Comparison of the drift spectra generated using continuous ...scientiairanica.sharif.edu/article_1511_6c88ed55318558cc...The response spectrum [1,2] is an important tool in seismic

Scientia Iranica A (2013) 20(5), 1337{1348

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Comparison of the drift spectra generated usingcontinuous and lumped-mass beam models

F.R. Rofooei� and A.H. ShodjaDepartment of Civil Engineering, Sharif University of Technology, Tehran, P.O. Box 11155-9313, Iran.

Received 16 April 2012; received in revised form 9 March 2013; accepted 14 May 2013

KEYWORDSTime history analysis;Near-�eld records;Drift spectrum;Approximationmethods.

Abstract. The drift spectrum, as a new measure of earthquake-induced demands instructures, was developed using a uniform-sti�ness continuous beam model. Practicallimitations in the procedures based on such models have caused structural engineers topay little attention to drift spectra. In this paper, a method is proposed to estimate theseismic-induced inter-story drift demands, using the modal analysis technique of lumpedmass, non-uniform beam models. The proposed approach is simple and can overcome manyde�ciencies of the previous methods. It can take into account the very important e�ect ofheight-wise sti�ness reduction on drift demands, without the numerical di�culties usuallyencountered in computing the drift spectra. The drift spectra for high-rise shear buildingsare computed using both continuous and lumped mass beam models. The comparisonsindicate that the results of the continuous uniform-sti�ness beam models in some cases arenot acceptable. The e�ects of various parameters such as the structural system, the height-wise variation of structural sti�ness, the number of stories, higher modes and damping ratioson inter-story drift demands are investigated. Also, the inuences of the structural systemand height-wise variation of structural sti�ness on the height-wise distribution of maximuminter-story drift demands are evaluated using a number of building models.c 2013 Sharif University of Technology. All rights reserved.

1. Introduction

The response spectrum [1,2] is an important tool inseismic design of structures. Using modal analysisprocedures, the peak elastic response of a Multi-DegreeOf Freedom (MDOF) system during an earthquakeexcitation can be approximately determined from theearthquake response spectrum, without performing adynamic time history analysis. The response spectrumanalysis can also provide an estimate of the overall dis-placement demands of the building models. However,extensive research has shown that the inter-story driftratio has the best correlation with the inicted damages

*. Corresponding author. Tel: 98-21-66164233;Fax: 98-21-66164233E-mail addresses: [email protected] (F.R. Rofooei),[email protected] (A.H. Shodja)

to buildings under earthquake excitations [3-5]. Thedistribution of the inter-story drift demands along theheight of the buildings is not uniform, and thereforecannot be obtained directly from the displacementresponse spectrum.

The responses of the structures during near-�eldexcitations cannot be characterized by a resonancebuildup, so the underlying assumptions of the modalanalysis method are not satis�ed for structures sub-jected to these ground motions [6]. Iwan [6] introducedthe so-called \drift spectrum" as a new measure ofthe induced seismic demands on structures for suchsituations. Iwan's drift spectrum was based on thesolution of the problem for non-dispersive dampedwaves in a one-dimensional continuous medium, astreated by Courant and Hilbert [7]. He used the shearstrain in a continuous shear beam as an analogy to theinter-story drift ratio in a building structure.

1338 F.R. Rofooei and A.H. Shodja/Scientia Iranica, Transactions A: Civil Engineering 20 (2013) 1337{1348

Using modal analysis of a continuous shear beam,Chopra and Chintanapakdee [8] showed that the dif-ferences between drift and response spectra are notunique for near-�eld strong ground motions. Thesedi�erences simply reect the e�ects of higher modes onthe response that are larger due to the characteristics ofnear-�eld ground motions [8]. Also, the existing modalcombination rules are equally accurate for near-�eldand far-�eld ground motions, although their main as-sumptions are not satis�ed by near-�eld excitations [8].

