10. seismic performance evaluation of steel moment resisting frames through incremental dynamic...

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2006; 35:1853–1873Published online 2 August 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.605

Approximate incremental dynamic analysis using the modalpushover analysis procedure

Sang Whan Han1,‡ and Anil K. Chopra2,∗,†,§

1Architectural Engineering, Hanyang University, Seoul 133-791, Korea2Civil and Environmental Engineering, University of California, Berkeley, CA 94720-1710, U.S.A.

SUMMARY

Incremental dynamic analysis (IDA)—a procedure developed for accurate estimation of seismic demandand capacity of structures—requires non-linear response history analysis of the structure for an ensembleof ground motions, each scaled to many intensity levels, selected to cover the entire range of structuralresponse—all the way from elastic behaviour to global dynamic instability. Recognizing that IDA ofpractical structures is computationally extremely demanding, an approximate procedure based on the modalpushover analysis procedure is developed. Presented are the IDA curves and limit state capacities for theSAC-Los Angeles 3-, 9-, and 20-storey buildings computed by the exact and approximate procedures foran ensemble of 20 ground motions. These results demonstrate that the MPA-based approximate procedurereduces the computational effort by a factor of 30 (for the 9-storey building), at the same time providingresults to a useful degree of accuracy over the entire range of responses—all the way from elastic behaviourto global dynamic instability—provided a proper hysteretic model is selected for modal SDF systems. Theaccuracy of the approximate procedure does not deteriorate for 9- and 20-storey buildings, although theirdynamics is more complex, involving several ‘modes’ of vibration. For all three buildings, the accuracyof the MPA-based approximate procedure is also satisfactory for estimating the structural capacities forthe limit states of immediate occupancy, collapse prevention, and global dynamic instability. Copyrightq 2006 John Wiley & Sons, Ltd.

Received 7 February 2006; Revised 25 May 2006; Accepted 26 May 2006

KEY WORDS: incremental dynamic analysis; modal pushover analysis; performance-based earthquakeengineering

∗Correspondence to: Anil K. Chopra, Civil and Environmental Engineering, University of California, Berkeley,CA 94720-1710, U.S.A.

†E-mail: [email protected]‡Associate Professor.§Horace, Dorothy, and Katherine Johnson Chair.

Contract/grant sponsor: SRC/ERC; contract/grant number: R11-2005-056-04002-0

Copyright q 2006 John Wiley & Sons, Ltd.

1854 S. W. HAN AND A. K. CHOPRA

1. INTRODUCTION

Performance-based earthquake engineering (PBEE) requires accurate estimation of the seismicdemand and capacity of structures. One of the methods that has been proposed to tackle this taskis incremental dynamic analysis (IDA) [1]. This procedure requires non-linear response historyanalyses (RHA) of the structure for an ensemble of ground motions, each scaled to many intensitylevels, selected to cover a wide range of structural response—all the way from elastic behaviour toglobal instability. From the results of such computation, it is possible to determine structural capac-ities (or ground motion intensities) corresponding to various limit states—immediate occupancy(IO), collapse prevention (CP), or global instability (GI).

Recognizing that IDA of practical structures is computationally extremely demanding, thedevelopers of IDA have devised a simplified, approximate method [2]. In this method, a single-degree-of-freedom (SDF) system is defined to approximate the static pushover curve (SPO) fora multi-degree-of-freedom structure. Restricted to first-mode dominated structures, this proceduredefines the force–deformation curve in initial loading of the SDF system to match the SPO curveof the MDF structure, and determines the peak deformation of the non-linear SDF system byempirical equations. Because the SPO curve depends on the choice of the height-wise distributionof forces, SPO curves are determined for several force distributions and the ‘worst’ SPO is chosen,but the choice is not unique. This procedure also requires estimating the elastic stiffness of theSDF system from the IDA curve, because several vibration modes contribute to the responsesplotted in the IDA curve.

Another approach to reduce the computational effort required for IDA is to estimate seismicdemands for the structure by modal pushover analysis (MPA), an approximate procedure [3, 4],instead of non-linear RHA. Thus, each of the many non-linear RHA required in IDA is replacedby a MPA. The objective of this paper is to explore the potential of MPA in estimating the IDAcurves for buildings.

2. INCREMENTAL DYNAMIC ANALYSIS FUNDAMENTALS

The differential equations governing the planar response of a symmetric-plan building to horizontalground motion, ug(t), along one axis of symmetry are

mu + cu + fS(u, sign u) = − miug(t) (1)

where u is the vector of N lateral floor displacements relative to the ground; m and c are themass and damping matrices; and fS(u, sign u) represents the non-linear, hysteretic relation betweenlateral force and displacements. Each element of the influence vector i is equal to unity. Non-linearRHA involves direct solution of these coupled equations to determine response as a function oftime.

