Assessment of MIDA method for mid-rise steel structures ...scientiairanica.sharif.edu/article_21356_702935d... · higher modes of structure on its failure mechanism. By providing

Scientia Iranica A (2019) 26(5), 2703{2711

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringhttp://scientiairanica.sharif.edu

Assessment of MIDA method for mid-rise steelstructures with non-symmetrical plan

M. Zanjanchi and M. Mo�d�

Department of Civil Engineering, Sharif University of Technology, Tehran, Iran.

Received 24 May 2017; received in revised form 9 May 2018; accepted 11 March 2019

KEYWORDSModal incrementaldynamic;Incremental dynamicanalysis;Asymmetric-planbuilding;Unconventional plan;Seismic demand.

Abstract. Determination of nonlinear dynamic behavior of structures has always beenone of the main goals of both structural and earthquake engineers. One of the newestmethods for analyzing seismic behavior of structures is Modal Incremental DynamicAnalysis (MIDA). In fact, this method is an alternative to the Incremental DynamicAnalysis (IDA), which is a di�cult and time-consuming method. Despite the MIDA'sapproximate results, advantages such as adequate accuracy, high speed, and low cost makethis method an e�cient and appropriate approach. In all the previous studies, the proposedmodels have had a regularized plan; hence, all the analyses have been carried out ona frame. In this study, the MIDA analysis was developed in an asymmetric-plan-typebuilding by considering three structures with 4, 7, and 10 stories having irregularity inplan. Accuracy of the work was examined. Also, through a simpli�cation, instead ofconsidering an unconventional plan, we used a rectangular plan with an eccentricity of 15%between the center of mass and the center of rigidity. Comparing the results of this studyand the IDA method proved the high level of accuracy of this method in assessing seismicdemands.© 2019 Sharif University of Technology. All rights reserved.

1. Introduction

Accurate estimation of seismic demand and capacityof structures is the main basis of Performance-BasedEarthquake Engineering (PBEE). However, severalmethods are being proposed for investigation. Explicitattention to the inelastic behavior of structures isrequired for estimating the seismic demands at lowperformance levels such as life safety and collapseprevention. Nonlinear static processes are utilized asa part of the conventional performance evaluation ofstructural engineering, although Non-Linear Response

*. Corresponding author. Tel.: +98 21 66014828;Fax: +98 21 66014828E-mail addresses: [email protected] (M.Zanjanchi); mo�[email protected] (M. Mo�d)

doi: 10.24200/sci.2019.21356

Time History Analysis (NL-RHA) is one of the mostprecise methods for seismic demand computations.

Many researchers have attempted over the lastfew years to complete the pushover analysis and toovercome the disadvantages of the previous methods.Sasaki et al. (1988) tried to apply the e�ects of thehigher modes of structure on its failure mechanism.By providing the MMP method, they were able todetect the failure mechanism under higher modes,in addition to using the advantages of the pushovermethod [1]. In 2002, PRC method was presentedby Moghadam and Tso. In this method, the e�ectof higher modes was considered. The structure wascovered under di�erent load distributions that werebased on di�erent shape modes [2]. In 2002, Chopraand Goel presented a method for calculating the seismicresponses of structures. This method was derived fromthe concepts of multi-mode pushover (MMP) analysisand spectral response analysis, in which the e�ect

2704 M. Zanjanchi and M. Mo�d/Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 2703{2711

of higher modes was also considered [3]. Moreover,methods such as Incremental Modal Pushover Analysis(IMPA), Modi�ed Modal Pushover Analysis (MMPA),and evaluation of energy-eased modal pushover analysishave been presented by researchers to improve thepushover method [4-6].

All the studies mentioned here are restrictedto two-dimensional frames and three-dimensionalsymmetric-plan structures. In recent years, researchershave attempted to generalize pushover analysis meth-ods to asymmetric-plan 3D-type building structures forconsidering torsional e�ects. The following studies areamong the most important ones carried out in this �eld.

Chopra et al. developed the Modal PushoverAnalysis (MPA) method, �rst, in a regular 3D structureand then, in a structure with irregularity in plan [7-9]. Poursha et al. developed the consecutive modalpushover procedure �rst for a structure having irregu-larity in one direction and then, considering irregularityin both directions [10-12]. Many researches have beenconducted to investigate and estimate the behavior ofasymmetric-plan buildings (e.g., [13-16]).