Miranda and Akkar [9] extended Iwan's driftspectrum for buildings with exural or exural-sheartype of behavior. They proposed the so-called \gen-eralized inter-story drift spectrum" that was basedon shear-exural continuous uniform sti�ness beam.A modi�cation factor was employed by Miranda andAkkar to consider any arbitrary combination of exuraland shear types of behavior for the structural system.The proposed generalized inter-story drift spectrumwas based on modal analysis techniques. To computethe inter-story drift ratio, Miranda and Akkar used acode-based relationship to estimate the height of thestructures from their fundamental periods.

As it was already mentioned, Iwan's drift spec-trum was based on the propagation of non-dispersivedamped waves, a concept not known by many struc-tural engineers. Also, Kim and Collins [10] showed thatthe formulation used in the Iwan's drift spectrum couldresult in residual drifts for certain ground motionsthat are not consistent with the assumed linear elasticmodel. On the other hand, for shear beams or beamswith predominant shear deformation, the computationof exact formulations of Miranda and Akkar [9] in mostcomputers may include considerable amount of roundo� error. So, they provided some approximate relationsfor the problem in such beams. In this work, theimportance of the error caused by the application ofthese approximate relations and their e�ects on theaccuracy of the method will be discussed.

Since the introduction of the concept of drift spec-tra by Iwan [6] in 1997, not many engineers have paidattention to these new spectra due to the di�cultiesexisted in treating the continuous beam models. Inthis paper, a new method is proposed to determinethe drift spectra, using the well-established modalanalysis technique for lumped mass, non-uniform beammodels. This method, unlike the previous ones, canconsider the e�ect of height-wise sti�ness reduction ondrift demands, which is shown to be very important.Application of Code-based approximate formulas, torelate the height of the structural models to their fun-damental periods, which was among the shortcomingsof previous methods, does not exist in the proposedapproach.

The mean drift spectra for high-rise buildingmodels are computed for both continuous and lumped

mass beam models, using ten near-�eld strong groundmotions. The e�ects of various parameters such asthe type of structural system, the ratio of the lateralsti�ness at the top of the building model to the lateralsti�ness at its base, number of stories, higher modesand damping ratio on the inter-story drift demands areinvestigated. Moreover, the e�ects of structural systemand height-wise variation of structural sti�ness on thedistribution of maximum inter-story drift demands areevaluated using a number of building models.

2. Miranda and Akkar's \generalized driftspectrum"

Miranda and Akkar [9] used the Csonka's [11] exural-shear beam model, shown in Figure 1, to obtainestimates of drift demands in linearly elastic buildings.It assumes that these beams are connected by anin�nite number of axially rigid members that onlytransmit horizontal forces. They used the derivatives ofthe mode shapes of the continuous shear-exural beamto approximate the drift ratios. When the undampeduniform model is subjected to a horizontal accelerationug(t), the Inter-story Drift Ratio (IDR) at the jth storyis estimated by [9]:

IDR(j; t) � 1H

mXi=1

�i�0i(x)Di(t); (1)

where H is the total height of the building, �i is themodal participation factor of the ith mode of vibrationof the continuous beam model, and �0i(x) is the �rstderivative of the ith mode at a non-dimensional heightx of the beam model which is the average height ofthe j + 1 and j oors. Also, Di(t) is the relative

Figure 1. Flexural-shear beam used in Miranda andAkkar's generalized drift spectrum.

F.R. Rofooei and A.H. Shodja/Scientia Iranica, Transactions A: Civil Engineering 20 (2013) 1337{1348 1339

displacement response of a SDOF (Single-Degree ofFreedom) system with period Ti and modal dampingratio �i corresponding to those of the ith mode ofvibration, subjected to a ground acceleration ug(t), andm is the number of modes considered in the analysis.

The details of computing values of �0i(x) and �ican be found in Miranda and Taghavi [12]. The modeshapes of vibration and their derivatives of continuousbeams depend only on the lateral sti�ness ratio �,de�ned as:

� = Hr

GAEI

; (2)

where GA and EI are the shear and exural rigiditiesof the shear beam model, respectively.

