200313

Bulletin of Earthquake Engineering 1: 336, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands.

3

An Incremental Response Spectrum AnalysisProcedure Based on Inelastic SpectralDisplacements for Multi-Mode SeismicPerformance Evaluation

M. NURAY AYDINOGLUDepartment of Earthquake Engineering, Bogazii University, Kandilli Observatory and EarthquakeResearch Institute (KOERI), 81220 engelky, Istanbul, Turkey

Received 17 January 2003; accepted 30 January 2003

Abstract. The so-called Nonlinear Static Procedure (NSP) based on pushover analysis has been de-veloped in the last decade as a practical engineering tool to estimate the inelastic response quantitiesin the framework of performance-based seismic evaluation of structures. However NSP suffers froma major drawback in that it is restricted with a single-mode response and therefore the procedure canbe reliably applied only to the two-dimensional response of low-rise, regular buildings. Recognizingthe continuously intensifying use of the pushover-based NSP in the engineering practice, the presentpaper attempts to develop a new pushover analysis procedure to cater for the multi-mode responsein a practical and theoretically consistent manner. The proposed Incremental Response SpectrumAnalysis (IRSA) procedure is based on the approximate development of the so-called modal capacitydiagrams, which are defined as the backbone curves of the modal hysteresis loops. Modal capacitydiagrams are used for the estimation of instantaneous modal inelastic spectral displacements in apiecewise linear process called pushover-history analysis. It is illustrated through an example analy-sis that the proposed IRSA procedure can estimate with a reasonable accuracy the peak inelasticresponse quantities of interest, such as story drift ratios and plastic hinge rotations as well as thestory shears and overturning moments. A practical version of the procedure is also developed whichis based on the code-specified smooth response spectrum and the well-known equal displacementrule.

Key words: incremental response spectrum analysis, inelastic spectral displacements, modal capac-ity diagrams, multi-mode pushover analysis, nonlinear static procedure, performance-based seismicevaluation, piecewise linear mode-superposition

1. Introduction

It has been long recognized that the structural behavior and damageability of struc-tures during earthquakes is essentially controlled by the inelastic deformation ca-pacities of the ductile structural elements. This eventually led to a notion that theseismic evaluation and design of structures should be based on displacements (ormore correctly on deformations) demanded by the earthquake action, not on thestresses induced by the assumed equivalent seismic forces. In spite of this recogni-tion the current seismic design practice is still governed by the force-based design

4 M. NURAY AYDINOGLU

principles. Nevertheless significant attempts have been made in the last decade toincorporate the displacement-based evaluation and design concept into the seismicengineering practice. Those attempts are developed in two interrelated but differ-ent directions yielding the displacement-based design methods aiming at directdesign of new structures (e.g., Priestley, 2000) and the displacement-based evalu-ation methods dealing with the seismic performance evaluation of pre-designed orexisting structures.

The present paper is concerned with the displacement-based evaluation meth-ods, which were formally introduced during the last decade within the frameworkof the performance-based seismic engineering (ATC, 1996; FEMA, 19972000).In this context, the Nonlinear Static Procedure (NSP) is primarily based on a staticnonlinear analysis called pushover analysis, which is performed for the monotonicincrements of equivalent seismic loads with prescribed-invariant or adaptive pat-terns. The outcome of the pushover analysis is the pushover curve, which repre-sents the inelastic variation of the base shear with respect to the roof displacement.The selection of the coordinates of the pushover curve is somewhat arbitrary; nev-ertheless they are indicative of the overall strength and deformation capacities ofa given structure. However the ultimate objective of NSP is the estimation of peakinelastic deformations of individual structural elements, such as the plastic hingerotations, demanded by the seismic action. For this purpose use is made of the peakinelastic displacement, i.e., inelastic spectral displacement of an equivalent single-degree-of-freedom (SDOF) system, the properties of which are defined from thecoordinates of the pushover curve.

Nonlinear static procedure based on pushover analysis rapidly became popularin structural earthquake engineering community. The practicing engineers, whoare traditionally trained for the linear response under reduced equivalent seismicforces, were given a new chance of gaining valuable insight to the nonlinear seis-mic behavior of structures at the system and element levels. At the same time,successful applications made on relatively simple structural systems encouragedthe engineers for a wider use of the new procedure. However it should be admittedthat pushover-based NSP still remains intuitive rather than mathematical (Elnashai,2002) and it suffers from a number of limitations and problems (Krawinkler andSeneviratna, 1998) to be resolved. An ongoing project being conducted by theApplied Technology Council (ATC, 2002) is expected to shed light to at least someof those problems towards the improvement of NSP.

The major drawback of NSP in its existing form (ATC, 1996; FEMA, 19972000) lies in the fact that it is basically restricted with a single-mode response. Itmeans that the procedure can be reliably applied only to two-dimensional responseof low-rise structures regular in plan, where the response is effectively governed bythe first mode. Consequently, the application of NSP to a high-rise building regularin plan or any building irregular in plan involving three-dimensional response isprone to produce erroneous results. Thus, given the continuously intensifying useand irreversible popularity of the pushover-based NSP in the engineering practice,

INCREMENTAL RESPONSE SPECTRUM ANALYSIS 5

there is an urgent need for improvement of the procedure to cater for the multi-mode response in a practical and theoretically consistent manner. The objective ofthe present paper is to contribute to the ongoing research efforts directed to achievethis goal.

2. Critical Review of Current Procedures for Multi-mode Pushover Analysis

Given the challenge that displacement-based approach will provide the engineerwith a better understanding of the real nonlinear behavior of a structure comparedto the conventional force-based approach, advocating the use of NSP in its ex-isting form for all types of buildings is unacceptable. However, the most recentpublication on NSP, i.e., FEMA 356 (FEMA, 2000), which is now accepted as apre-standard by the American Society of Civil Engineers (ASCE), explicitly statethat NSP is not restricted with the low-rise buildings governed by a single-moderesponse. In fact, FEMA 356 indicates that NSP is applicable to buildings withmore than 10 stories. This is rather surprising, because the invariant or adaptivelateral load patterns specified in FEMA 356 are all associated with a single-moderesponse with only one exception, which is defined as an invariant load pattern tobe obtained from the modal combination of story shears through an elastic responsespectrum analysis. Clearly implying the multi-mode response, FEMA 356 requiresthat this load pattern should be used when the fundamental period exceeds 1.0 secand a sufficient number of modes are to be considered to capture at least 90%of the total mass. The equivalent seismic loads are then applied to the structureincrementally according to this invariant pattern and the pushover curve is plotted.

It should be pointed out that defining the seismic loads through elastic spectralaccelerations has no theoretical basis, as they are not consistent with the inelasticdeformation of the structure during the pushover process. However the major draw-back in this procedure is that although the resulting pushover curve is assumed tocontain the effects of higher modes, eventually it has to be treated as the repre-sentative curve of a SDOF system to estimate the peak response quantities. It isworth repeating that the pushover analysis as described in FEMA 356 is essentiallybased on the representation of inelastic multi-degree-of-freedom (MDOF) systemby an equivalent SDOF system. This is achieved by converting the coordinatesof the pushover curve to the modal pseudo-acceleration and displacement of theequivalent SDOF system. The conversion parameters are the effective participatingmass and the participation factor times the roof displacement, respectively, whichare defined on the basis of either the elastic first mode shape or the displacedconfiguration of the structure at the peak response (ATC, 1996; FEMA, 2000).Therefore it is clear that the peak response quantities associated with the multi-mode effects cannot be correctly estimated with a conversion technique based on asingle-mode response.