The IDA requires a series of non-linear RHA of the structure for an ensemble of groundmotions and each record is scaled to several levels of intensity to encompass the full range ofstructural behaviour: from elastic to yielding that continues to spread, finally leading to globalinstability. The results of these analyses for one ground motion lead to one IDA curve. This isa plot of the ground motion intensity against a seismic demand parameter. The ground motionintensity is characterized by, say, A(T1, 5%), the spectral pseudo-acceleration corresponding to thefirst-mode elastic vibration period and 5% damping ratio. The demand parameter may be �roof,

Copyright q 2006 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2006; 35:1853–1873DOI: 10.1002/eqe

APPROXIMATE INCREMENTAL DYNAMIC ANALYSIS 1855

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Figure 1. IDA curves and limit-state capacities for SAC-Los Angeles 9-storey building: (a) IDA curvesfor 20 ground motions; and (b) fractile IDA curves.

the peak roof drift ratio, defined as the roof displacement divided by building height, or �max, themaximum over all stories of the peak interstorey drift ratio, defined as the storey drift divided bythe storey height.

Shown in Figure 1(a) are the IDA curves for the SAC-Los Angeles 9-storey building for anensemble of 20 ground motions. From these data, the 16, 50, and 84% fractile values of the demandgiven the ground motion intensity are computed to obtain these fractile IDA curves in Figure 1(b).

Limit states can be defined on the IDA curves [5], as shown in Figure 1. Immediate occu-pancy (IO) is violated when �max exceeds 2% and CP is reached when the local slope of theIDA curve is 20% of the slope of the IDA curve in the elastic range or �max = 10%, whicheveroccurs first, i.e. at lower intensity. Finally, global dynamic instability (GI) is identified by char-acteristic flattening of each IDA, termed the flatline, where the seismic demand increases greatlywith the slightest increase in ground motion intensity. The 16, 50, and 84% fractile values ofstructural capacity corresponding to these limit states are identified in Figure 1(b). Combin-ing these intensity measure (IM) capacities with probabilistic seismic hazard analysis [1, 6],the mean annual frequencies of exceeding each limit state can be computed, one of the goalsof PBEE.

To minimize the number of non-linear RHAs necessary to generate an IDA for a singleground motion, the hunt-and-fill-tracing algorithm has been developed [1]. With this algorithmit was possible to generate one IDA curve by 12 non-linear RHAs corresponding to 12 groundmotion intensity levels. An ensemble of 20 ground motions then required 240 non-linear RHAsof the structure, which is an onerous task for actual buildings with hundreds, even thousandsof structural elements. For example, considering 12 excitation intensity levels, the calculation ofthe full, 20-record IDA for one frame of the SAC-Los Angeles 9-storey building requiresabout 5 h of computing on a single Pentium 4 processor, with 3.0GHz of CPU and DDR512MB RAM.



3. MODAL PUSHOVER ANALYSIS FUNDAMENTALS

In the MPA-based approximate procedure to determine IDA curves, the MPA procedure is usedto estimate seismic demands due to each ground motion at each intensity level instead of non-linear RHA. Although modal analysis theory is strictly not valid for inelastic systems, the factthat elastic modes are coupled only weakly in the response of inelastic systems [3, 4] permitteddevelopment of the MPA procedure. The MPA procedure provides a computationally efficient,although approximate, alternative to non-linear RHA. In MPA, the effective earthquake forces [seeEquation (1)],

peff(t) = − miug(t) (2)

are expanded into their modal components. This spatial (height-wise) distribution of the effectiveearthquake forces over the building is defined by the vector s≡mi and their time variation byug(t). This force distribution can be expanded as a summation of modal inertia force distributionssn [7, Section 13.2]:

s=N∑

n = 1sn, sn ≡ �nm/n (3)

where / n is the nth-mode of natural vibration and �n =/Tnmi//Tnm/n . Thus,peff,n(t) = − snug (t) (4)

is the nth-mode component of effective earthquake forces.In the MPA procedure, the peak response of the building to peff,n(t)—or the peak ‘modal’

demand rn—is determined by a non-linear static or pushover analysis using the modal forcedistribution s∗n =m/n [based on Equation (3b)] at the peak roof displacement urn associatedwith the nth-mode inelastic SDF system. The peak modal demands rn are then combined byan appropriate modal combination rule to estimate the total demand. This procedure is directlyapplicable to the estimation of deformation demands (e.g. floor displacements and storey drifts).

The MPA procedure has been described in a convenient step-by-step form [3, 8]. This approx-imate procedure has been shown to estimate seismic demands to a useful degree of accuracyfor the SAC 9- and 20-storey buildings, generic frames (vertically ‘regular’ as well as vertically‘irregular’) of height varying from 3 to 18 stories [8–10].

Based on structural dynamics theory, the MPA procedure is computationally attractive becauseit avoids non-linear RHA of the structure. Instead, computing each modal demand rn requiresone non-linear static analysis of the structure and a non-linear RHA of a ‘modal’ SDF system;and ‘modal’ demands need to be determined only for the first few (generally 2 or 3) ‘modes’of the structure. Because the MPA procedure leads to a unique SPO for each mode, it by-passes the search for the ‘worst’ SPO mentioned earlier [2]. Furthermore, the elastic stiffnessof the force–deformation curve for the modal SDF system is uniquely defined as the modal fre-quency squared, thus avoiding the complications in the simplified IDA procedure mentioned inReference [2].

