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An adaptive modal pushover procedure for asymmetric-plan buildings

Kazem Shakeri , Karim Tarbali, Mohtasham MohebbiFaculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran

a r t i c l e i n f o

Article history:Received 22 February 2011Revised 23 November 2011Accepted 25 November 2011Available online 3 January 2012

Keywords:3D pushoverNonlinear analysisAsymmetric-planAdaptiveModalStory shearTorqueEnergy

a b s t r a c t

An innovative single-run adaptive modal pushover procedure based on the modal story shear and torquehas been developed for seismic assessment of asymmetric-plan buildings. In which the load pattern con-

sists of the lateral forces and torques which are derived from the instantaneous combined modal storyshear and torque proles in each step. The contribution of the higher and torsional modes and the effectsof the changes in the structural properties during the inelastic domain are considered. Although the qua-dratic combination rule is used in dening the load pattern, the effects of reversal signs in the highermodes are simulated in the load pattern. The interaction between modes in the inelastic phase isreected in the single load pattern and the structural responses during the analysis are easily tracedthrough the single-run procedure. Furthermore the capacity spectrum curve is obtained from a formula-tion based on the modal energy concept and the ambiguity of choosing controlling point to obtain thetarget displacement in asymmetric-plan buildings has been eliminated. Accuracy of the proposed methodhas been veried on three asymmetric-plan buildings under different earthquake records. The resultsindicate that this method could predict the results of the nonlinear time history analysis appropriately.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

During the last decade, performance based design (PBD) proce-dure has become one of the most important research area in theearthquake engineering. Since structures exhibits nonlinear behav-ior during earthquakes, using the nonlinear analysis in PBD proce-dure is inevitable to observe whether the structure is meeting thedesirable performance or not.

Nonlinear time history analysis (NTHA) is the most accurateanalysis method however, it is too demanding since it takes toomuch time and computational efforts and the resulted responsescannot be used easily in the design practice. Also choosing a properacceleration record to run the analysis is a serious debatable issue.Therefore in order to solve these problems, nonlinear static proce-dure (NSP) based on pushover analysis has been developed as a

practical method [13] , in which the inelastic seismic demand isestimated through the inelastic design spectra.

However NSP in its current form in the seismic guidelines andcodes [1,2] suffers from some drawbacks. The major shortcomingof the conventional NSP is that it is restricted to a single-moderesponse,and cannot consider thecontributionsof thehigher modesand the effects of the progressive changes in the modes shapesbecause of structural yielding [4] . In order to remedy these draw-backs and to consider the effects of the higher modes, someresearchers have proposed multi-mode procedures [59] . Also to

consider the effects of the progressive changes in the structuralproperties, adaptive modal procedures have been proposed[1016] . However, since theconventional NSPand themost of theseadvanced procedures originally have been developed for two-dimensional (2D) frames, extending the application of NSP tothree-dimensional (3D) buildings with asymmetric plan is anongoing research.

2. Overview of 3D pushover procedures

In order to consider the torsion effects in the nonlinear static re-sponses of the asymmetric-plan buildings, Tso and Moghadam [17]have simply extended the application of planar pushover procedureto asymmetric-planbuilding. In theextendedversion of their meth-od [18] , the target displacement for each resisting element (e.g.planar frames, walls) is determined through the 3D elastic responsespectrum analysis of structuralmodel andthen the planarpushoveranalysis is performed for each resisting element independently.Kilar and Fajfar [19] have proposed using complete 3D model of structures in the nonlinear static procedure of asymmetric-planbuildings. They proposed to apply incrementally a height-wise dis-tribution of lateral forces at the mass centers of oors. Also Faellaand Kilar [20] have investigated the effect of applying lateral forcesat differentpoints in theplanof asymmetric-planbuildings.They donot propose any solution for obtaining the target point and in theirstudy, pushover analyses are performed until the mass center of top story reaches the maximum displacement obtained by theNTHA.Furthermore De Stefano and Rutenberg [21] haveconsidered

0141-0296/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi: 10.1016/j.engstruct.2011.11.032

Corresponding author.E-mail addresses: [email protected] , [email protected] (K. Shakeri).

Engineering Structures 36 (2012) 160172

Contents lists available at SciVerse ScienceDirect

Engineering Structures

j o u r na l h om e p a ge : www.e l s e v i e r. c o m / l oc a t e / e ngs t r uc t
http://dx.doi.org/10.1016/j.engstruct.2011.11.032mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.engstruct.2011.11.032http://www.sciencedirect.com/science/journal/01410296http://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructhttp://www.sciencedirect.com/science/journal/01410296http://dx.doi.org/10.1016/j.engstruct.2011.11.032mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.engstruct.2011.11.032

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theinteraction between wallsand framesin pushover analysisof 3Dwall-frame structures.

Moghadam and Tso [22,23] have proposed a 3D pushover pro-cedure in which the capacity curve of structure is obtained byapplying a lateral load vector in the mass centers of oors. TheV D (base shear versus displacement of mass center in roof) curveresulted from pushover analysis is considered as the capacity

curve. Based on the assumed fundamental mode shape corre-sponding to the assumed load vector and considering the govern-ing equation of motion, the capacity curve of the multi degree of freedom system (MDOF) is transformed to the capacity curve of an equivalent single degree of freedom (SDOF) system. The targetdisplacement is obtained from NTHA of the equivalent SDOF sys-tem. Ayala and Tavera [24] have also proposed a similar procedurein which the structural model is loaded simultaneously by forces intwo orthogonal directions and torques are applied in the mass cen-ters of oors. The contributions of the higher modes are consideredin this load pattern.