Miranda and Akkar [9], assuming a T1 =0:0853H0:75 relationship between fundamental periodand height of steel moment-resisting frames, obtainedthe ordinates of the generalized inter-story drift spec-trum. In a previous study, Miranda and Reyes [13]concluded that the maximum inter-story drift demandswere not signi�cantly inuenced by reduction of sti�-ness along the height of the structural system. Basedon this study, Miranda and Akkar [9] computed thedrift spectra for uniform (sti�ness) beams, and sug-gested the following equation describing the maximuminter-story drift ratio:

IDRmax � max8t;x�� 1H mX

i=1

�i�0i(x)Di(t): (3a)

Since application of any assumption for determiningthe height of the building models could reduce theaccuracy of their proposed method, this assumption isnot considered in this paper. Instead, a drift spectrumis generated using the value of the \inter-story driftratio � total height", as its ordinates. Therefore,Eq. (3a) can be re-arranged as:

IDRmax �H � max8t;x�� mXi=1

�i�0i(x)Di(t): (3b)

3. Problem formulation

The response of an MDOF system with N degrees offreedom subjected to horizontal ground acceleration isgoverned by the following di�erential equation:

[M ]f Ug+ [C]f _Ug+ [K]fUg = �[M ]f1gug(t); (4)where [M ], [C], and [K] are mass, damping, andsti�ness matrices, respectively, fUg is the relativedisplacement vector of the dynamic system, and ug(t) isthe input ground acceleration. The response of elasticstructures with classical damping can be obtained using

the expansion theorem. The relative displacementvector of MDOF systems can be computed as:

fU(t)g =NXi=1

fUi(t)g; (5)

in which fUi(t)g is the contribution of the ith modeto the displacement vector of the system fU(t)g,determined as:

fUi(t)g = �if�igDi(t); i = 1; 2; � � � ; N; (6)in which:

�i =f�igT [M ]f1gf�igT [M ]f�ig ; i = 1; 2; � � � ; N; (7)

where �i and f�ig are the modal participation factorand the ith mode shape, respectively. The relativedisplacement history of the ith mode shape of thesystem, Di(t), can be determined from:

Di(t) + 2�i!i _Di(t) + !2iDi(t) = �ug(t);i = 1; 2; � � � ; N; (8)

in which !i and �i are the ith mode un-damped naturalfrequency and damping ratios, respectively. The inter-story drift history of the jth story can be obtained as:

Driftj =NXi=1

fUi(j)g � fUi(j � 1)g

=NXi=1

�i[�i(j)� �i(j � 1)]Di(t);

i = 1; 2; � � � ; N; (9)in which a smaller number of modes can be used forinter-story drift estimation.

Now, consider the lumped mass shear beam rep-resenting an N -story building shown in Figure 2. Themasses of di�erent oors that are considered to beequal, are lumped at each oor level. The mass matrix[M ] can therefore be written as:

[M ] = m[I]; (10)

where m is the mass of each story and [I] is the identitymatrix. For a building with uniform mass, the modalparticipation factor given by Eq. (7) reduces to:

�i =

NPj=1

�i(j)

NPj=1

�2i (j); i = 1; 2; � � � ; N: (11)


Figure 2. The lumped mass, non-uniform beam modeland variation of sti�ness along its height.

The non-uniform distribution of sti�nesses along theheight of beam model is assumed to be estimated from:

ki =

1� (1� �)

�i� 1N � 1

��!k; (12)

where ki and k are the sti�ness of the ith and the �rststories, respectively. The parameter � is the ratio ofthe lateral sti�ness at the top story to that of the �rststory of the beam model, and � is a non-dimensionalparameter that controls variation of the lateral sti�nessalong the beam. This relationship is similar to theone used by Miranda and Reyes [13] for continuousbeams. As they noted, values of � equal to 1 or 2correspond respectively to linear or parabolic variationof lateral sti�nesses along the height of the beam model,as shown in Figure 2. All elements of the sti�nessmatrix of the system can be determined in terms ofthe �rst story sti�ness k, so this matrix can be writtenas:

[K] = k[S]; (13)

where non-dimensional matrix [S] is a function of N , �and �.