Recognizing the fact that invariant load patterns are not compatible with the pro-gressive yielding of the structure during pushover analysis, alternative procedures


based on adaptive load patterns have been proposed (Elnashai, 2002; Antoniouet al., 2002). In these procedures, equivalent seismic loads are calculated at eachpushover step using the mode shapes based on instantaneous (tangent) stiffnessmatrix and the corresponding elastic spectral pseudo-accelerations. The seismicloads are then combined with a modal combination rule, normalized and appliedto the structure at each step to obtain the increments of the pushover curve co-ordinates. It may be argued that such an adaptive scheme better represents theinelastic behavior compared to invariant load pattern, yet it suffers from the sameproblems mentioned above for the FEMA 356 procedure, as the final output is stillthe conventional pushover curve to be represented by an equivalent SDOF system.The use of the instantaneous values of the elastic spectral pseudo-accelerationsappears at the first glance to be an improvement, but still they are not compatiblewith the instantaneous inelastic response. Another pitfall inherent in this procedureis the application of the modal combination in defining the equivalent seismic loadsinstead of combining the response quantities induced by those loads in individualmodes (Chopra, 2001, p. 569).

Another adaptive procedure developed by Gupta and Kunnath (2000) starts witha similar approach, i.e., the equivalent seismic loads are calculated at each pushoverstep again using the instantaneous mode shapes, and the associated elastic spectralpseudo-accelerations are used for scaling. However the above-mentioned pitfallis avoided in this procedure where the seismic loads are not combined at eachstep, instead they are applied to the structure in each mode independently and theincrements of the modal response quantities of interest including pushover curvecoordinates are calculated. They are then combined with SRSS (square-root-of-sum-of-squares) rule and added to the quantities calculated at the previous step.This approach seems to be more meaningful, as the conventional response spec-trum analysis (RSA) is actually being applied at each pushover step. Howeverthe main problem remains: the elastic spectral accelerations associated with theinstantaneous free vibration periods are not consistent with the inelastic behaviorof the structure. Thus, as in the previously described procedures, it is not possiblewith this procedure to correctly estimate the peak response quantities that are repre-sentative of the multi-mode inelastic response. It is for this reason that in the abovereferenced paper Gupta and Kunnath had to compare the story drifts estimatedby their procedure with those obtained from the nonlinear time-history analysisat a roof displacement equal to the peak roof displacement of the latter analysis.In order to estimate the peak demand quantities by the proposed procedure itself,it becomes inevitable again to resort to the equivalent SDOF system based on asingle-mode response.

Regarding the above-described multi-mode adaptive procedures, two criticalconclusions can be drawn:(a) Loading characteristics based on elastic instantaneous spectral accelerations arenot compatible with the inelastic instantaneous response,


(b) The conventional pushover curve combining multi-mode effects is not an ap-propriate tool to estimate the peak response quantities.

Recently a notable contribution to the multi-mode pushover analysis is achievedwith the development of the Modal Pushover Analysis (MPA) procedure (Chopraand Goel, 2001). The basic idea behind the procedure was in fact proposed inearlier studies (Paret et al., 1996; Sasaki et al., 1998), which may essentially beregarded as a simple extension of the conventional single-mode pushover analysisto the multi-mode response with the following steps:(1) Run pushover analysis and plot pushover curves independently for each modewith invariant lateral load patterns associated with the linear (initial) mode shapes,(2) Convert the pushover curve in each mode to a capacity diagram (capacity spec-trum ATC, 1996) of the corresponding equivalent SDOF system using the modalconversion parameters based on the same linear (initial) mode shapes,(3) Calculate peak inelastic displacement of the equivalent SDOF system in eachmode for a given earthquake using the bilinear form of the capacity diagram as abackbone curve (alternatively calculate inelastic spectral displacement using smoothresponse spectrum FEMA, 2000),(5) Calculate peak inelastic response quantities of interest, such as story drifts andplastic hinge rotations independently in each mode,(6) Apply SSRS rule to estimate the combined peak response quantities.

It is noticed at the first glance that running the pushover analysis independentlyin each mode and neglecting the contribution of other modes in the plastic hingeformation is the weakest point of MPA procedure. In fact in a frame analysis,different sets of plastic hinges are developed at different locations independentlyin each mode and generally linear behavior governs even in high-rise structuresexcept in the first few modes. This results in unacceptably large errors in plastichinge rotations, however errors are found relatively smaller in story drifts, thanksto the participation of the elastic higher modes. This finding led to a questionablesuggestion that story drifts could be considered in lieu of the plastic hinge rotationsas the representative demand parameter in the acceptance criteria of NSP (Chopraand Goel, 2001). Since the inelastic behavior in higher modes is poorly estimated,a modified version of MPA may be proposed, in which the pushover analysis is runonly in the first mode and the resulting peak inelastic response quantities are com-bined with the peak elastic quantities developed in the higher modes (Aschheim,personal communication, 2002).

In view of the above discussion, the aim of the present paper is to develop analternative multi-mode pushover analysis procedure in an attempt to better estimatethe main inelastic response quantities, i.e., the peak displacements, story drifts aswell as the plastic hinge rotations.


3. Piece-wise Linear Mode-Superposition for Nonlinear Time-HistoryAnalysis: Definition of Modal Capacity Diagrams

In this section, the nonlinear time-history analysis is treated on the basis of a piece-wise linear mode-superposition procedure. The aim is to establish a theoreticalbasis for the multi-mode pushover analysis procedure to be proposed in this paper.

3.1. INCREMENTAL EQUATIONS OF MOTION

When multi-linear hysteretic models are used to represent the nonlinear behavior ofstructural members, such as the plastic hinges, the dynamic response is essentiallylinear in an incremental step (i) between a time t and a previous time station ti1at which the response is already determined. Thus, piecewise linear incrementalequations of motion of a nonlinear 3-D structure subjected to a uni-directionalearthquake can be written for t > ti1 as

M[u(t) u(ti1)

] + C(i) [u(t) u(ti1)] + K(i) [u(t) u(ti1)

]

= MIgx[u

gx(t) ugx(ti1)

] (1)

in which u(t) represents the relative displacement vector and ugx(t) refers to theground acceleration of a given earthquake in x direction. Igx is a kinematic vec-tor representing the pseudo-static transmission of the ground acceleration to thestructure, whose components associated with the degrees of freedom in x earth-quake direction are unity and others are zero. In Eq. 1, M denotes the mass matrixand K(i) represents the instantaneous (tangent) stiffness matrix in the incrementalstep (i). The instantaneous damping matrix C(i) is generally expressed as a linearcombination of mass and stiffness matrices (Rayleigh damping).

3.2. PIECEWISE LINEAR MODE-SUPERPOSITION

Eq. 1 is solved by means of step-by-step integration methods, such as Newmarksbeta methods, details of which can be found in standard textbooks (Clough andPenzien, 1993; Chopra, 2001). However, one can think that the conventional mode-superposition method may be equally applied to the piecewise linear solution ofEq. 1. It may be argued that this method would be inefficient for nonlinear systems,as it would require an eigenvalue analysis to be performed nearly at every solutionstep. However it may be envisioned that the very rapid developments taking placein the computer industry in terms of hardware speed and capacity may soon in-validate this argument. Note that mode-superposition method may be attractivebecause of its two important advantages, namely, the freedom in assigning themodal damping ratios in each mode and the superior accuracy obtained in thesolution of the modal SDOF systems.

