tjhin2006

TECHNICAL NOTES

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ity O

f W

isco

nsin

-Milw

auke

e on

10/

17/1

4. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Observations on the Reliability of Alternative Multiple-ModePushover Analysis Methods

T. Tjhin1; M. Aschheim2; and E. Hernández-Montes3

Abstract: Although multiple-mode pushover analysis methods have been proposed for general use in the seismic analysis of moment-resisting frames, difficulties have been encountered in their implementation with specific structures. Two alternative multiple modemethods were developed to overcome these difficulties. Estimates of four response quantities determined with the alternative methods arecompared herein for a set of five buildings subjected to suites of scaled ground motions. The uneven accuracy of the estimates, relativeto the range of values determined by nonlinear dynamic analysis, suggests that results obtained by both alternative methods should beregarded with caution, until such time that the scope of applicability of the methods has been clearly established.

DOI: 10.1061/�ASCE�0733-9445�2006�132:3�471�

CE Database subject headings: Earthquake engineering; Seismic effects; Nonlinear response; Displacement.

Introduction

The popularity of Nonlinear Static Procedures for the evaluationand design of buildings to resist seismic motions has grown sub-stantially since the publication of the Capacity Spectrum Methodin ATC-40 �ATC 1996� and the Displacement Coefficient Methodin FEMA-273 �BSSC 1997�. In these procedures, a pushovercurve is developed on the basis of a nonlinear static analysis ofa model of the structure. The pushover curve, which consists of aplot of base shear as a function of roof displacement, is then usedto estimate the peak displacement �or demand displacement� atthe roof in conjunction with an elastic spectrum, which representsthe hazard at the site. In many cases, additional response quanti-ties needed in the detailed evaluation of a building, such as inter-story drifts and story shears, are also based on the results of thenonlinear static analysis, neglecting the potential contributions ofhigher modes �or multidegree of freedom �MDOF� effects, in thecase of nonlinear response�. The importance of higher modes wasdiscussed in the ATC-40 publication and is recognized in various

1Univ. of Illinois at Urbana-Champaign, Urbana, IL 61801; formerly,Graduate Student.

2Associate Professor, Civil Engineering Dept., Santa Clara Univ., 500El Camino Real, Santa Clara, CA 95053 �corresponding author�. E-mail:[email protected]

3Associate Professor, Dept. of Structural Mechanics, Univ. ofGranada, Campus de Fuentenueva, 18072 Granada, Spain.

Note. Associate Editor: Gregory A. MacRae. Discussion open untilAugust 1, 2006. Separate discussions must be submitted for individualpapers. To extend the closing date by one month, a written request mustbe filed with the ASCE Managing Editor. The manuscript for this techni-cal note was submitted for review and possible publication on October 5,2004; approved on June 27, 2005. This technical note is part of theJournal of Structural Engineering, Vol. 132, No. 3, March 1, 2006.
©ASCE, ISSN 0733-9445/2006/3-471–477/$25.00.
JOURN

J. Struct. Eng. 2006.

proposals �e.g. Reinhorn 1997; Sasaki et al. 1998; Chopra andGoel 2001; and Jan et al. 2004�.

The multimode pushover analysis �MPA� procedure �Chopraand Goel 2001� was developed primarily for estimating interstorydrifts in frame structures. The recent ATC-55 project attempted toapply this procedure to estimate story shears and overturning mo-ments, in addition to floor displacements and interstory drifts, butencountered reversals in the third mode pushover of a three-storysteel moment-resistant frame building �Fig. 1� and in the secondmode pushover of a weak-story variant of this three-story frame.These “reversals” signify that increments in roof displacement arein a direction opposite to the base shear, which may happen de-pending on the mechanism that develops within the structure. Forexpediency, the representation of higher mode contributions wasachieved in the ATC-55 project by including elastic contributionsof the higher modes together with a potentially inelastic contribu-tion from the first mode, by means of an SRSS �square root of thesum of squares� combination. This method was subsequentlyevaluated for use in estimating peak interstory drifts by Chopraet al. �2004�.

