higher mode effects on seismic behavior of mdof steel ...response of multi degree of freedom (mdof)...

Higher Mode Effects on Seismic Behavior of MDOF Steel Moment Resisting Frames

JSEE: Fall 2003, Vol. 5, No. 3 / 41

ABSTRACT: In this paper, the effects of higher modes on seismicresponse of multi degree of freedom (MDOF) steel moment-resistingframes (SMRF) are investigated. Modification factors to the responseof first mode SDOF system are presented in order to estimate seismicMDOF system demands. The study is based on spectral analysis andlinear and nonlinear dynamic time history analysis of 4, 10, 15, 20 and25 storey SMRFs under Elcentro, Tabas, Naghan and Manjilearthquake loading. A modification factor el

dispα is defined in order toestimate roof elastic displacement demands of an MDOF frame fromfirst mode elastic displacement spectra. Base shear modification factor

VMH.α (to compute MDOF strength reduction factors), maximum storyy

drift demand modification factor ,.d

MHα and maximum story dynamicductility demand modification factor µα MH. are defined and presentedfor SDOF system responses in order to estimate the main MDOF systemnonlinear seismic responses, including higher mode effects.

Keywords: Seismic response; Demand; Higher mode effects; Dynamicanalysis; SMRF

Higher Mode Effects on Seismic Behavior of MDOF Steel

Moment Resisting Frames

1. Introduction

Different responses of MDOF frames under seismicmotions are influenced by higher mode effects andthus the overall response of the structure will beusually significantly different from the first moderesponse. The amount of this effect depends onvarious parameters such as response type, earthquakespecifications, structural configurations and level ofductility. Higher mode effects are clearly moreimportant for high rise structures and thus eliminatinghigher modes may lead to incorrect results. Thesensitivity of different structural responses to highermode effects is also different.

Several studies on strength reduction factors,have been conducted in SDOF systems, the mostimportant one is presented by Miranda and Bertero[1] however only few researchers have studied theMDOF effects on strength reduction factors. Therelationship between MDOF and SDOF systems wasfirst studied by Veletsos and Vann [2]. The objective oftheir study was to identify the significant parameters

F. Daneshjoo1 and M. Gerami 2

1. Associate Professor, Civil Engineering Department, Tarbiat Modarres University,Tehran, Iran, email: [email protected]

2. Ph.D. Student, Tarbiat Modarres University, Civil Engineering Department, email:[email protected]

in the response of MDOF elastoplastic systems.The relationship between the response of thesesystems with the equivalent linear ones was alsostudied. Nassar and Krawinkler [3] studied threetypes of simplified MDOF models to estimate themodifications required to the inelastic strengthdemands, obtained from bilinear SDOF systems inorder to limit the story ductility demand in the firststory of MDOF systems to a predetermined value.The three types of MDOF models were composed oftwo-dimensional regular frames named “beam-hinge”,“column-hinge” and “weak-story” models.

Humar and Rahgozar [4] studied a ten-story shearframe with a uniform distribution of mass and storyheights. In the study, it was shown that for highlevels of ductility, the displacement ductility demandin many stories of MDOF frame can be much higherthan the equivalent SDOF system. They also concludedthat the critical story in most of MDOF systems isthe lowest; however, the upper stories can also exhibit

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F. Daneshjoo and M. Gerami

larger ductility demands due to the participation ofhigher modes. Krawinkler and Seneviratna [5] studiedthe modifications to SDOF system responses inorder to estimate seismic demands of MDOF framesfrom elastic and inelastic spectra. They consideredtwo types of lateral load resisting systems ofmoment-resisting frames and isolated structuralwalls. This study concluded that except for shortperiod frames, the maximum story ductilitydemand for MDOF models was higher than thetarget ductility ratio of the first mode SDOF system.This amplification increases with increasingperiods, which indicates the importance of highermode effects.