The application of the mode-superposition method to the piecewise linear so-lution of nonlinear systems will be treated in this paper merely from a conceptual


point of view. The aim is to provide an analytical background to the multi-modepushover analysis to be proposed by introducing the approximate modal hysteresiscurves and their backbone curves called modal capacity diagrams. To this end, theinstantaneous displacement response during the piecewise linear incremental step(i) can be expanded to the modal coordinates as

u(t) =Ns

n=1un(t) (2a)

un(t) = (i)n (i)xndn(t) (2b)where Ns refers to the sufficient number of modes to be considered in the modalexpansion, (i)n represents the instantaneous nth mode shape vector, dn(t) is themodal displacement in the nth mode and (i)xn denotes the instantaneous participa-tion factor for an earthquake in x direction, which is defined as

(i)xn =L(i)xn

(i)Tn M(i)n

; L(i)xn = (i)Tn MIgx (3)

Substituting Eq. 2 and time derivatives into Eq. 1, pre-multiplying with (i)Tn ,making use of the modal orthogonality conditions and considering Eq. 3 resultin an uncoupled instantaneous modal equation of motion in the nth mode:

dn(t)+ 2 (i)n (i)n dn(t)+ ((i)n )2dn(t) = [ugx(t) ugx(ti1)

] + dn(ti1)+ 2 (i)n (i)n dn(ti1)+ ((i)n )2dn(ti1) (4)

in which (i)n and (i)n represent the instantaneous natural frequency and modal

damping ratio, respectively, while dn(ti1) is expressed as

dn(ti1) = (i)Tn Mu(ti1)

L(i)xn

=(i)Tn M

Nsm=1

(i1)m (i1)xm dm(ti1)

L(i)xn

(5)

which represents the initial modal displacement to be considered at t = ti1 for thenew, modified system at t > ti1. Similar relationships can be written for the timederivatives of dn(ti1).

Considering Eq. 5 and time derivatives, the single-step solution of Eq. 4 fordn(t) is simple and it can be achieved by any integration method. In this respect,superior accuracy can be obtained by the Piecewise Exact Method, which is re-cently reformulated in a unified format including the P-delta effect (Aydinoglu and


Fahjan, 2003). In each step modal displacements are evaluated for ti = ti1 + tfollowed by the determination of displacements and other response quantities ofinterest at time station ti. To detect the yielding and unloading points in plastichinges, the regular time step t is appropriately reduced.

It is clear from Eq. 5 that dn(ti1) is different from the modal displacementdn(ti1) as the former includes the effects of all modes belonging to the previousstep (i 1), which are different from those at step (i).

3.3. MODAL CAPACITY DIAGRAMS

Had it been actually implemented, the solution of Eq. 4 followed by the use ofEq. 2b at each step would have provided a very valuable insight to the modalnonlinear behavior of the MDOF system in different modes. In particular, it wouldalso be very instructive to observe the individual behavior of the modal SDOFsystems themselves. However the behavior of each modal SDOF system is discon-tinuous, because a different structural system is actually being considered at eachincremental step. As indicated above, it is for this reason that the initial modaldisplacement dn(ti1) defined in Eq. 5 for step (i) is different from the modal dis-placement dn(ti1) calculated at the end of the previous step (i1). However it maybe assumed that this difference would not be very significant, because mode shapeswould change only slightly in consecutive incremental steps (especially in highlyredundant systems). Although such changes in mode shapes have been consideredin the piecewise modal transformation at time t, for the sake of simplicity they maybe ignored in determining dn(ti1). Applying modal orthogonality relationships inEq. 5 and considering Eq. 3, this approximation leads to

dn(ti1) = dn(ti1) (6)Hence modal response in each mode is now assumed continuous and Eq. 4 can bewritten in an incremental form as

d(i)n + 2 (i)n (i)n d(i)n + ((i)n )2d(i)n = ug(i)x (7)where ug(i)x represents the ground acceleration increment, i.e., the first term on theright-hand side of Eq. 4. Modal displacement increment at the (i)th incrementalstep is expressed as

d(i)n = dn(t) dn(ti1) (8)Eq. 7 can be appropriately rewritten as

d(i)n + 2 (i)n (i)n d(i)n +a(i)n = ug(i)x (9)where the third term on the left-hand side represents the modal pseudo-accelerationincrement:

a(i)n = ((i)n )2d(i)n . (10)


Figure 1. Schematic representation of hypothetic modal hysteresis loops and modal capacitydiagrams (solid curves).

Now hypothetically it is possible to construct the continuous modal displacementversus modal pseudo-acceleration diagrams governed by Eq. 9. Those diagramsrepresent the modal hysteresis loops, which are schematically depicted in Figure 1.The outer hysteresis loops are the fattest in the first mode as indicated in the figureand get thinner and steeper as the mode number increases. According to Eq. 10, theinstantaneous slope of a given diagram is equal to the eigenvalue (natural frequencysquared) of the corresponding mode at the piecewise linear increment concerned.

The backbone curves of the hypothetical modal hysteresis loops in the firstquadrant may be appropriately called the modal capacity diagrams, which areindicated by solid curves in Figure 1. Note that although those diagrams are repre-sentative of the structures strength capacity in each mode, they are also dependentupon the seismic demand. It means that modal capacity diagrams would be dif-ferent for each earthquake considered. The only exception is the case where thefirst mode alone is assumed to represent the dynamic response. In this case modalcapacity diagram is demand-independent and by definition it is identical to the so-called capacity spectrum used in the Capacity Spectrum Method (ATC, 1996). Theterm modal capacity diagram is preferably used in this paper by adding the wordmodal to the terminology proposed by Chopra and Goel (1999).


Figure 2. Linear modal capacity diagrams associated with elastic response.

The multi-mode pushover analysis procedure proposed in this paper is basedon an approximate development of the modal capacity diagrams utilizing modalinelastic spectral displacements. Note that the procedure does not require the con-ventional pushover curve be plotted. It is pointed out earlier that the pushover curveis not an appropriate tool for estimating the peak response quantities of interest,which is the ultimate goal of the pushover analysis.

In the remainder of the paper, the lumped plasticity approach is adopted forthe sake of simplicity, which means that the nonlinear behavior of the structuralelements is assumed to be represented by the plastic hinges. However the proposedmethod is also applicable to distributed plasticity approach provided that reason-ably small incremental steps are used in between the two consecutive configurationof the nonlinear system. In each incremental step or in between the formation oftwo consecutive hinges, a piecewise linear behavior is considered.

4. Development of Incremental Response Spectrum Analysis (IRSA)Procedure for Multi-Mode Pushover Analysis

The basic motive behind the proposed procedure stems from the question whetherthe modal capacity diagrams defined above could be constructed in a practical man-ner. Admittedly this is a difficult task because, regardless of whether the response islinear or nonlinear, different modes are randomly excited in MDOF systems due torandom nature of the seismic action. In linear systems, a practical answer has beenfound by applying the approximate Response Spectrum Analysis (RSA) procedurewhere the relative magnitudes of modal displacements are estimated in each modewith respect to spectral displacements, which actually represent the peak points ofthe linear modal capacity diagrams, as shown in Figure 2. Peak response quantitiesof interest are then estimated using an appropriate modal combination rule.


It is known that RSA is approximate, but it is the only practical procedure thatever found to replace the linear MDOF time-history analysis. It is the authorsopinion that in a practical nonlinear analysis, i.e., in multi-mode pushover analysis,the same concept is bound to be used in one way or the other in order to avoidthe tedious nonlinear MDOF time-history analysis. However, in nonlinear systemsRSA needs to be implemented in a piecewise linear fashion at each pushover stepin between the formation of two consecutive plastic hinges.

In such an Incremental Response Spectrum Analysis (IRSA) procedure, the firsttask is the scaling of the modal response increments in each mode in such a way thatthe progressive development of the inelastic behavior is appropriately represented.Then the increments of the response quantities of interest including the plastichinge moments can be combined with an appropriate modal combination rule toestimate the response at the next plastic hinge formation. The square-root-of-sum-of-squares (SRSS) rule appears to be the obvious choice for modal combination,although complete quadratic combination (CQC) rule (Chopra, 2001) may be moreappropriate when close modes are present as in the case of coupled lateral-torsionalresponse of three-dimensional systems.