In applying MPA to obtain IDA curves for all fractiles, an nth-mode pushover analysis of thestructure is implemented only once. The resulting database provides all the response informationneeded to estimate seismic demands due to any ground motion scaled to any intensity level. The



‘modal’ response is extracted from this database at the roof displacement urn due to the selectedground motion at the selected intensity level.

4. BUILDINGS, GROUND MOTIONS, AND RESPONSE STATISTICS

The SAC commissioned three consulting firms to design 3-, 9-, and 20-storey model buildingswith symmetric square plan according to the local code requirements of three cities: Los Angeles,Seattle, and Boston. Described elsewhere in detail [11], the structural systems of these modelbuildings consist of special moment-resisting frames (SMRF) along the plan perimeter and interiorgravity frames; the N–S perimeter frames of the three buildings designed for Los Angeles, areused as examples in this paper. The frame is idealized by the M1 model, a basic centreline modelin which the panel zone size, stiffness, and strength are not presented [11]. Plastic hinges atthe beam (or column) ends are modelled by bilinear, kinematically hardening moment-rotationsprings with post-yield stiffness equal to 3% of the initial stiffness, consistent with the resultsof the tests on WUF-B connections [11, 12] P–� effects due to gravity loads are included inthe analysis.

The first three natural periods of the structures vibrating within the elastic range are: 1.03, 0.33,and 0.17 s for the 3-storey building; 2.34, 0.88, and 0.50 s for the 9-storey building; 3.98, 1.36, and0.79 s for the 20-storey building. The damping matrix in Equation (1) is chosen as c= a0m+ a1k,where k is the initial elastic stiffness matrix for the building and the constants a0 and a1 aredetermined from specified damping ratios at two periods. For 3- and 9-storey buildings, dampingratios of 2% are specified at the first-mode period and at 0.2 s. For the 20-storey building, dampingratios of 2% are specified at the first and fifth-mode periods [11].

An ensemble of 20 ground motions was selected. Listed in Reference [2], these motions wererecorded on firm soil, during three earthquakes of M 6.5–6.9 (Loma Prieta, 1989; SuperstitionHills, 1987; and Imperial Valley, 1979) at distances ranging from 15 to 32 km. Thus, no near-faultmotions with directivity effects are included.

The dynamic response of each structural system to each of the 20 ground motions scaled tothe selected intensity A(T1, 5%) was determined by two procedures: non-linear RHA and MPA.The seismic demand parameters monitored were �max and �roof. For a specified ground motionintensity A(T1, 5%), the fractile values (16, 50, and 84%) of �max and �roof were computed bythe counting method. For this purpose, the 20 data values were sorted in ascending order. Themedian is estimated as the average of 10th and 11th values starting from the lowest value, the16% fractile is given by the 5th value, and the 84% fractile is given by the 17th value. Plots ofA(T1, 5%) against the fractile value of �max and �roof provide the three IDA curves. Using thedemand data computed by non-linear RHA or MPA gives the exact or approximate IDA curves,respectively.

5. MODAL SDF SYSTEMS

As mentioned earlier, computing each modal demand rn requires two analyses: (1) non-linearstatic analysis of the building subjected to lateral force distribution s∗n =m/n , leading to the baseshear-roof displacement, Vbn − urn , pushover curve; and (2) non-linear RHA of the nth-modeinelastic SDF system.



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Figure 2. Static pushover curves for the first three natural vibration ‘modes’ for SAC-Los Angeles buildings:(a) 3-storey building; (b) 9-storey building; and (c) 20-storey building.

5.1. Pushover curves

Figure 2 shows the pushover curves for the first few modes of each of the three selected buildings.Observe that the post-yield slope of the first-mode pushover curves is negative because of P–�effects. The force–deformation relation for initial loading of the modal SDF system is determinedby appropriately scaling both axes of the idealized modal pushover curve and plotting Vbn/M∗

nversus urn/�n�rn , where M∗

n is the effective modal mass and �rn is the value of /n at the roof[3, 4].

5.2. Selection of force–deformation model

Several models are available to describe the hysteretic behaviour of steel moment resisting frames(SMRFS) for unloading and reloading during oscillatory response. The three models consideredare: (1) a bilinear hysteretic model, (2) trilinear hysteretic model; and (3) strength-limited bilinearhysteretic model [13]; see Appendix A. Note that the third model is called bilinear, althoughthe initial loading curve may be trilinear because of the strength limit. The force–deformationbehaviour of the modal SDF systems during initial loading, determined from the pushover curveof Figure 2, is idealized using these three models, and the three hysteretic models are comparedwith the cyclic behaviour of the building determined by non-linear static analysis of the buildingsubjected to lateral force distribution s∗1.



