generalized interstory drift spectrum

Generalized Interstory Drift SpectrumE. Miranda, M.ASCE1; and S. D. Akkar2

Abstract: The recently developed drift spectrum is extended to buildings that do not deform laterally like pure shear beams. Similarlyto Iwan’s drift spectrum, the proposed generalized interstory drift spectrum uses a continuous linear-elastic model to obtain estimates ofinterstory drift demands in buildings. However, the generalized interstory drift spectrum is based on a continuous model that consists ofa combination of a flexural beam and a shear beam, rather than only a shear beam. By modifying one parameter the model used in theproposed generalized interstory drift spectrum can consider lateral deformations varying from those of a flexural beam to those of a shearbeam. Hence, it permits us to account for a wide range of modes of deformation that represent more closely those of multistory buildings.The proposed generalized interstory drift spectrum is based on modal analysis techniques that are familiar to structural engineers and usesa damping model that avoids the problems that can occur in the drift spectrum. The generalized interstory drift spectrum is illustrated withvarious recorded ground motions. The effects of damping, higher modes, and lateral stiffness ratio are investigated and discussed.

DOI: 10.1061/�ASCE�0733-9445�2006�132:6�840�

CE Database subject headings: Drift; Approximation methods; Buildings, multistory; Building design; Beams.

Introduction

The response spectrum is a simple and powerful tool that pro-vides an indication of overall displacement and acceleration de-mands on structures subjected to earthquake ground motions.Since its development �Biot 1941� and its first use in aseismicdesign �Housner 1947; Housner et al. 1953�, the response spec-trum has been routinely used as a measure of the intensity ofearthquake ground motion on structures with different periods anddifferent damping ratios. In particular, response spectrum dis-placement ordinates provide a measure of overall displacementdemands on buildings, and hence provide some indication of theaverage interstory drift in buildings. However, interstory drift de-mands are practically never uniformly distributed in multistorybuildings and therefore response spectrum ordinates do not pro-vide a direct indication of maximum interstory drift demands.Furthermore, response spectrum ordinates are based on a single-degree-of-freedom �SDOF� system and therefore cannot accountfor the influence of higher modes on interstory drift demands inmultistory buildings.

Many studies have shown that the interstory drift ratio, definedas the difference in lateral displacements in between two consecu-tive floors normalized by the interstory height, is the responseparameter that is best correlated with damage in buildings �Algan1982; Sozen 1983; Moehle 1984; Qi and Moehle 1991; Moehle

1Assistant Professor, Dept. of Civil and Environmental Engineering,Stanford Univ., Stanford, CA 94305-4020 �corresponding author�.E-mail: [email protected]

2Assistant Professor, Dept. of Civil Engineering, Middle EastTechnical Univ., 06531 Ankara, Turkey.

Note. Associate Editor: Rakesh K. Goel. Discussion open untilNovember 1, 2006. Separate discussions must be submitted for individualpapers. To extend the closing date by one month, a written request mustbe filed with the ASCE Managing Editor. The manuscript for this paperwas submitted for review and possible publication on May 3, 2004; ap-proved on September 8, 2005. This paper is part of the Journal of Struc-tural Engineering, Vol. 132, No. 6, June 1, 2006. ©ASCE, ISSN 0733-
9445/2006/6-840–852/$25.00.
840 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006

1994; Gülkan and Sozen 1999; Bozorgnia and Bertero 2001�. In1997, Iwan introduced a simple and direct measure of drift de-mand for earthquake ground motions called the drift spectrum�Iwan 1997�. Like the response spectrum, the drift spectrum isbased on a relatively simple linear model. However, the driftspectrum differs from the response spectrum in that it is based ona continuous shear beam rather than a SDOF system. The mostimportant advantages of this new powerful tool are that it takesinto account the fact that interstory drift demands are not uni-formly distributed along the height of buildings and considers thecontribution of higher modes. Therefore, the drift spectrum resultsin more accurate estimations of maximum interstory drift de-mands than does the response spectrum. In his study, Iwanstrongly recommended the use of drift spectrum in structural de-sign and concluded that the drift spectrum was particularly usefulin estimating drift demands in buildings subjected to pulse-likeground motions.

Despite its revolutionary concept and its many advantages, thedrift spectrum has received very little attention from structuralengineers. One possible explanation is that Iwan’s drift spectrumis based on the propagation of nondispersive damped waves trav-eling up and down the shear beam, and unlike seismologists, moststructural engineers are not familiar with wave propagation.Chopra and Chintanapakdee �2001� showed that drift spectracould also be computed using conventional modal analyses tech-niques that are familiar to structural engineers. In particular, theyshowed that the solution computed with modal analysis will con-verge to the wave propagation solution as more modes are in-cluded. More recently, Kim and Collins �2002� indicated that theoriginal mathematical formulation used by Iwan corresponds toan analytical model in which the cantilever shear beam is attachedto external springs and dampers anchored to a fixed referencepoint. Furthermore, they showed that the formulation used in thedrift spectrum, inconsistently with the assumed linear elasticmodel, could result in residual drifts for certain ground motions.

For many buildings, the shear beam model can lead to reason-able estimations of interstory drift demands. This is particularly
true in the case of moment-resisting frame buildings whose beams

are significantly stiffer than the columns and where axial defor-mations in the columns are negligible. In such cases, modes ofvibration will be relatively similar to those of a shear beam. How-ever, in buildings with bracing or shear walls or for moment-resisting buildings where the lateral stiffness provided by thecolumns is significant relative to that provided by the beams orwhere axial deformations are significant, the use of a shear beammay not be adequate.