In fact incremental response spectrum analysis approach was the basic ideabehind the adaptive procedure developed by Gupta and Kunnath (2000). Howeverthe most critical issue in a spectral approach is the scaling procedure to be appliedand the selection of the modal response quantity to be scaled. Gupta and Kunnathused the elastic instantaneous spectral accelerations to scale the modal pseudo-accelerations, which in turn are used to define the modal seismic load patterns.However, as pointed out earlier, elastic instantaneous spectral accelerations arenot consistent with the inelastic response. Therefore in the present developmentof IRSA, modal displacement increments are scaled at each pushover step usinginelastic spectral displacements associated with the instantaneous configurationof the system. Such a scaling also permits the consistent estimation of the peakresponse quantities at the last pushover step, which is not possible in other adaptiveprocedures.

The proposed procedure traces the development of inelastic response as theplastic hinges yield sequentially in a process called pushover-history analysis, whichis based on the approximate construction of the modal capacity diagrams of allmodes simultaneously, as described in the following. It is important to note thatIRSA is not restricted to two-dimensional response and it is applicable to three-dimensional (3-D) response of any structure.

4.1. AN APPROXIMATE PROCEDURE FOR PUSHOVER-HISTORY ANALYSISWITH SIMULTANEOUS DEVELOPMENT OF MODAL CAPACITY DIAGRAMS

Based on the approximation made in Eq. 6 leading to the incremental form ofmodal equations of motion given in Eq. 9, the participation of the nth mode to thedisplacement vector defined in Eq. 2b can also be expressed in an incremental form


as

u(i)n = (i)n (i)xnd(i)n (11)

d(i)n = d(i)n d(i1)n (12)where the superscript (i), which represented the instantaneous incremental stepin the time-history analysis formulation now indicates the static pushover step inbetween the formation of two consecutive plastic hinges at (i 1) and (i) dur-ing a pushover-history analysis. For the sake of completeness, the correspondingequivalent seismic load increments in the same mode can be written as

f(i)Sn = K(i)u(i)n = M(i)n (i)xna(i)n (13)in which modal pseudo-acceleration increment, a(i)n , is already defined in Eq. 10.The displacement increment given in Eq. 11 can be appropriately expressed as

u(i)n = u(i)n d(i)n (14)where u(i)n represents the displacement vector due to unit modal displacement in-crement in the nth mode:

u(i)n = (i)n (i)xn (15)Accordingly, participation of the nth mode to the increment of any response quan-tity of interest, such as a story drift or the plastic rotation of a previously developedhinge may be written as

r(i)n = r (i)n d(i)n (16)in which r (i)n refers to the response quantity obtained from u

(i)n .

As mentioned above, the increments of the response quantities can be reason-ably estimated by an appropriate modal combination rule. Utilizing for exampleSRSS rule, the increment of the combined response quantity can be estimated as

r(i) = Ns

n=1(r

(i)n )2 =

Nsn=1

(r(i)n d

(i)n )2 (17)

At this point a critical assumption is to be made to scale the modal displacementincrements, d(i)n . The scaling should be such that it must reflect the progressivedevelopment of the inelastic behavior in the system, but at the same time it mustbe applicable to the piecewise linear pushover steps. Accordingly, referring to themodal capacity diagram of the nth mode shown in Figure 3, it is assumed that thefollowing relationship holds for all modes at the ith pushover step:

d(i)n = F (i)(S(i)din d(i1)n ) (18)


Figure 3. Scaling procedure for a modal displacement increment at the ith pushover step.

where F (i) is a constant scale factor applicable to all modes at the ith pushover stepand d(i1)n denotes the nth modal displacement obtained at the end of the previouspushover step.

In Eq. 18, S(i)din refers to the peak inelastic modal displacement, i.e., inelasticspectral displacement obtained from the solution of Eq. 7 where the hystereticbehavior is represented by the modal capacity diagram corresponding to the in-stantaneous configuration of the plastic hinges at the beginning of the ith pushoverstep. In other words, S(i)din represents the peak inelastic modal displacement as if theprevious hinge at (i 1) were the final hinge yielded in the system, thus reflectingapproximately the progressive development of inelastic behavior in the structure.

The approximate scaling procedure defined in Eq. 18 assumes that at the firstpushover step (i = 1) where d(0)n = 0, modal displacement increments are scaledwith respect to elastic spectral displacements and the scale factor (F (1) < 1) isdetermined at the first plastic hinge formation (Note that in a fully elastic RSA,F (1) = 1). At a typical pushover step (i) after the first step, the origins of thecapacity diagrams may be assumed shifted to (i 1), which corresponds to theprevious hinge, and again the same scaling procedure is actually applied, wherethe modal displacement increments are scaled with respect to the inelastic spectraldisplacements measured from the shifted origins, i.e., S(i)din d(i1)n .

In computing S(i)din, the modal capacity diagram at the ith pushover step is ide-alized as a bilinear diagram for mathematical convenience. As shown in Figure 4,the post-yield slope is taken equal to the eigenvalue, ((i)n )

2, which is calculated for


Figure 4. Bi-linearization of a modal capacity diagram at the ith pushover step.

the ith pushover step after the formation of the last hinge at (i 1). The effectiveyield point coordinates, d(i)yn and a

(i)yn , are then obtained by equating the areas under

the original and bilinear diagrams, which also define the effective initial slope tobe considered in the solution of Eq. 7 as ((i)En)

2 = a(i)yn/d(i)yn . Note that this processdoes not apply to the first two pushover steps, as the modal capacity diagrams arelinear in the first step and already bilinear in the second step.

Instead of using Eq. 18 directly for scaling, it is preferred to express the modaldisplacement increment in a given mode in terms of a reference modal displace-ment increment. Appropriately selecting the first mode as the reference mode,Eq. 18 may be alternatively expressed as

d(i)n = (i)n d(i)1 (19)where (i)n , which may be called inter-modal scale factor, is defined as

(i)n =S(i)din d(i1)n

S(i)di1 d(i1)1

(20)

It is clear that at every pushover step inter-modal scale factor associated with thefirst mode is unity:

(i)1 = 1 (21)

while those of the higher modes are likely to be smaller, however exceptions arepossible.


Now Eq. 19 can be substituted in Eq. 17 to obtain the increment of the combinedresponse quantity of interest in terms of the first modal displacement incrementonly:

r(i) = r (i)d(i)1 (22)in which r (i) is defined as

r (i) = Ns

n=1(r

(i)n

(i)n )2 (23)

It is worth noting that Eq. 23 is actually analogous to the estimation of a responsequantity of interest through a standard response spectrum analysis applied at eachpushover step based on a fictitious pseudo-acceleration spectrum, the ordinate ofwhich is given for the nth mode as

S(i)an,fict = ((i)n )2(i)n (24)

In order to locate the next hinge to develop at the end of the ith pushover step,the general expression given in Eq. 22 is specialized for the bending moment of apotential plastic hinge at joint j:

M(i)j = M(i)j d(i)1 (25)

in which M(i)j is defined through Eq. 23 as

M(i)j =

Nsn=1

(M(i)jn

(i)n )2 (26)

where M(i)jn is the bending moment obtained from u(i)n defined by Eq. 15. At the end

of ith pushover step, the bending moment at the potential hinge location j can becalculated as

M(i)j = M(i1)j +M(i)j = M(i1)j + M(i)j d(i)1 (27)

where M(i1)j represents the bending moment obtained at the end of the previousstep (in the first step it is equal to the bending moment due to gravity loads).When M(i)j reaches the yield moment, M

(y)j , of the plastic hinge, the first modal

displacement increment can be extracted from Eq. 27 as

d(i)1 =

M(y)j M(i1)j

M(i)j

(28)

Since which plastic hinge will develop at the end of the pushover step is not knowna priori, in practical applications d(i)1 is calculated from Eq. 28 for all potential


hinge locations and their minimum value determines the yielding hinge. Note thatthe effect of axial forces on yield moments is omitted in the above derivation forthe sake of simplicity. However, the bending moment-axial force interaction can bereadily considered through a piecewise linear representation of the yield surfaceswith planes in biaxial bending and lines in uni-axial bending. Since the signs of thebending moments and axial forces are lost in the modal combination, in the caseof unsymmetrical yield surfaces, such as in reinforced concrete sections, the signsmay be assumed to be the same as those in the first mode response.