Gupta [6] studied the behavior of moment framesunder different levels of seismic hazard. They consid-ered 3, 9 and 20 story frames and evaluated highermode effects on maximum story drift demand. Theemphasis of their study was on quantification ofglobal and local deformation demands for varioushazard levels. It was concluded that for long periods,as higher mode effects become more important, thedistribution of drift demands in the height of theframe would be no more uniform and the differencebetween global and story drift demands increasewith period. Santa-Ana and Miranda [7] studiedstrength reduction factor of MDOF systems bydefining modification factors to SDOF systems. Theyalso investigated soil condition effects on thesefactors. Their study concluded that the modificationfactors increase for long periods and high levels oftarget ductility. This implies the importance of highermodes in strength (base shear) response of MDOFframes. Daneshjoo and Gerami [8-9] studied thereasons contributing in overstrength of highriseSMRFs and higher mode effects importance onseismic behavior. They concluded that higher modeeffects in seismic evaluation of MDOF frames resultin the amplification of ductility demand and thusreduction of behavior factor. Therefore neglectingthese effects, specially in the case of tall buildingswould lead to unconservative results.

The main objective of this paper is to presentmodification factors to the response of first modeSDOF system in order to estimate seismic MDOFsystem demands. In this context, seismic response offive steel moment resisting frames models under fourdifferent earthquakes are studied using spectrumanalysis, linear and nonlinear dynamic time historyanalysis and DRAIN-2DX software [20]. Theinfluences of period and ductility level on higher mode

effects are also investigated.

2. Steel Moment Resisting Frame Models

Two dimensional 4, 10, 15, 20 and 25 storey steelmoment resisting frames with three bays have beendesigned according to Iranian 519 loading code andIranian 2800 code for seismic resistant design ofbuildings and are used in this study. The configura-tions of the frames together with their designresults are shown in Figure (1a). The inter-storeyheight and span length of the frames are all constantand equal to 4m and 5m, respectively. Storey massesare calculated using dead load plus 20% of live load.High seismic relative hazard for site region with designbase acceleration A=0.35g, beds of gravel and sandwith weak cementation for site soil classifications,median importance factor (such as residual buildingframes) for the frames, and a behavior factor of 6have been assumed in the designs. In mathematicalmodeling strain hardening ratio of 3% and dampingratio of 5% of critical one have been assumed. Thecriteria for the formation of plastic hinge and theelements behavior, was an interaction of Moment-Axial force (M-P) curve. The inter-storey drift anglecapacity at the serviceability Limit State has beenassumed equal to 0.006rad. The stiffness requirementhas governed the design, as is usually the case for steel

Figure 1a. Steel Moment Resisting Frame Models.


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structures. The beam-to-column joints were assumedto be full-strength and rigid in the design procedure.Rigid and full strength connections are characterizedby moment-rotation relationships with strengthdegradations. Nonlinear dynamic time historyanalyses have been performed by using DRAIN-2DXsoftware, where the proposed hysteretic modelshave been implemented.

3. The Seismic Input Characteristics

Four ground acceleration time-histories (Tabas,Naghan, Manjil and Elcentro earthquakes) have beenselected for performing numerical analysis. Tabasearthquake has a peak ground acceleration (PGA) ofabout 0.93g and strong motion duration of about 25seconds. Its predominant period is 0.769 seconds.Naghan earthquake has been recorded for 20.98seconds and has a PGA of about 0.72g and predomi-nant period of 0.764 seconds. The duration ofManjil record is 53.52 seconds. The PGA andpredominant period of this earthquake are 0.5g and0.556 seconds respectively. And finally Elcentroearthquake has a strong motion duration of about12.1 seconds. Its PGA and predominant period are0.32g and 0.555 seconds, respectively. Predominantperiods of the ground motions are calculated frompredominant frequency content of the same groundmotions, which is quantified by plotting FourierAmplitude Spectrum (FAS) and Power SpectrumDensity (PSD), using Finite Fourier Transform (FFT).Figures (1b) and (1c) show the acceleration timehistory of these earthquake components andcorresponding spectrums together with appropriateIranian 2800 design spectrum.

4. Method of Estimating Inelastic Demands

Few researchers have investigated the methods ofestimating MDOF inelastic seismic demands. Theirefforts can be categorized into following three maingroups:v) Methods based on equivalent linear structures.

