribed before in Section 2. This shows that in the FAP method,not only the effect of the sign changes in the higher modes is notconsidered, but also the applied loadpattern is not compatiblewithany effectivemode shape (first mode) and it loses the advantage ofusing the fundamental first mode as a load pattern.The errors of the SSAP procedure for the SAC-20 building under

all of the records are significantly less than the errors of the othersingle-run procedures (e.g. FAP,M1) and theMMPAprocedure. Theeffects of the higher modes in the SAC-9 model are less than thosein the SAC-20 model. Thus the accuracy of the M1 method in theprediction of the inter-story drift of the SAC-9model is comparablewith the accuracy of the SSAP procedure. Furthermore, as shown inFig. 5 in both buildings, the predicted inter-story drift by the SSAPprocedure is close to the one resulting from the NTH analysis in the

upper stories where the effects of the higher modes are importantwhile the peak inter-story drift is predicted accurately in the lowerstories by the M1 procedure. As a result, combination of both SSAPand M1 methods could lead to an efficient method to accuratelypredict the inter-story drift in all stories. This combination can bedone conservatively by selection of the maximum drift resultingfrom the SSAP and M1 methods at each story. This combinedmethod is called SS–M1. The errors of the SS–M1 are also presentedin Figs. 3 and 4where in all cases they are significantly less than theerrors of the other methods.In Fig. 6 the displacement profiles resulting from each NSP are

compared to the responses of the NTH analysis. Since differentprocedures use different load patterns, the characteristics ofthe corresponding SDOF systems are different. Also in the SSAPprocedure the equivalent SDOF system is developed based onthe energy concept which is different from the other procedures.Consequently the target displacements and the predicted roofdisplacements in the different procedures are not equal as shownin Fig. 6.

4.3.2. Clarification of the load pattern and target displacement effectsIt is clear that the response of each procedure is affected by the

load pattern as well as by the target displacement. To evaluate the

180 K. Shakeri et al. / Engineering Structures 32 (2010) 174–183

Fig. 6. Peak story displacement profiles resulting from the different NSPs and the NTH analysis for the SAC-20 and SAC-9 buildings under the sample ground motions.

FAP

M1

SSAP

SS-M1

MMPA

5

10

15

20

Loma Prieta Landers Kobe Erzican

Err

or

on

dri

ft (

%)

0

25

SAC 20 - Equal roof displacement

TabasNorthridge

Fig. 7. Observed errors in the different NSPs using equal roof displacement for the SAC-20 building under the different records.

FAP

M1

SSAP

SS-M1

MMPA

5

10

15

20

25

SAC 9 - Equal roof displacement

Err

or

on

dri

ft (

%)

0

30

Loma Prieta Landers Kobe Erzican TabasNorthridge

Fig. 8. Observed errors in the different NSPs using equal roof displacement for the SAC-9 building under the different records.

efficiency of the SSAP procedure, it will be more informative if theeffects of the proposed load pattern and target displacement in thefinal improvement of the responses are separated.To remove the effects of different target displacements and to

evaluate only the effects of the load patterns, the structural modelsin all of the considered procedures have been pushed until the roof

displacement becomes equal to the response resulting from theNTH analysis. The error indices resulting from the different loadpatterns are computed for the SAC-20 and SAC-9 using Eq. (14)as summarized in Figs. 7 and 8. Furthermore, in Fig. 9 the inter-story drift profiles resulting from the different load patterns arecompared to the responses of the NTH analysis.

K. Shakeri et al. / Engineering Structures 32 (2010) 174–183 181

Fig. 9. Peak inter-story drift profiles resulting from the different NSPs using equal roof displacement and the NTH analysis for the SAC-20 and SAC-9 buildings under thesample ground motions.

NTHA

FAP

M1

SSAP

SS-M1

MMPA

NTHA

FAP

M1

SSAP

SS-M1

MMPA

1

NTHA

FAP

M1

SSAP

SS-M1

MMPA

0.02 0.04 0.06 0.08 0.1

Drift (m)

0 0.12

2468

101214161820

Sto

rey

0

22

2468

101214161820

Sto

rey

0

22

SAC 20 SAC 20

SAC 9 SAC 9

-50 500

Error (%)-100 100

-50 500

Error (%)-100 100

1

2

3

4

5

6

7

8

9

Sto

rey

0

10

1

2

3

4

5

6

7

8

9

Sto

rey

0

10

0 0.05 0.1 0.15 0.2

Drift (m)

a b

Fig. 10. (a) Mean of the inter-story drifts resulting from all of the records for the NTH analysis and the different NSPs. (b) Error of the mean inter-story drift of the differentNSPs with respect to the mean inter-story drift of the NTH analysis using all of the records.

