push 15 2012.pdf

Journal of Constructional Steel Research 72 (2012) 155–167

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Journal of Constructional Steel Research

Assessment of modified consecutive modal pushover analysis for estimating theseismic demands of tall buildings with dual system considering steel concentricallybraced frames

Faramarz Khoshnoudian ⁎, M. Mehdi B. KashaniFaculty of Civil Engineering, AmirKabir University of Technology, Hafez St., Tehran, Iran

⁎ Corresponding author.E-mail address: [email protected] (F. Khoshnoudia

0143-974X/$ – see front matter © 2011 Elsevier Ltd. Aldoi:10.1016/j.jcsr.2011.12.002

a b s t r a c t
a r t i c l e i n f o
Article history:Received 22 February 2011Accepted 3 December 2011Available online 11 January 2012

Keywords:Modified consecutive modal pushover(MCMP) procedureCMP procedureTall buildingsHigher-mode effectsSeismic demands

According to the previous researches, conventional nonlinear static procedure (NSP), which is limited to sin-gle mode response, cannot predict the seismic demands of tall buildings with reliable accuracy. To estimatethe seismic demands in upper stories for tall buildings the effects of higher modes should be included. In therecent years, developing traditional pushover analysis to consider the effects of higher modes conducted re-searchers to propose several methods, such as N2, MPA and MMPA procedures, that have a specific approachto estimate seismic demands of structures but the accuracy of them is doubtable for estimating of hinge plas-tic rotations. Recently consecutive modal pushover (CMP) procedure was proposed to consider the effects ofhigher modes with acceptable accuracy especially in prediction of hinge plastic rotations. The CMP procedurewas limited to include two or three modes, and use of higher modes might cause some inaccuracy at resultsof upper stories. In CMP procedure, estimation of modal participating factors is important and choosing inad-equate modes may cause large errors. In this paper some changes have been applied to the CMP procedure toimprove accuracy of the results and the modified method is proposed and named modified consecutivemodal pushover (MCMP) procedure. In this modified method the contribution of mode is used of effectivemodal participating mass ratio. The comparison of MCMP procedure to exact values derived by nonlinear re-sponse history analysis (NL-RHA) demonstrated the reliable predictions and it can overcome the limitationsof traditional pushover analysis.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Nonlinear static analysis, or pushover analysis, has been developedover the past decades and has become the commonprocedure for build-ing evaluation and design verification; however the procedure involvescertain approximations and simplifications. According to literature, theaccuracy of pushover analysis in predicting seismic demands has beenthe controversial discussion, so the proposed approaches are leading re-searchers to achieve more accurate and reliable method; however non-linear static procedures suffer a lot of limitations especially for high-risebuildings. The invariant load pattern is one of the most significant lim-itations of traditional methods, because the actual inertia force distribu-tion changes continuously during seismic events due to higher modecontribution and structural degradation, which modifies the stiffnessof individual structural elements and, consequently of the structure asa whole [1]. Therefore the effects of higher modes should be consideredfor estimating seismic demands of tall buildings.

n).

l rights reserved.

Generally, usingmodal properties of the structure in nonlinear staticanalysis is most accessible approach to take into account the dynamiccharacteristics of the system. MMP (Multi Mode Pushover procedure)[2] was proposed to involve higher modes effects in pushover analysis.MMP provides better estimation of seismic demands comparing to tra-ditional pushover methods based on load pattern using first mode.Although higher modes have being participated in MMP analysis, re-sponses estimation anddistribution of themover height of the structurewere inadequate. More recently, PCR (Pushover Results Combination)[3]was proposed to consider the effects of highermodes. In thismethodseveral load patterns using mode shapes should be applied and thenfinal responses would be determined as a weighted (using modal par-ticipation factors) summation of the results from each pushover analy-sis. MPA (Modal Pushover Analysis) [4] was developed to considerhigher modes effects by analyze each mode as an equivalent single-degree-of-freedom (SDOF) system including nonlinear properties relatedto that mode. The MPA procedure was able to predict seismic demandswith reliable errors at the level of displacements, but in order to calcu-late hinge plastic rotations MPA procedure had underestimate predic-tions. Then a modified version of the MPA (MMPA) [5] was proposed.The seismic demands of the structure were obtained by combiningthe inelastic response of first-mode pushover analysis with the elastic

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Notations

[c] Damping matrix of structure[k] Lateral stiffness matrix of structure[m] Diagonal mass matrix of structure{i} Unit vector{Peff(t)} The effective earthquake forces{s} Spatial distribution of effective forces{sn} Modal inertia force distribution of the nth mode{sn*} Incremental lateral force distribution for the nth stage

of multi-stage analysis{u(t)} Displacement of NDOF system{φn} nth mode shape of the structuremi Lumped mass of ith floorM* Total mass of the structureMn* Effective modal mass of the nth modeqn(t) Modal co-ordinater the peak response of the structure in the CMP

procedureüg(t) Acceleration of ground motionαn Effective modal participating mass ratio of the nth

modeδt Target displacement of the roofωn Natural frequency of the nth modeζn Damping ratio of nth modeΓn Modal participating factor of the nth mode

156 F. Khoshnoudian, M.M.B. Kashani / Journal of Constructional Steel Research 72 (2012) 155–167

response of higher modes. Upper-bound pushover analysis [6] was an-other method which was able to overcome the invariant load patternlimitation. Incremental response spectrum analysis (IRSA) [7] was de-veloped by other researchers in which whenever a new plastic hingeoccurs, elastic modal spectrum analysis was executed. More recently,an adaptive modal combination (AMC) [8] procedure was proposed,in which the applied lateral forces are updated in accordance with thechanges in the dynamic characteristics during inelastic analysis, foreach mode.