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Excitation: 2.5 x LP89-SCA [A(T1, 5%) = 0.53g]

Excitation: 5xLP89-SCA [A(T1, 5%) = 1.05g]

Time, sec

-5

10

-10

10 20 30 40 10 20 30 40 10 20 30 40

(a) (b) (c)

Figure 3. Roof-drift history of 9-storey building due to LP89-SCA ground motion scaled by three factors,computed by two methods: non-linear RHA of the building denoted as ‘exact’ and one-mode MPAdenoted as ‘approximate’, using three different SDF-system hysteretic models: (a) bilinear; (b) trilinear;

and (c) strength-limit bilinear.

Because the objective is to estimate the IDA curves, the selected model should be capable ofcomputing to a satisfactory degree of accuracy the structural responses over a wide range—allthe way from elastic behaviour to global dynamic instability. To identify the most appropriatemodel, Figure 3 presents the roof-drift history for the ground motion recorded at Sunnyvale ColtonAvenue during the 1989 Loma Prieta earthquake (LP89-SCA), scaled by factors 1.0, 2.5, and 5.0to three different intensity levels: A(T1, 5%) = 0.21g, 0.53g, and 1.05g. For each intensity level,the approximate results obtained by MPA considering only the first mode are compared with theexact results obtained by rigorous non-linear RHA of the building.

Figure 3 demonstrates that all three SDF system models can satisfactorily predict the roof-displacement history of a building to the LP89-SCA ground motion; however, as the intensityof the excitation increases, the accuracy of the bilinear hysteretic model starts to deteriorate.Its estimate of response to 2.5×LP89-SCA ground motion is unsatisfactory. At this intensitylevel, the trilinear hysteretic model is fairly accurate in estimating the response. But its accuracy



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Roof drift ratio, % (a) (b) (c) (d)Roof drift ratio, % Roof drift ratio, % Roof drift ratio, %

Figure 4. Base shear versus roof drift plots determined by four analyses: one-mode MPA using threehysteretic models: (a) bilinear; (b) trilinear; (c) strength-limited bilinear; and (d) non-linear RHA of the

building. Excitation is LP89-SCA ground motion scaled by a factor of 1.0.

V/W

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(a) (b) (c) (d)Roof drift ratio, % Roof drift ratio, % Roof drift ratio, %



begins to deteriorate with further increase of excitation intensity and becomes unsatisfactory for5.0×LP89-SCA ground motion. The strength-limited bilinear hysteretic model turns out to be thesuperior model over the entire range of excitation intensity.

To understand better the reasons underlying these observations, Figures 4–6, plot the base shearagainst roof drift, as determined by four analyses: MPA procedure considering only the first modeusing the three hysteretic models and rigorous non-linear RHA of the building. Comparing thefour sets of results reveals that for A= 0.21g, non-linear RHA reveals that the building respondsessentially within the elastic range (Figure 4(a)) and all three hysteretic models used in MPA areadequate (compare Figures 4(a)–(c) with (d)). For A= 0.53g, the bilinear model for the first-modeSDF system (Figure 5(a)) is unable to follow the hysteretic behaviour of the building (Figure 5(d)),but the trilinear and strength-limited bilinear models seem adequate (compare Figures 5(b)–(c) with5(d)). However, for A= 1.05g, hysteretic behaviour of only the strength-limited bilinear model(Figure 6(c)) is close to the ‘exact’ result from non-linear RHA (Figure 6(d)).

Based on the preceding discussion of the results presented in Figures 3–6, it is concluded that,if plastic hinges at the beam (or column) ends are modelled by bilinear, kinematically hardeningmoment-rotation springs, the strength-limited bilinear hysteretic model best replicates the hystereticbehaviour of the building and is expected to be most accurate in estimating IDA curves. This



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MPA - Based IDA : 3 models

Bilinear

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Figure 7. Fifty per cent fractile IDA curves determined by MPA-based approximateprocedure using three SDF-system hysteretic models—bilinear, trilinear, and strength-

limited bilinear—versus exact non-linear RHA.

expectation is confirmed in Figure 7 that presents the 50% fractile IDA curves computed by usingthese three hysteretic models for modal SDF systems in the MPA-based approximate procedureand compares them with the exact IDA curve. Both the bilinear hysteretic, and trilinear hystereticmodels are inadequate in estimating the IDA curve, whereas the strength-limited bilinear hystereticmodel results in an IDA curve that is in good agreement with the exact IDA curve. Therefore, this



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Figure 8. First-mode static pushover curves and their idealization for three buildings: (a) 3-storey building;(b) 9-storey building; and (c) 20-storey building.

is the hysteretic model selected for modal SDF systems in all MPA analyses presented in the restof this paper.

Figure 8 shows the first-mode pushover curve and its idealized version used in the final analysisof the three buildings. The idealized pushover curve is bilinear with a negative post-yield slopefor the 3-storey building, but trilinear with a positive slope for the first branch in the post-yieldrange, and a negative slope for the second branch in the post-yield range for the 9- and 20-storeybuildings.