As can be observed, the results shown in Figs. 7 and 8 aresimilar to Figs. 3 and 4. Also the inter-story drift profiles shownin Fig. 9 are similar and comparable to those in Fig. 5. In all of theprocedures, the target displacements are equal to the exact valueretrieved from the NTH analysis, however, the errors of the SSAPprocedure in all cases are less than the errors of the FAP procedure.Furthermore, the errors of the SS–M1 method in most cases areless than the errors of the other methods. This shows that theload pattern proposed in the SSAP procedure is more efficient than

the other load patterns. Even though the roof displacement is notpredicted accurately in the SSAP procedure using the energy-basedtarget displacement (see Fig. 6), but comparing the Figs. 5 and9 shows that using this energy-based target displacement couldimprove the inter-story drift response of the SSAP procedure.

4.3.3. Comparison of the mean inter-story driftsFor the purpose of general evaluation, in Fig. 10(a) the mean of

the inter-story drifts resulting from all of the records for the NTH

182 K. Shakeri et al. / Engineering Structures 32 (2010) 174–183

Fig. 11. ‘‘Mean− Standard deviations’’ (16 percentile) of the inter-story drifts resulting from all of the records for the NTH analysis and the different NSPs.

Fig. 12. ‘‘Mean+ Standard deviations’’ (84 percentile) of the inter-story drifts resulting from all of the records for the NTH analysis and the different NSPs.

2

4

6

8

10

12

14

16

FAP

M1

SSAP

SS-M1

MMPA

P 84%0

18

Err

or

on

dri

ft (

%)

P 84%P 50% P 16%

SAC-9

P 50% P 16%

SAC-20

Fig. 13. Error indices of the Mean and Mean± Standard deviations (16–84 percentiles) profiles resulting from the different NSPs.

analysis and the different NSPs are compared. Also in Fig. 10(b)the error of the mean inter-story drift of the different NSPs withrespect to the mean inter-story drift of the NTH analysis usingall of the records are shown. Furthermore, in Figs. 11 and 12 theMean± Standard deviations (16–84 percentiles) of the inter-storydrifts resulting from all of the records for the NTH analysis andthe different NSPs are compared. As presented in Figs. 10–12, theprofiles resulting from the SS–M1 procedure are close to thoseresulting from the NTH analysis.The error index of the Mean and Mean ± Standard deviations

(16–84 percentiles) profiles resulting from the different nonlinearstatic procedures (FAP, SSAP, M1, SS–M1 and MMPA) with respectto the corresponding profiles resulting from the NTH analyses areshown in Fig. 13. These error indices are calculated by a formulasimilar to Eq. (14) except that here the 1i-NTHA and the 1i-NSP arethe Mean or Mean ± Standard deviations of the inter-story driftsresulting from all of the records for the NTH analyses and the NSPs,respectively. As shown in Fig. 13, the errors of the SS–M1 methodin both cases (SAC-9 and SAC-20) for the different percentiles aresignificantly less than the errors of the other methods.

5. Conclusions

An innovative story shear-based adaptive procedure (SSAP) isproposed for nonlinear static analysis,which takes into account theeffects of the higher modes, interaction between modes in the in-elastic phase, progressive changes in the vibration characteristicsduring the inelastic response, period elongation and the frequencycontent of a design or particular response spectra while consid-ering the sign reversal of the story forces in the higher modes.Furthermore a solution is proposed to create the capacity curveof the equivalent single degree of freedom (SDOF) system in theadaptive nonlinear static procedure. The proposed SSAP proce-dure is evaluated through the SAC-20 and SAC-9 buildings undersix different near-fault ground motions and is compared to theother nonlinear static procedures. Themain results of this study aresummarized as follow:

1. The accuracy of the SSAP in drift is increased whenever theeffects of the highermodes are significant as in the upper stories

K. Shakeri et al. / Engineering Structures 32 (2010) 174–183 183

of both SAC-9 and SAC-20 buildings, and also in the SAC-20model with respect to the SAC-9 model.

2. The resulting inter-story drift profiles show that, in general, theaccuracy of the conventional nonlinear static procedure basedon the first mode (M1) in the lower story levels is better wherethe effects of the higher modes are less. On the contrary, theperformance of the SSAP in the upper story levels is better thanthe other procedures. Therefore, the combined method whichselects the maximum drift resulting from each SSAP and M1methods (SS–M1) leads to a conservative and more accurateprediction for all the stories. The errors of the SS–M1 methodin all cases of the analyzed samples are less than the errors ofthe other nonlinear static procedures.

Acknowledgments

The first author wishes to thank Professor Toshimi Kabeyasawafrom the Earthquake Research Institute at the University of Tokyowho hosted him as a visiting researcher in his lab. Finally,the authors would like to thank Professor Athol Carr from theUniversity of Canterbury, for reviewing this paper and they arealso grateful to the anonymous reviewers of this paper whosecomments have contributed to the improvement of the paper.

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