Although nonlinear static analysis is basically developed for struc-tures with the predominant first vibrationmode, many nonlinear staticapproaches, such as MPA [4], MMP [2], PRC[4] are proposed to considerhigher elastic modes as lateral load pattern to take into account highermodes effects. The accuracy of the mentioned approaches was demon-strated by comparing to nonlinear dynamic analysis as an exact solutionand it was confirmed the possibility to use nonlinear static analysis forpredicting seismic demands of tall buildings which are influenced byhigher modes effects. However there are some errors in these ap-proaches, the simplicity and time-consuming of these methods leadprofiting them. In addition, the aim of new pushover analysis is to re-duce these errors. Beside in this paper, the newproposed pushover pro-cedure is more reliable than the previous pushover analysis.

To improve pushover analysis, Consecutive Modal Pushover (CMP)procedure was proposed [9], in which the incremental forces were ap-plied to structure continuously. This method was examined on build-ings with moment-resistant frame system. The achievement of thismethod was demonstrated by having better estimation of hinge plasticrotations. The distributions of Lateral forces in ConsecutiveModal Push-over (CMP) procedure are calculated by using mode shapes which areobtained from Eigen-analysis of linearly elastic structure. Using elasticmodal properties are suggested formost thenonlinear static proceduresmerely because of simplicity and reduction of time-consuming of anal-ysis. The reason of how higher modes are going to be represented byelastic modes after deterioration can be explained by comparison ofthe results obtained from the proposed nonlinear static procedure tothose obtained from nonlinear time history analysis as an exact

solution. The CMP procedure includes two types of pushover analysis;single-stage andmulti-stage pushover analyses. Earlier the contributionof eachmode inmulti-stage analysis of the CMP procedure was derivedfrom effective modal participating mass ratio of that mode.

Afterwards, the accuracy of the CMP procedure has been verified foranother lateral force-resisting system, dual systemwith steel concentri-cally braced frames and in addition for improving the CMP procedure toestimate seismic demands of tall buildings without need of the funda-mental period of structure for performing two-stage or three-stagepushover analysis, consequently authors suggest modified consecutivemodal pushover analysis (MCMP) as it will be explained in details inthe following.

2. Basic theoretical concepts

Modal analysis principles are used in the proposed procedure;therefore it would be useful to understand the definition of theseprinciples. Eq. (1) shows the differential equation governing the re-sponse of a multi-degree-of-freedom (MDOF) system under earth-quake ground motion [10]:

m½ � €uf g þ c½ � _uf g þ k½ � uf g ¼ − m½ � if g:€ug tð Þ ð1Þ

where [m], [c] and [k] are diagonal matrices of mass, damping andstiffness of structure respectively and {i} is unit vector. Right handside of previous equation represents the effective earthquake forces,{Peff(t)}, and can be written as:

Peff tð Þn o

¼ − m½ � if g:€ug tð Þ ¼ − sf g€ug tð Þ: ð2Þ

Vector of {s} is the distribution of effective earthquake forces overbuilding's height and can be expanded as a summation of the modalinertia force distributions, sn, as follows:

sf g ¼ m½ � if g ¼XN

n¼1

snf g ¼XN

n¼1

Γn m½ � φn

� � ð3Þ

where, Γn is the nth modal participation factor and {φn} is the cor-responding mode shape. The displacement vector of a N degree offreedom system is defined as:

u tð Þf g ¼XN

n¼1

φn

� �qn tð Þ ð4Þ

where qn(t) is the modal coordinates and the following equationwill be obtained as follows:

€qn þ 2ζnωn _qn þω2nqn ¼ −Γn€ug tð Þ: ð5Þ

qn(t) presents a property of a multi degrees of freedom system, whichis defined by an elastic single degree of freedom system. ωn and ζn arenatural frequency and damping ratio of nth mode, respectively. Γn isobtained as follows:

Γn ¼ φn

� �T m½ � if gφn

� �T m½ � φn

� � : ð6Þ

To solve Eq. (5) the substitutionDn(t) instead ofqn(t) could be usefuland the relation between them is described as follows:

qn tð Þ ¼ ΓnDn tð Þ ð7Þ

157F. Khoshnoudian, M.M.B. Kashani / Journal of Constructional Steel Research 72 (2012) 155–167

where Dn(t) is governed by the equation of motion for a single-degree-of-freedom system subjected to üg(t):