All the above-mentioned studies have been de-voted to the development of pushover methods. Onthe other hand, since the Modal Incremental DynamicAnalysis (MIDA) method is a combination of MPA andIncremental Dynamic Analysis (IDA), it is also neces-sary to express the history of IDA method. NonlinearIDA, which seems to be the most accurate structuralanalysis method under earthquake stimulation, was�rst presented by Bertero in 1977 [17]. Afterwards,many researchers attempted to develop and completethe approach, e.g., Brunesi et al. [18], Zhou et al. [19],and Jalili Sadr Aabad et al. [20]. High accuracyand actual simulation of the behavior of the structureare among the advantages of this method. In thismethod, the actual nonlinear behavior of the materialas well as the actual dynamic nature of the earthquakestimulation is considered. The main method presentedas an alternative to the IDA is the MIDA method,which is called MPA-based IDA by some researchers.This method was �rst presented by Mo�d et al. in2005 [21] and then, it was utilized by Han & Chopra in2006 [22]. In 2011, this method was developed by Mo�det al. by means of the trilinear idealization model [23].

This study is aimed at covering a broader rangeof structures by applying MIDA as a quick and ac-curate method to steel structures with medium- andrelatively-low-rise buildings that are irregular in theplan.

2. The equation of motion

As we know, solving the di�erential equation for aSingle-Degree-Of-Freedom (SDOF) structure is sim-pler and faster than solving the same equation for

Multi-Degree-Of-Freedom (MODF) structures. Thisissue is most important for doing nonlinear analyses.Thus, the idea of using an SDOF structure with thecharacteristics of an MDOF structure is a familiarone. Now, the only important issue is to obtainthe characteristics of the SDOF structure, includingperiod, damping, yielding load, and sti�ness strain.Additionally, the conversion factor for the responsesfrom the SDOF structure to the MDOF structureshould be determined.

Since in this study, the structures are modeledin 3D form, in order to obtain the characteristics ofthe SDOF structure, the equations of motion are �rstexpressed and then, the MIDA method is interpreted.

In the case of considering x and y componentsof the motion of the Earth, the equation of motiongoverning the structure response will be as follows:

M �u+ fs(u; sign _u) = �M�x�ugx(t)�M�y�ugy(t); (1)

which can also be written in matrix form as follows:24m 0 00 m 00 0 Io

358<:�ux�uy�u�

9=;+ fs(u;sign _u) =

�24m 0 0

0 m 00 0 Io

35 ix�ugx(t)�24m 0 0

0 m 00 0 Io

35iy�ugy(t);(2)

where, m is a diagonal matrix equal to the lumped massof each story oor and Io is a mass moment of inertia ofeach story. ux and uy are the horizontal displacementsin x and y directions, respectively, and u� is related tothe torsional motion of the oor of the structure. Thehysteretic and nonlinear forces, fs, including fsx, fsy,and fs�, represent the horizontal components of forcein x and y directions and torsional force, respectively.

In the governing equation, the e�ect vector as-sociated with the motion of the Earth along x and ydirections is as follows:

�x =

24100

35 ; �y =

24010

35 : (3)

The second side of the governing equation can berewritten as the e�ective earthquake force, s�ug, alongx and y directions, where s is the spatial distributionof the e�ective earthquake force:

Pe�(t) = �s�ug(t) = �8<:m1

00

9=; �ugx(t); and

�8<: 0

m10

9=; �ugy(t); (4)

M. Zanjanchi and M. Mo�d/Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 2703{2711 2705

s =3NXn=1

sn =3NXn=1

�nM�n =3NXn=1

�n

24m�xnm�xyIo��n

35 ; (5)

where, �xn, �yn, and ��n indicate the modal shapes ofthe structure vibration in two transitional and torsionaldirections and:

�n =LnMn

Mn = �TnM�n Ln =

(�Tnxm1�Tnym1 (6)

3. MIDA method

IDA is capable of well illustrating the demand ofstructures at di�erent levels of earthquake. Assessingseismic demand under di�erent levels of earthquake byusing an approximate approach with su�cient accuracywill save time. In the MIDA method, non-lineardynamic analysis of the MDF structure has not beenused to obtain IDA curves so far. To achieve this goal,we �rst considered the entire structure equivalent toan SDOF one and then, evaluated it through a modalpushover analysis method. Changes can be made inthe calculation speed when the analysis of an MDOFstructure is not nonlinear dynamic.

The MIDA procedure to estimate the seismicdemand of an asymmetric-plan multistory building isa sequence of steps found below:

Step 1. Design and modeling of structures: First,the structures are linearly analyzed and carefullydesigned. The dynamic properties of the structures,including periods and mode shapes, are obtained inall the three components of x, y, and �.