6. EVALUATION OF MPA

6.1. Higher mode contributions in IDA curve

Figures 9–11 compare the MPA-based approximate IDA, including contributions of a variablenumber of modes and the exact IDA for 3-, 9-, and 20-storey buildings, respectively. Each figureis presented for two demand parameters: (a) �roof and (b) �max, and each part contains results for16, 50, and 84% fractile IDA curves.

These results demonstrate that, as expected, the higher-mode contributions to the approximateIDA are small for the 3-storey building, and they increase progressively for the 9- and 20-storeybuildings. For a given building, these higher-mode contributions are smallest for the 16 percentileIDA, and progressively increase for the 50 and 84 percentile cases.



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1 mode2 modes3 modes

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84% IDA

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16% IDA

Figure 9. Sixteen, 50, and 84% fractile IDA curves for 3-storey building from MPA, including variablenumber of modes, versus exact IDA from non-linear RHA.

However, the higher mode contributions in the roof drift and the maximum interstorey drift arenot especially significant for the 9-storey building or even for the 20-storey building with someexceptions in the latter case. To understand this result, Figure 12 plots the 50 percentile floordisplacements and storey drifts over the height of the 9-storey building, corresponding to groundmotion intensities defined by A= 0.3g and 0.7g. It is clear that the higher mode contributions havelittle influence on the one-mode estimate of the roof displacement (and, hence, roof drift ratio) andthe maximum interstorey drift, which occurs in the first storey for A= 0.7g and in the second storeyfor A= 0.3g. This does not necessarily imply that the MPA-based result including several modesis always close to the exact IDA. For example, the approximate result for maximum interstoreydrift is quite accurate for A= 0.3g but not for A= 0.7g (Figure 12). Furthermore, higher modesmay be important if a different engineering demand parameter is selected or the limit states werebased on a damage index that considers the inelastic action throughout the building.

It should be noted, however, that as demonstrated earlier [8], the one-mode result is grosslyinaccurate in estimating the storey drifts in upper stories of this building, and including higher-mode contributions improves significantly the results (Figure 12). To estimate the storey drifts inupper stories, it would be necessary to include the first three modes in MPA, consistent with earlierresults for this building for a different ensemble of ground motions [8].



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6.2. Accuracy of MPA-based IDA curves

Figures 13–15 present IDA curves corresponding to 16, 50, and 84% fractiles for the 3-, 9-, and 20-storey buildings. The approximate IDA curves estimated by the MPA-based approximate procedureincluding three modes is presented together with the exact IDA curves determined by non-linearRHA. The preferred choice for ground motion intensity is A(T1, 5%), the pseudo-acceleration spec-tral ordinate at the fundamental vibration period, T1, of the structure in its elastic range for 5% damp-ing ratio. Figures 13 and 14 present IDA curves for 3- and 9-storey buildings using this intensitymeasure, wherein the authors computed both the exact and approximate results. However, Figure 15,which presents the IDA curves for the 20-storey building, uses a different intensity measure: the trueacceleration spectrum ordinate at period T1. The exact IDA results by Vamvatsikos and Cornell [2]were available for this intensity measure against which the approximate MPA-based IDA results arecompared.

Comparison of the curves demonstrates that the MPA-based approximate IDA curve is fairlyaccurate over the entire range of drift ratios, even close to collapse. There seems to be no systematicbias in the approximate procedure; i.e. it does not always underestimate or overestimate capacities(or demands). This set of results indicates that the approximate procedure is generally less accurate



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84% IDA

15 20 1510 20 0 5 1510 20


in estimating IDA curves plotted against maximum interstorey drift ratio compared to the roof driftratio. For a given structure, as expected, the approximate procedure is more accurate in estimatingthe 50% fractile curve compared to the 16 and 84% fractile curves. Among the three buildings, theaccuracy of the MPA procedure is about the same, i.e. it does not deteriorate significantly forthe 9- and 20-storey buildings where higher-mode contributions to structural response are moresignificant.

The MPA-based approximate procedure, which provides good estimates of IDA curves, requiresa small fraction of the computational effort compared to that required in the ‘exact’ procedurethat uses non-linear RHA to compute demands. If a Pentium 4 processor with 3.0GHz CPU andDDR 512MB RAM is used the computer time for analysis of one frame of the SAC Los Angeles9-storey building is reduced from 5 h for the exact result to only 10min for the approximateprocedure including three modes. Thus, a fast estimate of the IDA curves for multistorey buildingsis achieved at only a small loss in accuracy.