Recognizing that the roof displacement may not always be thebest index as a basis for establishing the properties of so-called“equivalent” single degree of freedom �SDOF� �or ESDOF� sys-tems, Hernández-Montes et al. �2004� developed an alternativeindex, known as an energy-based displacement. Tjhin et al.�2005� recently published results suggesting that first mode esti-mates of peak roof displacement are improved when the energy-based displacement is used in place of the roof displacement toestablish the properties of the first mode ESDOF system.

This technical note compares estimates of response quantitiesobtained with the modified MPA method and a new MPA methodthat uses the energy-based displacement for five buildingmodels subjected to 11 ground motions. The response quantitiesconsidered are peak roof drift, interstory drift, story shear, and
overturning moment.
AL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2006 / 471

132:471-477.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ity O

f W

isco

nsin

-Milw

auke

e on

10/

17/1

4. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Methodology

Buildings

Five building models used in the ATC-55 project were used in thisstudy. These are a three-story steel moment-resistant frame, anine-story steel moment-resistant frame, an eight-story reinforcedconcrete wall building, and weak-story variants of the three- andnine- story moment frames, in which the strengths of the loweststories were reduced in order to create weak-story behavior inthese frames. Bilinear hysteretic models were used for the mem-bers of the steel frames, while a fiber element was used to modelthe flexural behavior of the wall. P-Delta effects were included in

Table 1. Ground Motions Used in this Study

Number IdentifierEarthquake and date

station location

1 ICC000 Superstition Hills November 24, 198701335: El Centro Imperial County Center

2 LOS000 Northridge January 17, 199416628: W. Canyon Rd., Canyon Country

3 G02090 Loma Prieta October 18, 198947380: Gilroy Array Number 2

4 TCU122N Chi-Chi, Taiwan September 20, 1999TCU122

5 G03090 Loma Prieta October 18, 198947381: Gilroy Array Number 3

6 CNP196 Northridge January 17, 19947769 Topanga Canyon Blvd., Canoga Park

7 CHY101W Chi-Chi, Taiwan September 20, 1999CHY101

8 ICC090 Superstition Hills November 24, 198701335: El Centro Imperial County Center

9 CNP106 Northridge January 17, 19947769 Topanga Canyon Blvd., Canoga Park

10 E02140 Imperial Valley October 15, 19795115: El Centro Array Number 2

11 E11230 Imperial Valley October 15, 19795058: El Centro Array Number 11

aClosest distance to fault rupture.bCDMG�California Division of Mines and Geology; USC�Univers

Fig. 1. Reversal of third mode pushover curve for three-story steelmoment-resistant frame

CWB�Central Weather Bureau, Taiwan.

472 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2006


all models, in both the dynamic and static analyses. These build-ings are more fully described in FEMA-440 �Building SeismicSafety Council 2005�, which presents the results of the ATC-55project.

Ground Motions and Scaling

Eleven ground motions were used in the analyses, as listed inTable 1. The ground motions were recorded at NEHRP Site ClassC sites at epicentral distances of 9–16 km, and originated fromearthquakes having magnitudes �Ms� between 6.6 and 7.6. Theground motions were iteratively scaled to achieve peak roof driftsequal to 0.5, 2, and 4% of the building height for the frames, and0.2, 1, and 2% of the building height for the walls. The target driftlevels were selected to represent practical ranges of interest,wherein the lowest roof drift corresponds to elastic responsewhile the intermediate and highest values of drift correspond tomoderate and greater degrees of inelasticity in the structures.Note that this scaling approach is equivalent to an incrementaldynamic analysis �Vamvatsikos and Cornell 2002� in which ana-lytical results are retained only for those cases in which thedesired peak roof drift is obtained.