Step 2. Pushover analysis and obtaining capacitycurve of the structures: In order to map an MDOFstructure onto SDOF structure, capacity curve ofthe structure relative to its seismic properties isrequired. The pushover analysis of the lateral load,S�n (Eq. (7)), is performed for structures in each modeand the base shear-roof displacement curve, which is

same as the frame capacity curve, is obtained:

S�n =

8<:m�xnm�ynIo��n

9=; : (7)

Step 3. Idealizing the pushover curve as a bilinearcurve: In order to perform an SDOF structuralanalysis, the force-displacement model is required.Considering software limitations and keeping simplic-ity of the wok in this method, the force-displacementmodel should be bilinear. Therefore, the MDOFstructure capacity curve in the dominant directionshould be approximated by a bilinear model in orderto obtain the characteristics of an SDOF structurefrom it.

Step 4. Obtaining the SDOF system characteris-tics: To convert the main MDOF structure to anequivalent SDOF structure, it is applied in sucha way that the SDOF structure has period anddamping equal to the period and damping of the mainMDOF structure. Also, its load-deformation curveis obtained from the MDOF structure as shown inFigure 1.

Consequently, the following relationships can bewritten:

FsnyLn

=VbnyM�n

; (8)

Dny =urny

�n�rn; (9)

where:Fsn The scaled sti�ness of the system

proportional to time response of thestructure;

V ybn Yielding strength;uyrn Yielding displacement of the roof of

the ith mode of vibration;� The strain hardening angle of the

material;

Figure 1. Properties of the nth-mode inelastic SDF system in the pushover curve.


�rn The �n at the roof in the direction ofthe selected pushover curve.

M�n and �n correspond to the direction of theground motion under consideration (x or y) and areequal to:

M�n = �nLn (10)

Step 5. Analyzing SDOF structure: At this step, theobtained SDOF structure is subjected to the intendedearthquake record and nonlinear time-history anal-ysis is performed in order to calculate a maximumdisplacement (Dn). The SDOF structure acts as anonlinear spring the force-displacement characteristicof which has been obtained in Step 4.

Step 6. Calculating the maximum displacement ofMDOF structure: By multiplying the coe�cients�ny and �yn from the displacement obtained in theprevious step, the maximum displacement of theMDOF structure will be calculated:

uxn = �nx�xnDn uyn = �ny�ynDn: (11)

Step 7. Performing pushover analysis and extractingthe results: The main MDF structure will be pushed,according to S�n lateral loading pattern, to the targetdisplacement urn, which was obtained in the previ-ous step. Then, the maximum drift of each storywill be computed for the considered level of scaledearthquake record.

Step 8. Combination of the e�ects of modes: At theend, the results obtained from di�erent modes shouldbe combined. For this purpose, the maximum roofdisplacement, the relative story displacement, andthe maximum plastic rotation obtained in each modeat the identical seismic level are combined using theSRSS rule.

4. Modeling and record selection

The studied structures are 4-, 7-, and 10-story buildingsthe structural system of which is moment-resistingframe. In addition, the height of each story is 3 m,and the building has 5 spans of 4 m in the x directionand 3 spans of the same length in the y direction. The3D view of the 4-story building is shown in Figure 2.

The structures were primarily designed by assum-ing linear behavior for the materials, in uencing P ��e�ect, perfectly rigid beam and column connections,and disregarding the dimensions of the panel. Thefollowing were also considered in the design:

� The buildings were placed on soil type II;

� The building gravity loading was in accordance withthe American standard for loading (ASCE7-10) [24];

Figure 2. 3D view of the 4-story structure.

� The building lateral loading was according to theAmerican standard for loading (ASCE7-10);

� Designing of the steel members was performedbased on American standard for steel structures(AISC360-10) [25].

To create an asymmetry, the story masses weredistributed in an unbalanced way so that each story hadan eccentricity of 15% between the centers of gravityand rigidity along the y axis (Figure 3).

All nonlinear analyses were performed by OpenSees software (Computers and Structures, 2004).

Scaling acceleration records is one of the most im-portant challenges in nonlinear time history analyses.Di�erent codes and approaches have been proposed forscaling the acceleration records. In this study, theacceleration records were scaled up according to thefourth edition of the Iranian standard no. 2800 (codeof practice for earthquake-resistant design of build-ings) [26] and noting the considerations suggested byCharney [27]. These three asymmetric-plan buildingswould experience coupled x-horizontal and torsionalmotions due to the x-component of earthquake, whichwas the focus of this paper. In this regard, three ground

Figure 3. Plan of the selected asymmetric-plan buildings.