In his conclusions, Iwan �1997� wrote that he hoped his paperwould promote further discussion and investigation of the driftspectrum as a measure of seismic demand on buildings. In thespirit of those remarks, the objectives of this paper are to extendthe drift spectrum buildings where the shear beam model may notbe adequate. The new spectrum, referred to as generalized inter-story drift spectrum, is based on a continuous model that consistsof a combination of a flexural beam and a shear beam, rather thanonly a shear beam. By modifying one parameter this model canconsider lateral deformations varying from those of a flexuralbeam to those of shear beam. Hence, it permits us to account fora wide range of modes of deformation that represent more closelythose of multistory buildings. Furthermore, based on Chopra’sobservations �Chopra and Chintanapakdee 2001�, the proposedgeneralized interstory drift spectrum is computed using conven-tional modal analysis techniques that are very familiar to struc-tural engineers and assumes classical modal damping, thus alsoavoiding the residual drift problems in Iwan’s drift spectrum thatwere identified by Kim and Collins �2002�.

Simplified Continuous Model

The use of continuous cantilever models to estimate force anddeformations in buildings subjected to lateral loads is by nomeans new. For example, Westergaard �1933� and Jennings andNewmark �1960� used a continuous shear beam model with uni-form mass and stiffness to estimate the lateral deformations inbuildings subjected to earthquake ground motions. Rosenbluethet al. �1968� used shear beams and response spectrum analysis toestimate story shears and story overturning moments in buildings.In the drift spectrum Iwan �1997� also used a uniform shear beam.However, instead of using response spectrum analysis, he usedresponse history analyses using wave propagation techniques.More recently, Heidebrecht used uniform shear beam model toestimate the maximum interstory drift of moment-resisting framebuildings responding primarily in the fundamental mode �Heide-brecht and Naumoski 1997; Heidebrecht and Rutenberg 2000�.Similarly, Gülkan and Akkar �2002� proposed an alternativesimplified formulation to the drift spectrum by considering onlythe fundamental mode of vibration and using conventional re-sponse spectrum displacement ordinates. Flexural beams �Euler–Bernoulli beams� have also been used for a long time to estimateforces and deformation in structures. For example, Montes andRosenblueth �1968� used flexural beams to estimate shear andoverturning moment demands along the height of chimneys.

While shear beams and flexural beams can provide adequatemodels for certain types of buildings, there are many types ofbuildings for which these two extreme modes of lateral deforma-tion may not be appropriate. The need for considering intermedi-ate modes of deformation has been recognized since the 1960s�Rinne 1960; Blume 1968�. The continuum model used in thegeneralized drift spectrum consists of a flexural cantilever beamand a shear cantilever beam deforming in bending and shear con-
figurations, respectively �Fig. 1�. The flexural and shear cantilever
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beams are assumed to be connected by an infinite number ofaxially rigid members that transmit horizontal forces, thus, theflexural and shear cantilevers in the combined system undergo thesame lateral deformation at all heights. Floor masses and lateralstiffnesses are assumed to remain constant along the height of thebuilding.

Models that combined shear and flexural elements were firstproposed by Khan and Sbarounis �1964� to evaluate the interac-tion of shear walls and frames. Rosman �1967� also studiedlaterally loaded systems consisting of frames and walls. A modelsimilar to that shown in Fig. 1 was first used by Csonka �1965�,who derived the differential equation controlling displacements inthe model when it is subjected to static lateral loads. The differ-ential equation controlling the static behavior of the model wasindependently derived by Traum and Zalewski �1970� who foundan analogy between the lateral deformations of the system tothose of a tensioned cable under transverse loading. Heidebrechtand Stafford Smith �1973� derived closed form solutions of thelateral displacements, bending moments, and overturningmoments when the model is subjected to static lateral loads thathave an inverted triangular and uniform distributions. Fajfar andStrojnik �1981� obtained approximate story shear and story over-turning moments using a linear combination of shears andoverturning moments in the shear and flexural beams, Miranda�1999� used the same model to obtain estimates of maximum roofdisplacements and maximum interstory drifts in buildings re-sponding primarily in the fundamental mode. Models that canaccount for shear, flexural, and combined modes of deformationhave also been used to study the behavior of buildings underlateral forces by Zalka �2000� and by Potzta and Kollar �2003�.More recently, Miranda and Reyes �2002� extended Miranda’smethod �1999� to cases in which the lateral stiffness does not

Fig. 1. Continuum model used in generalized interstory driftspectrum

remain constant along the height. The study concluded that maxi-

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mum interstory drift demands were not significantly influenced byreductions in stiffness along the height, provided that no abruptreductions in stiffness occur.

The response of the undamped uniform shear-flexural modelshown in Fig. 1 when subjected to a horizontal acceleration at thebase ug�t� is given by the following partial differential equation

�

EI

�2u�x,t��t2 +

1

H4

�4u�x,t��x4 −

�2

H4

�2u�x,t��x2 = −

�

EI

�2ug�t��t2 �1�

where �=mass per unit length in the model; H=total height of thebuilding; u�x , t�=lateral displacement at nondimensional heightx=z /H �varying between zero at the base of the building and oneat roof level� at time t; EI=flexural rigidity of the flexural beam,and �= lateral stiffness ratio defined as

� = H�GA

EI�2�

where GA=shear rigidity of the shear beam. The lateral stiffnessratio, ��dimensionless parameter that controls the degree ofparticipation of overall flexural and overall shear deformations inthe continuous model and thus, it controls the lateral deflectedshape of the model. A value of � equal to zero represents a pureflexural model �Euler–Bernoulli beam� and a value equal to �corresponds to a pure shear model. Intermediate values of � cor-respond to multistory buildings that combine overall shear andflexural deformations. The lateral stiffness ratio � is similar toBlume’s joint rotation index, �, defined as the ratio of flexuralstiffness of the beams to the flexural stiffness of the columns�1968�. However, the former is used in continuous models such asthat shown in Fig. 1, while the latter is used in frame buildings.The influence of the joint rotation index has been studied exten-sively by Chopra �2001�.