Furthermore Chopra and Goel [25] have extended the applica-tion of the well-known modal pushover analysis (MPA) to asym-metric-plan buildings. In which multiple pushover analyses areconducted with the load pattern corresponding to the considered3D elastic mode shapes, and then the total seismic response is esti-mated by combining the responses due to each modal load. In eachrun, the structure is loaded according to the spatial distribution of modal forces in two translational directions and torques in theheight of the building. The contribution of the higher modes andthe effects of torsion issue are considered in the extended versionof the MPA procedure. However, the analysis for each mode is runindependently so the effects of yielding in one mode are not re-ected in the other modes and the interaction between modes inthe nonlinear phase is neglected. Furthermore the total responsesof the structure are only estimated at the end of the procedureby combining the modal responses so the total response andbehavior of the structure during the procedure could not be tracedeasily. Recently Reyes and Chopra [26] have extended the applica-tion of the MPA method to analysis of symmetric and asymmetric-plan buildings under bi-directional excitation. Also, Poursha et al.[27] have proposed a consecutive modal pushover procedure forone-way asymmetric buildings.

Fajfar et al. [28] have also extended the well-known N2 methodto asymmetric-plan buildings in which the pushover analysis for3D model is conducted independently in two orthogonal directionsand the target point in each direction is dened through N2 meth-od. In order to consider the torsion effects, the resulted responsesfrom the pushover analysis are amplied based on the responsesresulted from the 3D elastic dynamic (spectrum) analysis of thestructural model. The simplicity of the original N2 method is keptwhile the major drawbacks mentioned for the conventional proce-dure exist still. Also the applicability of the N2 and MPA procedures

on 3D models have been evaluated by Erduran [29] .Penelis and Kappos [30,31] have proposed an interesting single-

run 3D modal pushover procedure in which the load vector is de-rived fromthecombinedspectralmodal storydisplacementand tor-sion proles. The required forces and torques which are applied inthe mass center of oor to produce the combined spectral modalstorydisplacement andtorsion proles areconsidered as loadvectorcomponents.The effects of thehigher andtorsional modes as well asthe interaction between modes in the nonlinear phases are consid-ered in the single load vectors. However, the load vector in thismethod is still invariant duringthe analysis.Also, since thedamage-ability of structures is controlled by the inter-story drifts, it seemstheload vector shouldbe derived from thecombined spectral modalinter-story drifts prole instead of the modal displacement proles.

In order to consider the effect of the changes in the structuralmodal properties of asymmetric-plan buildings during the

pushover analysis, Pinho and his co-workers [32,33] have studiedthe application of the displacement-based adaptive pushover pro-cedure [13] on a 3D building case. Bento et al. [34] employed theapplication of adaptive capacity spectrum method (ACSM) [35]along with other methods on an asymmetric-plan building. AlsoAdhikari and Pinho [36] proposed new versions of ACSM for 3Dasymmetric-plan buildings. To pursue the contribution of these

researchers, an innovative single-run adaptive modal procedure,referred to as story-shear-and-torque-based adaptive (STA) proce-dure has been developed in this paper.

3. Development of the STA procedure

The proposed STA procedure is originally an extension of theSSAP (story shear-based adaptive pushover) procedure [16] to 3Dbuildingframes with asymmetric plan. At each step of the proposedmethod, the load pattern is updated based on the instantaneousmodal properties of structure.The effects of thehigher andtorsionalmodes and the interaction between modes are considered in thesinglemodal load pattern. Themain advantageof theSTA procedurelies in the methodologies used in dening the load pattern andobtainingthe capacity spectrum curvewhich are explained in detailin the following sections. Since the theoretical background of theproposed procedure lies in the modal analysis concept, here a brief reviewof the modal response historyanalysis and spectral dynamicanalysis (SDA) of linear 3D structures has been presented.

3.1. Equation of motion for 3D model

The approach and notation presented here is similar to the pro-cedure explained in reference [37] . In all of the equations of mo-tion, it is assumed that the all oors of the building are rigid inplane and exible in out of plane direction. The equation of motion,governing the MDOF system with 3 N degree of freedom ( N : num-ber of stories) for uni-directional excitation is:

M U C

_U KU Ml yu g yt 1

where M, C and K are, respectively, the mass, damping and stiffnessmatrices. U , _U and U are, respectively, acceleration, velocity and dis-placement vectors. u g yt is the earthquake acceleration in y direc-tion. l y is the excitation inuence vector:

l y h0 1 0 i 2

where vector 1 is N 1 the vector with elements equal to unity andvector 0 is N 1 vector with zero elements. Displacement vectorconsists of three parts, as shown below:

U hU x U y U h iT 3

where U x and U y are N 1 displacement vectors of N stories in x and y translational directions of rigid diaphragm stories and U h is N 1rotation vector of them.

From modal decomposition concept, total response of the struc-ture can be decomposed into responses of individual modes whichtheir summation yields the total response [Eq. (4)]:

U XU n XU n qn 4 where U n and qn are, respectively, mode shape vector and general-ized modal coordinate for n th mode.