€Dn þ 2ζnωn_Dn þω2

nDn ¼ −€ug tð Þ: ð8Þ

Displacement vector for the elastic system is indicated as follows:

u tð Þf g ¼XN

n¼1

Γn φn

� �Dn tð Þ ¼ Γ1D1 tð Þ φ1

� �þ Γ2D2 tð Þ φ2

� �

þ Γ3D3 tð Þ φ3

� �þ :::: ð9Þ

Response history analysis (RHA) and response spectrum analysis(RSA) could be used to estimate the peak response of the system, ro.However, in the case of usage in pushover analysis, the simplicity ofRSA leads us to use it instead of RHA. Generally, in modal responsespectrum analysis, the peak response of the corresponding nthmode is determined by Eq. (10), and total response of the system isestimated by some combination rules such as SRSS and CQC methods. InRSA the value of Dnocould be derived by using standard or design spectra.

rno ¼ rstn An ð10Þ

Where rnoand rn

st are the peak response and equivalent static re-sponse of nth mode, respectively, and An is derived from pseudo-acceleration response (or design) spectrum which is determined byTn and ζn.

Table 1General specification of structures.

Periods (s) Distributedlive load(kg-f/m)

Distributeddead load(kg-f/m)

T3 T2 T1

0.126 0.241 0.835 1000 32500.179 0.374 1.452 1000 32500.233 0.468 1.809 1000 32500.361 0.794 3.119 1000 3250

a) configuration of structure

Fig. 1. Configuration of structure and fra

It is notable that each mode shape in the CMP procedure is normal-ized relative to its roof component, φrn. Therefore, total displacement ofroof is determined as follows:

ur tð Þ ¼ Γ1D1 tð Þ þ Γ2D2 tð Þ þ Γ3D3 tð Þ þ :::: ð11Þ

Also there are some parameters which are used in the procedureand they are defined in Eqs. (12) to (15):

M�n ¼ LnΓn ð12Þ

αn ¼ M�n

M� ð13Þ

where;

Ln ¼ φn

� �m½ � if g ð14Þ

M� ¼ ∑mj ð15Þ

where M* is the total mass of the structure obtained by summa-tion of the lumped masses, mj, over all floor levels.

3. Modified consecutive modal pushover (MCMP)

Consecutive Modal Pushover (CMP) procedure was proposed to es-timate the seismic demands of structures. The process in the CMP anal-ysis was, so that loads in each stage were applied to structure at the

Lumpedmasses(kg-fs2/m)

b(m)

h(m)

No. ofstories

Structuretype

5538 15 32 10 F15636 15 48 15 F25693 15 64 20 F35794 15 96 30 F4

b) frame position

me position (N: Number of stories).

Table 6Used ground motions.

PGA (g) Component (deg) Station number Station name

0.134 E 1061 Lamont0.239 90 24,157 LA—Baldwin Hills0.147 270 1498 Rio Dell Overpass, F0.621 45 6604 Cerro Prieto0.196 271 1028 Hollister City Hall0.204 315 5051 Parachute Test Site0.174 180 135 LA Hollywood Stor L

Fig. 2. Generalized load-deformation curve for hinges [13].

Table 5Geometric properties of braces.

a (cm) Section Brace type

2 2UNP140 Br72 2UNP100 Br9

Table 4Geometric properties of beams.

tw (cm) tf (cm) bf (cm) h (cm) Beam type

0.8 1.5 20 32.5 B50.8 1.5 20 30 B6

Table 3Geometric properties of columns.

t (cm) d (cm) Column type

2.5 35 C42 30 C51.5 25 C6

Table 2Used section of 10-story structure.

Frametype

Levels Column type Beamtype

Bracetype

Interior col. Exterior col.

F1 1 C4 C4 B5 Br72 C4 C5 B5 Br73 and 4 C5 C5 B5 Br95 C5 C5 B6 Br96 C5 C6 B6 Br97 to 10 C6 C6 B6 Br9


deformed shape state resultant of previous stage and initial condition(stress and deformation) of each stage was the same as the state atthe last step of analysis in the previous stage. In the CMP procedure,the value of roof displacement increment, uri, at each stage of themulti-stage analysis was the factor of target displacement. This fac-tor was determined based on the effective modal participatingmass ratio. According to previous research on utilizing CMP proce-dure for steel moment-resisting frames, the procedure had accept-able accuracy in estimating hinge plastic rotations. The CMP procedureconsists of single-stage and multi-stage pushover analyses. Firstapply the gravity loads and then perform these displacement-control pushover analyses using consecutive pushover analysis;however the number of stages depends on fundamental period ofstructure [9].

In this paper authors propose to use the modal response analysis,to estimate contribution of each mode in the CMP procedure. Accord-ingly, numbers of modes that are participating in the CMP analysis arenot limited to two or three modes any more. Theoretically, usinggreater number of modes in the modal response analysis shouldcause reduction of structural response errors. So here it is proposedto use of all modes which the sum of their effective modal massesshould be at least 90% of the total structure mass.