Response Quantities

The response quantities extracted from the dynamic analyses andestimated using the two alternative pushover methods over theheight of each building are peak displacement, peak interstorydrift, peak story shear, and peak floor overturning moment.

ude

Epicentraldistance�km�a Component

PGA�g�

PGV�m/s�

Char.period

�s� Sourceb

.6 13.9 000 0.358 46.4 0.60 CDMG

.7 13.0 000 0.410 43.0 0.59 USC

.1 12.7 090 0.322 39.1 0.69 CDMG

.6 9.0 000 0.261 34.0 0.85 CWB

.1 14.4 090 0.367 44.7 0.40 CDMG

.7 15.8 196 0.420 60.8 0.61 USC

.6 11.1 270 0.353 70.6 1.27 CWB

.6 13.9 090 0.258 40.9 1.03 CDMG

.7 15.8 106 0.356 32.1 0.45 USC

.9 10.4 140 0.315 31.5 0.29 USGS

.9 12.6 230 0.380 42.1 0.27 USGS

Southern California; USGS�United States Geological Survey; and

Magnit

Ms=6

Ms=6

Ms=7

Ms=7

Ms=7

Ms=6

Ms=7

Ms=6

Ms=6

Ms=6

Ms=6

ity of

132:471-477.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ity O

f W

isco

nsin

-Milw

auke

e on

10/

17/1

4. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Modified MPA Method

A modified version of the MPA method �Chopra and Goel 2001�was developed in the ATC-55 studies and was used herein. In themodified method, elastic contributions associated with the secondand third modes were combined with contributions from the firstmode, which may be inelastic, using an SRSS combination. Theelastic contributions of the higher modes are evaluated using themean spectrum determined for the ground motions as scaled forthe roof drift level of interest for each building. The demanddisplacement used to determine the first mode contribution corre-sponded to the predetermined roof drift.

Energy-Based MPA Method

In the energy-based pushover approach of Hernández-Monteset al. �2004�, the capacity curve associated with each modal push-over analysis is determined based on the work done in the analy-sis. The work is computed incrementally, typically for each stepin the pushover analysis. The increment in the energy-based dis-placement of the nth mode ESDOF system, �De,n, is obtained as

�De,n =�En

Vb,n�1�

where �En=increment of work done by the lateral forces actingthrough the displacement increment associated with one step ofthe nth mode pushover analysis and Vb,n=base shear at that step ofthe pushover analysis, which is equal to the sum of the lateralforces at that step. The incremental displacements, �De,n, are ac-

Fig. 2. Story shears and overturning moments for three-s

cumulated �summed� to obtain the displacement, De,n, of the

JOURN


ESDOF system at any given step in the modal pushover analysis.The capacity curve of each modal ESDOF system is a plot of

Vb,n / ��nW� as a function of De,n, where �n=the modal mass co-efficient for the nth mode and W=weight of the structure. Thiscapacity curve is used to characterize the load–deformation be-havior of each modal ESDOF system for the purpose of estimat-ing the peak response of the structure. As for the modified MPAmethod, the first mode ESDOF system was pushed until the roofdisplacement was equal to the predetermined roof drift. The peakdisplacements of the second and third mode ESDOF systemswere estimated by applying an R-C1-T relation to the mean elasticspectrum, computed using the ground motions as scaled forthe particular building and drift level under consideration. TheR-C1-T relation applied to the mean spectrum is the one devel-oped in the ATC-55 project, which is

C1 = 1 +R − 1

90Te2 �2�

where C1=peak displacement of the inelastic system divided bythe peak displacement of an elastic system having the same pe-riod, Te�s� and R=ratio of the elastic strength demand to the yieldstrength of the inelastic system. The estimated peak displacementfor the second and third mode ESDOF systems corresponds to aparticular step in the nonlinear static pushover analysis of theMDOF system, and it is at this step that the modal responsequantities are extracted for the second and third modes. �Note thatthe energy-based modal pushover analysis method could haveused this approach for the first mode contribution as well�. Con-

oment-resistant steel frame at peak drifts of 0.5 and 4%
tory m
tributions from the three modes to each response quantity of in-


132:471-477.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ity O

f W

isco

nsin

-Milw

auke

e on

10/

17/1

4. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Fig. 3. Response quantities for nine-story moment-resistant steel frame at peak drift of 4%


J. Struct. Eng. 2006.132:471-477.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ity O

f W

isco

nsin

-Milw

auke

e on

10/

17/1

4. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Fig. 4. Response quantities for weak-story variant of nine-story moment-resistant steel frame at peak drift of 4%

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2006 / 475

J. Struct. Eng. 2006.132:471-477.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ity O

f W

isco

nsin

-Milw

auke

e on

10/

17/1

4. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

terest were combined using an SRSS approach, to obtain theenergy-based estimates.