Figure 4. Natural periods and modes of vibration for 4-, 7-, and 10-story asymmetric-plan systems: (a) Asymmetric-planof 4-story, (b) asymmetric-plan of 7-story, and (c) asymmetric-plan of 10-story.

motion records were used for the time history analysis,including Imperial Valley (1940), Northridge (1994),and Park �eld (1966).

5. Numerical results

After verifying the modeling, the mode shape of eachstructure was drawn. For this purpose, it was takenfrom the participation of the �rst six mods of eachstructure. Figure 4 illustrates the mode propertiesof each structure including the natural period of eachmode, the horizontal components of x-direction, andthe torsional component. Having the mode shape andits values, the equations explained in detail in theprevious section were solved.

Thereafter, the structures were pushed with re-gard to force distribution, S�n, to obtain the baseshear curves based on the roof displacement andthen, the curves were idealized with bilinear curves.The pushover curves for the four modes of the 10-story building, where x component of displacementsis dominant compared to the y component, are shownin Figure 5.

Figures 6, 7, and 8 compare the MIDA and theexact IDA for 4-, 7-, and 10-storey buildings with threeground motion records, respectively. Each �gure ispresented for maximum inter-story drift.

In order to estimate the seismic demand, thecontribution of the �rst six modes was included in theanalysis of the asymmetric building. As expected, the


Figure 5. Bilinear idealization of the pushover curve for the 10-story building.

Figure 6. Comparing the single IDA and MIDA curves for maximum inter-story drift with three records for the 4-storybuilding.




Figure 9. In uence of higher modes on the accuracy of MIDA method.

Figure 10. Story drifts at CM for the �rst �ve steps of Imperial Valley earthquake.

�rst mode had an insigni�cant impact in estimatingthe story drifts, especially in the upper step of theearthquake (Figure 9). Also, considering the responsecontribution of higher modes signi�cantly enhanced thestory drifts. For checking the precision of this method,story drifts at CM for Imperial Valley earthquake aredisplayed in Figure 10 only for maximum displacement.

6. Conclusion

The purpose of present study is to extend the ModalIncremental Dynamic Analysis (MIDA) method for 3Dstructures with irregularity in plan. Performance ofthe MIDA procedure has been investigated on threemoment-resisting structure with 4, 7, and 10 stories.Seismic responses from MIDA procedure have beencompared with those from IDA method.

� From the single and multiple curves of IDA andMIDA as well as their di�erence percentages, it

was found that the MIDA method was capable ofcorrectly assessing the behavior of the structure withan error not exceeding 15%;

� The error of MIDA method at lower levels of earth-quake intensity was much lower and the di�erencebetween the two diagrams increased by increasingearthquake intensity. This was due to the nonlin-ear behavior of the structure in higher earthquakeintensities. In fact, in smaller scale factors, sincethe structure was in the linear region, the accu-racy of the MIDA method was greater and themethod was more consistent with the results of IDA.However, by increasing the earthquake intensity,the structure began to leave the linear region andmove towards the nonlinear region. Moreover, thisnonlinear behavior led to approximations in thework. Therefore, as the intensity of the earthquakeincreased and more structural elements entered thenonlinear region, inaccuracy of the MIDA approach


increased. However, it was still accurate enough inhigh-intensity earthquakes;

� The di�erence between the MIDA and IDA di-agrams in the 4-story structure was very small;however, the greater the number of stories, thegreater the di�erence would be. The reason was thatthe e�ect of higher modes, especially the torsionalmodes, increased with increase in the number ofstories. Also, inaccuracy of MIDA increased;

� In the analysis of 3D structures using the MIDAmethod, solely the �rst mode of a structure cannotgive us its correct response. Therefore, in suchstructures, the e�ect of higher modes should beconsidered. We used the �rst six modes in thepresent study;

� The most important issue to note is that using anMDF systems to perform dynamic analysis (IDAmethod) is very time-consuming. This issue isintensi�ed with a steep slope with increase in thenumber of stories, while the time demanded by theMIDA method for the analysis of a structure isconsiderably low in comparison. For instance, thecomputation time required for 4-storey buildings isreduced from two days for an accurate result to onlytwo hours for the approximate result.

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Biographies

Moein Zanjanchi completed his MSc studies at SharifUniversity of Technology in 2016. His research interestsare in the areas of structural dynamics and earthquakeengineering, seismic design of steel and concrete struc-tures, �nite element analysis, and rehabilitation.

Massood Mo�d is a Professor in the Civil Engineer-ing Department at Sharif University of Technology. Hecompleted his PhD studies at Rice University.

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Assessment of MIDA method for mid-rise steel structures ...scientiairanica.sharif.edu/article_21356_702935d... · higher modes of structure on its failure mechanism. By providing