6.3. Limit state capacities

The structural capacities, defined in terms of the IM, are calculated for the limit states (IO, CP,and GI) defined earlier, directly on the fractile IDAs instead of estimating them individually on



1

2

3

4

5

6

7

8

9


Floor Displacement / Building Height (%)

1

2

3

4

5

6

7

8

9

0 2 4 6

Story Drift ratio (%)

Flo

or

Ground0 1 2 3 0 2 4 6

A = 0.3g

ExactMPA

A = 0.7g

A = 0.3g A = 0.7g

Flo

or

Ground0tory drift ratio( 3 6s) 9r

(a)

(b)

Figure 12. Median floor displacements and storey drifts for 9-storey building from MPA, includingvariable number of modes, versus exact non-linear RHA. Excitations are LP89-SCA ground motion

scaled to A(T1, 5%)= 0.3g and 0.7g: (a) floor displacements; and (b) storey drifts.

the IDA for each ground motion and then determining the fractiles. This approach has beendemonstrated to be theoretically correct for IO and GI limit states [2], and the intensity measure(IM) values corresponding to �max = 0.02 and ∞ are determined from the fractile IDA curves.However, the CP limit-state points do not lie on the fractile IDAs (see Figure 1(b)), but theyare close to the IDA curves [2]. Therefore, the fractile point corresponding to the CP limit stateis estimated as the lower of the two values of IM, the first corresponding to the point wherethe tangent stiffness of the IDA is 20% of the elastic stiffness, and the second corresponding to�max = 10%.

Table I presents the structural capacities defined in terms of the selected IM [A(T1, 5%) for3- and 9-storey buildings, but true spectral acceleration at (T1, 5%) for the 20-storey building].



0 5 10 15 200

1

2

3

Peak roof drift ratio, � roof (%)

Firs

t-m

ode

pseu

do a

ccel

erat

ion,

A(T

1, 5

%)/

g

2.5

1.5

0.5


Exact IDA

MPA-based IDA

16% IDA

50% IDA

84% IDA

0 5 10 15 20

50% IDA

16% IDA

84% IDA

Figure 13. Sixteen, 50, and 84% fractile IDA curves for 3-storey building from non-linear RHA (exact)versus MPA-based approximate procedure.


Firs

t-m

ode

pseu

do a

ccel

erat

ion,

A(T

1, 5

%)

0

0.5

1

1.5

2

0 5 10 15


Firs

t-m

ode

pseu

do a

ccel

erat

ion,

A(T

1, 5

%)

/g

20 0 5 10 15 20

16% IDA

50% IDA

84% IDA

Exact IDA

MPA-based IDA

16% IDA

50% IDA

84% IDA


Presented are two values of structural capacity, one determined from the exact IDA curves and theother from MPA-based approximate IDA curves.

The accuracy of the MPA-based approximate method is satisfactory for a range of buildingheights and for each of the three limit states. Even for the 20-storey building, with its complexdynamics due to significant responses in higher modes, the MPA results are remarkably accurate.For all three buildings, they are more accurate than the approximate results from the simplifiedprocedure in Reference [2] corresponding to the three limit states. Because of the approximationsunderlying this procedure, the developers of this procedure recognized its limitation in estimating



0 5 10 15 20

16 % IDA

0 2 4

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Firs

t mod

e tr

ue s

pect

ral a

ccel

erat

ion

/ g


84% IDA

50% IDA

16% IDA

Exact IDA

MPA-based IDA


6


Table I. Exact and approximate values of 16, 50, and 84% fractile IM capacities corresponding to threelimit states for three buildings.

16% (g) 50% (g) 84% (g)

Building Limit state Exact Approx. Exact Approx. Exact Approx.

IO 0.48 0.47 0.58 0.56 0.74 0.703 Storey CP 1.06 0.90 1.20 1.40 2.42 2.25

GI 1.28 1.20 1.60 1.60 2.98 2.85

IO 0.14 0.22 0.24 0.26 0.28 0.309 Storey CP 0.70 0.66 1.00 1.02 1.38 1.34

GI 0.76 0.72 1.26 1.26 1.50 1.44

IO 0.12 0.10 (0.08)∗ 0.16 0.12 (0.10) 0.21 0.17 (0.15)20 Storey CP 0.23 0.26 (0.22) 0.34 0.33 (0.35) 0.53 0.51 (0.57)

GI 0.26 0.29 (0.26) 0.39 0.35 (0.40) 0.63 0.56 (0.61)

∗Data in parentheses is from Reference [2].

the IM capacity for the IO limit state. The MPA-based approximate procedure overcomes thislimitation because it included all significant ‘modes’ of vibration.

An alternative format, as in FEMA-350, is to express the seismic demand at the capacity point[14]. For IO and GI limit states, by definition the interstorey drift is 0.02 and ∞, respectively.For the CP limit state, the values of �max are listed in Table II, which demonstrate that theMPA-based approximate method provides results that in most cases are accurate to a usefuldegree.



Table II. Exact and approximate values of 16, 50, and 84% fractile �max capacities corresponding tocollapse prevention (CP) limit state for three buildings.

16% 50% 84%

Building Limit state Exact Approx. Exact Approx. Exact Approx.

3 Storey CP 0.10 0.05 0.06 0.10 0.10 0.099 Storey CP 0.10 0.10 0.10 0.10 0.10 0.1020 Storey CP 0.05 0.08 (0.10) 0.06 0.07 (0.10)∗ 0.07 0.07 (0.10)

∗Data in parentheses is from Reference [2].