Results

Because higher mode contributions to floor displacements tend tobe relatively small, both alternative multimode approaches �aswell as many of the quasi-first mode approaches investigated inATC-55� provided excellent estimates of mean floor displace-ments over the height of all five buildings, for the three driftlevels. Both alternative multimode approaches also provided verygood estimates of interstory drift for the three-story frame and forthe eight-story wall for the three drift levels. However, the accu-racy of the estimates in other cases depended on the responsequantity of interest, the building considered �and its mechanism�,and the drift level. Examples of this are provided as follows.

Fig. 2 shows story shears and overturning moments over theheight of the three-story frame for peak drift levels of 0.5% �elas-tic� and 4%. The bar symbol indicates the minimum, maximum,and mean result of the 11 dynamic analyses at each level, as wellas the median and mean +/−1 SD values. The dash-dot and solidlines indicate the estimates made using the modified and energy-based multimode analysis methods, respectively. Fig. 2 illustratesthat elastic estimates coincide at 0.5% drift �elastic response�while the estimates diverge at 4% drift. However, at 0.5% driftthe estimates of story shears are somewhat below the mean dy-namic results, while those for overturning moments are closer tothe mean dynamic results. This difference may be due to the useof 2% damped spectra for the estimates, whereas less dampingwas present in the higher modes of the dynamic analyses. Alsopossible is that the SRSS combination is not an upper bound tothe mean dynamic results. At 4% drift, the energy-based approach

Fig. 5. Story shears and overturning moments for

gives better estimates of story shears and overturning moments.



Fig. 3 presents results for the nine-story moment-resistantframe at a drift level of 4%. Estimates made using the twomultimode approaches differed from each other, but to differentdegrees depending on the response quantity. While the modifiedmethod provided generally better estimates of interstory drift,the energy-based method provided generally better estimates ofstory shears and overturning moments. If only one method wereused to estimate all four response quantities, substantial errorscould occur, depending on the response quantity and location ofinterest.

Fig. 4 presents results for the weak-story variant of the nine-story moment-resistant frame, at a peak roof drift of 4%. Whileboth multimode methods capture the overall trend of interstorydrift demands, estimates at the third, fourth, and fifth stories weresubstantially below the dynamic results. As well, neither methodcould provide consistently reliable estimates of the dynamic storyshears and overturning moments determined for this building.

Fig. 5 presents results for the eight-story reinforced concretewall building at a drift of 1%. While both methods provided verygood estimates of floor displacement and interstory drift �notshown�, the modified method tended to underestimate story shearswhile providing very good estimates of overturning moments,while the energy-based method tended to overestimate both ofthese quantities.

Conclusions

The conventional multimode procedure, which was developedprimarily for estimating interstory drifts and was recently ex-tended for estimating plastic hinge rotations on the basis ofassumed plastic hinge distributions �Goel and Chopra 2004�, ishampered by the occurrence of reversals in higher mode pushover

story reinforced concrete wall building at 1% drift
eight-
curves in some cases. Alternative MPA methods were developed

132:471-477.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ity O

f W

isco

nsin

-Milw

auke

e on

10/

17/1

4. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

to overcome this problem. On the basis of the data presented, itappears that the modified and energy-based MPA methods haveutility for estimating mean floor displacements and interstorydrifts for some buildings, but neither is able to provide consis-tently reliable estimates of story shears and overturning momentsover the height of the buildings considered. Consequently, therobustness of these procedures for general use in evaluation anddesign is open to question. Further clarification of the circum-stances under which these methods produce reliable estimates ofthe broad range of response quantities of interest in the practice ofstructural engineering is needed.