Once d(i)1 is determined, the increment of any response quantity of interest canbe obtained from the general expression given in Eq. 22 and added to the responsequantity obtained at the end of the previous step, i.e.,

r(i) = r(i1) + r (i)d(i)1 (29)Subsequently, modal displacement increments of other modes are obtained fromEq. 19 and the increments of modal pseudo-accelerations are then calculated fromEq. 10. Adding to those calculated at the end of the previous step, the coordinatesof all modal capacity diagrams at the end of the ith pushover step are determinedas

d(i)n = d(i1)n +d(i)n = d(i1)n + (i)n d(i)1 (30a)

a(i)n = a(i1)n +a(i)n = a(i1)n + (i)n ((i)n )2d(i)1 (30b)

4.2. ESTIMATION OF PEAK RESPONSE QUANTITIES

The above-described procedure, which is termed the pushover-history analysis, isrepeated until any of the modal displacements, say, the first modal displacement ob-tained at the end of a pushover step exceeds the inelastic spectral displacement cal-culated for that step. It means that the peak response has been reached somewherewithin this step. When such a step is detected, which is indicated by superscript(p), modal displacements are set equal to the inelastic spectral displacements:

d(p)n = S(p)din (31)and their last increments are calculated from Eq. 30a as

d(p)n = S(p)din d(p1)n (32)which means that modal displacements reach their peaks in all modes simulta-neously with a scale factor of F (p) = 1 (see Eq. 18). Finally the peak responsequantities of interest are obtained from Eq. 29 as

r(p) = r(p1) + r (p)d(p)1 (33)


4.3. SINGLE-MODE ADAPTIVE PUSHOVER ANALYSIS: A SPECIAL CASE OFIRSA

It is known that in low-rise structures regular in plan where the response is ef-fectively governed by the first mode, the single-mode pushover analysis providessatisfactory results (e.g., Krawinkler and Seneviratna, 1998).

The single-mode adaptive pushover analysis can be readily performed as aspecial case of the Incremental Response Spectrum Analysis (IRSA) proceduredeveloped in this paper. Although the response spectrum approach is not relevantto a single-mode response, it is clear that the expressions given for multi-modeIRSA in Eqs. 1133 are equally applicable to the single-mode case when they arewritten for n = 1. Thanks to the selection of the first mode as a reference mode,the inter-modal scale factor defined in Eq. 20 is always unity at all pushover stepsand the response is solely controlled by a single independent parameter, namelythe first modal displacement, d(i)1 . It means that the course of development of themodal capacity diagram, i.e., the pushover-history is independent of the earthquakespecified, which needs to be considered only at the last pushover step to estimatethe peak response quantities according to Eqs. 32, 33.

As in the multi-mode IRSA, the peak response quantities can be estimated inthe single-mode analysis for a given earthquake. The peak value of the first modaldisplacement, i.e., the inelastic spectral displacement, Sdi1, of the equivalent SDOFsystem is obtained from the solution of Eq. 7 for n = 1 based on a bilinear capacitydiagram. When the seismic action is defined through the smooth elastic responsespectrum, which will be considered in the next section, the displacement modifica-tion factor, i.e., C1 coefficient of FEMA 356 (FEMA, 2000) may be appropriatelyused to estimate the inelastic spectral displacement.

As it is known, in the two different implementations of NSP (FEMA, 2000;ATC, 1996), plotting the conventional pushover curve in terms of base shear versusroof displacement is the essential requirement to start with the procedure. In theCapacity Spectrum Method (ATC, 1996), the pushover curve is converted to theso-called capacity spectrum, i.e., the modal capacity diagram of the first mode,through modal conversion parameters already described above. On the other hand,the Displacement Coefficient Method of FEMA 356 (FEMA, 2000) works directlywith the pushover curve, but essentially the same conversion parameters are used todefine the coefficients of the method. On the contrary, the above-described single-mode pushover analysis procedure does not require the conventional pushovercurve be plotted. The modal capacity diagram is obtained directly and it is suf-ficient to estimate the peak response quantities of interest. However if required,roof displacement and base shear increments may be obtained from the generalexpression given in Eq. 22 to plot the conventional pushover curve.

In passing note that in the Capacity Spectrum Method (ATC, 1996) the peakmodal displacement is estimated with a different procedure, which is based on anempirical response of a linear substitute SDOF system represented by its secant


Figure 5. Half-fishbone generic frame.

stiffness and an equivalent viscous damping associated with the hysteretic energydissipation at the peak modal displacement. Although this procedure has been verypopular in the engineering community due to its graphical appeal, comparisonswith the inelastic response spectra have yielded contradictory results (Chopra andGoel, 1999).

4.4. ILLUSTRATIVE EXAMPLE

As mentioned earlier, the Incremental Response Spectrum Analysis (IRSA) proce-dure developed in this paper is applicable to any structure including three-dimen-sional systems provided that appropriate hysteretic models are used. However atthe initial stage of development, the procedure is tested only on two-dimensionalsystems. In the example presented below, a simple half-fishbone generic frameshown in Figure 5 is considered. Such generic frames are occasionally used toapproximate the lateral response of multi-story, multi-bay frames with equal spans(e.g., Nakashima et al., 2002).

The generic frame shown in Figure 5 is derived from the nine-story bench-mark steel building designed for the Los Angeles area as part of the SAC project(Gupta and Krawinkler, 1999). In the present analysis, the basement of the frameis not considered and the first story column is fixed at its base. Beam cross-sectioncharacteristics of the generic frame are the same as the perimeter frames of theSAC building while the story masses are taken equal to one tenth of the total storymasses. In the analysis model, centerline-to-centerline dimensions are consideredfor beams and columns. Rigid-plastic point hinges with an elastic-ideal plastichysteretic behavior are used to represent the nonlinear behavior. Gravity loads and


P-delta effects are neglected in the analysis. The first three elastic natural periodsof the generic frame are calculated as 2.191 s, 0.832 s and 0.484 s.

The analysis is performed for the N-S component of El Centro record of Impe-rial Valley earthquake (1940), which is amplified by a factor of 1.5 to augment theinelastic deformations. Four modes are considered in the analysis with IRSA. Non-linear time-history analysis is also performed using the recently developed analysismodule of SAP 2000 software incorporating the plastic hinge element (CSI, 2002).Rayleigh damping model is used in both types of analysis with 2% damping in thefirst and second linear (initial) modes. Since the damping matrix is updated in SAP2000 according to the instantaneous stiffness matrix, compatible modal dampingratios are calculated for IRSA according to the instantaneous frequencies.

It must be emphasized that the performance of any approximate procedureincluding IRSA under a given individual earthquake heavily depends on the se-lection of earthquake record itself. In this respect the classical El Centro recordis intentionally selected in this paper because of its broad frequency band, leadingto a balanced excitation in different modes. Note that same individual record isalso used by Chopra and Goel (2001) in introducing the Modal Pushover Analy-sis (MPA) procedure. The overall performance of a given procedure can only beverified by statistical studies based on a suite of representative structures and ap-propriately selected earthquake records (e.g., Chopra and Chintanapakdee, 2002).

Figure 6 shows the modal capacity diagrams developed by IRSA for 1.5 timesthe El Centro record. Circles on the diagrams denote the plastic hinges and trianglesindicate the peak modal response points in each mode. For this particular example,the beam plastic hinges are sequentially developed at stories 7, 8, 1, 3, 2, 4, 5, 6, 9.