By transforming the total response into modal coordinate theEq. (1) is decomposed into 3 N (N is the number of stories) equa-tions [Eq. (5) ]:

qn 2 nnx n x 2n qn C n y u g y 5

where C yn is the modal participation factor for n th mode in excita-tion direction.

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C yn L ynM n

; M n U T n M U n ; L yn UT n Ml y 6

The U n vector consists of two translational vectors in x and y direc-tions and rotational vector as shown in Eqs. (7)(10) :

U n hU xn U yn U hn iT 7

U xn h/ x1 n / x2 n . . . / xnn iT

8 U yn h/ y1 n / y2 n . . . / ynn i

T 9 U hn h/ h1 n / h2 n . . . / hnn i

T 10

The maximum induced modal forces and torques in each mode,because of excitation in one direction (the y-direction) is computedby Eqs. (11)(13) . These loads are applied in the mass center of thestories:

F xij C y j / xij m xi S a yj 11

F yij C y j / yij m yi S a yj 12

T hij C y j / hij I hi S a yj 13

where i is the story number, j is the mode number, m xi is the trans-lational mass of the ith story in x direction, m yi is the translationalmass of the ith story in y direction, I hi is the rotational mass of the ith story and S a yj is the spectral acceleration in y direction corre-sponding to the jth mode.

In the well-known spectral-dynamic-analysis (SDA) themaximum total responses of the structure is approximated bythe combination of responses due to maximum modal forces[Eqs. (11)(13)]. Chopra and Goel [25] have extended this conceptto nonlinear analysis by proposing the MPA procedure.

3.2. Load pattern

As mentioned before, in the multi-run procedures the effects of yielding in one mode are not reected in the other modes and theinteraction between modes in the nonlinear phase is neglected. Toresolve these drawbacks, in this paper an innovative single-runmodal adaptive pushover procedureis proposed in which themodal

combination concept isused to denethe load pattern ratherthantocombine the nonlinear responses due to each mode. Then the struc-ture is loaded according to this combined modal load pattern. Thecontribution of the higher modes and theinteraction between themin the inelastic phase are reected in the united load pattern. Sincethe story drift is an important index in the structural damage, itseems rational to obtain the load pattern based on the modal driftprole of the structure. The story drift of 3D structures consist of the translational and torsional relative displacements and theseparameters are directly related to the story shears and total storytorques, therefore in the proposed STA procedure the load vectorassociated with the combined modal story shear and torque proleis selected as the load pattern which would be the best representa-tive ofthe damagein structures. Theprocessof dening theloadpat-tern at eachstep of theSTA procedureis illustrated through a samplenine-story building in Fig. 1. This sample building is asymmetricabout x-axis and y-axis (two horizontal directions) which is sub- jected to horizontalgroundmotion in onedirection(the y-direction).

At each step, based on the instantaneous structural properties,the modal story forces in two translational (the x and y) directionsand modal torque applied in the mass centers of stories, are calcu-lated for each considered modes by Eqs. (11)(13) [Fig. 1(a)].

Thestoryshears in translationaldirections andstory torque asso-ciated with each mode arecalculated by Eqs. (14)(16) [Fig. 1(b), (c)and (d)]. Then the combined modal story shears and combined

Fig. 1. The process of dening the load pattern at each step of the STA procedure. (a) Modal story forces in each mode. (bd) Modal story shears and torques. (eg) Combinedmodal story shear and torques. (hj) Component of the proposed load pattern ( x, y and h direction) in each story.

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modal story torques are calculated using the complete-quadratic-combination (CQC) or square-root-of-the-sum-of-the-squares(SRSS) rule [Eqs. (17)(19), Fig. 1(e), (f) and (g)].

SS xij Xn

ki

F xkj 14

) CSS xi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXm

j1 SS 2

xijr 17 SS yij X

n

ki

F ykj 15

) CSS yi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXm

j1SS 2 yijr 18

ST hij Xn

ki

T hkj 16

) CST hi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXm

j1ST 2hij

r 19

where SS xij and SS yij are the story shears in oor i associated withmode j in x and y directions, respectively. ST hij is the story torquein oor i associated with mode j. CSS xi and CSS yi are the combinedmodal story shears in oor i associated with all the modes consid-ered. CST hi is the combined modal story torque in oor i associatedwith all the modes considered.

The lateral forces (in two translational directions) and torquesrequired to generate CSS xi and CSS yi (the combined modal storyshears) and CST hi proles are assumed as the load pattern. The re-quired story forces in each direction and the required story torqueare calculated by subtracting the combined modal story shears andcombined modal story torques of consecutive stories using Eqs.(20)(22) [Fig. 1(h), (i) and (j)]:

F xi CSS xi CSS xi 1 i < nF xn CSS xn i n

20

F yi CSS yi CSS yi 1 i < nF yn CSS yn i n

21

T hi CST hi CST hi 1 i < nT hn CST hn i n

22

Byapplyingtheseloads (Eqs. (20)(22) )inthemasscenterofeachstory,the resultedstoryshear andtorquewould beequal to thecom-bined modalstoryshears andtorque ineachstory.Consequently,the

induced drift in each story would be equal to the combined modalstory drift in linear-elastic structures. Therefore the authors believethat these loads (Eqs. (20)(22) ) could bean efcientloadpattern ineach step of pushover analysis of nonlinear structures.

The components of this load pattern are normalized with re-spect to the summation of the force components (base shear) inthe earthquake excitation direction (the y-direction) using Eqs.