Substituting Eqs. (10) and (13) into the charac-teristic equation of the system yields:

det(�!2i [M ] + [K]) = 0; i = 1; 2; � � � ; N; (14)or:

det��!2i mk [I] + [S]� = 0; i = 1; 2; � � � ; N: (15)

The mode shapes (eigenvectors) can then be deter-mined from:��!2i mk [I] + [S]� f�ig = 0; i = 1; 2; � � � ; N: (16)De�ning the non-dimensional parameter `�i' by:

�2i = !2imk: (17)

Eqs. (15) and (16) can be written as:

det(��2i [I] + [S]) = 0; (18)and:

(��2i [I] + [S])f�ig = 0: (19)Eq. (19) shows that all N -story buildings with uniformmass and a similar variation pattern for sti�ness, withno dependency on the value of sti�ness, mass andheight, would have identical mode shapes. UsingEq. (17) for the 1st and the ith modes of vibrationproduces:

�2i�21

=!2i!21: (20)

Therefore, the ith mode circular frequency or periodcan be obtained from the frequency or the period ofthe �rst mode as follows:

!i =�i�1!1; Ti =

�1�iT1: (21)

This means that if the fundamental period of vibra-tion of a building with particular sti�ness variationis known, its higher mode natural periods can becalculated using Eq. (21).

For a given ground motion time history, the maxi-mum inter-drift distribution for anN -story shear build-ing with the height-wise sti�ness variation describedby Eq. (12) can be obtained from its fundamentalperiod of vibration. The procedure is simple: �rstthe matrix [S] for the speci�ed values of � and � isgenerated, and then Eqs. (18) and (19) can be usedto �nd the non-dimensional parameters �i and modeshapes f�ig. The periods of the higher modes and themodal participation factors are calculated by Eqs. (21)and (11), respectively. If the period Ti of any modeis known, the response history of the equivalent SDOFsystems, Di(t), for the selected modal damping ratios�i, under any earthquake record can be determined bysolving Eq. (8). The drift history of all stories can thenbe calculated from:

Driftj =NXi=1

fUi(j)g � fUi(j � 1)g

=NXi=1

�i[�i(j)� �i(j � 1)]Di(t);

j = 1; 2; � � � ; N: (22)Considering two buildings with n1 and n2 number ofstories, and the same fundamental period T , one couldwrite:n1h1 = H1; (23)

n2h2 = H2; (24)


in which h1 and h2 are the story heights and H1and H2 are the building heights, respectively. Thus,the maximum inter-story drift ratios (MIDR) of thesebuildings will be equal to:

MIDR1 =�1;maxh1

=�1;maxn1

H1; (25)

MIDR2 =�2;maxh2

=�2;maxn2

H2; (26)

where �i;max is the maximum inter-story drift demandof the ith building.

Tall buildings can be approximated by continuousbeams. In a continuous beam model, the MIDRcan be obtained using Eq. (3a). It is also shown inEq. (19) that all N -story buildings with uniform massand a similar variation pattern for sti�ness, with nodependency on the value of sti�ness, mass and height,would have identical mode shapes. It is thereforereasonable to assume that tall uniform-mass buildingswith identical natural periods and similar variationpattern for sti�ness have nearly the same mode shapes.Hence, from Eq. (3b) it can be concluded that:

MIDR1 �H1 � MIDR2 �H2; (27)or \maximum inter-story drift ratio � total height" fortall buildings with uniform mass and similar variationpattern for sti�ness would be nearly the same iftheir fundamental periods of vibration be identical.Increasing the height of the buildings will improve this

approximation. The maximum inter-story drift ratio �total height is an upper bound for the roof displacementof the building models. So, two tall buildings withuniform distribution of mass, same fundamental periodand similar height-wise variation of sti�ness, wouldhave the same upper bound for roof displacement,which is consistent with the concept of displacementspectrum. This result will be numerically veri�ed inthis work, and then will be used in presentation of thedrift spectra.