Peak floor displacements, story drift ratios, beam plastic hinge rotations, storyshears and story overturning moments estimated by IRSA are shown in Figure 7.Plotted on the same graphs are the results obtained from the nonlinear time-historyanalysis (NLTHA). Figure 7 indicates that in this particular example IRSA esti-mates all response quantities of interest with a reasonable accuracy and the errorsare within the acceptable limits for an approximate method. It appears that themain source of error is the application of RSA at each pushover step, which leadsto the errors of similar magnitude or even greater in the purely elastic response. Infact such errors can be seen in Figure 8 where the response quantities obtained forthe same earthquake record from the elastic RSA and linear time-history analysis(LTHA) are compared with those shown in Figure 7. Since no plastic hinge oc-curs in the linear response, the elastic rotations of the rigid beam-column jointsare shown together with the plastic hinge rotations. Note that errors obtained inthe elastic response are implicitly accepted by the engineering profession in thestandard code applications based on RSA.

In Figure 9, the modal capacity diagrams obtained from IRSA are plotted to-gether with those obtained from the Modal Pushover Analysis (MPA) procedure(Chopra and Goel, 2001) using again four modes and the same damping model.Note that in MPA the beam plastic hinges develop in the first mode at stories 3, 1, 4,


Figure 6. Modal capacity diagrams developed by IRSA for 1.5 times the 1940 El Centro (N-S)record.

2, 5, however those hinges do not affect the independent formation of hinges at 8th

and 7th stories in the second mode, as indicated by asterisks in Figure 9 (in the firstmode hinges at 4th and 2nd stories are almost coincident and the asterisks overlap).Fully elastic behavior is observed in the third and fourth modes. Peak responsequantities obtained by both procedures are shown in Figure 10 together with thenonlinear time-history analysis (NLTHA) results. For this particular example, thesuperior performance of IRSA is clearly observed, especially in the estimation ofplastic hinge rotations. Also plotted in Figure 10 are the peak response quantitiesestimated by a single-mode adaptive pushover analysis (single-mode IRSA), whichdemonstrate the significance of the higher modes particularly in the story driftratios and plastic hinge rotations of a nine-story structure.

5. Practical Version of IRSA Using Smooth Elastic Response Spectrum

In the preceding section the Incremental Response Spectrum Analysis (IRSA) pro-cedure is developed and preliminarily tested for a given earthquake ground motion.However the new method is ultimately intended for the practical applications wherethe seismic input is defined through smooth elastic response spectra. In fact the di-rect use of real or simulated ground motion time-histories is still impractical in the


Figure 7. Peak response quantities including floor displacements, story drift ratios, beamplastic hinge rotations, story shears and story overturning moments estimated by IRSA andnonlinear time-history analysis (NLTHA) for 1.5 times the 1940 El Centro (N-S) record.

earthquake engineering practice because of several reasons, such as the difficultiesin specifying the appropriate ground motions, scaling problems, large scatter inresults, difficulties in interpreting the design response quantities, etc. Therefore, astandardized elastic response spectrum is preferable in identifying the earthquakeinput, which ideally suits to the response spectrum analysis (RSA) procedure rec-ommended in all current seismic codes. Since the proposed procedure is essentiallybased on an incremental application of the same approach, a practical version ofIRSA based on smooth response spectrum can be readily developed as explainedin the following.


Figure 8. Peak response quantities obtained from elastic response spectrum analysis (RSA)and linear time-history analysis (LTHA) with their inelastic counterparts obtained from IRSAand NLTHA for 1.5 times the 1940 El Centro (N-S) record (elastic rotations of the rigidbeam-column joints are shown together with the plastic hinge rotations).

5.1. IRSA BASED ON EQUAL DISPLACEMENT RULE

It is well known that inelastic spectral displacements may be estimated by appro-priately modifying the elastic spectral displacements defined through code-basedsmooth response spectrum. In fact this is the practical approach adopted in FEMA356 (FEMA, 2000) as well as by others (e.g., Fajfar, 19992002). The approach isessentially based on the well-known equal displacement rule, which means that thespectral displacement of an inelastic SDOF system and that of the correspondingelastic system are practically equal to each other provided that the effective initialperiod, T (i)En = 2/(i)En (see Figure 4), is longer than the characteristic period ofthe elastic response spectrum. The characteristic period is approximately defined asthe transition period from the constant acceleration segment to the constant velocitysegment of the pseudo-acceleration spectrum. For periods shorter than the charac-


Figure 9. Comparison of modal capacity diagrams of four-mode IRSA and Modal PushoverAnalysis (MPA) for 1.5 times the 1940 El Centro (N-S) record.

teristic period, the elastic spectral displacement is amplified using a displacementmodification factor, i.e., C1 coefficient given in FEMA 356 (FEMA, 2000). In shortperiod range, C1 coefficient is a function of the effective initial period and theyield reduction factor, R, the latter of which is defined as the ratio of the elasticspectral pseudo-acceleration and the effective yield pseudo-acceleration, i.e. a(i)yn inFigure 4.

However it is worth reminding that the equal displacement rule may not bevalid in the case of near-fault records with forward directivity where high amplifi-cations may be observed in inelastic spectral displacements at certain pulse periods.Note that the standard elastic response spectrum defined in FEMA 356 is also notapplicable to the near-fault earthquakes.

With the exception of near-fault records with forward directivity, the equaldisplacement rule may be efficiently exploited in the practical implementation ofIRSA. It is clear that in mid- to high-rise structures, the effective initial periodsof the first few modes are likely to be longer than the above-defined characteristicperiod of the elastic spectrum and therefore those modes automatically qualify forthe equal displacement rule. On the other hand, effective post-yield slopes of thebilinear modal capacity diagrams get steeper and steeper in the higher modes withgradually diminishing inelastic behavior see Figure 6 (Note that in those stiffmodes C1 coefficient of FEMA 356 would not be applicable, since it is based onzero or very small values of post-yield slopes). Therefore it can be comfortably


Figure 10. Comparison of peak response quantities estimated by four-mode IRSA, four-modeMPA, single-mode IRSA and NLTHA for 1.5 times the 1940 El Centro (N-S) record.

assumed that the peak modal displacement response in higher modes would notbe different from the peak elastic response. Hence, the smooth elastic responsespectrum as a whole may be effectively used in IRSA without modification for theinelastic behavior. Accordingly, the inter-modal scale factor, (i)n , defined by Eq. 20can now be modified as

(i)n =S(i)den d(i1)n

S(i)de1 d(i1)1

(34)

where instantaneous inelastic spectral displacement in Eq. 20 is now replaced withS(i)den, which represents the elastic spectral displacement of the nth mode based on

the above-defined effective initial period. The same spectral displacements are usedin Eqs. 31, 32 in place of S(p)din to estimate the peak response quantities of interest.The rest of the implementation of IRSA is the same as described in the previoussection for a given earthquake record.


With the near-fault records excluded, it can be concluded that the equal dis-placement rule and hence the code-specified smooth elastic response spectrum maybe effectively used for the practical version of IRSA, as illustrated in the examplepresented below.

5.2. MONOTONIC SCALING OF THE SMOOTH RESPONSE SPECTRUM

In the above development of the practical version of IRSA, the effective initialperiods are used in estimating the inelastic spectral displacements according tothe equal displacement rule. It has to be admitted that the exact definition of aneffective initial period is always problematic, since bi-linearization is not a well-defined process. As an engineering approach, a further simplification can be madein IRSA where the initial elastic periods obtained at the first pushover step may beused instead of the effective initial periods. In this case the spectral displacementscalculated at the first pushover step can be constantly used in all subsequent steps,i.e.,

S(i)den = S(1)den (i = 2, 3, . . . ) (35)

Accordingly, the basic scaling expression given by Eq. 18 can now be simplified as

d(i)n = F (i)S(1)den (36)where F (i)is a constant scale factor replacing F (i) in Eq. 18, which is again ap-plicable to all modes at the ith pushover step. Consequently, the inter-modal scalefactor (Eq. 20) of each mode becomes constant for all pushover steps throughoutthe pushover-history analysis:

(i)n = (1)n =S(1)den

S(1)de1

(i = 2, 3, . . . ) (37)

Thus it is seen that the use of the initial elastic periods greatly simplifies the ap-plication of the practical version of IRSA. Following Eq. 36, it is further possibleto write a general scaling expression, which is applicable to the cumulative modaldisplacement at the end of the ith pushover step as

d(i)n = F (i)S(1)den (38)where F (i) refers to the cumulative scale factor. Note that Eq. 38 actually representsthe monotonic scaling of the entire smooth response spectrum at each pushover stepas illustrated in the following example.