Demands of an inelastic MDOF system areestimated from an elastic analysis of anequivalent MDOF system called “substitutestructure” [10]. In order to gain such a linearstructure, stiffness of potential inelastic elementsof the original structure is reduced and theireffective damping is increased. Empiricalrelations are presented to compute theseequivalent properties.

v Methods based on equivalent SDOF systems.Figure 1b. Acceleration time histories of earthquake

components.

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Figure 1c. Acceleration spectrum of the related earthquakeand design spectrum of Iran seismic code (2800).

The response of MDOF system is controlled bya single mode and the shape of this moderemains constant throughout the time history.Although both assumptions are not alwayscorrect, some studies conducted by severalresearchers [11, 12, 13, 14, 15] have indicatedthat they may lead to rather good predictions ofmaximum seismic response of many MDOFsystems.

v Methods based on using inelastic responsespectral, with elastic modal combinationmethods (i.e. SRSS and CQC) to compute theseismic demands of inelastic MDOF systems[16, 17]. Although this method has no theoreti-cal justification, it is expected to approximatethe overall inelastic behavior of MDOF systems.The studies carried by some researchers [18, 19]have shown that the results of this method mayneed to be modified to take into account thelocal concentration of inelastic effects andspecial effects due to near source motions.

In this paper, the quantification of seismic demandsfor an MDOF system is conducted through acomparative evaluation of inelastic dynamic responseof MDOF frames and their equivalent SDOF systemsand presenting modification factors to the responseof SDOF systems. Thus for each of the MDOFframes, an equivalent SDOF system is defined. Theproperties of these equivalent SDOF systems are setsuch that the weight of the SDOF system is the sameas the total weight of the original MDOF frame andthe period of vibration and damping ratio of SDOFsystem are the same as the fundamental modeproperties of the MDOF frame. The main reason ofdifference between the response of an inelasticMDOF frame and its equivalent SDOF system is the

contribution of higher modes in the response of MDOFsystem. However, other structural characteristicssuch as the global mode of deformation, distributionof strength and stiffness over the height of thestructure, structural system redundancy, mode offailure at both element and global levels and finallythe torsional affects can also affect the differenceof responses between MDOF systems and theirequivalent SDOF ones.

5. Higher Mode Effects in Elastic Analysis

The maximum effect of each mode is determined byconsidering the period of vibration of various modesof the frames and making use of the design responsespectrum. Then these maximum effects are combinedusing square root of sum of squares (SRSS) method toobtain the required response in the frames. In each oneof the perpendicular directions at least the first threemodes of vibration, or all the modes of vibration with aperiod of more than 0.4 seconds, or all first N modesthat sum of their modal mass participation factors ismore than 90 percent. The greater one is taken intoconsideration. Modal properties of the frames areshown in Table (1).

It can be observed that normalized first modemodal masses (represented by W1*/W in percent)decrease with increasing number of stories (period)which indicates the importance of higher mode effectsin highrise frames. For the same reason, the numberof modes required for analysis, increases with numberof stories or the fundamental periods.

Various linear seismic responses of a frame R are

Table 1. Designed MDOF Frame Modal Properties.

Frame Type N4b3 N10b3 N15b3 N20b3 N25b3 T1(Sec) 1.24 1.95 2.51 3.12 3.70 T2(Sec) 0.436 0.74 1.00 1.22 1.43 T3(Sec) 0.264 0.43 0.59 0.76 0.86 T4(Sec) 0.176 0.30 0.41 0.53 0.60 T5(Sec) ------ 0.23 0.32 0.40 0.45 PF1 1.305 1.397 1.484 1.533 1.556 PF2 -0.489 -0.587 -0.729 -0.867 -0.880 PF3 0.289 0.309 0.382 0.513 0.504 PF4 -0.105 0.212 0.285 0.318 0.312 PF5 ------ -0.180 -0.181 -0.212 -0.278 W1

*/W(%) 79.94 75.99 70.98 67.15 66.28 W2

*/W(%) 13.65 12.41 14.43 14.58 15.28 W3

*/W(%) 3.85 4.55 5.02 6.01 5.93 W4

*/W(%) 2.56 2.62 2.71 3.55 3.11 W5

*/W(%) ------- 1.22 1.59 1.84 1.81 Number of Modes Required for Analysis

3 3 4 5 5


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influenced by higher mode effects differently. Inthis paper, higher mode effects on roof shear Vroof,base shear Vbase and roof displacement Uroof areinvestigated. The higher model Eq. (1) is consideredas a criteria to evaluate the error caused by eliminat-ing modes higher than one.