The influence of the lateral stiffness ratio on roof displacementdemands and on interstory drift demands of buildings respondingprimarily in the first mode of vibration was studied by Miranda�1999�. Miranda and Reyes �2002� indicated that lateral deflectedshapes of buildings whose lateral resisting system consists only ofstructural walls can usually be approximated by using values of �between 0 and 2. The same study indicated that for buildings withdual lateral resisting systems consisting of a combinationof moment-resisting frames and shear walls or a combination ofmoment-resisting frames and braced frames, values of � are typi-cally between 1.5 and 6, while for buildings whose lateral resist-ing system consists only of moment-resisting frames values of �are typically between 5 and 20.

Generalized Interstory Drift Spectrum

Earthquake Response History Modal Analysis

Assuming that the system is classically damped, the contributionof the ith mode of vibration to the lateral displacement at nondi-mensional height x=z /H at time t is given by

ui�x,t� = �i�i�x�Di�t� �3�

where �i=the modal participation factor of the ith mode of vibra-tion; �i�x�=amplitude of the ith mode at nondimensional height x;and Di�t�=relative displacement response of a SDOF system, withperiod Ti; and modal damping ratio �i corresponding to those of

¨
the ith mode of vibration, subjected to a ground acceleration ug�t�.

The total response of the system can be computed by modalsuperposition as

u�x,t� = �i=1

�

ui�x,t� �4�

The derivative of Eq. �4� with respect to x provides the rota-tion response history at nondimensional height x

��x,t� =�u�x,t�

�x=

1

H�i=1

�

�i�i��x�Di�t� �5�

where �i��x�=first derivative of the ith mode shape �i�x� withrespect to nondimensional height x.

In buildings the interstory drift ratio IDR is defined as thedifference of displacements of the floors above and below thestory of interest normalized by the interstory height. Therefore,the interstory drift ratio at the jth story is given by

IDR�j,t� =1

h�i=1

n

�i��i�j + 1� − �i�j��Di�t� �6�

where �i�j+1� and �i�j�=amplitudes of ith mode of vibration atthe j+1 and j floors; respectively; h=interstory height; andn=number of stories. In the original formulation of the drift spec-trum, the maximum interstory drift ratio is approximated by thepeak shear strain at the base of the shear beam �Iwan 1997�. Here,the interstory drift ratio at the jth story is approximated by therotation in the model at the height corresponding to the middle ofthe story of interest as follows:

IDR�j,t� � ��x,t� =1

H�i=1

�

�i�i��x�Di�t� �7�

where x=average height of the j+1 and j floors.Although Eq. �7� indicates that an infinite sum is required to

obtain the rotation response history, as it will be shown later, inmost cases only a relatively small number of modes is sufficientto obtain good estimates of the peak rotation demand in themodel. Hence, the interstory drift ratio at nondimensional heightx, can be approximated as

IDR�x,t� �1

H�i=1

m

�i�i��x�Di�t� �8�

where m=number of modes considered in the analysis.It can be shown that in the limit, when h→0 �or n→�� and

m→�, Eqs. �6� and �8� are exact. However, because buildingshave a finite number of stories these equations are only approxi-mate. In cases where the variation of the rotation computed withthe continuous model is significant over the story height, an im-proved estimation over the interstory drift ratio can be obtainedby averaging the rotation computed with Eq. �8� over the storyheight.

The ordinates of a generalized interstory drift spectrum aredefined as the maximum peak rotation over the height of thebuilding and are computed as

IDRmax � max"t,x

��x,t� �9�

The generalized interstory drift spectrum is a plot of the fun-damental period of the building versus IDRmax. Hence, it providesa direct indication of the peak interstory drift in buildings with
different periods of vibration.

Dynamic Properties of Continuous Model

As shown by Eq. �8�, computation of the generalized interstorydrift spectrum requires knowledge of the product of modal par-ticipation factor �i and the derivative of the mode shape, �i��x�, aswell as modal periods Ti and the modal damping ratios �i.

Setting ug�t� and c equal to zero in Eq. �1�, the undamped freevibrations of the continuous model shown in Fig. 1 are controlledby

�

EI

�2u�x,t��t2 +

1

H4

�4u�x,t��x4 −

�2

H4

�2u�x,t��x2 = 0 �10�

As shown in Miranda and Taghavi �2005� the mode shapes ofa cantilever beam satisfying Eq. �10� are given by

�i�x� = sin�ix� −i

isinh�xi� − �i cos�ix� + �i cosh�ix�

�11�

where i and �i=nondimensional parameters for the ith mode ofvibration and given by

i = ��2 + i2 �12�

�i =i

2 sin�i� + ii sinh�i�i

2 cos�i� + i2 cosh�i�

�13�

and i=eigenvalue of the ith mode of vibration corresponding tothe ith root of the following characteristic equation

2 + 2 +�4

i2i

2�cos�i�cosh�i� + �2

ii�sin�i�sinh�i� = 0

�14�

Once the eigenvalues i have been computed, the circular fre-quencies �i of each mode of vibration are computed as follows

�i2 =

EI

�H4i2�i

2 + �2� �15�

If the fundamental period of the building is known, periods ofvibration of higher modes can be computed using period ratiosgiven by

Ti

T1=

1

i

�12 + �2

i2 + �2 �16�

The derivative of the mode shapes with respect to nondimen-sional height x is obtained by taking the derivative of Eq. �11�with respect to x as follows

�i��x� =d��x�

dx= i cos�ix� − i cosh�xi� + �ii sin�ix�

+ �ii sinh�ix� �17�

For a continuum model with uniform mass distribution, modalparticipation factors �i can be computed with the followingequation

�i =

�0

1

�i�x�dx

�1

�i2�x�dx

�18�

0

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Both integrals in Eq. �18� can be solved in closed form. Hence�i can be computed in closed form only as a function of � oncethe eigenvalues i have been determined �see the Appendix forthe equations�.