It is worth noting that the monotonic spectral scaling defined in Eq. 38 maybe regarded analogous to the scaling of an individual earthquake record as appliedin the Incremental Dynamic Analysis (IDA) procedure (Vamvatsikos and Cornell,2002). In other words, when the initial elastic periods are used in the implemen-tation of the equal displacement rule, IRSA can be looked upon as the spectral


Figure 11. Smooth elastic response spectrum with 2% damping.

analog of IDA. Hence it is possible to plot IRSA intensity/demand curves that areanalogous to the IDA curves as illustrated in the following example.

5.3. ILLUSTRATIVE EXAMPLE WITH SMOOTH ELASTIC RESPONSE SPECTRUM

The nine-story generic frame shown in Figure 5 is again considered to illustratethe application of the practical version of IRSA based on smooth elastic responsespectrum. The standard FEMA spectrum with 5% damping is considered wherethe short period and one-second spectral accelerations are taken as SS = 1.1 g andS1 = 0.64 g, respectively. For an application to a steel building, this spectrum isthen modified to a spectrum with 2% damping according to FEMA 356 (FEMA,2000) and plotted in Figure 11.

Figure 12 shows the modal capacity diagrams of the first four modes considered,where the initial elastic periods are used instead of the effective periods in theimplementation of the equal displacement rule. Plastic hinges are again denotedwith circles and the peak modal response points with triangles. Comparison ofFigure 12 with Figure 6 reveals the significant effect of the seismic input on thedevelopment of modal capacity diagrams. In this particular example, all beam-ends yield in sequence at stories 1, 3, 2, 4, 7, 8, 5, 6, 9 and the final yielding occursat the column base, indicating that the system has reached the global mechanismconfiguration. Consequently, the slope of the first-mode capacity diagram in the lastpushover step, i.e., in between the last circle and the triangle in Figure 12 is zero,which is actually equal to the first-mode eigenvalue of the system represented by asingular stiffness matrix. It is interesting to note that the well-known Jacobi methodbased on matrix transformation (Bathe, 1996), which is used in the implementationof IRSA, is able to handle a singular stiffness matrix on a regular basis, resulting ina zero eigenvalue in the first mode. The corresponding rigid mode shape is obtained


Figure 12. Modal capacity diagrams developed by the practical version of IRSA based onsmooth elastic response spectrum.

as a straight-line shape for the half-fishbone system considered. Other deformationmode shapes and the corresponding eigenvalues are calculated as usual.

The application of the equal displacement rule is illustrated in Figure 13. Theabove-defined smooth response spectrum is plotted in Acceleration-DisplacementResponse Spectrum (ADRS) format together with the modal capacity diagrams pre-viously given in Figure 12. The estimation of the modal spectral displacements atthe peak response according to the given spectrum as well the estimations at theformation of the beam plastic hinge at 5th story with a monotonically scaled-downspectrum are shown on the same figure.

Figure 14 shows the profiles of the pushover-histories (dashed lines) and thepeak values (solid lines) of the floor displacements, story drift ratios, beam plastichinge rotations, story shears and overturning moments estimated by IRSA usingthe above-defined elastic response spectrum. Note that the amplifications of storydrifts and plastic hinge rotations at 7th and 8th stories due to higher mode effectsexhibit the same trend observed in Figure 7. Those amplifications disappear inthe single-mode response as shown in Figure 15 where peak response quantitiesgiven in Figure 14 are compared with those obtained from the single-mode adaptivepushover analysis (single-mode IRSA).

Finally, two intensity/demand curves are plotted in Figure 16, which may beinterpreted as the spectral analogs of the IDA curves of the Incremental Dynamic


Figure 13. Illustration of equal displacement rule based on initial elastic periods to estimatemodal displacements associated with peak response and those at the formation of beam plastichinge at 5th story from smooth elastic response spectrum plotted in ADRS format.

Analysis (Vamvatsikos and Cornell, 2002). The vertical axis in both curves indi-cates the so-called ground motion intensity measure (IM), which is defined hereas the monotonic scale factor applied to the smooth response spectrum shown inFigure 11, while the horizontal axes refer to the damage measures (DM), whichare selected as the maximum story drift ratio and the maximum beam plastic hingerotation developed in the structure. These curves can be efficiently used in esti-mating the allowable ground motion intensity level corresponding to an acceptablelevel of a damage measure for a selected performance objective. For example, if0.02 radian maximum plastic hinge rotation is accepted as a performance limit forbeams, the allowable spectrum scale factor is read from the figure as 1.24. Howeverif the performance limit for the maximum story drift is specified as 2%, then theallowable spectrum scale factor is obtained as 0.90.

6. Summary of IRSA

Before concluding the paper it is deemed useful to summarize the analysis stagesof the Incremental Response Spectrum Analysis (IRSA) procedure to be appliedat each pushover step during the pushover-history process. Notes for single-modeanalysis are written in italics.


Figure 14. Pushover-histories and peak values of response quantities including floor dis-placements, story drift ratios, beam plastic hinge rotations, story shears and story overturningmoments estimated by the practical version of IRSA based on smooth response spectrum.

(1) Condense out massless degrees of freedom from the instantaneous (tangent)stiffness matrix modified at the end of the previous pushover step.(2) Run free vibration analysis (preferably use the Jacobi method to handle globalor local mechanisms). Obtain instantaneous eigenvalues with the correspondingeigenvectors and calculate the participation factors for the number of modes con-sidered (Eq. 3). In the case of single-mode analysis, consider the first mode only.(3) In each mode, calculate unit modal response quantities of interest, r (i)n , includingthe bending moments of the potential plastic hinges, M(i)jn , induced by u

(i)n defined

in Eq. 15.(4) Convert each modal capacity diagram to a bilinear diagram according to Fig-ure 4 and calculate the initial effective period. Skip this stage in the first and secondpushover steps (in the first step modal capacity diagrams are linear while in thesecond they are already bilinear). In the case of single-mode analysis both this


Figure 15. Comparison of peak response quantities estimated by four-mode IRSA andsingle-mode IRSA based on smooth response spectrum.

stage and the next stage are optional, which are required only to estimate the peakresponse quantities, see Stage (9). In the practical version of IRSA based on smoothresponse spectrum with initial elastic periods, skip this stage.(5) For each mode calculate spectral displacement either from the solution of Eq. 7for a given earthquake using the bilinear modal capacity diagram or from the spec-ified smooth elastic response spectrum using the initial effective period obtained atStage (4). If the initial elastic period is used in the latter case for simplicity, spectraldisplacements are calculated only once at the first pushover step.(6) Calculate inter-modal scale factors for all modes considered from Eq. 20 orEq. 34 as appropriate. In the practical version of IRSA with initial elastic periods,inter-modal scale factors are calculated from Eq. 37 only once at the first pushoverstep and thereafter constantly used in all steps. In the case of single-mode analysisskip this stage, as the inter-modal scale factor is always unity.


Figure 16. Intensity/demand curves associated with maximum story drift ratio and maximumbeam plastic hinge rotation.