4. Near-�eld ground motions

As mentioned in Section 1, the drift spectrum wasoriginally developed to provide a new measure ofinduced seismic demands in building models for situa-tions in which the ground motion cannot be appropri-ately speci�ed by the maximum spectral displacement.The fault-normal component of a near-�eld groundmotion that is characterized by a forward directivitye�ect is such a case [14]. The forward directivity-related velocity pulse could impart a large amountof energy to the buildings at the onset of an earth-quake episode [15]. Table 1 shows the 10 near-�eldrecords used in this study. The earthquake recordswith forward directivity have signi�cant PGV, andexhibit long period velocity pulses due to directivitye�ects. To provide a rational basis for comparison, theground motion records listed in Table 1 are scaled to0.5 g. All these records were obtained from the PEERwebsite [16].

Table 1. Details of the near-�eld ground motions.

No. Earthquake MW a Station DateVs(30)b

(m/s)NEHRPsite class

Tpc

(sec)PGAd

(g)

SourceDist.Rrupte

(km)

Comp.

1 Cape Mendocino 7.01 Petrolia 92/04/25 712.8 C 3 0.662 8.2 090

2 Erzincan 6.69 Erzincan 92/03/13 274.5 D 2.7 0.496 4.4 EW

3 Imperial valley 6.53EC Meloland

Overpass79/10/15 186.2 D 3.3 0.296 0.1 270

4 Kobe 6.90 JMA 95/01/17 312.0 D 1.0 0.821 1.0 000

5 Kobe 6.90 Takatori 95/01/17 256.0 D 1.6 0.611 1.5 000

6 Landers 7.28 Lucerne 92/06/28 684.9 C 5.1 0.721 2.2 275

7 Loma Prieta 6.93 LGPC 89/10/18 477.7 C 3.0 0.563 3.9 000

8 Northridge 6.69Sylmar-Olive

View94/01/17 440.5 C 3.1 0.843 5.3 360

9 Northridge 6.69 Rinaldi 94/01/17 282.2 D 1.2 0.838 6.5 228

10 Tabas 7.35 Tabas 78/09/16 766.8 B Backward 0.836 2.0 LNa Mw: moment magnitude; b Vs(30): 30 m divided by the travel time of an S-wave to 30 m depth;c Tp: period of pulse; d PGA: peak ground acceleration.; e Rrupt: closest distance to rupture plane.


5. Drift spectrum based on lumped massnon-uniform beam model

5.1. Computation of drift spectrumIt was shown in Section 3 that if the sti�ness variationalong the height of a building model is representedby Eq. (12), it is possible to estimate its maximuminter-story drift under earthquake excitation, using itsfundamental period of vibration. Using the describedlumped mass beam model for a given earthquakerecord, the drift spectrum for N -story building modelswith speci�c values of �, � and sti�ness variation givenby Eq. (12), can be computed by simply repeating theprocedure for the desired range of fundamental periods.Although a large number of drift spectra could begenerated using the above approach, it will be shownthat just a few spectra would be su�cient to addressmost practical cases. The drift spectra for buildingmodels with shear and exural behavior can also beproduced by the proposed method. However, only theresults for shear beam models are presented here.

5.2. The inuence of the number of stories onmaximum inter-story drift demands

Figure 3(a) and (b) represent the e�ect of a building'snumber of stories on its maximum inter-story driftdemands for a 0.02 damping ratio for a uniform sti�ness

beam and a beam with � = 0:35 and � = 2,respectively. Clearly, the number of stories has animportant e�ect on inter-story drift demands. Forany given period, the inter-story drift demands aredecreased by increasing the number of stories. Thistrend suggests that di�erent presentations of the driftspectra, i.e. the product of maximum inter-story driftand number of stories, or product of Maximum Inter-story Drift Ratio (MIDR) and total height of building(H) (see Eqs. (25) and (26)) versus the fundamentalperiod, as shown in Figure 4(a) and (b), may be moreuseful. These �gures show that for the shear buildingswith more than 20 Degrees Of Freedom (DOFs), thevalue of inter-story drift ratio multiplied by the totalheight is not very sensitive to the number of stories. Tobetter quantify the di�erence between these spectra,Figure 5(a) and (b) show the same spectra normalizedby the spectrum of 50-DOFs. These �gures showthat except for periods less than 0.5 sec (which arenot common for high-rise buildings), the di�erencesbetween drift spectra for shear buildings with morethan 20 degrees of freedom are less than 4 percent.Therefore, for tall buildings with speci�c values of �and �, a single \maximum inter-story drift ratio �total height" spectrum can be used to approximatelydescribe their behavior. This result is supported byEq. (27). One can conclude that it is acceptable to

Figure 3. The e�ect of number of stories on the maximum inter-story drifts demands.