(23)(25) . The incremental applied load prole at each step is com-puted by Eqs. (26)(28) :

F xi F xi

PF yi23

F yi F yi

PF yi24

T hi T hi

PF yi25

D F xi D V b y F xi 26 D F yi D V b y F yi 27 D T hi D V b y T hi 28

where DV b y is the incremental amount of the base shear in theearthquake excitation direction (the y-direction).

Even though the quadratic combination rule (e.g. SRSS) is usedin dening the load pattern, the calculated applied forces or torquein each story could be negative whenever the value of the com-bined modal shears or torque in one story is less than the one inthe upper story [ Fig. 1(e), (f) and (g)]. Therefore, this method is ableto simulate the force or torque distribution proles with reversalsigns as observed in dynamic analysis [ Fig. 1(h), (i) and (j)]. Sothe effects of the reversal signs in the higher modes are also simu-lated in the proposed load pattern.

3.3. Transforming the MDOF system into SDOF system

The main idea in the NSP is the extension of using the designspectrum in the nonlinear phase. To achieve this purpose theMDOF system is transformed to an equivalent nonlinear SDOF sys-tem by running the pushover analysis. The maximum response of the MDOF system is estimated from the maximum displacementof the SDOF system using the inelastic or equivalent elastic designspectrum. In the conventional NSP the pushover analysis is run byapplying the lateral forces corresponding to the assumed funda-mental mode [Eq. (29) ] and the base shear versus top displacementcurve of the MDOF system is transformed to force versus displace-

Fig. 2. Plans of SAC-3, SAC-9 and SAC-20 buildings.

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ment F D curve of the equivalent SDOF system with unit massusing Eqs. (30) and (31) .

F M U 29

F S a V bM

30

D S d ur C / r

31

where U is the assumed fundamental mode shape, M is the massmatrix, V b is the base shear, u r is the roof displacement, / r is the

component of the U in the roof level, C U T Ml=U T M U is theparticipation factor, M C L is the effective mass and L U T M U .

However in the proposed STA procedure the load pattern isadapted in each step, so the assumed fundamental mode is changedin eachstepand Eqs. (30) and(31) shouldbe presentedbased on theinstantaneous assumed fundamental mode shape [33,34] . There-foreat each step( k) of theSTA procedure, theassumed fundamentalmode shape is derived from the instantaneous load prole using Eq.(32) (inverted form of the Eq. (29) ). This assumed fundamental

mode shape is compatible with the load pattern in each step andalso reects the contribution of the higher and torsional modes:

Table 1

Ground motions characteristics.

Earthquake Year M Ms Station Component Closest distanceto fault (km)

PGA (g) PGV (cm/s) Site condition

CWB USGS

Loma Perieta 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 BChichi 1999 7.6 7.6 TCU071 N 4.94 0.655 69.4 CTabas 1978 7.4 7.4 9101 Tabas LN 0.836 97.8 C

0

5

10

15

20

S t o r y

Drift (m)

Landers -CM

0

5

10

15

20

Drift (m)

Landers -SS

0

5

10

15

20

Drift (m)

Landers -FS

NTHA

STA

MPA

M1

0

5

10

15

20

0 0.03 0.06

S t o r y

Drift (m)

Tabas -CM

0

5

10

15

20

0 0.03 0.06

Drift (m)

Tabas -SS

0

5

10

15

20

0 0.03 0.06

Drift (m)

Tabas -FS

NTHA

STA

MPA

M1

0

5

10

15

20

S t o r y

Drift (m)

Chichi -CM

0

5

10

15

20

Drift (m)

Chichi -SS

0

5

10

15

20

0 0.03 0.06 0 0.03 0.06 0 0.03 0.06

0 0.03 0.06 0 0.03 0.06 0 0.03 0.06

Drift (m)

Chichi -FS

NTHA

STA

MPA

M1

Fig. 3. Inter-story drift proles of SAC-20 building in CM, SS and FS for Tabas, Chichi and Landers records.

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U k M 1 F k 32

where F k is the vector of the total forces applied on the structure atstep k, U k is the equivalent fundamental mode shape at step k, M isthe mass matrix.

Based on this instantaneous fundamental mode shape the force

versus displacement F D curve of the equivalent SDOF system isspecied for the direction of excitation (the y-direction).

3.3.1. Dening F coordinate of the F D curve of the SDOF systemAt each step ( k) of the STA procedure, the instantaneous effec-

tive mass associated with the direction of excitation (the y-direc-tion) is calculated based on the instantaneous fundamental modeshape using Eq. (33) :

M k

y U k

T M l y

2

U kT

M U k33

By dividing the base shear of the structure in the excitationdirection to the instantaneous effective mass, the base shear of

the MDOF system is converted to the equivalent force of the equiv-alent SDOF system with unit mass, F k

y [Eq. (34)].

F k

y S k a y

V kb yM

k

y

34

3.3.2. Dening D coordinate of the F D curve of the SDOF system

In the 3D models with asymmetric-plan because of the torsionissue, different points in the roof plan have different amount of translation in the direction of excitation, so choosing the locationof the controlling point on the roof plan to establish the pushovercurve is encountered with critical ambiguity. On the other side inthe dened equivalent fundamental mode shape, the increase of the roof displacement is not proportional to the other stories dis-placement and may even be reversals. Furthermore the total struc-tural deections resulted from the 3D pushover analysis consist of the torsional and transitional components in the both horizontaldirections. Therefore, considering only the roof displacement asthe conversion parameter to convert the displacement coordinatesof the MDOF system to the displacement coordinate of the SDOFsystem could not be rational.

