Kim 2008 Engineering-Structures

Engineering Structures 30 (2008) 3793–3807

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Engineering Structures

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An alternative pushover analysis procedure to estimate seismicdisplacement demandsSun-Pil Kim a, Yahya C. Kurama b,∗a Hyundai Development Institute of Construction Technology, South Koreab Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, IN, 46556, USA

a r t i c l e i n f o

Article history:Received 25 July 2006Received in revised form24 May 2008Accepted 4 July 2008Available online 8 August 2008

Keywords:Modal pushover analysisSeismic displacement demandsSeismic design and analysis

a b s t r a c t

An alternative pushover analysis procedure is proposed to estimate the peak seismic lateral displacementdemands for building structures responding in the nonlinear range. As compared with other pushoveranalysis procedures, the main advantage of the proposed procedure is that the effects of higher modeson the lateral displacement demands are lumped into a single invariant lateral force distribution thatis proportional to the total seismic masses at the floor and roof levels. The applicability and validityof the proposed procedure, which is referred to as the Mass Proportional Pushover (MPP) procedure,are critically evaluated through comparisons with multi-degree-of-freedom nonlinear dynamic time-history analysis results for a set of benchmarked three-story, nine-story, and twenty-story steel momentresisting building frame structures. The estimated demands are also compared with results from aModalPushover Analysis (MPA) procedure. The comparisons demonstrate that the proposed Mass ProportionalPushover procedure provides, on average, better roof and floor lateral displacement demand estimatesthan the Modal Pushover Analysis procedure. The improvement from the proposed procedure is largerfor the nine-story and twenty-story structures than the improvement for the three-story structure andis also larger for the Design Basis Earthquake (DBE) ground motion set than the Maximum ConsideredEarthquake (MCE) set.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

As performance-based considerations become more commonin the seismic design and evaluation of building structures, theestimation of peak lateral displacement demands under specifiedseismic intensity levels (e.g., Design Basis Earthquake, MaximumConsidered Earthquake) has gained utmost importance [1]. Themost accurate analytical procedure to estimate the seismicdisplacement demands of a structure responding in the nonlinearrange is to conduct nonlinear multi-degree-of-freedom (MDOF)dynamic time-history analyses. However, since this procedurerequires the consideration of a large number of earthquakes andis relatively complicated and time-consuming, the use of staticnonlinear pushover analysis procedures [1] is generally consideredto bemore suitable for the seismic design and evaluation of regularbuilding structures in standard practice.Among the static pushover analysis procedures that have been

previously developed to estimate the seismic displacement de-mands of building structures, ‘‘adaptive’’ pushover methods [2,3]

∗ Corresponding author.E-mail address: [email protected] (Y.C. Kurama).

0141-0296/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.07.008

aim to capture the changes that occur in the vibration propertiesof a structure, and the associated variations in the inertia forces, asthe structure is displaced into the nonlinear range during an earth-quake. For each step of an adaptive pushover analysis, the lateralforce distribution is evaluated and adjusted as necessary based onthe nonlinear behavior of the structure. This approach can providegood estimates of the nonlinear seismic displacement demands;however, it is relatively complicated for use in common structuralengineering design practice.A second static approach, the Modal Pushover Analysis

procedure proposed by Chopra and Goel [4–7], uses invariantmodal lateral force distributions in accordance with linear-elastictheory. This method is in general simpler, and thus, more practicalthan the adaptive pushover method for use in seismic design.For structures with increased higher mode effects, the ModalPushover Analysis procedure allows the use of multiple modesin the estimation. The primary limitations of this approach are:(1) as many static pushover analyses of the structure need to beconducted as the number of modes considered; (2) reversal in thepushover load versus displacement relationships of the structureunder higher-mode lateral force distributions is possible [5],resulting in an unstable solution; and (3) a modal combinationprocedure is needed to estimate the total demands, often resultingin an overestimation of the peak demands. The Mass Proportional

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3794 S.-P. Kim, Y.C. Kurama / Engineering Structures 30 (2008) 3793–3807