Acknowledgments

The writers extend their sincere appreciation to Professor Anil K.Chopra and Professor Rakesh K. Goel for their insightful contri-butions to modal pushover analysis. Much of the present workwas stimulated by the discussions and developments that tookplace under the auspices of the recent ATC-55 project. Thisproject, which focused on the use of inelastic analysis proceduresfor design and rehabilitation, was conducted by the Applied Tech-nology Council with funding from the Federal Emergency Man-agement Agency. However, the writers are solely responsible forthe results and conclusions presented herein, which do not neces-sarily represent the views of these organizations.

Notation

The following symbols are used in this technical note:C1 � peak displacement of inelastic system divided by

peak displacement of elastic system having sameTe;

De,n � energy-based displacement of ESDOF system;R � ratio of elastic strength demand and yield

strength of inelastic system;Te � period of vibration;

Vb,n � base shear at that step of pushover analysis innth mode;

W � weight of structure;�n � modal mass coefficient for nth mode;

�De,n � incremental energy-based displacement ofESDOF system; and

JOURN


�En � increment of work done by lateral forces actingthrough displacement increment associated withone step of pushover analysis in nth mode.

References

Applied Technology Council �ATC�. �1996�. “Seismic evaluation andretrofit of concrete buildings.” Rep. No. ATC-40, Volumes 1 and 2,Applied Technology Council, Redwood City, Calif.

Building Seismic Safety Council �BSSC�. �1997�. NEHRP guidelines forthe seismic rehabilitation of buildings, FEMA-273, Federal Emer-gency Management Agency, Washington, D.C.

Building Seismic Safety Council. �2005�. Improvement of inelasticseismic analysis procedures, FEMA-440, Federal EmergencyManagement Agency, Washington, D.C.

Chopra, A. K., and Goel, R. K. �2001�. A modal pushover analysisprocedure to estimate seismic demands for buildings: Theory andpreliminary evaluation, PEER-2001/03, Pacific Earthquake Engineer-ing Research Center, Univ. of California, Berkeley, Calif.

Goel, R. K., and Chopra, A. K. �2004�. “Evaluation of modal and FEMApushover analyses: SAC buildings.” Earthquake Spectra, 20�1�,225–254.

Chopra, A. K., Goel, R. K, and Chintanapakdee, C. �2004�. “Evaluationof a modified MPA procedure assuming higher modes as elastic toestimate seismic demands.” Earthquake Spectra, 20, 757–778.

Hernández-Montes, E., Kwon, O.-S., and Aschheim, M. �2004�. “Anenergy-based formulation for first- and multiple-mode nonlinear static�pushover� analyses.” J. Earthquake Eng., 8�1�, 69–88.

Jan, T. S., Liu, M. W., and Kao, Y. C. �2004�. “An upper-bound pushoveranalysis procedure for estimating the seismic demands of high-risebuildings.” Eng. Struct.; 26�1�, 117–128.

Reinhorn, A. �1997�. “Inelastic analysis techniques in seismic evalua-tions.” Seismic design methodologies for the next generation of codes,Proc. of the International Workshop, P. Fajfar and H. Krawinkler, eds.,Bled, Slovenia.

Sasaki K. K., Freeman, S. A., and Paret, T. F. �1998�. “Multi-mode push-over procedure �MMP�—A method to identify the effects of highermodes in pushover analysis.” Proc., 6th U.S. National Conf. on Earth-quake Engineering, Earthquake Engineering Research Institute,Seattle.

Tjhin, T., Aschheim, M., and Hernández-Montes, E. �2005�. “Estimatesof peak roof displacement using ‘equivalent’ single degree of freedomsystems.” J. Struct. Eng., 131�3�, 517–522.

Vamvatsikos, D., and Cornell, C. A. �2002�. “Incremental dynamic analy-sis.” Earthquake Eng. Struct. Dyn., 31�3�, 491–514.


132:471-477.

Documents

tjhin2006