Regarding these issues, in this paper a formulation based on theenergy concept [38] is developed for 3D models to dene the

0

1

2

3

4

5

6

7

8

9

10

S t o r y

Drift (m)

Landers -CM

0

1

2

3

4

5

6

7

8

9

10

Drift (m)

Landers -SS

0

1

2

3

4

5

6

7

8

9

10

Drift (m)

Landers -FS

NTHA

STA

MPA

M1

0

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1

S t o r y

Drift (m)

Tabas -CM

0

1

2

3

4

5

6

7

8

9

10

Drift (m)

Tabas -SS

0

1

2

3

4

5

6

7

8

9

10

Drift (m)

Tabas -FS

NTHA

STA

MPA

M1

0

1

2

3

4

5

6

7

8

9

10

S t o r y

Drift (m)

Chichi -CM

0

1

2

3

4

5

6

7

8

9

10

Drift (m)

Chichi -SS

0

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0 0.05 0.1 0 0.05 0.1

0 0.05 0.1 0 0.05 0.1

0 0.05 0.1 0 0.05 0.1 0 0.05 0.1

Drift (m)

Chichi -FS

NTHA

STA

MPA

M1



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displacement coordinate of the corresponding equivalent SDOFsystem. The amounts and signs of the displacements and rotationin all stories are considered in the formulation. The total work doneby the applied forces (in two translational directions) and torquesin all stories is assumed to be equal to the work done by the baseshear in the direction of excitation [Eq. (35) ]. In each step the incre-mental displacement, D Dk in the equivalent SDOF system is calcu-

lated from Eqs. (35) and is added to the value of the displacementin the previous step [Eq. (36) ]:

Xn

i1 F k 1 i x

1

2D F ki x D dki x

Xn

i1 F k 1 i y

1

2D F ki y D dki y

Xn

i1 T k 1 ih

1

2D T kih D dkih

Xn

i1 F k 1 i y

1

2D F ki y D Dk 35

Dk Dk 1 D Dk 36

where F k 1i x , F

k 1i y and T

k 1ih are, respectively, the existing forces (inthe x and y directions) and existing torque in the story i at the end of

step k 1. DF ki x , D F ki y and DT

kih are, respectively, the incremental

applied forces and torque in the story i at step k. D dki x , Ddki y and

D dkih are, respectively, the incremental displacements (in x and ydirections) and incremental rotation in the story i due to the incre-mental applied load at step k. D Dk is the incremental displacementof the equivalent SDOF system at step k. Dk is the displacementcoordinate of the equivalent SDOF system at step k.

3.4. Target displacement

By plotting the dened displacement coordinate, Dk of theequivalent SDOF system against the equivalent force, F

k

y associ-ated with excitation direction (the y-direction) at each step ( k),the capacity curve of the equivalent SDOF system ( F D curve)is established. Since in this study the evaluation of the consideredpushover procedures is done by comparing the results of the push-over analysis with the result of the NTHA for each considered re-cord. The capacity curve of the SDOF system is plotted againstthe inelastic demand spectra of each considered record in ADRS(accelerationdisplacement response spectra) format (Howeverfor engineering practice, the site-specic nonlinear response spec-

tra could be used instead of the response spectrum of a specic re-cord). The intersection points between the capacity curve and the

0

1

2

3

S t o r y

Drift (m)

Landers -CM

0

1

2

3

Drift (m)

Landers -SS

0

1

2

3

Drift (m)

Landers -FS

NTHA

STA

MPA

M1

0

1

2

3

0 0.05 0.1 0.15

S t o r y

Drift (m)

Tabas -CM

0

1

2

3

Drift (m)

Tabas -SS

0

1

2

3

Drift (m)

Tabas -FS

NTHA

STA

MPA

M1

0

1

2

3

S t o r y

Drift (m)

Chichi -CM

0

1

2

3

Drift (m)

Chichi -SS

0

1

2

3

0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15

0 0.05 0.1 0.15 0 0.05 0.1 0.15

0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15

Drift (m)

Chichi -FS

NTHA

STA

MPA

M1



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inelastic demand spectra for a specic record would yield the tar-get displacement (the performance point) of the system for that re-cord. In this study the peak displacement of the equivalentinelastic SDOF system for each record (target displacement) is ob-tained directly through the NTHA of the idealized bilinear SDOFsystem. However the existing methods in literature for estimatingthe inelastic displacement demand of the SDOF system (e.g. over-damped elastic spectrum and inelastic deformation ratio) [1,39]could be used to obtain the performance point of the system.

4. The STA procedure

Sequential steps of the STA procedure are presented as follows:

1. Creating the 3D model of the structure.2. Running eigenvalue analysis to obtain mode shapes and

modal periods.3. Computing the modal story forces and modal story torque

for selected number of modes using Eqs. (11)(13) .4. Computing the modal story shear and total modal story tor-

que using Eqs. 14, 16 and 18 .5. Calculating the combined-story-shear and combined-total-

story-torque proles by quadratic combination rule (e.g.CQC or SRSS rule) using Eqs. 15, 17 and 19 .

6. Obtaining force components (in two translational directions)and torque component of the incremental load pattern vec-tor in each story by subtracting the combined-story-shearand combined-total-story-torque of consecutive storiesusing Eqs. (20)(22) .