Pushover procedure proposed in this paper offers a possibleimprovement to address these limitations.Themain simplification and difference of theMass Proportional

Pushover procedure as compared to modal pushover analysis isthat the effects of higher modes on the lateral displacement de-mands are lumped into a single invariant lateral force distributionthat is proportional to the total seismic masses (or weights) as-signed to the structure at the floor and roof levels. Thus, only onepushover analysis of the structure is required with no need to con-duct a modal analysis or modal combination, even when highermode effects are considerable.The applicability and validity of the newprocedure are critically

evaluated through comparisons with MDOF nonlinear dynamictime-history analysis results and with modal pushover analysisresults using a set of benchmarked three-story, nine-story, andtwenty-story steel moment resisting building frame structures.It is shown that especially for the nine-story and twenty-storystructures under theDBE groundmotion set, theMass ProportionalPushover procedure results in considerably better roof and floorlateral displacement demand estimates than the Modal PushoverAnalysis procedure while also requiring a relatively simplerprocess. The method is currently limited to the estimation of peaklateral displacement demands, and the estimation of other seismicdemand quantities is out of its scope. Note also that the proposedprocedure should not be used for structures that have significantplan and/or height irregularities. Furthermore, an evaluation of theprocedure for wall structures is not yet available; and thus, theresults presented below are limited to frame structures only.

2. Proposed pushover procedure

This section describes the Mass Proportional Pushover proce-dure proposed in this paper and its relationship with the ModalPushover Analysis procedure in Chopra and Goel [4–7].

2.1. Uncoupled modal response history analysis for nonlinear struc-tures

The governing differential equation of motion for a nonlinearMDOF structure under horizontal earthquake ground acceleration,ug(t), is as follows:

[m]{u} + [c]{u} + {fs} = −[m]{1}ug(t) (1)

with

{fs} = {fs}({u}, sign{u}) (2)

where, [m] and [c] are the mass and classical damping matrices ofthe structure, {fs} is the internal resisting force vector, and {u} isthe floor/roof relative lateral displacement vector with {u} and {u}representing the first and second time derivatives (i.e., floor/roofvelocity and acceleration vectors), respectively. Eq. (2) indicatesthat the internal resisting force vector depends not only on thelateral displacements of the structure, but also on the displacementhistory.Even though linear modal analysis is not valid for nonlinear

structures, it is used in the formulation below to transform Eq. (1)into a set of single-degree-of-freedom (SDOF) equations ofmotion.According to modal analysis theory, the vertical distribution,[m]{1}, of the lateral force vector on the right hand side of Eq.(1) can be expanded as the summation of the modal lateral forcedistributions, {sn} as:

[m]{1} =N∑n=1

{sn} =N∑n=1

Γn[m]{φn} (3)

where,

Γn =LnMn={φn}

T[m]{1}

{φn}T[m]{φn}(4)

and {φn} is the nth mode shape.Then, the governing differential equation of motion for the

structure can be written as:

[m]{u} + [c]{u} + {fs}({u}, sign{u}) = −N∑n=1

{sn}ug(t). (5)

Also, because any set of N independent vectors can be used asa basis for representing any other vector of order N , the modalexpansion of the lateral displacement vector, {u}, has the form

{u} =N∑n=1

{un} =N∑n=1

{φn}qn (6)

where, qn are the modal amplitudes.Finally, substituting Eq. (6) and its derivatives into Eq. (5),

pre-multiplying by {φn}T, and using the mass and dampingorthogonality of themode shapes result in the following governingequation for a SDOF system:

Dn + 2ζnωnDn +FsnLn= −ug(t) (7)

where, Fsn = {φn}T{fs(Dn, sign Dn)} is the modal internal resistingforce, Dn is the modal displacement (with Dn and Dn representingthe modal velocity and acceleration, respectively), and ζn and ωnare the modal damping and frequency, respectively.Eq. (7) can be solved either by conducting a nonlinear

SDOF dynamic time-history analysis or by using a nonlinearSDOF displacement response spectrum [8]. More details andassumptions related to the above formulation can be found inChopra [9]. Note that the coupling that occurs between the modalresponses of the structure in the nonlinear range is neglected inthis procedure.