XNs

i¼1

M�i ≥0:9M� ð16Þ

where Ns is the numbers of stages (or numbers of modes) that theyshould be used in the process of analysis. Some of the principles of push-over analysis are used in consecutivemodal pushover analysis procedure.In this approach at least two nonlinear static analyses with modal loaddistribution are performed. Load distributions in modified consecutivemodal pushover are based on modal shapes derived from Eigen-analysis of the linearly elastic structure. It is noteworthy to imply thatchanges in the modal properties of the structure are ignored when thestructure experiences nonlinear yielding under increasing lateral loadsduring pushover analysis [9]. Roof displacement increment, uri, for eachstage of themulti-stage pushover analysis is the factor of target displace-ment (Eq. (17)). The coefficient value is calculated based on the modalresponse spectrum analysis for elastic system. The displacement incre-ment, uri, at the roof in the ith stage of multi-stage pushover analysis, istherefore calculated as:

uri ¼ βiδt ð17Þ

where;

βi ¼Γ iDi

PNs

n¼1ΓnDn

ð18Þ

where δt is target displacement and βi is the contribution of ithmode. Several approaches can be used to estimate the total target dis-placement at the roof level. This displacement can be determined byusing the capacity spectrum method [11], the displacement coefficientapproach [12,13], the N2 method [14,15], or dynamic analysis of the

Magnitude Date E.Q. name No.

Ms(7.3) 1999/11/12 Duzce, Turkey 1Ms(6.7) 1994/01/17 Northridge 2

F Ms(7.2) 1980/11/08 Trinidad, California 3Ms(6.4) 1980/06/09 Victoria, Mexico 4Ml() 1961/04/09 Hollister 5Ms(6.9) 1979/10/15 Imperial Valley 6

ot Ms(6.6) 1971/09/02 San Fernando 7

Unlabelled image

Unlabelled image

Unlabelled image

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1.5 2.51 2 3 3.5 4

Natural Vibration Period,T(Sec)

Psu

do-A

ccle

rati

on(A

/g)

Duzce

Northridge

Trinidad

Victoria

San Fernando

Imprial Vally

Hollister

Average Spec.

Fig. 3. Pseudo-acceleration spectra.


structure [16,3,17,18]. Γi is derived from Eq. (6) and Di is the peak re-sponse of a single-degree-of-freedom (SDOF) system equivalent to ithmode which can be calculated by Eq. (8) and also can be estimated bystandard (or design) spectra, based on the modal response spectrumanalysis.

a) Story Drift Ratios of the 10-Story stru

1 2 3 4 5

Dri

ft r

atio

Single-Stage

Multi-Stage

NLTH

b) Story Drift Ratio of the 15-story struc

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0.9%

Dri

ft r

atio

Single-Stage

Multi-Stage

NLTH

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%Single-Stage

Multi-Stage

NLTH

1 2 3 4 5 6 7

Single-Stage

Multi-Stage

NLTH

Fig. 4. Peak values of story drift ratios derived from pushover analysis used in

In addition to multi-stage analysis also there is single-stage push-over analysis which is proposed to be performed by uniform distribu-tion for the dual systems.

The details of the MCMP procedure for a single plane frame areexpressed as a sequence of the following steps:

cture

6 7 8 9 10Floor

ture

Floor8 9 10 11 12 13 14 15

the MCMP procedure and from NLTH for the 10 and 15-story structures.

image of Fig.�4


1. Calculation of natural frequencies and modes shapes. These prop-erties are determined by Eigen-analysis of the linearly elasticstructure. The mode-shapes should be normalized with roof com-ponent so that the roof component of {φn} equals unity.

2. Compute {sn*}=[m]{φn}, where {sn*} shows the distribution of in-cremental lateral forces over the height of the structure for thenth stage of multi-stage pushover analysis.

3. Compute the total target displacement of the structure at the roof,δt.

4. Calculating coefficients for each mode based on the Eq. (18).5. As it was mentioned before, MCMP procedure is included single-

stage and multi-stage pushover analysis. First of all, gravity shouldbe applied to the structure and then according to the steps listedbelow displacement-controlled pushover analysis should be per-formed. It is noted that the vertical loads are applied to the build-ings according to the FEMA-356 regulations:5.1 Single-stage analysis, with uniform distribution over height of

building, until the control node, gradually reaches the totaltarget displacement.

5.2 Multi-stages analysis, that the number of stages (Ns) is equalto number of first modes which summation of their effectivemodal mass (Mn) reaches at least 90% of total mass (M*).