1001 ×−

=αN

NelR R

RR (1)

elRα is the percentage of higher mode effect in

response R of an N story frame. R1 and RN are thelinear spectral response R of the frame (i.e. Vroof,Vbase and U roof ) considering 1 and N modes ofvibrations, respectively.

For each of the frames, linear seismic spectralresponses i.e. Vroof, Vbase and Uroof are calculatedonce using all the N required modes of vibrationsand another time using only the first mode ofvibration. The percentage of higher mode effect ineach response for each of the frames is thencalculated using Eq. (1). Figure (2) shows how thehigher mode effect factor el

Rα varies with the firstperiod T1 of each frame for different spectralresponses.

Figure 2. Higher mode effects on roof shear, base shear androof displacement of frames (%), linear spectrumanalyses.

It can be seen that the percentage of highermode effects in different responses of the framesincreases with increasing period which indicatesthe importance of higher modes in high rise frames.On the other hand for a certain frame, roof shearis affected by higher modes more than the othertwo responses and roof displacement takes theleast affect (i.e. ).el

UelV

elV roofbaseroof

α>α>α For exampleabout half of the overall roof shear response inthe frame n25b3 (45.3%) is allocated to higher

modes while this percentage for base shear and roofdisplacement responses decreases to 13.06% and1.16%, respectively.

The above results were only presented for 3 bayframes. Higher mode effects modification factorsare also calculated for 20 storey frames with 1, 2, 3and 5 bays in order to investigate the effects ofnumber of bays. The higher mode effect modificat-ion factor is computed for each response of theseframes (n20-b1, n20-b2, n20-b3 and n20-b5). Theresults are shown in Figure (3).

It is observed that the trend of higher modeeffects on different responses holds true for allnumber of bays. Higher mode effects in responses ofroof shear and base shear is reduced with increasingnumber of bays. An opposite trend was concludedfor roof displacement response.

Elastic first mode spectral displacement can be usedto estimate the roof elastic displacement demands ofMDOF frames using the elastic displacement demandmodification factor el

dispα as defined by Eq. (2)

elSDOF

eltel

dispδ

δ=α (2)

Where eltδ is the maximum roof displacement

demand of an elastic MDOF frame under a particularearthquake calculated by linear dynamic time historyanalysis. And el

dispα is the elastic first mode displace-ment obtained from response spectrum of the sameearthquake. Thus the maximum roof displacement ofan elastic MDOF frame is related to the elastic firstmode spectral displacement. Variation of el

dispα

computed for each of the five frames under thedifferent four earthquakes, together with variation of

Figure 3. Influence of number of bays in higher mode effectson roof shear, base shear and roof displacement of20-story frames.

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the first mode participation factors (PF1 s) of theframes are shown in Figure (4).

It can be seen that the mean values of eldispα ampli-

fication factors for different records are in goodagreement with PF1 factors. On the other hand, el

dispαfactors increase with increasing fundamentalperiod, and their variation is similar to PF1 factors.This means that maximum elastic roof displacementdemand of an MDOF frame can be estimated bythe product of elastic first mode spectral displacementby first mode participation factor. PF1 factors areactually applied to account for higher mode effects.The compatibility of PF1 and el d

MH,.α factors was also

concluded in other studies [21-22].

The important note in this process is to evaluatefundamental period of the structures. Design codesusually underestimate the period of SMRFs andconsequently elastic spectral displacements (Sd) areunderestimated too. Thus, it is strongly recommendedto compute fundamental period of the structuresthrough eigen value analysis in order to get relativelyjustified estimations of maximum roof displacementdemands in elastic MDOF frames.