Ordinates in the generalized interstory drift spectrum are com-puted with Eqs. �8� and �9� using undamped dynamic propertiescomputed using Eqs. �16�–�18� and user-specified modal dampingratios �i. which in general will be different for each mode. How-ever, Taghavi and Miranda �2005� showed that in many cases it ispossible to obtain relatively good estimations of the seismic re-sponse of buildings by assuming the same damping ratio for all mmodes. Eqs. �11�–�17� are exact, however because computerstypically use floating-point standard arithmetic with a small num-ber of digits, for large values of �, their use in most computerscan lead to wrong results due to the loss of significant digits.Alternative equations to be used in computers using floating-pointstandard arithmetic when � takes large values are provided in theAppendix.

Influence of � on Dynamic Properties of ContinuousModel

As indicated by Eqs. �11�–�14� and by Eq. �18�, modes of vibra-tion and modal participation factors in the continuous modeldepend only on a single parameter, the lateral stiffness ratio �.Fig. 2 shows the product of the modal participation factor andmodal shape for the first four modes of vibration. Products corre-sponding to five values of � are shown for each mode to examinethe influence of the lateral stiffness ratio on �i�i�x�. In thesefigures, a value of �=0 corresponds to a flexural beam while

Fig. 2. Effect of � on product of mode shape and participation factor

a value of �=650 corresponds approximately to a shear beam.


Intermediate values of �, represent structural behaviors that com-bine flexural and shear deformations. It can be seen that the in-fluence of � represented on �i�i�x� is significant. For example, atmidheight the product of the modal participation factor and themode shape for the first mode of vibration of a shear beam isapproximately 50% larger than that of a flexural beam. Similarly,at midheight the product of the modal participation factor and themode shape for the second mode of vibration of a flexural beam ispractically twice that of a shear beam.

Fig. 3 shows a similar comparison but now for the product ofthe modal participation factor and the derivative of the mode

Fig. 3. Effect of � on product of derivative of mode shape andparticipation factor

Fig. 4. Influence of � on g


shape with respect to x. Again, the comparison is presented forthe first four modes of vibration and for five values of �. It can beseen that in this case the lateral stiffness ratio not only has asignificant influence on this product, but also on the amplitudeand location of the maximum value along the height. For the firstmode, the influence of � is particularly large near the top andbottom of the structure, while at midheight, the influence of � ismoderate. Examination of this figure indicates that for large val-ues of the lateral stiffness ratio �� 10�, maximum contributionsof the first mode to the interstory drift ratio will tend to occurtoward the bottom of the structure while for small values of ��smaller than 3� maximums tend to occur in the upper portion ofthe structure. It can also be seen that, as expected, for the shearbeam ��=650� the contribution of the first mode has its maxi-mum at the base of the structure, while for a flexural beam��=0� the contribution of the first mode has its maximum at thetop. The effect of � on the product of the modal participationfactor and the derivative of the mode shape is also significant forhigher modes.

It should be noted that, although the modal participation factordepends on how the mode shapes are normalized, the product ofthe modal participation factor and the mode shape, or the productof the modal participation factor and the derivative of the modeshape, are independent of how the modes are normalized, so re-sults presented in Figs. 2 and 3 are general and do not depend onhow modes are normalized. This study deals with interstory driftdemands, however the proposed model can also be used to esti-mate acceleration demands �e.g., Miranda and Taghavi 2005� orforce and overturning moment demands. As an example, expres-sions to compute lateral forces and shear forces at any point alongthe height of the building are given in the Appendix. Similarly, anequation to compute base shears is also presented.

Influence of � on Interstory Drift Demands

Fig. 4 presents generalized interstory drift spectra for undampedmodels with four different values of � values when subjected tothe NS component of the ground motion recorded at Rinaldi Re-ceiving Station during the 1994 Northridge earthquake and theN49W component of the ground motion recorded at the TakatoriStation during the 1995 Hyogo-ken-Nambu �Kobe� earthquake.Results presented in this figure were computed using Eqs. �8� and�9� considering the first six modes of vibration �i.e, m=6�. For agiven fundamental period of vibration, the total height of the

ized interstory drift spectra
eneral

model needed in Eq. �8� was computed using the relationshipsuggested for steel moment-resisting frames in the 1997 UBCcode �ICBO 1997�, namely, T1=0.0853H0.75, where H is inmeters. It can be seen that the influence of � is relatively small forfundamental periods of vibration smaller than 1.5 s. However, forlonger fundamental periods of vibration the differences becomelarger. In particular, it can be seen that interstory drift demandscan be larger or smaller than those computed with a model cor-responding approximately to a shear beam ��=650�.