In this step, lateral force distribution in nth stage is {sn*} that is de-rived from {sn*}=[m]{φn}. These incremental lateral forces are

a) Story Drift Ratio of the 20-story struc

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

1 2 3 4 5 6 7 8 9 1

Dri

ft r

atio

Single-Stage

Multi-Stage

NLTH

b) Story Drift Ratio of 30-story structur

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0.9%

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Dri

ft r

atio

Single-Stage

Multi-Stage

NLTH

Single-Stage

Multi-Stage

NLTH

Single-Stage

Multi-Stage

NLTH

Fig. 5. Peak values of story drift ratios derived from pushover analysis used in

applied till displacement increment of each stage reaches the valueof urn=βnδt in which βn is calculated by Eq. (18).As it was demon-strated previously, initial conditions of each stage are the conditionsof last step of previous stage. It is notable that the nonlinear behaviorof the structure depends on the loading path, and separation of theloading input and the structural response is far from real behavior[19]. Consequently, modal pushover analysis must be carried out con-secutively in the order of modes, from the first to the higher ones.6. Calculating the peak values of desired responses such as, displace-

ment, drift ratios, plastic hinge rotations from last step (step 5). Re-sponses of single- and multi-stage analyses, have been shown by rSand rM, respectively.

7. Calculate the envelope, r, of the peak responses as follows:

r ¼ Max rS; rMf g: ð19Þ

This shows that the seismic demands of the inelastic structure isobtained by enveloping the peak responses in this procedure.

4. System and excitation considered

To evaluate MCMP procedure, the dual system considering steelconcentrically braced frames has been used. The configuration andbraces distribution over height of frame are illustrated in Fig. 1. Also

ture

0 11 12 13 14 15 16 17 18 19 20Floor

e

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Floor

the MCMP procedure and from NLTH for the 20 and 30-story structures.


the position of frame in whole building is illustrated in Fig. 1. The con-sidered structures are three-bay frames with four different heights of10, 15, 20, and 30 stories, covering a wide range of fundamental pe-riods. All the frames have 5 m bays. A story height of 3.2 m was as-sumed throughout. Specifications such as dimensions and naturalvibration periods are listed in Table 1. Assumption of rigid diaphragmhas been used for all frames, and lumped masses have been applied incenter of mass at each level. As an example, used sections of 10-storybuilding and their geometric properties of columns, beams and bracesare listed in Tables 2, 3, 4 and 5 respectively.

Allowable stress design has been used in order to design structures[20]. The dead and live loads have been assumed to be 650 and 200 kg-f/m2, respectively and loading width of beams was assumed to be equalto 5 m. The dead load (total dead load) and 20% of live load has been con-sidered to calculate the mass of each floor. The plastic hinge propertieswere determined according to FEMA-356 [13] to consider the nonlinearbehavior of structure (Fig. 2) which are located at each end of members.The post-yield slope (BC) was assumed to be 3% of the elastic slope.

P-Δ (second-order) effects have been included, but the panel zonesize, strength, and deformation have been neglected. The effect of p-δ,as the geometric nonlinear behavior of themembers has been consideredwhich is effective to determine themember stiffnessmatrix and then fol-lowed by the stiffness matrix of whole structure.

To evaluate the MCMP procedure, nonlinear time history analysishas been performed. Seven ground motions have been selected from

a) Story Drift Ratios of the 10-Story stru

0

1

2

3

4

5

6

7

8

9

10

0.000 0.001 0.002 0.003Drift

Flo

or

NLTH

MPA

MCMP

b) Story Drift Ratio of the 15-story struc

0123456789

101112131415

0.0% 0.1% 0.2% 0.3% 0.4%Drift

Flo

or

NLTH

MCMP

MPA

NLTH

MPA

MCMP

NLTH

MCMP

MPA

Fig. 6. Height-wise variation of the story dr

the strong ground motion database of the Pacific Earthquake Engineer-ing Research (PEER) Centre (http://peer.berkeley.edu). The distance tothe fault (more than 12 km) and the soil at the site have been the cri-teria to select these records. The ground motion records are selectedto be far field records; the soil site corresponds to NEHPR class C andalso covers large frequency contents of earthquakes. The seismic effectswere determined in accordance with the requirements of the Iraniancode of practice for the seismic-resistant design of buildings [21]. To en-sure that the structures respond into the inelastic rangewhen subjectedto groundmotions, the records were scaled up to 0.7 g. More character-istics of the used ground motion records are given in Table 6. Pseudo-acceleration spectrum with the corresponding the mean spectrum, arepresented, for 5% damping ratio, in Fig. 3. The mean spectra are shownby a thicker line.

5. Types of analysis

To evaluate the MCMP procedure, Non-Linear Time History (NLTH)analysis has been used to achieve the exact value of responses. Themean value of the responses has been determined over the set of usedground motions. Also the MCMP procedure has been compared toMPA (Modal Pushover Analysis) procedure as a well-known nonlinearstatic procedure. P-Δ effects have been included in the MCMP andMPA procedures for all modes. In the MCMP procedure, the target dis-placement at the roof has been derived by averaging the maximum

cture

0.004 0.005 0.006 0.007Ratio

ture

0.5% 0.6% 0.7% 0.8% 0.9% ratio

ifts for the 10 and 15-story structures.

http://peer.berkeley.edu


values of roof displacements resulting from the NLTH (Non-Linear TimeHistory) analysis for the selected ground motions. The target displace-ments that have been calculated are equal to values 13.36, 24.13,30.57 and 46.02 cm for structures F1, F2, F3, and F4, in the same orderaccording to Table 1. The nonlinear time history analysis has been per-formed using theWilson- time integrationmethod, in which the stabil-ity and accuracy characteristics are determined by the parameter . Thisparameter has been assumed a value of 1.4. A damping ratio of 5% hasbeen considered for the first and the mode which summation of its ef-fective modal masses and lower modes reaches at least 90% of totalmass of the structure, in order to define the Rayleigh damping matrix.It is noted that the nonlinear version of the computer programSAP2000 [22] was used to perform these analysis.