6. Higher Mode Effects on Strength ReductionFactors of MDOF Frames

Linear analyses are unable to predict the nonlinearbehavior of the structures under strong earthquakeexcitations. A rational design concept should comparethe evaluated ductility reserves of the structure withductility demands evaluated for appropriate groundmotion by nonlinear time history analysis. Seismiccodes allow the structures to behave inelasticallywhen subjected to strong earthquake ground motions.Nonlinear behavior of the structures in inelastic rangeresults in the reduction of effective lateral forces. Thisreduction in design forces is applied through strength

Figure 4. Elastic normalized roof displacement demands-frame structures.

reduction factors Rµ. In this paper, at first, nonlineareffects of SDOF systems are investigated using Rµfactors and then a modification factor V

MH.α isdefined to account for higher mode effects instrength reduction factor of MDOF frames.

SDOF strength reduction factors Rµ that takeinto account the nonlinear behavior of these systemsare defined by Eq. (3)

µµ =SDOF

elSDOF

V

VR (3)

Where elSDOFV is the elastic strength demand of

SDOF system and µSDOFV is the minimum strength

demand required to keep displacement ductilitydemand of SDOF system less than a predeterminedvalue of target ductility µ. Figure (5a) illustrates howRµ varies in different frames with the period of firstmode of vibrations of the frames under differentearthquake loading and for different values of µ.Figure (5b) shows the variation of the mean values.

The results indicate that Rµ factors increase withthe increase in first natural period of vibration T1 andwith increase in level of ductility µ. The greater Rµfactors for large ductility values are due to the fact thatthese factors account for inelasticity in SDOF systems.Thus, they must increase in high levels of ductility tocause more reductions in design forces.

Due to the participation of higher modes, Rµfactors as defined by Eq. (3) can not be used asMDOF strength reduction factors. In order toaccount for higher modes effect, V

MH.α modificationfactor is defined by Eq. (4)

µ

µ

=αSDOF

MDOFVMH

V

V. (4)

Where µMDOFV and µ

SDOFV are the minimum lateralstrengths required to limit the ductility ratio of MDOFand SDOF systems to target values.

The strength demand of an SDOF system withductility µ, is multiplied by the modification factor

VMH.α to get the lateral strength demand of the

original MDOF frame with the same level of ductilityµ. On the other hand, 1/ V

MH.α represents a modifica-tion factor to SDOF strength reduction factors Rµ,to be applied to original MDOF frames. Thus theintroduced modification factors must be applied toRµ, Y and .µ factors, which are drawn from analyticalresults, in order to account for higher mode effectsand not to the presently used reduction factors incodes of practices.


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Figure 5a. SDOF strength reduction factors.

Figure 5b. SDOF mean strength reduction factors.

In this study, VMH.α modification factors for each

of the five frames are computed under the fourdifferent earthquakes and with three different targetductility values of 3, 5 and 7. Figure (6a) illustrateshow V

MH.α modification factors varies with the firstnatural period of each frame T1 for different ductilityµ. Figure (6b) shows the variation of the meanvalues.

It can be observed that VMH.α factors always

increase with increase in period T1, indicating theimportance of higher modes in strength demands ofMDOF frames. They are also usually incremental

with level of ductility. These factors are higher thanunity in all cases and for n25b3 frame, values aslarge as 4 have also been observed under certainrecords.

Estimation of interstory and global drift demandsis of great significance in seismic design of SMRFsbecause their excess from allowable limits wouldcause great damage in both structural and nonstruct-ural components. The distribution of drift demands inframes' height and the relationship between maximumstory drift and global drift demands are investigated.The frames with three different levels of structuralyield strength µSDOF of 3, 5 and 7 are subjected to thefour different earthquakes. Nonlinear dynamic timehistory analysis are conducted and interstory driftangles (i.e. storey drift over storey heigth) and globaldrift angles (Roof drift over total heigth of frame)are calculated. The corresponding results for the20-story frame are shown in Figures (7a) and (7b).