7. Normalizing the load pattern vector with respect to thesummation of force components of the load pattern vectorin excitation direction (the y direction) using Eqs.(23)(25) .

8. Scaling the load pattern vector with respect to the incrementof base shear in excitation direction using Eqs. (26)(28) .

9. Applying the scaled load pattern on the structure and com-puting the desired responses.

10. Computing the assumed equivalent fundamental modeshape using Eq. (32) and obtaining the force and displace-ment coordinates of the equivalent SDOF system using Eqs.(33)(36) .

11. Returning to step 2 and continuing with the steps until thedesired maximum value of base shear in excitation directionis reached.

12. Developing the forcedisplacement curve ( F D of theequivalent inelastic SDOF system, based on the computedvalues in step 10 for each cycle and idealize it as a bilinear

curve.

0

10

20

30

40

50

60

70

80 CM -SAC 9

Chichi Lander Tabas KobeLomaPrieta

0

10

20

30

40

50

60

70

80SS -SAC 9

0

10

20

30

40

50

60

70

80FS -SAC 9

0

20

40

60

80

100

120

140

160

180 CM -SAC 3

STA

MPA

M1

Chichi Landers Tabas KobeLomaPrieta

020

40

60

80

100

120

140

160

180SS -SAC 3

STA

MPA

M1

0

20

40

60

80

100

120

140

160

180 FS -SAC 3

STA

MPA

M1

0

5

10

15

20

25

30

T o t a

l E r r o r o n

D r i

f t ( % )

CM -SAC 20

0

5

10

15

20

25

30

T o

t a l E r r o r o n

D r i

f t ( % )

SS -SAC 20

0

5

10

15

20

25

30

T o

t a l E r r o r o n

D r i

f t ( % )

FS -SAC 20

Chichi Landers Taba KobeLomaPrieta



Chichi Landers Tabas KobeLoma Prieta

Chichi Landers Tabas K o beLomaPrieta


Chichi Landers Taba KobeLomaPrieta

Fig. 6. The total error on inter-story drift values derived for STA, MPA and M1 methods for ve near-fault records.

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13. Plot the capacity curve ( F D, against the inelastic demandspectra in ADRS format and dene the target displacement.In this study, in order to verify the method against the NTHA,the peak displacement of the equivalent inelastic SDOF sys-tem for each record (target displacement) is obtained

directly through the NTHA.14. Determining the seismic demand of the structure in the step

corresponded to the target displacement.

5. Validation study

The proposed STA method has been veried through the pro-gramming of the aforementioned steps in OpenSees [40] software.Three steel moment resistant frame buildings with three, nine and20 stories denoted as SAC-3, SAC-9 and SAC-20 are selected for thevalidation study. These buildings are designed for phase II of SACproject [41] and meet the 1994 UBC seismic design code specica-tions for Los Angeles, California region. Complete details of thesebuildings are presented in reference [42] . To study the effects of

torsion issue, the center of the mass (CM) of each story is relocatedtoward one side of the plan (along the x direction) to induce

eccentricity equal to 10% of plan dimension. These plans are pre-sented in Fig. 2 which shows only the moment resistant frames.In the modeling of the buildings it is assumed that the all oorsof the building are rigid in plane and exible in out of plane direc-tion. Displacement based beam-column elements with ve inte-

gration points and ber sections are used for nonlinear modelingof the structures.

Since the responses of the 3D asymmetric buildings are affectedby torsion issue, different points in their plans undergo differentamount of displacement. Therefore to observe the response of eachstory of the structures, center of the mass (CM) of each story andtwo sides of the plan in the extremities are selected to be moni-tored during the analysis. Since one side of the plan can undergomore displacement than other side, this side is referred to as ex-ible side (FS) and the other side is referred to as stiff side (SS)(Fig. 2).

In order to evaluate the accuracy of the STA method against thenonlinear time history analysis, the selected buildings aresubjected to ve near-fault records along Y direction of the plan

(Fig. 2). The characteristics of the selected records are listed inTable 1 .

0123456789

10

S t o r y

Mean Drift (m)

CM -SAC9

0123456789

10

Mean Drift (m)

SS -SAC9

0123456789

10

Mean Drift (m)

FS -SAC9

NTHA

SSAP

MPA

M1

0

1

2

3

S t o r y

Mean Drift (m)

CM -SAC3

0

1

2

3

Mean Drift (m)

SS -SAC3

0

1

2

3

0 0.05 0.1 0 0.05 0.1 0 0.05 0.1

0 0.05 0.1 0.15 0 0.05 0.1 0.15 0 0.05 0.1 0.15

Mean Drift (m)

FS -SAC3

NTHA

STA

MPA

M1

0

5

10

15

20

0 0.03 0.06

S t o r y

Mean Drift (m)

CM -SAC20

0

5

10

15

20

0 0.03 0.06

Mean Drift (m)

SS -SAC20

0

5

10

15

20

0 0.03 0.06

Mean Drift (m)

FS -SAC20

NTHA

SSAP

MPA

M1

Fig. 7. Mean inter-story drift prole of SAC-20, SAC-9 and SAC-3 buildings in CM, SS and FS.