2.2. Pushover lateral force distribution

In order to solve Eq. (7), equivalent nonlinear SDOF representa-tions of the structure need to be determined using pushover anal-ysis to define the relationship between Fsn and Dn. According tolinear-elastic uncoupled modal response history analysis theory,the only lateral force distribution that can produce displacementsproportional to {φn} is {sn}. Therefore, the Modal Pushover Analy-sis procedure uses {sn} as the force distribution for the pushoveranalysis in each mode [4–7].In contrast with this approach, the proposed Mass Proportional

Pushover procedure uses the total seismic mass (or weight)distribution assigned to the structure at the floor and roof levels,[m]g{1} = [w]{1} as the lateral force distribution. Furthermore,the linear-elastic floor/roof lateral displacement vector resultingfrom the application of [w]{1} on the structure is used as themodeshape of the structure.There are several reasons and advantages for the above choice

of the lateral force distribution rather than {sn}:(1) According to classical modal analysis theory, it is impossible

to superpose modal responses of nonlinear structures not onlybecause of the coupling that exists between the governingdifferential equations for the N modes but also because of thechanges that occur in the vibration properties of the structure. Inotherwords, the choice for the use of {sn} as themodal lateral forcedistribution vector in the Modal Pushover Analysis procedure isnot based on nonlinear theory (i.e., is not unique). Thus, the onlyway to show the accuracy of an assumed force distribution is by

S.-P. Kim, Y.C. Kurama / Engineering Structures 30 (2008) 3793–3807 3795

(a) Idealized MDOF pushover base shear force versus roofdisplacement relationship.

(b) SDOF pseudo-acceleration versus displacement relationship.

Fig. 1. Equivalent single-degree-of-freedom representation.

comparing the results from SDOF analyseswith results fromMDOFnonlinear dynamic time-history analyses.In the proposedMass Proportional Pushover procedure, a single

pushover analysis is conducted with the lateral force distributiongiven by the total seismic mass (or weight) vector, [m]g{1} =[w]{1}. Based on Eq. (3), this implies that the effects of highermodes on the lateral displacement demands are lumped into thefirst mode. Note that this is a reasonable approximation onlywhen the displacements of the structure are primarily governedby the firstmode; and thus, structures that developweak/soft storymechanisms are outside the scope of the proposed procedure. Notealso that different lateral force distributionsmay result in differentpushover yield/failure mechanisms in the structure and that theproposed lateral force distribution may tend to exacerbate weakstory mechanisms relative to a first mode approach.(2) In order to define a relationship between Fsn and Dn in each

mode, the Modal Pushover Analysis procedure requires as manystatic pushover analyses of the structure as the number of modesconsidered.(3) As a possible limitation of the Modal Pushover Analysis

procedure, reversal in the pushover load versus displacementrelationships of the structure under higher-mode lateral forcedistributions is possible [5], resulting in an unstable solution.(4) If nonlinear SDOF displacement response spectra (instead of

dynamic time-history analyses) are used to solve Eq. (7), a differentresponse spectrum may be needed for use with each mode sincenonlinear response spectra depend, among other factors, on thepost-yield stiffness ratio of the structure [10], which would bedifferent for each mode.(5) The Modal Pushover Analysis procedure requires a modal

analysis of the structure to determine the mode shapes and othermodal properties such as {sn} and Γn based on Eqs. (3) and (4).In contrast, the Mass Proportional Pushover procedure does notrequire a modal analysis since the linear-elastic floor/roof lateraldisplacement vector resulting from the application of [w]{1} on thestructure is used as the mode shape.(6) Finally, the Modal Pushover Analysis procedure uses

approximate rules to combine the peak modal responses, suchas the square-root-sum-of-squares (SRSS) rule and the completequadratic combination (CQC) rule. TheMass Proportional Pushoverprocedure does not require the combination of peak responsessince Eq. (7) is solved only once using a single equivalent SDOFrepresentation of the structure as described below.