6. Discussion of the results

As it was described previously, the MCMP procedure uses two typesof pushover analysis. The peak values of story drift ratios for 10, 15, 20and 30 stories structure outcome from single- and multi-stage analysisand also nonlinear time history analysis has been illustrated in Figs. 4and 5, separately. Generally, according to the obtained results, it couldbe concluded that single-stage analysis and multi-stage analysis coverthe seismic demands of lower and higher stories, respectively. This con-clusion is acceptable for the all seismic demands which have been con-sidered in this study, except for displacement results single-stageanalysis has been governed for all stories. Actually, it was expectedthat using higher modes strongly affects on upper stories results.

a) Story Drift Ratio of the 20-story struc

0

2

4

6

8

10

12

14

16

18

20

0.0% 0.1% 0.2% 0.3% 0.4

0.0% 0.1% 0.2% 0.3% 0.4%

Drift

Drift

Flo

or

NLTH

MCMP

MPA(3 Modes)

b) Story Drift Ratio of 30-story structur

0

3

6

9

12

15

18

21

24

27

30

Flo

or

Fig. 7. Height-wise variation of the story dr

These mentioned figures show which pushover analysis, single-stageor multi-stage analysis, is dominated in the seismic demands of tallbuildings. Story drift ratios of all four Structures have been shown inFigs. 6 and 7.

As it was mentioned previously single- and multi-stage analysesare important to predict seismic demands of lower and upper stories,respectively. Therefore, it seems better to separate stories of eachstructure to two parts to compare the results of MCMP and MPAwith NLTH analysis. This separation is caused by experienced behav-iors of structures in respect to how incremental forces are applied.

Both the procedures (MPA and MCMP) are predicting drift ratioswith sufficient accuracy. However, MPA procedure is more accuratethan MCMP especially in 10-story structure. Drift ratios over upperfloors, where multi-stage analysis is governing, estimated by MCMPprocedure are as accurate as MPA procedure; however, in 10-storybuilding results are overestimated using MCMP procedure becauseof system characteristics behave rigidly.

The errors in story drift ratios are illustrated in Figs. 8 and 9. Asshown in these figures the MCMP procedure provides better estima-tions than the MPA procedure at some higher floors, whereas the er-rors from the MPA procedure are less than those obtained from theMCMP procedure at some lower floors. Maximum error for MCMPprocedure happens in lower stories which are obtained and domi-nated by single-stage analysis and choosing adequate load patternfor single-stage could omit these errors over lower stories. The errorsfor MPA procedure are 9.65, 4.37, 32.29 and 4.23% for 10-, 15-, 20- and30-story buildings in the same order, and also for MCMP procedure

ture

% 0.5% 0.6% 0.7% 0.8%

0.5% 0.6% 0.7% 0.8%

ratio

ratio

e

0.9%

NLTH

MCMP

MPA

ifts for the 20 and 30-story structures.

a) Story Drift Ratios of the 10-Story structure

0

1

2

3

4

5

6

7

8

9

10

-10% -5% 0% 5% 10% 15% 20%Error

Flo

or

MCMP

MPA

b) Story Drift Ratio of the 15-story structure

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-40% -35% -30% -25% -20% -15% -10% -5% 0% 5% 10%Error

Flo

or

MCMP

MPA

MCMP

MPA

MCMP

MPA

Fig. 8. Errors in the story drifts for the 10 and 15-story structures.


are 26.47%, 14.24%, 36.94% and 16.97%. It is notable that 15-story build-ing has large displacement and deformation at first two stories thatcaused by some used records in NLTH analysis. These deformations can-not be predicted by such nonlinear static analysis procedures.

Therefore the accuracy of MCMP procedure as well as MPA proce-dure in predicting story drift ratios of studied buildings is acceptable.

Hinge plastic rotations of structures are obtained by considering themaximum plastic rotations of middle spans. The value of hinge plasticrotation related to each floor is obtained from software. The results ofhinge plastic rotations and their errors are shown in Figs. 10 to 13.