0 5 10 15 200.0

0.1

0.2

0.3

0.4

0.5

0.6

0 4 6

Firs

t mod

e tr

ue s

pect

ral a

ccel

erat

ion

/ g

0.7

2Maxmimum interstorey drift ratio, � max (%)Peak roof drift ratio, �roof (%)

16% IDA

50% IDA

84% IDA

Exact IDA

MPA-based IDA

SPO2IDA

16% IDA

50% IDA

84% IDA

Figure 16. Sixteen, 50, and 84% fractile IDA curves for 20-storey building determined by MPA-basedand SPO2IDA-based approximate procedures [2] versus exact non-linear RHA.

6.4. Comparisons of MPA with SPO2IDA approach

The accuracy of the MPA-based approximate procedure is compared against the SPO2IDA-basedsimplified procedure presented in Reference [2]. This comparison for the 20-storey building ispresented in Figure 16 and Tables I and II. These data demonstrate that the MPA-based approximateprocedure generally gives superior results. A similar comparison was not possible for the 3- and9-storey buildings because, as mentioned earlier, the intensity measure selected herein is differentthan the one used in Reference [2].

7. CONCLUSIONS

Recognizing that IDA of practical structures is computationally extremely demanding, this attemptto develop an approximate method that requires much less computational effort has led to thefollowing conclusions:

1. Estimating seismic demands for the structure by MPA, an approximate procedure, instead of‘exact’ non-linear RHA leads to a highly efficient procedure. The MPA-based approximate



IDA procedure requires a small fraction (roughly 3%) of the computational effort compared tothat required in the ‘exact’ procedure. If a Pentium 4 processor with 3.0GHz CPU and DDR512MB RAM is used, the computer time required for one frame of the SAC-Los Angeles9-storey building is reduced from 5 h for the exact result to only 10min for the approximateresult.

2. The MPA-based approximate IDA curve is fairly accurate over the entire range of drift ratio,even close to collapse; there seems to be no systematic bias in the procedure. Among thethree buildings considered, the accuracy of the MPA procedure is about the same, i.e. it doesnot deteriorate for the 9- and 20-storey buildings with their more complex dynamics involvingsignificant contributions from several ‘modes’. For a given structure, the approximate procedureis more accurate in estimating the 50% fractile curve, compared to the 16 and 84% fractilecurves.

3. For all three buildings, the accuracy of the MPA-based approximate procedure is satisfactoryalso for estimating structural capacities for the immediate occupancy, collapse prevention, andglobal instability limit states.

4. To obtain accurate results by the MPA-based approximate procedure, it is important to select anappropriate model for the hysteretic behaviour of the modal SDF systems that best replicatesthe hysteretic behaviour of the building, which obviously depends on how its plastic hingebehaviour is idealized. If the plastic hinges at the beam (or column) ends are modelled by bilin-ear, kinematically hardening, hysteretic moment-rotation springs, the strength-limited bilinearhysteretic model is the superior model over a wide range of excitation intensities.

5. Surprisingly, the higher mode contributions in the roof drift (or displacement) and the maximuminterstorey drift, the demand parameters conventionally chosen for constructing IDA curves,are not especially significant for the 9-storey building or even the 20-storey building. Thus,the IDA curves can be estimated by including very few ‘modes’, in many cases only onemode, in the MPA-based approximate procedure. It should be noted, however, that the one-mode results are seriously inaccurate in estimating the storey drifts in upper stories of tallerbuildings [8].

APPENDIX A

Three different hysteretic models are considered to represent dynamic behaviour of SAC steelbuildings using approximate SDF systems; bilinear model, trilinear model, and strength-limitedbilinear model.

The bilinear model (Figure A1) has been often used to idealize pushover curves for steel momentframes. This model can be identified using two parameters: elastic stiffness k and post-elasticstiffness �k, the latter can be either positive or negative. The trilinear model has essentially thesame properties as the bilinear model except that this model is defined by three different stiffnessvalues for the three segments, as shown in Figure A2. Neither of these models can represent cyclicdeterioration in strength or stiffness.

However, strength and stiffness generally deteriorate with the number and amplitude of cyclesduring the cyclic tests. For simulating such behaviour, Ibarra et al. [15] developed three differenthysteretic models; strength limited bilinear model, peak-oriented model, and pinching model. Asshown in Figure A3(a), the first two branches of the initial loading curve are defined by threeparameters: elastic initial stiffness ke, yield strength Vy , and strain-hardening stiffness �ske. The



Figure A1. Bilinear hysteretic model.

Figure A2. Trilinear hysteretic model.

(a) (b)

Figure A3. (a) Initial loading curve on the left; and (b) strength-limited bilinear model on the right(adapted from Reference [15]).

softening branch is defined by stiffness �cke, peak strength Vc, and corresponding deformation uc.After defining the initial loading curve, the hysteretic model and its parameters should be selectedthat best represent cyclic deterioration in strength and stiffness.