2.3. Equivalent single-degree-of-freedom representation

In both the Modal Pushover Analysis procedure and the MassProportional Pushover procedure, the base shear force versus roof

displacement relationship obtained from the pushover analysisof the structure is converted into an equivalent SDOF pseudo-acceleration versus displacement relationship as shown using theidealized bi-linear relationships in Fig. 1.In the case of the Modal Pushover Analysis procedure, the base

shear force, Vbn from each modal pushover analysis is convertedinto a modal equivalent SDOF pseudo-acceleration, An using

An =VbnM∗n

(8)

where,M∗n = ΓnLn is the effective modal mass.For the conversion from modal pushover roof displacement,

urn to modal SDOF displacement, Dn, the Modal Pushover Analysisprocedure uses

Dn =urnΓn

(9)

where, the modal participation factor Γn is calculated from Eq. (4)based on the linear-elastic mode shape {φn} normalized withrespect to the roof.In the Mass Proportional Pushover procedure, the conversion

from the pushover base shear force, Vb to SDOF acceleration, A isaccomplished by using the total mass,M of the structure as:

A =VbM. (10)

Note that the use of the total mass, M in Eq. (10) is done soas to include the representation of the entire seismic mass ofthe structure in the seismic demand estimates (i.e., it lumps alleffective modal masses into the first mode). This selection isconsistent with the use of the total seismic mass (or weight)distribution, [m]g{1} = [w]{1} as the lateral force distribution inthe proposed procedure.The SDOF displacement, D is given by

D =urΓ

(11)

with the participation factor calculated as:

Γ ={ue}T[m]{1}{ue}T[m]{ue}

(12)

where, {ue} is the floor/roof lateral displacement vector (normal-ized with respect to the roof displacement) obtained from thelinear-elastic response range of the pushover analysis under the[m]g{1} = [w]{1} lateral force distribution.A summary of the Mass Proportional Pushover procedure

proposed in this paper is as follows:


1. Conduct a single nonlinear pushover analysis of the structureunder the lateral force distribution given by [m]g{1} = [w]{1}.

2. Determine theMDOF base shear force versus roof displacement(Vb–ur ) relationship from thepushover analysis of the structure.

3. Determine the equivalent SDOF pseudo-acceleration versusdisplacement (A–D) relationship using Eqs. (10) and (11).

4. Find the maximum SDOF displacement, Dmax by solving Eq. (7)with Fs/L = A.

5. Calculate the maximum MDOF roof and floor lateral displace-ments of the structure as:

{umax} = DmaxΓ {ue}. (13)

3. Numerical approach

This section provides an overview of the numerical approachused to critically evaluate the proposed Mass ProportionalPushover procedure as follows: (1) prototype structures; (2)analytical modeling; (3) nonlinear SDOF models for prototypestructures; and (4) ground motion records.

3.1. Prototype structures

A set of benchmarked three-story, nine-story, and twenty-storyprototype steelmoment resisting building frame structures is usedin this investigation. These buildings were designed by Brandow &JohnstonAssociates, LosAngeles, California as part of the SACPhaseII Steel Project [11]. Although the SAC buildings were not actuallyconstructed, they were designed in compliance with the gravity,wind, and seismic building requirements of the 1994 UniformBuilding Code [12] and represent typical low-rise, medium-rise,and high-rise buildings on stiff soil sites (Soil Type S2 in the 1994Uniform Building Code) in Los Angeles, California.As shown in Fig. 2, the prototype structures have no significant

plan or height irregularities. The moment resisting frames weredesigned only along the perimeter of the buildings, with theinterior system consisting of gravity load resisting members. Thedesign of the moment resisting frames in all three structures wasgoverned by seismic loads.The moment resisting frames in the north–south (N–S)

direction of the buildings are used in this paper. Table 1 providesthe total heights, H for the SAC buildings as well as the buildingseismic mass assigned to each frame, M (assumed to be equallydivided between the two moment resisting frames in eachorthogonal direction of the building), the distribution of theseismic mass over the height of each frame, and the periods for thefirst three linear-elastic modes, T1, T2, and T3 (determined usingthe analytical model described below). More details on the SACprototype buildings can be found elsewhere [11].