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As mentioned before, since inter-story drift values play a crucialrole in the amount of damage induced in the structure duringearthquake, this quantity has been observed for CM, SS and FS of each story of selected structures in excitation direction. The in-ter-story drift proles resulted from the STA, MPA and M1 (theMPA procedure using only the rst mode) and the NTHA for threesample records are compared in CM, SS and FS of SAC-20, SAC-9and SAC-3 buildings in Figs. 35 , respectively.

As shown in Figs. 35 the trend of inter-story drift proles re-sulted from STA method follow the trend of NTHA results in each

record. While the trend of inter-story drift proles resulted fromthe other pushover procedures have constant shape for all of the

records. The results of the MPA procedure in lower stories are sim-ilar to the results of the M1 method and including the highermodes in MPA procedure has effect only on the estimations of the higher stories drifts. In the most cases the inter-story drift pro-les resulted from the STA procedure are very close to those re-sulted from the NTHA in comparison with MPA and M1procedures (see Figs. 3 and 4 ; e.g. the drift proles of SAC-20 andSAC-9 buildings under Chichi and Landers records). The successfulperformance of the STA method in following the trend of the NTHAresults, lies within the basic concepts of the proposed method in

which the variation of dynamic characteristics of the structure,the effects of the higher modes and the characteristics of the

-125

-75

-25

25

75

125

E r r o r

%

SS CM FS

Story 2

STAMPAM1

-125

-75

-25

25

75

125Story 4

-125

-75

-25

25

75

125Story 6

-125

-75

-25

25

75

125Story 10

-125

-75

-25

25

75

125

E r r o r %

Story 12

-125

-75

-25

25

75

125Story 14

-125

-75

-25

25

75

125Story 16

-125

-75

-25

25

75

125Story 18

-125

-75

-25

25

75

125Story 8

-125

-75

-25

25

75

125

SS CM FS SS CM FS SS CM FS

SS CM FS SS CM FS SS CM FS SS CM FS

SS CM FS

SS CM FS

Story 20

Fig. 8. Error on mean inter-story drift in CM, SS and FS for stories of SAC-20.

-120

-80

-40

0

40

80

120

E r r o r

%

SS CM FS

Story 1

STAMPA

M1 -120

-80

-40

0

40

80

120

Story 2

-120

-80

-40

0

40

80

120Story 3

-120

-80

-40

0

40

80

120Story 5

-120

-80

-40

0

40

80

120

E r r o r %

Story 6

-120

-80

-40

0

40

80

120

Story 7

-120

-80

-40

0

40

80

120

Story 8

-120

-80

-40

0

40

80

120Story 9

-120

-80

-40

0

40

80

120

SS CM FS SS CM FS SS CM FS

SS CM FS SS CM FS SS CM FS SS CM FS

SS CM FS

Story 4

Fig. 9. Error on mean inter-story drift in CM, SS and FS for stories of SAC-9.


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frequency content of the records (using the response spectrum of the record in dening the load pattern) are all considered in theload pattern.

In order to evaluate the accuracy of the considered pushoverprocedures in estimating the results of the NTHA, the total errorof each pushover procedure on the inter-story drift prole is calcu-lated using Eq. (37) in which the results of the NTHA are assumedto be the exact responses [43] :

Total Error % 1001

n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i1

D i NTHA D i NSP D i NTHA

2s 37 where D i NTHA is the maximum inter-story drift of ith story resultedfrom NTHA, D i NSP is the inter-story drift of ith story in the consid-ered NSPs at the target displacement, and n is the number of thestories of the building.

The calculated total error could accommodate the differences of the drift values in all stories of a structure for each NSP with respect

to NTHA. Thetotal error ofSAC-20,SAC-9andSAC-3buildingsfor theconsidered pushover methods in CM, SS and FS under each recordareshownin Fig. 6 . Theamount of total error calculated for STApro-cedurein most of thecases arelessthantotalerrorof theotherpush-over methods especially in SAC-20 and SAC-9 buildings, where theeffect of the higher modes are signicant. So that the total error of the STA method for SAC-20 and SAC-9 buildings in all cases are lessthan theerror of theother pushover methods, except in Kobe recordin which the amount of total error in worst case are less than 17%.Even though the amount of total error on drift for SAC-3 buildingin Chichi record is more than the total error of the other methods,the results of STA method in this case are overestimated (see Fig. 5).

To establish an overall overview in comparison of STA proce-dure with NTHA and other pushover procedures, mean drift prole

resulted from NTHA under ve different records and mean driftprole of each pushover method are presented in Fig. 7. The mean

inter-story drift proles resulted from STA procedure in the se-lected buildings especially in SAC-20 and SAC-9 are very close tothose resulted from the NTHA (see mean drift prole in CM andSS in Fig. 7for SAC-20 and SAC-9). Despite the fact that the meandrift proles resulted from the STA method are slightly underesti-mated in exible side, the trend of these proles match the trend of the proles resulted from NTHA.

Also in order to evaluate the accuracy of the considered push-over procedures in estimating the drift value of each story, an errorindex is used according to Eq. (38) which is calculated in CM, SSand FS of each story.