In order to quantify the effect of � on interstory drift demands,Fig. 5 shows interstory drift demands computed with �=0, 5, and30 normalized by interstory drift demands computed using�=650. Ordinates greater than one in this figure indicate inter-story drift demands larger than those computed for shear beamswhile values less than one indicate interstory drift demands lowerthan those of shear beams. It can be seen that, in general, forfundamental periods of vibration smaller than 1.5 s, interstorydrift demands for different values of �, do not change more than15% with respect to those computed using a shear beam. Further-more, in this period range interstory drift ratios computed with aflexural beam ��=0� tend to be higher than those computed witha shear beam. For fundamental periods of vibration larger than1.5 s, the influence of � can be very large. For example, for theTakatori record interstory drift demands computed with �=5,a lateral stiffness ratio that could correspond to buildings withdual lateral-resisting systems, at certain fundamental periods can

Fig. 5. IDRmax ratio

Fig. 6. Effect of damping on maximum

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be less than half or 75% more than those computed with ashear beam model. For fundamental periods longer than2.5 s, interstory drift ratios computed with a flexural beam��=0� tend to be lower than those computed with a shear beam.

Influence of Damping on Interstory Drift Demands

The generalized interstory drift spectra presented in Fig. 6 showchanges in interstory drift demands produced by changes indamping when using the near-fault Takatori record. The spectraon the left correspond to models with pure flexural behavior��=0�, while the spectra on the right correspond to models withpractically pure shear behavior ��=650�. In both cases spectra arecomputed using only the first six modes of vibration and themodal damping ratio is assumed to be the same in all six modes.Similarly to response spectra ordinates, a decrease is observed ininterstory drift demands with increasing damping and the influ-ence varies for different fundamental periods of vibration. Exami-nation of Fig. 6 also indicates that the effect of damping is morepronounced for buildings deforming laterally like shear beamsthan for buildings deforming laterally like flexural beams whentheir fundamental periods of vibration are longer than 3.0 s.

Fig. 7 presents interstory drift demands computed with flexuralbeams normalized by interstory drift demands computed withmodels deforming approximately like shear beams ��=650� forthree different levels of damping. It can be seen that for funda-

ifferent values of �

tory drift estimates for two values of �

s for d

inters


mental periods of vibration smaller than 2.0 s interstory drift de-mands in flexural and shear beams are very similar regardless ofthe level of damping. However, for fundamental periods of vibra-tion longer than 2.5 s, interstory drift demands computed in flex-ural beams tend to be smaller than those computed in shear beamsand it can be observed that the difference in demands decreases asdamping increases.

Influence of Higher Modes on Interstory Drift Demands

As indicated in the Introduction, drift spectrum ordinates have theadvantage over response spectrum ordinates of being able toconsider the contribution of higher modes in interstory driftdemands. By considering different number of modes in the sum-mation of Eq. �8� the generalized interstory drift spectrum canprovide indication of the contribution of higher modes to inter-story drift demands and, as indicated previously by Chopra andChintanapakdee �2001�, it also provides information on howmany modes need to be considered to obtain adequate estimatesof interstory drift demands.

Fig. 8 presents generalized interstory drift spectra of the NS

Fig. 7. Effect of damping on ratio of maximum interstory drift inflexural beams to maximum interstory drift in shear beams for 1995Takatori record

Fig. 8. Influence of higher modes on genera


component of the Rinaldi Receiving Station record considering afinite number of modes m=1,2 ,4, and 6. Spectra on the left arecomputed for flexural beam models ��=0� while spectra shownon the right-hand side of the figure are computed for �=100,which corresponds to a model that primarily deforms like a shearbeam but that it also includes some overall flexural deformations.In both cases a 5% damping ratio is assumed for all modes. It canbe seen that for this record, drift estimates are practically thesame regardless of the number of modes that are considered in theanalysis for models with fundamental periods of vibration smallerthan 2 s. For periods longer than 2.0 s, using only the fundamen-tal mode of vibration generally �but not always� results in under-estimation of maximum interstory drift demands. It can be seenthat for all fundamental periods of vibration, demands computedwith four modes are practically the same as those computed withsix modes, indicating that a relatively small number of modes isenough to capture maximum interstory drift demands. For thisrecord and for models deforming like flexural beams, using onlytwo modes provides adequate interstory drift estimates. The writ-ers have computed generalized interstory drift spectra as those

interstory drift spectra for �=0 and �=100

Fig. 9. Comparison of interstory drift estimates using procedureproposed in this study using �=650 with those proposed by Chopraand Chintanapakdee �2001� and by Iwan �1997� for undampedsystems subjected to Rinaldi record

lized

shown in Fig. 8 for many other recorded ground motions, and forpractically all cases, the number of modes required to capture themaximum interstory drift demand increases with increasing val-ues of �. Furthermore, considering only the first five or six modesprovides sufficient accuracy for fundamental periods smaller than5 s.

Comparison with Previous Studies

Previous studies by Iwan �1997� and by Chopra and Chintanapa-kdee �2001� computed drift spectra using a shear beam. In hisstudy, Iwan used a wave propagation approach, while Chopra andChintanapakdee used classical modal analysis. The ith mode ofvibration of a shear beam, as used in the latter study, is given by

�i�x� = �− 1�i−1 sin�2i − 1��x

2H�19�

As mentioned before, as �→� the modes of vibration in themodel used in the generalized interstory drift spectrum will tendto those of a shear beam. It is then interesting to compare theresult of the three approaches when estimating interstory driftdemands in elastic buildings that deform laterally approximatelylike shear beams. The drift spectra computed with the three meth-ods are shown in Fig. 9 for undamped models subjected to the NS

Fig. 10. Comparison of interstory drift estimates using procedureproposed in this study using �=650 with those proposed by Iwan�1997� and by Chopra and Chintanapakdee �2001� for 5% dampedsystems subjected to Rinaldi record

Fig. 11. Comparison of interstory drift response histories com

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component of the 1994 Rinaldi Receiving Station record. Theapproach used by Chopra and Chintanapakdee �2001� is referredto in the figure as shear beam. For the approach used in this study,equations given in the Appendix were used with �=650 for rep-resenting a shear beam. In both cases only the first six modes ofvibration were used. It can be seen that for the undamped modelthe three approaches yield the same results. In particular, as pre-viously noted by Chopra and Chintanapakdee �2001�, modalanalysis approach the results of the wave propagation formulationused by Iwan for undamped model, provided that sufficientmodes are included in the analyses.