3.2. Analytical modeling

Two-dimensional nonlinear MDOF analytical models for themoment resisting frames in the N–S direction of the prototypeSAC building structures were developed using the OpenSeesprogram [13]. To reflect flexural nonlinearity, the column andbeam members were modeled using nonlinear beam–columnelements that consider the spread of plasticity along the member.The beam and column cross sections were assumed to havebi-linear stiffness degrading [14] moment versus curvaturerelationships with a post-yield stiffness equal to 3% of the linear-elastic stiffness. The columns of the three-story structure wereassumed to be fixed at the base. For the nine-story and twenty-story structures, basement levels were used with assumed pinnedconditions for the column bases as shown in Fig. 2. Typicalhysteretic base shear force versus roof displacement (Vb–ur )

relationships obtained from the OpenSeesmodels of the SAC framestructures under the [w]{1} lateral force distribution are shown inFig. 3.For dynamic analysis, the building mass assigned to each frame

was assumed to be lumped at the horizontal degrees of freedomof the beam–column joint nodes. The dampingmatrix was definedusing Rayleigh damping [9] with a damping ratio of 3% for periodsof T1 and 0.1T1. The average acceleration method, which is one oftwo special cases of Newmark’s method [9], was used to conductthe nonlinear time-history analyses.

3.3. Equivalent SDOF models for prototype structures

The bi-linear dashed-line plots in Fig. 4 show the equivalentSDOF pseudo-acceleration versus displacement (A–D) relation-ships for the prototype SAC frame structures as obtained fromthe proposed Mass Proportional Pushover procedure. To obtainthese relationships, first, smooth pushover base shear force versusroof displacement (Vb–ur ) relationshipswere determinedusing theMDOF OpenSees models of the structures. Then, the MDOF Vb–urrelationshipswere transformed into SDOF A–D relationships as de-scribed previously. The smooth SDOF A–D relationships of the SACstructures as obtained using the proposed procedure are shownwith the smooth solid lines in Fig. 4.The idealized bi-linear A–D relationships for the structures in

Fig. 4 (i.e., the bi-linear slopes, ω2 and αω2, and the yield point,Dy and Ay) were determined by assuming: (1) the initial slope,ω2 of the bi-linear relationship is the same as the initial slopeof the smooth relationship; (2) the maximum displacement isapproximately equal to five times the yield displacement, Dy; and(3) the areas underneath the smooth and bi-linear relationships upto the maximum displacement are the same.The BISPEC program [15] was used to conduct nonlinear dy-

namic time-history analyses of the idealized bi-linear equiva-lent SDOF models for the SAC buildings. A stiffness degradingforce–displacement relationship [14] was assumed for the hys-teretic behavior of the SDOF models under cyclic loading.To facilitate comparisons with the nonlinear MDOF dynamic

analyses described above, the SDOF system damping ratio in theMass Proportional Pushover procedure was assumed to be equalto 3%. Similarly, the modal damping ratios used in the ModalPushover Analysis procedure were determined from Rayleighdamping with a damping ratio of 3% for periods of T1 and 0.1T1.While the damping ratios and the damping models used in theanalyses of the structures can certainly be debatable, the useof the same damping ratios between the MDOF analyses, theMass Proportional Pushover procedure, and the Modal PushoverAnalysis procedure ensures that the differences in the lateraldisplacement demands from these three approaches are notcaused by differences in damping.

3.4. Ground motion records

A total of 40 ‘‘far-fault’’ ground motion records (LA01–LA40)are used to evaluate the effectiveness of the Mass ProportionalPushover procedure in estimating the MDOF lateral displacementdemands. The design-level ground motion ensemble (twentyrecorded motions, LA01–LA20, as shown in Table 2) correspondsto a 10% probability of exceedance in 50 years representing theDesign Basis Earthquake (DBE) in IBC-2003 [16]. Similarly, thesurvival-level ensemble (ten recorded and ten generated motions,LA21–LA40, as shown in Table 3) corresponds to a 2% probabilityof exceedance in 50 years, representing the Maximum ConsideredEarthquake (MCE) in IBC-2003. These groundmotion records werecompiled by the SAC Phase II Steel Project [17] for a site in


(a) Three-story. (b) Nine-story.