Error i% 100 M D i NSP M D i NTHA

M D i NTHA 38 where M D i NTHA is the mean of maximum inter-story drift in ithstory resulted from NTHA under the ve considered records,M D i NSP is the mean of the inter-story drift in ith story resulted from

the pushover method for the ve considered records.The error of the mean inter-story drift resulted from the consid-

ered pushover methods in CM, SS and FS of each story are shown inFigs. 810 for SAC-20, SAC-9 and SAC-3 buildings, respectively. Asit is shown in these gures, in all stories of the SAC-3 building andalso in most of the stories of the SAC-20 and SAC-9 buildings accu-racy of the STA procedure in CM, SS and FS is higher than the othermethods. As shown in Fig. 8, in most of the stories of the SAC-20building, the results of the STA procedure have error on mean driftless than 25% (in any monitoring point) and in worst case only instory 20 of this building the error is near to 50% while the errorof the MPA and M1 methods in most of the cases is close to 80%and even though in some cases it is close to 125%. Also as shownin Fig. 9, in all stories of the SAC-9 building the error of the STA pro-

cedure in SS, CM and FS is less than 15%, 30% and 45%, respectively.While these errors for the MPA procedure is near to 100%, 90% and

-150

-100

-50

0

50

100

150

E r r o r

%

SS CM FS

Story 1

STAMPAM1

-150

-100

-50

0

50

100

150

SS CM FS

Story 2

-150

-100

-50

0

50

100

150

SS CM FS

Story 3

Fig. 10. Error on mean inter-story drift in CM, SS and FS for each stories of SAC-3.

0

10

20

30

40

50

60

70 SAC 3

STA

MPAM1

0

10

20

30

40

50

60

70SAC 9

CM SS FS CM SS FS0

10

20

30

40

50

60

70

T o

t a l E r r o r o n

M e a n

D r i

f t ( % ) SAC 20

CM SS FS

Fig. 11. Total error on mean inter-story drift at CM, SS and FS for SAC-20, SAC-9 and SAC-3.


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125%, respectively. It should be noted that unless the STA proce-dure is generally underestimated, the estimations of this procedureis close to the results of the NTHA and the corresponding errors areless than other considered methods. Furthermore Figs. 810 showthat the M1 method is overestimated in low-rise building (SAC-3)and in lower stories of high-rise buildings (SAC-20 and SAC-9);while this method is underestimated in upper stories of high-rise

buildings (SAC-20 and SAC-9), where the effects of the highermodes are signicant.The total error of mean inter-story drift proles are also repre-

sented in Fig. 11 for the selected structures. These errors are calcu-lated using Eq. (37) in which the D i NTHA and D i NSP are replacedwith corresponding mean values. As shown in this gure, the totalerror of the STA procedure on mean drift for SAC-20 and SAC-9buildings are less than 7% and 12%, respectively. The higher errorof the STA procedure appears in SS frame of SAC-3 building whichis about 25% while the other methods show an error about 60% inthis building. Furthermore the error of the STA procedure is lessthan the error of the other considered pushover methods in allcases (in CM, SS and FS for SAC-20, SAC-9 and SAC-3).

6. Conclusion

An innovative adaptive pushover procedure is proposed to esti-mate the seismic responses of asymmetric-plan buildings. Thisprocedure considers the effects of the higher and torsional modesand the progressive changes in the dynamic characteristics of thestructure in the nonlinear phase. Also in each step of analysis,the effects of the frequency content of a specic record and the ef-fects of the interaction between modes in the nonlinear phase areall considered in a single load pattern. Furthermore the response of the structure during the analysis can be traced easily by applyingthe proposed load pattern through the single-run pushover proce-dure. In order to resolve the ambiguity of choosing control point inasymmetric-plan buildings, energy based approach has been usedto establish the capacity curve of building. The proposed procedureand the related equations are derived to analyze asymmetric-planbuildings under uni-directional excitations and the procedurecould be extended to bi-directional excitation. The proposed meth-od has been evaluated through SAC-20, SAC-9 and SAC-3 buildingsunder ve different near-eld records. The estimates of the pro-posed method are compared to the results obtained from the non-linear time history analysis (NTHA) as the benchmark response.Also these results are compared to those obtained from the modalpushover analysis (MPA) and M1 (the MPA procedure using onlythe rst mode) methods.

The main results of this study are summarized as follows:

1. The inter-story drift proles resulted from STA method followthe trend of NTHA results in each record. While the trend of

inter-story drift proles resulted from the other consideredpushover procedures have constant shape for all of the records.

2. The proposed method could estimate the seismic demand of 3Dasymmetric-plan buildings more accurate than other consid-ered pushover procedures (especially in center of mass (CM)and stiff side (SS) frames of SAC-20 and SAC-9 buildings). Inmost of the stories of the SAC-20 building, the results of theSTA procedure have error on mean drift less than 25% (in anymonitoring point) and in all stories of the SAC-9 building theerror of the STA procedure in SS, CM and exible side (FS) is lessthan 15%, 30% and 45%, respectively. Furthermore, the totalerror of the STA procedure on mean drift (under ve records)for SAC-20 and SAC-9 buildings are less than 7% and 12%,respectively. The higher error of the STA procedure appears in

SS frame of SAC-3 building which is about 25% while the othermethods show an error about 60% in this building.

3. The results of the proposed method in the exible side framesare slightly underestimated however their trends are similarto the trend of NTHA results and the corresponding errors areless than the errors of other considered methods.

4. The M1 method is overestimated in low-rise building (SAC-3)and in lower stories of high-rise buildings (SAC-20 and SAC-9); while this method is underestimated in upper stories of

high-rise buildings (SAC-20 and SAC-9), where the effects of the higher modes are signicant.

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http://opensees.berkeley.edu/http://opensees.berkeley.edu/

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