Figure 4. Alternative representation of the e�ect of number of stories on the maximum inter-story drifts demands.


Figure 5. Quantifying the e�ect of number of stories on the maximum inter-story drifts demands.

Figure 6. The e�ect of � on the maximum inter-story drifts demands with the assumed H-T relationship.

consider a 50-DOFs lumped mass shear beam as anapproximation for a continuous shear beam. Thisresult will be veri�ed in Figure 6(a) by comparingthe drift spectra of continuous model and 50-DOFstick model for the Rinaldi NS component of 1994Northridge earthquake.

It should be noted that the proposed methodhas no assumption on the value of structural or storyheight. It is not possible to represent the inter-storydrift ratio spectra without doing such assumptions.Therefore, from now on in this manuscript, the term\drift spectrum" means \maximum inter-story driftratio � total height of building" spectrum.5.3. Comparison of drift spectra obtained

using continuous and lumped mass beammodels

The drift spectra obtained using the continuous andlumped mass beam models are compared to investigatetheir di�erences. For comparison purposes, the driftspectra obtained by the method of Miranda and Akkar(by using the code based relationship between theheight and the fundamental period of structural mod-els) for the Rinaldi NS component of 1994 Northridgeearthquake is shown in Figure 6(a). This �gure showsthat the lumped mass beam with 50-DOFs very closelyapproximates values obtained by the exact continuous-beam method by Miranda and Akkar [9]. This �gure

also shows that the approximate method may lead toerrors up to 33%.

Figure 6(b) shows the drift spectra obtained usingthe LGPC component of 1989 Loma Prieta earthquake.As this �gure indicates, the spectra for small values of �are not necessarily very di�erent from that of a shearbeam model. Furthermore, the maximum inter-storydrift demands are not necessarily very sensitive to thevalue of �. The next section discusses on this subjectin more detail.

5.4. Inuence of the structural system on thevalue and height-wise location ofmaximum inter-story drift demands

In Section 5.3, it is shown that the amount of maximuminter-story drift ratio is not necessarily dependent onthe structural system, but the drift spectra do notprovide any information about the height-wise locationof the maximum inter-story drift ratio. Thus, the e�ectof structural system on the position of maximum inter-story drift demands needs to be investigated. The driftspectra of continuous shear-exural beams for variousvalues of � and drift spectrum of 50-DOFs lumpedmass uniform beams are presented in Figure 7(a).The drift spectra for values of � less than 30 arecalculated using the exact relations derived by Mirandaand Akkar [9]. However, as mentioned in Section 1,for large values of �, the computation of these exact


Figure 7. The e�ect of � on the maximum inter-story drifts demands.

Figure 8. The e�ect of � on the height-wise variation of the maximum inter-story drifts demands for 20-DOFuniform-sti�ness beam models with � = 0:02.

relations in most computers may include considerableamount of round o� error. Thus, a lumped mass shearbeam with 50-DOFs is considered as an approximationfor a continuous beam. Also, Figure 7(a) representsthe results of the approximate relations of Miranda andAkkar for large values of � that corresponds to a shearbeam model. Figure 7(b) shows the same drift spectranormalized by the spectrum of a 50-DOFs beam mod-els. This �gure shows that the average drift spectraobtained, using the 10 earthquake records of Table 1,are not very sensitive to the value of �. Moreover,Figure 7(b) shows that the error caused by using theapproximate relations is not negligible. These resultsare not consistent with the results obtained by Mirandaand Akkar, as they examined only two ground motionrecords. It is observed that by increasing the numberof records, the sensitivity of the inter-story drift ratios

to the value of � is reduced and the error, due tousing approximate relations, becomes important. Thepresented results are obtained using the strong groundmotions of Table 1, and more earthquake recordsshould be studied before generalizing this �nding.