The distribution of drift demands in the structures,responding predominantly in their first mode, israther uniform over their height. As can be seenfrom Figures (7a) and (7b), the distribution of storydrift demands over the height of the 20-story frame is

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Figure 6a. MDOF modification factor for base shear (strength) demands.

Figure 6b. Mean MDOF modification factor for base shear(strength) demands.

not uniform, specially in higher stories that shows asudden amplification. Sudden amplification of storydrift in lower stories of a structure is due to ∆−Peffects, while the occurrence of such an amplificationin top stories is attributed to higher mode effects.Maximum story drift angle, has been observedlarger than global drift angle under all earthquakes andfor all values of SDOF ductility, in 20-story frame,

see Figures (7a) and (7b). The same result is observedin all the other frames.

7. Higher MODF Effects on Story Drift andDuctility Demands of MDOF Frames

Yield story drift iy,δ is computed using pushoveranalysis of story i such that all the nodes beneaththe considered story are fixed by hinge supports and alateral load is applied to the story roof incrementally.The consequent story drift corresponding to theformation of the first plastic hinge in the story isthen yield story drift ( ~

,iyδ dynamic story ductility ofthe frames under different earthquake records werecomputed for µSDOF values of 3, 5 and 7 usingEq. (5))

{ }iy

xmasxmas

is

xma

,

,,

,

,

δ

δδ=µ

−+

(5)

Where µs,i is dynamic ductility ratio of story iand −+ δδ max,max, , ss and iy,δ are the maximum positive,the maximum negative and the yield story drifts,respectively.


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Figure 7a. Inter story and global drift demands of 20-story frame.

These parameters are illustrated in Figure (8),schematically. In this study, the ductility of MDOFframes is defined as story ductility which wasdefined by Eq. (5). Yield story drift ( )iy,δ

.. is computedfrom pushover analysis of story i such that all thenodes beneath the considered story are fixed byhinge supports and a lateral load is applied to the storyroof, incrementally. The consequent story driftcorresponding to the formation of first plastic hingein the story is then yield story drift ( )iy,δ .

Figure 7b. Inter story and global drift demands of 20-storyframe (mean values).

Figure (9a) shows the distribution of storydynamic ductility demands over the height of 20-story frame under different earthquakes. Figure (9b)shows the variation of the mean values.

As can be seen there is a sudden amplificationin dynamic ductility demands of higher stories in20-story frame that can be ascribed to higher modeeffects. The same trend was observed for story driftdemands. Distribution of story drift and its variationtrend in the height of a structure is generally closely

Figure 8. Story shear vs. story drift response.

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Figure 9a. Story dynamic ductility ratios of 20-story frame.

Figure 9b. Story dynamic ductility ratios of 20-story frame(mean values).

related to the story ductility demands of the samestructure.

The difference between story and roof drift anglescan be a parameter for evaluation of higher modeeffects on maximum story drift demand. If a structureis not affected by higher modes and its behavior iscontrolled by one mode in linear state, the story androof drift angles are expected to be equal according tothe following equation

i

i

t

t

hhδ

=δ

(6)

It was observed from Figure (7) that thedistribution of story drift demands over the heightis not uniform, specially in higher stories that shows

a sudden amplification. Sudden amplification of storydrift in lower stories of a structure is mostly dueto P ∆− effects while the occurrence of such anamplification in top stories is mostly attributed tohigher mode effects. Therefore in reality Eq. (6) is justvalid in idealized situation. This equation doesn’thold true in reality, even in the linear range and for astructure vibrating in a single mode and as wasmentioned before it is just valid in idealized situation.The differences between roof drift and story driftangle cannot be attributed only to higher modeeffects, but it can be said that higher modes have adirect effect in this regard. The d

MH.α factor is definedby Eq. (7) in order to evaluate higher mode effectson maximum story drift demand

tt

isdMH h

h

/

/max,. δ

δ=α (7)

dMH.α factors were computed under different records

for each value of µSDOF (3, 5 and 7). The results areillustrated against T and µSDOF in Figure (10a),separately. Figure (10b) shows the variation of themean values.