Chopra and Chintanapakdee �2001� only compared their ap-proach to Iwan’s approach for undamped shear beams. Fig. 10shows a comparison of the three approaches similar to that shownin Fig. 9, but using a damping ratio of 5%. In both modal analysisapproaches modal damping ratios were assumed to be the samefor all modes. Results using the shear beam and those computedwith this study using �=650 are identical. However, it can beseen that the model used by Iwan predicts larger interstory driftdemands for periods smaller than 0.3 s and longer than 1.5 s. Forperiods longer than 1.5 the difference increases with increasingfundamental period. In some cases the differences between Iwan’smodel and the other two models are larger than 50%.

Fig. 11 compares interstory drift time histories computed withthe approach in this study using m=6 and �=650 to those com-puted using Iwan’s expression for 5% damped systems with afundamental period of vibration of 0.1 s when subjected to the NScomponent of the Rinaldi Receiving Station record. It can be seenthat the approach used by Iwan can lead to larger interstory driftdemands than those computed by the other two approaches.

The difference in results can become even larger for certainground motions. A comparison of drift spectra computed with thethree methods for 5% damped system excited by the S80W com-ponent of the motion recorded at the Lucerne Valley station dur-ing the 1992 Landers earthquake is shown in Fig. 12. Again,results computed using the shear beam and those computed withthe proposed approach with �=650 yield identical results. How-ever, it can be seen that, with the exception of fundamental peri-ods of vibration between 0.5 and 1.2 s, the model used by Iwanleads to larger interstory drift demands. Kim and Collins �2002�attributed the larger interstory drift demands when using Iwan’smodel to records where permanent ground displacement occurs.In particular, they reported that the model used by Iwan couldlead to residual drift demands for records with permanent grounddisplacement. According to Iwan �1997� this component of theLucerne record has a permanent displacement of nearly 2.0 m. Anexample of the residual interstory drifts that can be obtained withIwan’s model is illustrated in Fig. 13. This figure compares inter-story drift time histories computed with the approach presented in

for damped-short period structures subjected to Rinaldi record
puted

this study using m=6 and �=650 to those computed using Iwan’sapproach for 5% damped systems with a fundamental period ofvibration of 0.2 s when subjected to the Lucerne record. Althoughperiods of vibration shown in Figs. 11 and 13 are very short andof little practical significance �most buildings will have funda-mental periods of vibration longer than 0.2 s�, these figures illus-trate the difference in the results from different approaches.Fig. 13 shows that the approach used by Iwan in this case leads topeak interstory drift demands that are approximately twice ofthose computed with the other two approaches and produces aresidual drift of approximately 0.5%. As noted by Kim andCollins �2002�, the residual drift is inconsistent with the linearelastic behavior assumed in the development of the drift spec-trum. As shown in Figs. 11 and 13 the modal damping used in thegeneralized interstory drift spectrum does not produce residualdrifts.

Influence of Lateral Resisting System on InterstoryDrift Demands

The generalized interstory drift spectrum extends Iwan’s driftspectrum to estimate interstory drift demands in buildings whoselateral deformations differ from those of a shear beam. Therefore,the generalized interstory drift ratio can be used to study the

Fig. 12. Comparison of interstory drift estimates using procedureproposed in this study using �=650 with those proposed by Iwan�1997� and by Chopra and Chintanapakdee �2001� for 5% dampedsystems subjected to the Lucerne record

Fig. 13. Comparison of interstory drift response histories


influence of the lateral resisting system on interstory drift de-mands. Fig. 14 presents interstory drift demands computed forsteel moment-resisting frame �SMRF� buildings, for reinforcedconcrete moment-resisting frame �RCMRF� buildings, and for re-inforced concrete shear wall �RCSW� buildings with differentnumber of stories. For a given fundamental period of vibration,the number of stories has been computed using the mean relation-ships between height and period proposed by Chopra and Goel�2000� for these structural systems and by assuming a constantstory height of 3 m. For shear wall buildings Chopra and Goelrecommend estimating the fundamental period of vibration as afunction of the height of the buildings and the equivalent sheararea expressed as a percentage of the building plan area Ae. InFig. 14 this equivalent shear area was assumed to be 0.16%. Thedamping ratio was assumed 2% for steel buildings and 5% forreinforced concrete buildings, and again assumed a constantdamping ratio for all modes. Similarly, the lateral stiffness ratiowas assumed equal to 20 for moment-resisting frame buildingsand equal to one for shear wall buildings. It can be seen that thelateral strength resisting system has a very large effect on inter-story drift demands, with interstory drift demands in shear wallbuildings being significantly smaller than those of moment-resisting frame buildings. Furthermore, interstory drift demandscomputed for steel moment-resisting frame buildings are largerthan those computed for reinforced concrete moment-resisting

ted for damped-short period structures for Lucerne record

Fig. 14. Effect of lateral resisting system on interstory drift estimatesfor buildings with different number of stories

compu

frames because the latter are slightly stiffer and have more damp-ing. This figure highlights the possible use of generalized inter-story drift spectra in assisting the selection of the lateral resistingsystem for buildings located in seismic regions.