To investigate the e�ect of structural system onthe location of maximum inter-story drift demands, 15uniform-sti�ness structural models that are representa-tives of 5 di�erent structural systems are investigated.The 70 meters tall structural models have 20 DOFswith fundamental periods of vibration of 2, 3 and 4seconds. They exhibit pure exural behavior with� = 0, pure shear behavior with � = 1 and shear-

exural behaviors with � = 5, � = 10 and � = 20.The damping ratio is assumed to be � = 0:02. Thedistributions of inter-story drift ratio demands overthe height of these structural models are shown in


Figure 8. From this �gure it is concluded that thelocation of maximum inter-story drift ratio is verysensitive to the structural system, while its value isnot strongly dependent on the structural system, forthe considered ground motions. Figure 8 shows thatthe location of maximum inter-story drift ratio foruniform-sti�ness structures with exural behavior is atroof level, while for structures with shear behavior isat �rst story, and for systems with combined shear-

exural behavior is somewhere in the mid-height of thestructural models.

5.5. Inuence of the structural fundamentalperiod on maximum inter-story driftdemands

Figure 9 shows the e�ect of the fundamental periodof vibration of the structural models on maximuminter-story drift demands for uniform-sti�ness 20-DOFmodels with damping ratio � = 0:02. This �guredemonstrates the distribution of inter-story drift de-mands over the height of structural models for �vevalues of �. As this �gure indicates, by increasingthe period of vibration, the inter-story drift demandsusually increase, but as Figure 7(a) shows, smalldecreases are also possible. Figure 9 also shows that theheight-wise distribution of maximum inter-story driftdemands is mostly dependent on the structural system.For certain values of �, uniform height-wise changes inthe sti�ness do not signi�cantly a�ect the distributionpattern of maximum inter-story drift demands. Theseobtained results are due to the earthquake recordslisted in Table 1.

5.6. Inuence of the sti�ness reductionpattern on maximum inter-story driftdemands

Figure 10(a) shows the e�ect of the sti�ness reductionpattern � on maximum inter-story drift demands for20-DOFs beam models with � = 0:35 and dampingratio � = 0:02. The sensitivity of maximum inter-storydrift demands to changes in � is negligible, thus makingit possible to assign it a �xed value. Since, for most ofthe building models, � lies between 1 and 3, for theremainder of this work, this parameter is assumed tobe equal to 2. Figure 10(b) presents the same spectranormalized by the spectrum for � = 2. It is seen thatthe error due to ignoring � is in general less than 10%,while for a wide range of periods, it is less than 5%.

5.7. The e�ect of top to bottom story sti�nessratio on the maximum inter-story driftdemands

Figure 11(a) shows the e�ect of the lateral sti�nessratio of the top story to that of the �rst story, �,on maximum inter-story drift demands for 20-DOFsmodels with � = 2 and � = 0:02, while Figure 11(b)shows the same spectra normalized by the spectrumfor � = 1. These �gures demonstrate that the e�ect of� on maximum inter-story drift demands for values of� < 0:5 is not small. Reducing the lateral sti�ness ratiocauses inter-story drifts to increase signi�cantly. Thise�ect for values of � greater than 0.5 is insigni�cant. Avalue of � = 0:5 can be used for values of � � 0:5 withan error below 10%, while for values of � < 0:5, di�erentspectra should be computed. These �gures show that

Figure 9. The e�ect of structural fundamental period on the height-wise variation of the maximum inter-story driftsdemands for 20-DOF uniform-sti�ness beam models with � = 0:02.


Figure 10. The sensitivity of the maximum inter-story drift demands to the sti�ness reduction pattern.

Figure 11. The sensitivity of the maximum inter-story drift demands to the parameter, �.

the e�ect of lateral sti�ness ratio of the top stories, withrespect to the �rst ones, is considerable. For furtherinvestigation, the distribution of inter-story drift ratiodemands over height of two shear building modelswith 30 stories and fundamental periods of vibrationof 3 sec and 4 sec, is illustrated in Figure 12(a)and (b), respectively. In these �gures, drift demandsare computed for various values of � for � = 2 and� = 0:02. These �gures show that lateral sti�nessratio of the top story to that of the �rst story not onlya�ects the value of maximum inter-story drift demand,

but also inuences its location along the height of thebuilding.