(c) Twenty-story.

Fig. 2. SAC model frame structures.


Table 1Properties of SAC prototype frame structures

Structure Total height, H (m) Seismic mass,M per frame (kg) Floor/roof seismic mass per frame (kg) Period (s)First mode, T1 Second mode, T2 Third mode, T3

Three-story 11.88 1.48× 106 0.48× 106 (floors 2–3) 1.01 0.33 0.170.52× 106 (roof)

Nine-story 37.17 4.50× 106 0.51× 106 (floor 2) 2.26 0.84 0.490.49× 106 (floors 3–9)0.54× 106 (roof)

Twenty-story 80.73 5.55× 106 0.28× 106 (floors 2–20) 3.96 1.37 0.790.29× 106 (roof)


(c) Twenty-story.

Fig. 3. Base shear force versus roof displacement behaviors of SAC structures.

LosAngeleswith a stiff soil profile (Site ClassD in IBC-2003), similarto the site conditions used in the design of the SAC prototypestructures.For each ground motion, two horizontal components, rotated

45 degrees away from the fault-normal and fault-parallel orienta-tions, were provided by the SAC steel project. The ground motionswere scaled based on target linear-elastic smooth accelerationresponse spectra as described in Somerville et al. [17]. The prob-abilistic ground motion spectra published by the United StatesGeological Survey [18,19], modified to represent Site Class D, wereused as the target spectra. To preserve the variability in the char-acteristics of the individual ground motions, the shapes of the ac-celeration response spectra of the recordswere notmodified in thescaling procedure. Instead, for each groundmotion, a single scalingfactor was found that minimized the weighted sum of the squarederror between the average 5%-damped linear-elastic acceleration

response spectra of the two horizontal components and the corre-sponding target response spectrum in the period range of 0.3–4 s.This scale factor was then applied to both components of theground motion, thus retaining the ratios between the componentsat all periods. The weights used in the determination of the groundmotion scaling factors were 0.1, 0.3, 0.3, and 0.3 for periods of 0.3,1, 2, and 4 s, respectively. More details on the SAC ground motionrecords can be found in Somerville et al. [17].Tables 2 and 3 provide the following information for the

SAC LA01–LA40 ground motion records: (1) record site location;(2) earthquake magnitude; (3) epicentral distance; (4) SAC scalefactor; (5) peak ground acceleration, PGA; and (6) maximumincremental velocity, MIV. The MIV of a ground motion is equalto the maximum area under the acceleration time-history ofthe record between two successive zero-acceleration crossings.Previous research has shown that a strong correlation exists



(c) Twenty-story.

Fig. 4. SDOF pseudo-acceleration versus displacement relationships for SAC structures.

Table 2SAC DBE-level ground motions LA01–LA20

Record name Site location EQ magnitude Distance (km) Scale factor PGA (g) MIV (cm/s)