Because Ibarra et al. [15] showed that strength-limited bilinear model best predicts the cyclicresponse of the beam–column subassemblies of steel moment frames, this model was selected forinelastic modal SDF systems in this investigation. The strength-limited bilinear model denotes ahysteretic model with kinematic strain hardening. In this model, ‘bilinear’ refers to the standardbilinear model with kinematic strain hardening, but not to the number of segments used to model theinitial loading curve. As shown in Figure A3(b), the strength limit is enforced if the initial loadingcurve includes a softening branch with negative slope. Thus, in the strength-limited bilinear model,cyclic deterioration is represented by the strength limit and the softening branch. For example, theloading segment starting at Point 5 continues up to Point 6 at the strength limit, not Point 6′. Notethat Point 6 is the limit corresponding to the strength attained in earlier cycles defined by Point 3.Without imposing the strength limit, the strength could continue to increase in later loading cyclesNote that this model does not account for the cyclic deterioration in stiffness. More details on thismodel can be found in References [13, 15].

ACKNOWLEDGEMENTS

Several individuals provided assistance that was important for this investigation. Rakesh K. Goel providedthe DRAIN-2DX input files for the buildings analysed; Dimitrios Vamvatsikos provided the 20 groundmotions used in this investigation, as well as his ‘exact’ IDA data for the 20-storey building and offeredcomments on a final draft of the paper; and Luis F. Ibarra provided the software for dynamic analysis ofstrength-limited bilinear hysteretic SDF systems. Eungsoo Kim, a graduate student at Hanyang University,Seoul, Korea, assisted in the massive computing effort. After the research had been completed and thepaper submitted, we became aware of related work by Dolsek and Fajfar [16]. Finally the first authoracknowledges the financial support provided by SRC/ERC (R11-2005-056-04002-0) during this visit tothe University of California, Berkeley.

REFERENCES

1. Vamvatisikos D, Cornell CA. Incremental dynamic analysis. Earthquake Engineering and Structural Dynamics2002; 31:491–514.

2. Vamvatisikos D, Cornell CA. Direct estimation of seismic demand and capacity of multi-degree of freedomsystems through incremental dynamic analysis of single degree of freedom approximation. Journal of StructuralEngineering (ASCE) 2005; 131(4):589–599.

3. Chopra AK, Goel RK. A modal pushover analysis procedure for estimating seismic demands for buildings.Earthquake Engineering and Structural Dynamics 2002; 31:561–582.

4. Chopra AK, Goel RK. A modal pushover analysis procedure to estimate seismic demands for unsymmetric-planbuildings. Earthquake Engineering and Structural Dynamics 2004; 33(8):903–928.

5. Vamvatisikos D, Cornell CA. Applied incremental dynamic analysis. Earthquake Spectra 2004; 20(2):523–533.6. Krawinkler H, Miranda E. Performance-based earthquake engineering. In Earthquake Engineering: From

Engineering Seismology to Performance-Based Engineering, Bozorgnia Y, Bertero VV (eds). CRC Press: BocaRaton, FL, 2004.

7. Chopra AK. Dynamics of Structures: Theory and Applications to Earthquake Engineering (2nd edn). Prentice-Hall:New Jersey, 2001.

8. Goel RK, Chopra AK. Evaluation of modal and FEMA pushover analyses: SAC buildings. Earthquake Spectra2004; 20(1):225–254.

9. Chopra AK, Chintanapakdee C. Comparative evaluation of FEMA and modal pushover analyses: vertically regularand irregular generic frames. Earthquake Spectra 2004; 20(1):255–271.

10. Chintanapakdee C, Chopra AK. Seismic response of vertically irregular frames: response history and modalpushover analyses. Journal of Structural Engineering (ASCE) 2004; 130(8):1177–1185.

11. Gupta A, Krawinkler H. Seismic demands for performance evaluation of steel moment resisting frame structures(SAC Task 5.4.3). Report No. 132, John A. Blume Earthquake Engineering Center, Stanford University, Stanford,CA, 1999.



12. SAC. Experimental investigation of beam–column subassemblies. Report No. SAC/BD 96/01, SAC Joint Venture,Richmond, CA, 1996.

13. Ibarra LF. Global collapse of frame structures under seismic excitations. Ph.D. Thesis, Department of CivilEngineering, Stanford University, Stanford, CA, 2003.

14. Federal Emergency Management Agency. Recommended Seismic Design Criteria for New Steel Moment FrameBuildings. Washington, DC, 2000.

15. Ibarra L, Medina RA, Krawinkler H. Hysteretic models that incorporate strength and stiffness deterioration.Earthquake Engineering and Structural Dynamics 2005; 34:1489–1515.

16. Dolsek M, Fajfar P. Simplified non-linear seismic analysis of infilled concrete frames. Earthquake Engineeringand Structural Dynamics 2005; 34(1):49–66.


10. seismic performance evaluation of steel moment resisting frames through incremental dynamic...

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