Obviously, MCMP procedure has better accuracy in prediction ofhinges plastic rotations and however, the error of MPA procedure issignificant. This result is not only valid for the medium-rise structures(10 and 15 stories structures) but also the results for high-rise struc-tures (20 and 30 stories structures) are more accurate. As shown inthese figures MCMP procedure has more accurate results in upperfloors. However, the small value of hinge plastic rotations obtainedfrom NLTH analysis in lower floors cause 100% errors in MCMP andMPA procedures. In this case, at lower floors, MCMP even has betterprediction in comparing toMPAprocedure. As itwasdemonstrated pre-viously, effects of highermodes at higher floors have a significant role toachieve more accurate results. The main reason of this achievementwhich could also be considered as an advantage of MCMP procedureis the manner how incremental forces are applied. Applying incremen-tal forces consecutively in MCMP procedure takes into account the ef-fects of higher modes and nonlinearity simultaneously, whereas inMPA procedure these higher mode effects are not considered and

usually the nonlinearity of the system is considered just for the firstmode. The manner of applying incremental lateral forces in the multi-stage pushover analysis of the MCMP procedure, in comparison withthe MPA procedure, has significant effects in prediction of hinge plasticrotations at themiddle and upper floors. However, the pushover proce-dure suffers from the limitation that it is unable to take into account thecumulative rotation of hinges due to cyclic hysteretic behavior [19].

The errors of hinge plastic rotations forMCMPprocedure are 70.84%,39.22%, 23.34% and 72.81% for 10-, 15-, 20- and 30-story buildings in thesame order, while in MPA procedure these errors are 87.49%, 100%,100% and 100%. These results confirmed the accuracy of MCMP proce-dure in comparison with MPA procedure.

In addition to the above discussion for elaborating the explanationabout the hinge plastic rotation errors obtained from the MCMP proce-dure, some details are required. Regarding to Fig. 10, for 10-storybuilding in 3th floor and 10th floor, these errors are significant, becausethe exact values obtained from nonlinear time history analysis areso small for the mentioned floors. The same remarkable errors areextracted from Fig. 11 for 15-story building from 3th floor to 6th floorwhere exact hinge plastic rotations are small and consequently causeenormous errors. 4th floor and 20th floor of 20-story building (Fig. 12)demonstrated that the error of the MCMP procedure is 100%. Because inthese floors the exact value of hinge plastic rotations is almost zero,therefore the errors in these floors reach to 100%. For 30-story high risebuilding, the errors of the proposed procedure in 6th to 9th floors aresignificant due to small value of hinge plastic rotations obtained fromnonlinear time history analysis as an exact solution.

a) Story Drift Ratio of the 20-story structure

0

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b) Story Drift Ratio of 30-story structure

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MPA

Fig. 9. Errors in the story drifts for the 20 and 30-story structures.


7. Conclusions

The article focus on the assessment ofmodified version of consecutivemodal pushover analysis procedure for estimating the seismic demandsof tall building with dual system considering steel concentrically bracedframes. In the MCMP procedure, the seismic responses are evaluated byenveloping the peak responses obtained from the multi-stage andsingle-stage pushover analyses. Linearly-elastic modal properties areused in the multi-stage pushover analysis. The force distribution overthe height of the building in each stage of themulti-stage pushover anal-ysis is determined by mass matrix and relevant elastic mode shape. Thelateral forces are incrementally applied during the stages of the multi-stage pushover analysis. For this purpose, the MCMP procedure appliedto structure and then maximum responses from single- and multi-stagepushover are obtained and at the end the envelope of the peak responsesis obtained according to the mentioned details.

The MCMP procedure utilizes the basis of CMP procedure anduses single- and multi-stage pushover analyses. The contribution ofmodes to calculate incremental displacements of each mode is thedifference between the CMP and MCMP procedures. Actually, in theMCMP procedure, unlike to CMP procedure, more than two or threemodes could be considered. It does not need to know the fundamentalperiod of structure.

The accuracy of the MCMP procedure was evaluated, on the seismicresponse of tall buildings considering dual systemwith steel concentri-cally braced frames in this research and some changesweremade in theCMP procedure to gain more accurate and reliable results. The

advantage of the MCMP procedure is the manner of applying lateralforces which are applied to structure consecutively in order of modesnumber.

As it was mentioned previously, single- and multi-stage push-over analyses influence over lower and upper story responses, re-spectively. Indeed, in single-stage pushover analysis the accuracyof results extremely depends on applied load pattern and in theMCMP procedure choosing adequate load pattern for single-stageanalysis is very vital. In this investigation, uniform load pattern hasbeen selected to utilize in the single-stage pushover analysis.According to remarkable effects of higher modes over upper stories,the manner of calculating modes contribution is as important aschoosing adequate load pattern.

According to performed researches by authors, choosing effectivemodal participating mass ratios to calculate mode contributions inmulti-stage analysis have no accurate results. Using of the basis of Re-sponse Spectrum Analysis (RSA) is proposed to estimate themode con-tributions. In this case, the procedure hasmore reliable results especiallyfor upper stories.