(7) Using the information obtained at Stage (3) and Stage (6) calculate combinedunit response quantities of interest, r (i), including the bending moments of thepotential plastic hinges, M(i)j , from Eq. 23 and Eq. 26, respectively. In the caseof single-mode analysis skip this stage, as these quantities are equal to thosecalculated at Stage (3).(8) Calculate the first modal displacement increment from Eq. 28 and locate theplastic hinge yielded at the end of this pushover step. Then obtain the responsequantities of interest from Eq. 29 and the new coordinates of modal capacity dia-gram(s) from Eq. 30.(9) Check if the first modal displacement exceeded the first-mode spectral displace-ment obtained at Stage (5). If exceeded, calculate the peak response quantities fromEqs. 32, 33 and terminate the analysis. If not, continue with the next stage.(10) Considering the last yielded hinge determined at Stage (8), modify the currentstiffness matrix and return to Stage (1) for the next pushover step.

7. Conclusions

In this paper an attempt was made to develop a new pushover analysis procedure,which is able to consider the multi-mode effects in a practical and theoretically con-sistent manner. In this direction, first the nonlinear time-history response of MDOFsystems was treated through the conventional mode-superposition procedure in apiecewise linear fashion. The aim of this treatment was to show that the uncoupledmodal equations of motion actually exhibit a hysteretic behavior represented bymodal hysteresis loops. Making a reasonable approximation, it was further shownthat the backbone curves of those hysteresis loops could be defined as the modalcapacity diagrams. Those diagrams can be looked upon as the generalization of


first-mode capacity diagram, which is essentially identical to the capacity spectrumdefined in ATC-40 document (ATC, 1996).

The practical construction of the modal capacity diagrams was the basic motivebehind the development of IRSA. This required two critical approximations to bemade. First, the conventional Response Spectrum Analysis (RSA) procedure ap-plicable to linear systems was extended to the nonlinear systems as an incrementalprocedure to be implemented at each pushover step in between the two consecutiveconfiguration of the nonlinear system, or more specifically in between the forma-tion of the two consecutive plastic hinges. The second critical approximation wason the estimation of the relative magnitudes of the modal displacement incrementsat each pushover step. The novel scaling procedure proposed for this purpose in-corporated the inelastic spectral displacements associated with the instantaneousconfiguration of the nonlinear system.

It is important to note that the proposed multi-mode pushover procedure isessentially applicable to three-dimensional response of any structural system. How-ever at the present stage of development the application of IRSA is illustratedon a simple two-dimensional system under a specific earthquake ground motion.The preliminary results appear to be promising in terms of accuracy obtained inestimating the peak values of the main inelastic response quantities, such as lateraldisplacements, story drifts, plastic hinge rotations as well as the peak values of thestory shears and overturning moments. It is clear that the fidelity of the procedureshould be evaluated through statistical studies using different structural systemsand earthquake ground motions.

It needs to be emphasized that the ultimate objective of IRSA was its practicalapplication based on code-specified smooth response spectrum. This was achievedin the last section of the paper, where the well-known equal displacement rule waseffectively utilized to estimate the inelastic spectral displacements. An illustrativeexample was presented for the practical application. It was further shown thatif the initial elastic periods were used instead of the effective initial periods inthe implementation of the equal displacement rule, the practical version of IRSAcould be greatly simplified. In this particular case, the smooth response spectrumis monotonically scalable and IRSA effectively becomes the spectral analog of theIncremental Dynamic Analysis (IDA) procedure.

Finally, it is worth mentioning that the further development of IRSA is under-way, in which the procedure is being extended to include the P-delta effects.

Acknowledgements

The author is indebted to Professor Mustafa Erdik, chairman of Earthquake En-gineering Department of KOERI, for his continuous support and encouragementduring the course of development of the analysis procedure presented in this paper.The author is also thankful to his students Levent zden and Gktrk nem fortheir help in preparing figures and running time-history analysis.


References

Antoniou S., Rovithakis A., Pinho R. (2002) Development and verification of a fully adaptivepushover procedure. Paper No.822, Proceedings of 12th European Conference on EarthquakeEngineering, London.

ATC (1996) Seismic evaluation and retrofit of concrete buildings (ATC-40). Applied TechnologyCouncil, Redwood City, California.

ATC (2002) Evaluation and improvement of inelastic seismic analysis procedures (ATC-55). AppliedTechnology Council, www.atcouncil.org.

Aydnoglu M.N., Fahjan Y.M. (2003) A unified formulation of the piecewise exact method for inelas-tic seismic demand analysis including the P-delta effect. Earthquake Engineering and StructuralDynamics 32(6), 871890.

Bathe K-J. (1996) Finite Element Procedures, Prentice Hall, New Jersey.Chopra A.K. (2001) Dynamics of Structures, second edition, Prentice Hall, New Jersey.Chopra A.K., Chintanapakdee C. (2002) Modal pushover analysis on generic frames. Proceedings of

7th U.S. National Conference on Earthquake Engineering, Boston, Massachusetts.Chopra A.K., Goel R.K. (1999) Capacity-demand-diagram methods based on inelastic design

spectrum. Earthquake Spectra 15(4), 637656.Chopra A.K., Goel R.K. (2001) A modal pushover analysis procedure to estimate seismic demands

for buildings. PEER Report 2001/03, Pacific Earthquake Engineering Center, University ofCalifornia Berkeley.

Clough R.W., Penzien J. (1993) Dynamics of Structures, 2nd Ed., McGraw-Hill, New York.CSI (2002) SAP 2000 Version 8, Analysis reference manual. Computers and Structures, Inc.,

Berkeley, California.Elnashai A.S. (2002) Do we really need inelastic dynamic analysis? Journal of Earthquake

Engineering 6, 123130.Fajfar P. (1999) Capacity Spectrum Method based on inelastic demand spectra. Earthquake

Engineering and Structural Dynamics 28, 979993.Fajfar P. (2002) Structural analysis in earthquake engineering A breakthrough of simplified

non-linear methods. Paper No. 843, Proceedings of 12th European Conference on EarthquakeEngineering, London.

FEMA (1997) NEHRP guidelines for the seismic rehabilitation of buildings (FEMA 273). FederalEmergency Management Agency, Washington D.C.

FEMA (2000) Prestandard and commentary for the seismic rehabilitation of buildings (FEMA 356).Federal Emergency Management Agency, Washington D.C.

Gupta A., Krawinkler H. (1999) Seismic demands for performance evaluation of steel moment re-sisting frame structures. Report No. 132, The John A. Blume Earthquake Engineering Center,Stanford University, California.

Gupta B., Kunnath S.K. (2000) Adaptive spectra-based pushover procedure for seismic evaluation ofstructures. Earthquake Spectra 16(2), 367391.

Krawinkler H., Seneviratna G.D.P.K. (1998) Pros and cons of a pushover analysis of seismicperformance evaluation. Engineering Structures 20(46), 452464.

Nakashima M., Ogawa K., Inoue K. (2002) Generic frame model for simulation of earthquakeresponses of steel moment frames. Earthquake Engineering and Structural Dynamics 31,671692.

Paret T.F., Sasaki K.K., Eilbeck D.H. and Freeman S.A. (1996) Approximate inelastic procedures toidentify failure mechanisms from higher mode effects. Paper No. 966, Proceedings of 11th WorldConference on Earthquake Engineering, Acapulco, Mexico.

Priestley M.J.N. (2000) Performance based seismic design. Paper No. 2831, Proceedings of 12th

World Conference on Earthquake Engineering, New Zealand.


Sasaki K.K., Freeman S.A., Paret T.F. (1998) Multimode pushover procedure (MMP) A methodto identify the effects of higher modes in a pushover analysis. Proceedings of 6th U.S. NationalConference on Earthquake Engineering, Seattle, Washington.

Vamvatsikos D., Cornell C.A. (2002) Incremental dynamic analysis. Earthquake Engineering andStructural Dynamics 31, 491514.

200313

Documents

pushoverbased nsp

displacementbased evaluation

multimode pushover analysis

singlemode response

forcebased design4

procedure canbe

pushoverhistory analysis

smooth response spectrum