Relationships proposed by Chopra and Goel �2000� are basedon regression analyses on fundamental periods of vibration in-ferred from records obtained in instrumented buildings in Califor-nia. However, considerable scatter exists around the regressedrelations. The implication of this scatter is that for a given build-ing height there is a significant uncertainty in estimating the fun-damental period of vibration. Similarly, for a given fundamentalperiod of vibration there will be significant uncertainty in theestimation of the building height. In order to account for thisuncertainty Chopra and Goel also provided expressions corre-sponding to best-fits ±1 SD of the relationship between funda-mental period of vibration and building height. Here we use those±1 SD relationships to incorporate the uncertainty in the estima-tion of interstory drift demands in the generalized interstory driftspectrum. In particular, those relationships are used to obtainapproximate ±1 SD building heights corresponding to a givenfundamental period of vibration. Fig. 15 shows interstory driftdemands computed these building heights for reinforced concreteand steel moment-resisting frame buildings subjected to theLucerne record. In both cases a lateral stiffness ratio of 20 isassumed. Accordingly, maximum interstory drift demands formost frame buildings responding elastically would lie betweenthose two spectra.

Generalized Interstory Drift Demand Spectrum asMeasure of Damage Potential

Many studies have concluded that the interstory drift ratio is theparameter that is best correlated with damage in buildings. Hence,the generalized interstory drift spectrum can provide indication ofthe damage potential of different ground motions. For example,by comparing Figs. 10 and 12 it can be seen that even though thepeak ground accelerations and peak ground velocities of theRinaldi and Lucerne records are similar, the former has a muchlarger damage potential, especially for buildings with fundamen-tal periods of vibration smaller than 2.0 s. For tall buildings, withperiods of vibration longer than 3.0 s, the interstory drift demandsproduced by the Lucerne record are slightly larger than thoseproduced by the Rinaldi record.

In 1985 two large magnitude earthquakes occurred in Chile

Fig. 15. Influence of period uncertainty on maximum interstory driffunction of building height developed by Chopra and Goel �2000�

�Ms=7.8� and Mexico �Ms=8.1�. While significant damage oc-

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curred as a result of these earthquakes, the damage produced inthe Mexican earthquake was far greater than the damage pro-duced by the Chilean earthquake. The areas with the largestamount of damage were Valparaiso in Chile and the soft soil zoneof Mexico City, Mexico. The prevailing lateral resisting system inmultistory buildings in Valparaiso consisted of reinforced con-crete shear walls while in Mexico City it consisted of reinforcedconcrete moment-resisting frames. Fig. 16 shows the generalizedinterstory drift spectra for both structural systems computed withrecords representative of the ground motions at both locations. Ineach case, two spectra are computed using building heights com-puted with ±1 SD relationships proposed by Chopra and Goel�2000�. Although the relationships proposed by Chopra and Goelwere developed for buildings in California and probably are notentirely applicable to buildings in Valparaiso or Mexico City, nev-ertheless Fig. 16 provides information of the enormous differencein interstory drift demands imposed by the Secretaria de Commin-icaiones y Transportes �SCT� record in Mexico City compared tothose produced by the Valparaiso record. In particular, interstorydrift demands computed for RCMRF buildings with periods ofvibration between 2 and 3 s in Mexico City are more than 1 orderof magnitude larger than those computed for RCSW buildings inValparaiso.

Summary and Conclusions

A new measure of seismic demand of earthquake ground motionsis presented. The new measure, referred to as generalized inter-story drift spectrum provides an estimate of maximum interstorydrift demands in multistory buildings. Furthermore, its ordinatesincorporate the influence of higher modes of vibration. The pro-posed spectrum extends the drift spectrum developed by Iwan tobuildings that do not deform laterally like shear beams. The newspectrum is based on a model that consists of a combination of aflexural beam and a shear beam, rather than only a shear beam.By only modifying the lateral stiffness ratio, �, this model canconsider lateral deformations varying from those of a flexuralbeam to those of a shear beam. Hence, it permits to account for awide range of modes of deformation that represent more closelythose of multistory buildings.

The proposed generalized interstory drift spectrum is based onmodal analysis techniques that are very familiar to structural en-

ates. Computations are done using ±1 SD of fundamental period as
t estim
gineers and assumes a damping model that avoids the problems


that have been reported to occur in the model used in the originaldrift spectrum. The computational effort involved in computingthe proposed generalized interstory drift spectrum is very small.Computation of ordinates corresponding to a family of 50 differ-ent models �e.g., a spectrum with 50 different periods of vibra-tions� is less than 1 min in most personal computers.

It was shown that for undamped buildings deforming laterallylike shear beams, the results of this study when using �=650coincide with those of Iwan �1997� and with those of Chopra andChintanapakdee �2001�. For damped buildings, results coincidewith those of Chopra and Chintanapakdee but in many cases theyare lower than those of the original drift spectrum. For buildingswith fundamental periods of vibration smaller than 1.2 s resultsconsidering only the fundamental mode of vibration are, in gen-eral, relatively good. The effects of higher modes on maximuminterstory drift demands are more significant in buildings withpredominantly shear deformations than in buildings with pre-dominantly flexural deformations. For buildings with periodslonger than 1.2 s considering only five or six modes of vibrationis enough to obtain maximum interstory drift demands.

The effect of lateral stiffness ratio is particularly important forbuildings with periods of vibration longer than about 1.2 s. In thisperiod range, the effects of damping are more significant forbuildings with large values of �. It was shown that the uncertaintyin the estimation of the period of vibration can be easily incorpo-rated in the proposed spectrum.