The MCMP procedure has a reliable prediction of seismic demandson upper floors; especially in case of hinge plastic rotations the MCMPprocedure has better estimation in comparing to MPA procedure overall stories. However, in order to estimate displacements and drift ra-tios, the results of the MPA procedure for lower stories are more accu-rate than those obtained from the MCMP procedure. In addition, theassessment of the both procedure in evaluating of displacementsand drift ratios for upper stories in comparing to nonlinear time

a) Hinge Plastic Rotaions

0

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0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Hinge Plastic Rotaion (Rad)

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b) Errors of Hinge Plastic Rotaions

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MPA(3 Modes)

Fig. 10. Height-wise variation of the hinge plastic rotations and their errors for the 10-story structure.

a) Hinge Plastic Rotations

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loor

NLTH

MCMP

MPA(3 Modes)

b) Errors of Hinge Plastic Rotations

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MPA(3 Modes)

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MCMP

MPA



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0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

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MPA(5 Modes)


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MCMP

MPA(5 Modes)

MCMP

MPA(5 Modes)




history analysis demonstrated the reliability of both of them. To sumup, the time-consuming of the MCMP procedure for estimating seis-mic demands of tall building in comparing to the MPA procedure isthe another advantage.

Furthermore, the significant errors obtained from the MCMP proce-dure in predicting hinge plastic rotations happenwhere the exact valuesextracted from nonlinear time history as an exact solution are small andtherefore the application of the proposed method is on a safe side.

References

[1] Papanikolaou V, Elnashai AS, Pareja. Limits of applicability of conventional and adap-tive pushover analysis for seismic response assessment. Report 2005/02. Mid-America Earthquake Center Civil and Environmental Engineering Department Uni-versity of Illinois at Urbana-Champaign; 2005.

[2] Sasaki KK, Freeman SA, Paret TF. Multi-mode pushover procedure (MMP) a methodto identify the effects of higher modes in a pushover analysis. Proceedings of 6th USNat. Conf. on Earthq. Eng.; 1998. Seattle (Washington).

[3] Moghadam AS. A pushover procedure for tall buildings. Proc. 12th European Con-ference on Earthquake EngineeringLondon (United Kingdom): Elsevier ScienceLtd; 2002. p. 395.

[4] Chopra AK, Goel RK. A modal pushover analysis procedure for estimating seismicdemands for buildings. Earthquake Eng Struct Dyn 2002;31:561–82.

[5] Chopra AK, Goel RK, Chintanapakdee C. Evaluation of a modifiedMPA procedure as-suming higher modes as elastic to estimate seismic demands. Earthquake Spectra2004;20(3):757–78.

[6] Jan TS, LiuMW, Kao YC. An upper-bound pushover analysis procedure for estimatingthe seismic demands of high-rise buildings. Eng Struct 2004;26:117–28.

[7] Aydinoglu MN. An incremental response spectrum analysis procedure on inelasticspectral displacements for multi-mode seismic performance evaluation. BullEarthquake Eng 2003;1:3–36.

[8] Kalkan E, Kunnath SK. Adaptive modal combination procedure for nonlinear staticanalysis of building structures. ASCE. J Struct Eng 2006;132(11):1721–31.

[9] Poursha M, Khoshnoudian F, Moghadam AS. A consecutive modal pushover pro-cedure for estimating the seismic demands of tall buildings. Eng Struct 2009;31:591–9.

[10] Chopra AK. Dynamics of structures. Theory and applications to earthquake engi-neering, 2nd edEnglewood Cliffs (NJ): Prentice Hall; 2001.

[11] Applied Technology Council. ATC-40. Seismic evaluation and retrofit of concretebuildings. Vol. 1–2. Redwood City (California); 1996.

[12] Building Seismic Safety Council (BSSC). NEHRP guidelines for the seismic rehabilitationof buildings. FEMA-273. Washington (DC): Federal Emergency Management Agency;1997.

[13] Building Seismic Safety Council (BSSC). Pre-standard and commentary for the seismicrehabilitation of buildings. FEMA-356. Washington (DC): Federal EmergencyManagement Agency; 2000.

[14] Fajfar P. Capacity spectrum method based on inelastic demand spectra. Earth-quake Eng Struct Dyn 1999;28:979–93.

[15] Fajfar P. A nonlinear analysis method for performance based seismic design.Earthquake Spectra 2000;16(3):573–92.

[16] Fajfar P, Gaspersic P. The N2method for the seismic damage analysis of RC buildings.Earthquake Eng Struct Dyn 1996;25:31–46.

[17] Mwafy AM, Elnashai AS. Static pushover versus dynamic analysis of R/C buildings.Eng Struct 2001;23:407–24.

[18] TsoWK,MoghadamAS. Pushover procedure for seismic analysis of buildings. Progressin Struct Eng Mat 1998;1(3):337–44.

[19] Kim S, D'Amore E. Pushover analysis procedure in earthquake engineering. EarthquakeSpectra 1999;15:417–34.

[20] AISC-ASD. Manual of steel construction, allowable stress design. Chicago (IL):American Institute of Steel Construction; 1989.

[21] Standard No. 2800-05. Iranian code of practice for seismic resistant design of buildings.3rd ed. Building and Housing Research Centre, Iran; 2005.

[22] Computers, Structures Incorporated (CSI). SAP 2000 NL. USA, CA: Berkeley; 2004.

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