It is observed that dMH.α factors increase with

increasing period and level of SDOF ductility,indicating the importance of higher mode effects.The ratio of average interstory drifts to roof driftangles is usually greater than unity. The reason is thatmaximum story drifts of a frame under any earthquake


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do not occur simultaneously and sum of maximumstory displacements is always larger than roofdisplacement under a certain earthquake. Thus inorder to estimate maximum story drift of a frame bypushover analysis, target displacement is required tobe taken larger than roof displacement under designearthquake. This case is intensified for significantcontribution of higher modes. In other words,maximum story drift demand is underestimated byPushover analysis.

Due to the close relationship between story drift

Figure 10b. Higher mode effects on maximum story drift angledemands (mean values).

Figure 10a. Higher mode effects on maximum story drift angle demands.

and ductility demands, higher mode effects onmaximum story dynamic ductility are almost similarto those for maximum story drift demands. Thefollowing factor is defined to evaluate these effects

SDOF

xmasMH

µ

µ=αµ ,

. (8)

Where xmas ,µ is the maximum story dynamicductility ratio in MDOF frame and SDOFµ is theequivalent SDOF ductility ratio. Maximum storyductility of an MDOF frame can then be related toSDOF ductility through this factor which refers tohigher mode effects on maximum story dynamicductility ratio in an MDOF frame.

µα MH. factors were computed under differentearthquakes and for SDOFµ values of 3, 5 and 7. Thecorresponding results are shown in Figure (11a).Figure (11b) shows the variation of the mean values.

It can be seen that the amplification of maximumstory ductility demand in an MDOF frame, relative toequivalent SDOF ductility (represented by ),.

µα MHmostly increases with increasing period and SDOFductility. This indicates the importance of highermode effect on maximum story ductility demandin high rise frames and higher SDOF ductility levels.

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8. Conclusions

The purpose of this study was to investigate highermode effects on different responses of MDOF steelmoment-resisting frames under seismic motions. Inthis regard, some modification factors were defined toestimate seismic demands of MDOF frames fromequivalent SDOF systems. The following conclusionsare drawn from different parts of this study:A) Higher mode effects in elastic analysis:

l The percentage of higher modes' participation

in all responses of MDOF frames increaseswith number of stories (Fundamental period).

l Among different responses of an MDOFframe, roof shear is the most influenced byhigher modes and base shear response is inthe next order. Roof displacement is not thatmuch affected by higher modes. Forceresponses are generally more dependent onhigher mode affects rather than deformationresponses.

l The percentage of higher mode effects in forceresponses decrease with a rise in the numberof bays, while the opposite trend was observedfor deformation responses.

l The ratio of roof elastic displacement of anMDOF frame to elastic first mode spectraldisplacement, defined as eld

MH,.α can be

approximated by PF1 factors. These factorsalways increase with an increase in period.

B) Strength reduction factors of MDOF frames:l Rµ factor, which accounts for inelasticity in

SDOF systems, always increases withincreasing target ductility and mostly withstructural period.

l Strength reduction factors of MDOF frames,

Figure 11a. Higher mode effects on maximum story ductility demands.

Figure 11b. Higher mode effects on maximum story ductilitydemands (mean values).


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can be computed by multiplying Rµ by1/ V

MH •α . modification factor which takes the

higher mode effects into account.l Base shear higher modes modification factors

always increase with increase in fundamentalperiod of frames and mostly increase withincrease in target ductility.

C) Higher mode effects on maximum story drift andductility demand:l Vertical distribution of story drift and ductility

in height of the frames show sudden increasein higher stories. The reason is the contribu-tion of higher modes which is of more signifi-cance for long-period frames.

ld

MH.α modification factor, which is a criteriaof higher mode effects on maximum storydrift demands, mostly increases with a rise inperiod and level of SDOF ductility. In otherwords, the difference between maximum storyand global drift demands increases in long-period structures.

lµα MH. modification factors, implying higher

mode effects on maximum story ductilitydemand, mostly increase with period and levelof SDOF ductility, which indicates the impor-tance of higher modes.

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higher mode effects on seismic behavior of mdof steel ...response of multi degree of freedom (mdof)...

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