LA01 Imperial Valley, 1940, EI Centro 6.9 10 2.01 0.46 88.9LA02 Imperial Valley, 1940, EI Centro 6.9 10 2.01 0.68 81.0LA03 Imperial Valley, 1979, Array #05 6.5 4.1 1.01 0.39 103LA04 Imperial Valley, 1979, Array #05 6.5 4.1 1.01 0.49 74.6LA05 Imperial Valley, 1979, Array #06 6.5 1.2 0.84 0.30 106LA06 Imperial Valley, 1979, Array #06 6.5 1.2 0.84 0.23 81.6LA07 Landers, 1992, Barstow 7.3 36 3.20 0.42 59.1LA08 Landers, 1992, Barstow 7.3 36 3.20 0.42 71.5LA09 Landers, 1992, Yermo 7.3 25 2.17 0.52 135LA10 Landers, 1992, Yermo 7.3 25 2.17 0.36 76.4LA11 Loma Prieta, 1989, Gilroy 7.0 12 1.79 0.66 79.4LA12 Loma Prieta, 1989, Gilroy 7.0 12 1.79 0.97 86.8LA13 Northridge, 1994, Newhall 6.7 6.7 1.03 0.68 84.6LA14 Northridge, 1994, Newhall 6.7 6.7 1.03 0.66 132LA15 Northridge, 1994, Rinaldi RS 6.7 7.5 0.79 0.53 124LA16 Northridge, 1994, Rinaldi RS 6.7 7.5 0.79 0.58 165LA17 Northridge, 1994, Sylmar 6.7 6.4 0.99 0.57 102LA18 Northridge, 1994, Sylmar 6.7 6.4 0.99 0.82 139LA19 North Palm Springs, 1986 6.0 6.7 2.97 1.02 48.6LA20 North Palm Springs, 1986 6.0 6.7 2.97 0.99 122

between the MIV and the peak lateral displacement demandsresulting from a ground motion [20].Note that reliable nonlinear dynamic time-history analysis

results depend on the use of realistic ground motion records with

characteristics that are appropriate for the site soil conditions, siteseismicities, and seismic demand levels considered. According toSomerville et al. [17], the SAC ground motion records provide asample of the variability in earthquake characteristics through a


Table 3SAC MCE-level ground motions LA21–LA40

Record name Site location EQ magnitude Distance (km) Scale factor PGA (g) MIV (cm/s)

LA21 Kobe, 1995 6.9 3.4 1.15 1.28 274LA22 Kobe, 1995 6.9 3.4 1.15 0.92 242LA23 Loma Prieta, 1989 7.0 3.5 0.82 0.42 86.9LA24 Loma Prieta, 1989 7.0 3.5 0.82 0.47 211LA25 Northridge, 1994 6.7 7.5 1.29 0.87 202LA26 Northridge, 1994 6.7 7.5 1.29 0.94 269LA27 Northridge, 1994 6.7 6.4 1.61 0.93 166LA28 Northridge, 1994 6.7 6.4 1.61 1.33 226LA29 Tabas, 1974 7.4 1.2 1.08 0.81 92.2LA30 Tabas, 1974 7.4 1.2 1.08 0.99 128LA31 Elysian Park (simulated) 7.1 17 1.43 1.30 208LA32 Elysian Park (simulated) 7.1 17 1.43 1.19 260LA33 Elysian Park (simulated) 7.1 11 0.97 0.78 188LA34 Elysian Park (simulated) 7.1 11 0.97 0.68 161LA35 Elysian Park (simulated) 7.1 11 1.10 0.99 343LA36 Elysian Park (simulated) 7.1 11 1.10 1.10 329LA37 Palos Verdes (simulated) 7.1 1.5 0.90 0.71 263LA38 Palos Verdes (simulated) 7.1 1.5 0.90 0.78 302LA39 Palos Verdes (simulated) 7.1 1.5 0.88 0.50 117LA40 Palos Verdes (simulated) 7.1 1.5 0.88 0.63 279

set of time histories that are realistic not only in their averageproperties but also in their individual properties. Nevertheless, itshould be stated that the findings and conclusions presented inthis paper may be limited to the site and seismic characteristicsrepresented by the SAC ground motions.

4. Evaluation of demand estimates

This section critically evaluates the proposedMass ProportionalPushover procedure to estimate the seismic lateral displacementdemands for nonlinear building structures. Demand estimatesare obtained for the prototype SAC frame structures using theproposed procedure and are compared with estimates fromMDOFnonlinear dynamic time-history analyses aswell as from theModalPushover Analysis procedure described in Chopra and Goel [4–7].The demand estimates from the Mass Proportional Pushover

procedure were obtained by conducting nonlinear dynamic time-history analyses of the idealized equivalent bi-linear SDOF modelsin Fig. 4 to solve Eq. (7) under the DBE-level and MCE-level groundmotion records in Tables 2 and 3, respectively. MDOF models andequivalent bi-linear modal SDOF models (as obtained from theModal Pushover Analysis procedure) for the SAC structures werealso subjected to the same ground motion records. As describedpreviously, the BISPEC program was used to conduct the SDOFdynamic analyses assuming bi-linear stiffness degrading hystereticcharacteristics for the equivalentmodels. TheMDOF analyseswereconducted using theOpenSees analyticalmodels (see Figs. 2 and 3).The following seismic demand estimates determined using the