5.8. The e�ects of damping ratio and highermodes on the maximum inter-story driftdemands

Figure 13 shows the maximum inter-story drift spec-trum obtained using di�erent numbers of modes forthe 20-story building models with � = 0:35, � = 2, and� = 0:02. As this �gure shows, for the near-�eld groundmotions of Table 1, the spectral values obtained, using

Figure 12. E�ect of height-wise distribution of sti�ness on height-wise location of maximum inter-story drift demands.


Figure 13. The e�ect of higher modes on the maximuminter-story drift demands.

Figure 14. The e�ect of damping ratio on the maximuminter-story drift demands.

only one or two modes, are highly under-estimated,while those using the �rst 6 modes of vibration for 20-story buildings can produce acceptable estimates. Also,Figure 14 shows the e�ect of damping on maximuminter-story drift demands for 20-story buildings with� = 0:35 and � = 2. As expected, an increase inthe damping ratio leads to a decrease in the maximuminter-story drift demand.

6. Conclusions

A new method based on a lumped mass beam model forestimating the inter-story drift demands of buildings isproposed. It is shown that for the considered near �eldearthquake records, the average of the maximum inter-story drift ratios for high-rise buildings is mostly depen-dent on their fundamental periods, and its sensitivityto the structural system is not considerable, althoughthe location of its occurrence is strongly dependenton the structural system. The drift spectra for shear

buildings with di�erent numbers of stories and non-uniform distribution of sti�ness along the height arepresented. It is observed that the maximum inter-story drift is not strongly dependent on the sti�nessvariation pattern. However, its dependency on theratio of lateral sti�ness of the top story to that of the�rst story of the building models, that was not referredto in previous methods, is very important. For shearbuildings in which the lateral, top to bottom storysti�ness ratio is greater than 0.5, a single drift spectrumcan be used, while for ratios less than 0.5, di�erentspectra would be needed. Moreover, it is deduced thatthe product of maximum inter-story drift ratio andtotal height for tall buildings with similar variationpattern for sti�ness are mostly dependent on theirnatural fundamental periods. Based on this �nding, foreach value of lateral sti�ness ratio of top to base story,for buildings taller than 20 stories, a single spectrumis presented. In contrast to other available methods,no code-based relationship is used in this approach toestimate the height of the structural models from theirfundamental periods. The proposed method is simplebecause it uses well-established lumped mass beamconcept rather than complicated continuous shear-

exural beam representation with due computationaldi�culties. The proposed drift spectra are powerfultools for structures with regular height-wise sti�nessdistribution. They can be used to study the e�ect ofvarious parameters on inter-story drift demands andpreliminary assessment of drift demands, providingguidelines to optimum design of structures.

References

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Rept. No. 138, Dept. of Civil Eng., Stanford University,Stanford, CA (2001).

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Biographies

Fayaz Rahimzadeh Rofooei is a professor in theCivil Engineering Department of Sharif University ofTechnology, Tehran, Iran. Currently, he is the headof the Earthquake Engineering Research Center ofSharif University of Technology. He received his PhDfrom the Rensselaer Polytechnic Institute, Troy, N.Y.,USA. His research interests include structural control,nonlinear structural analysis, base isolation, seismicvulnerability analysis and retro�tting of structures andlifelines.

Amir Hossein Shodja is a PhD student in theDepartment of Civil Engineering at Sharif Universityof Technology, Tehran, Iran. He received his MSc andBSc degrees from Sharif University of Technology. Hisresearch interests include earthquake engineering, non-linear structural analysis, seismic vulnerability analysisand retro�tting of structures.

Documents

Comparison of the drift spectra generated using continuous ...scientiairanica.sharif.edu/article_1511_6c88ed55318558cc...The response spectrum [1,2] is an important tool in seismic