The generalized interstory drift spectrum is a powerful tool toprovide direct indication of interstory drift demands in buildings.The new generalized interstory drift spectrum is not recom-mended as a replacement of the response spectrum, but rather asan additional tool available to structural engineers to provide animproved measure of the intensity of earthquake ground motionon buildings.

Acknowledgments

This work was initiated during a postdoctoral stay of the secondauthor at Stanford University. Financial support for this postdoc-toral stay was partially provided by the Scientific Research andTechnical Council of Turkey. This financial support is greatlyappreciated. The writers thank Professor Lázló Kollár and

Fig. 16. Generalized interstory drift spectra as measures of damagearthquakes. Upper and lower bound IDRmax are computed using relatand Goel �2000�.

Dr. Gabriella Tarján for bringing to their attention the work on


continuous beams by Hungarian investigators. Similarly, they arealso grateful to Professor Andrei Reinhorn, Professor Polat Gül-kan, and Professor Peter Fajfar who also provided them with in-formation on previous research on continuous models. Commentsby two anonymous reviewers are also acknowledged.

Appendix

Eqs. �11�–�17� are exact, however since most computers usefloating-point standard arithmetic with a small number of digits,their use can lead to inaccurate results due to the loss of signifi-cant digits for large values of �. Here alternative equations arepresented where the loss of significant digits is prevented.

By replacing the hyperbolic functions in Eq. �11�, the modeshape equation can be rewritten as follows:

�i�x� = sin�ix� − �i cos�ix� +1

2�i −

i

i�eix +

1

2�i +

i

i�e−ix

�20�

where

�i −i

i=

ii2 sin�i� − i

3 cos�i� − ii2e−i

ii2 cos�i� + i

3 cosh�i��21�

From inspection of Eqs. �13� and �21� it can be seen that thelast three terms in Eqs. �11� and �20� tend to zero as �→�. Forthe first six modes these terms become negligible for ��650.

Multiplying Eq. �14� by 2i2i

2e−i and expressing hyperbolicfunctions in terms of exponentials, an alternate expression of thecharacteristic equation is obtained where no loss of significantdigits will occur. The alternate characteristic equation is given by

4i2i

2e−i + �2i2i

2 + �4�cos�i� + ��2ii�sin�i� + ��2i2i

2

+ �4�cos�i� − ��2ii�sin�i��e−2i = 0 �22�

As �→� the characteristic equation reduces to

2i2i

2 + �4

2 � + tan�i� = 0 �23�

ntial for different structural systems in 1985 Chilean and Mexicans between fundamental period and building height studied by Chopra

e poteionship

� ii

From inspection of Eq. �23�, it is clear that the first term goesto infinity as �→� �i.e., for a pure shear beam model�, hence theroots of the characteristic equation �the eigenvalues� will be val-ues of i that make tan i→−�, as follows:

i = �2i − 1��

2�24�

Dividing the numerator and denominator inside the square rootof Eq. �16� by �2, it can be seen that period ratios in a pure shearmodel ��→ � � are given by

Ti

T1=

1

i=

1

2i − 1�25�

An alternative expression to Eq. �17� where no loss of signifi-cant digits occurs is given by

�i��x� = i cos�ix� + �ii sin�ix� +1

2i�i −

i

i�eix

−1

2i�i +

i

i�e−ix �26�

As mentioned previously, integrals in Eq. �18� required tocompute the modal participation factors can be obtained in closedform. These integrals are given by

�0

1

�i�x�dx =1

2i2i

�2i2 + 2i

2 − 2i2 cos�i� − 2i

2�i sin�i�

+ ii�f1 − f2�� 27�

�0

1

�i2�x�dx =

1

2−

1

2isin�i�cos�i� −

�i

isin2�i� +

�i2

2

+�i

2

2isin�i�cos�i� +

i

i2 + i

2 �f1 − f2��sin�i�

− �icos�i�� −i

i2 + i

2 �f1 + f2��cos�i� + �isin�i��

+sinh�i�

4i f1�i −

i

i� + f2�i +

i

i�� +

f1f2

2

�28�

where f1 and f2= functions given by

f1 = ei�i −i

i� �29�

f2 = e−i�i +i

i� �30�

From Eqs. �27�–�30�, it can be shown that for a pure shear model��→ � � the modal participation factors are given by

�i =4�− 1�i−1

2�i�� − ��31�

This study has been aimed at developing a new type of spec-trum whose ordinates provide estimates of maximum interstorydrifts in buildings responding elastically, however, the continuousmodel used in the proposed generalized interstory drift has alsobeen shown to be useful in estimating peak floor accelerationdemands in buildings �Miranda and Taghavi, 2005�. Similarly, the
continuum model shown in Fig. 1 can also be used to estimate
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force, shear, and overturning moment demands. For example, thenormalized lateral force demand time history at nondimensionalheight x is given by

F�x,t�EI

� �i=1

m

�i�� j��x� + �2�i��x��Di�t� �32�

where �i��x�= third derivative of the ith mode shape. Similarly,the normalized lateral shear time history at nondimensional heightx is given by

V�x,t�EI

�1

EI�i=1

m

�i �x

1

�i��d� + �2�x

1

�i��d��Di�t�

�33�

It should be noted that integrals in Eq. �33� can be solved inclosed form. As an example, the normalized base shear time his-tory can be computed with

Vo�t�EI

�1

EI�i=1

m ��sin i − �i cos i + �i��2 − i2� +

i2 + �2

2

�i −i

i��ei − 1� + �i +

i

i��e−i − 1��Di�t�

�34�

It should be noted that for the model shown in Fig. 1, the onlyapproximation involved in Eqs. �32�–�34� is the truncation causedby considering only the contribution of the first m modes ofvibration.

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