Mass Proportional Pushover procedure are evaluated below: (1)peak roof lateral displacements; (2) peak floor lateral displace-ments; and (3) peak inter-story drift angles.

4.1. Peak roof lateral displacement demand estimates

The peak roof lateral displacement demands for the SAC framestructures were estimated using Eq. (13) with the peak SDOFdisplacement demands, Dmax, determined from the equivalent bi-linear SDOF dynamic time-history analyses described above. Theerrors between the estimated peak roof displacement demandsfrom the proposed Mass Proportional Pushover procedure and the‘‘exact’’ demands determined from the MDOF nonlinear dynamictime-history analyses of the SAC structures are shown by theNmarkers in Figs. 5 and 6. The error values were calculated as:

error in roof displacement =(ur,max)estimated − (ur,max)exact

(ur,max)exact(14)

where, (ur,max)estimated is the displacement demand estimatedfrom the pushover procedure and (ur,max)exact is the displacementdemand determined from the MDOF dynamic analysis. The meanand standard deviation values for the calculated errors in the peakroof displacement demands are provided in Tables 4 and 5. Thepositivemean errors indicate that the peak displacement demandsare, on average, conservatively over-estimated by the approximateprocedure.To provide a benchmark for the evaluation of the Mass

Proportional Pushover procedure, the peak roof displacementdemands for the SAC structures were also estimated using theModal Pushover Analysis procedure. Peak demand estimatesconsidering the first (i.e., fundamental) mode, first two modes,and first three modes (only for the twenty-story structure) weredetermined by combining the modal peak roof displacementdemands using the SRSS rule. The resulting errors with respectto the exact displacement demands from the MDOF analyses ofthe structures are given in Tables 4 and 5 and shown using the©markers in Figs. 5 and 6.As shown in Tables 4 and5, on average, both approximatemeth-

ods provide reasonable estimates for the peak roof displacementdemands from the MDOF analyses. The Modal Pushover Analysis(MPA) results in Figs. 5 and 6 are for demand estimates obtainedusing the first mode only. Themean errors increase asmoremodesare included in the Modal Pushover Analysis procedure (see Ta-bles 4 and 5). This is because, the first-mode estimates are, on aver-age, larger than the exact solution and the consideration of highermodes increases this over-estimation based on the SRSS combi-nation of the modal displacements. Note that this situation doesnot occur in the proposed procedure since the effects of highermodes are lumped into a single mode, and thus, there is no needfor modal combination. Note also that while the proposed lateralforce vector contains the highermodes, the degree towhich highermodes may be important for the demand estimation varies withthe number of stories and the excitation. This is discussed furtherbelow.It can be seen from the analysis results that the Mass

Proportional Pushover procedure provides, on average, smallerpeak roof displacement demand estimates than the estimatesfrom the Modal Pushover Analysis procedure considering the firstmode only. These reduced demand estimates are closer to themeanMDOF demandswhile still providing a conservative solution.One possible explanation for the reduced demand estimatesfrom the proposed procedure is the use of the linear-elastic



(c) Twenty-story.

Fig. 5. Errors in peak roof displacement demand estimates under LA01–LA20.

Table 4Percent errors in peak roof displacement demands under LA01–LA20

Structure Proposed procedure Modal Pushover Analysis (MPA) procedureMean (%) St. dev. (%) No. of modes Mean (%) St. dev. (%)

Three-story 4.5 22.0 One mode 6.3 23.0Two modes 6.1 22.0

Nine-story 3.1 14.8 One mode 4.6 16.2Two modes 6.8 15.4

Twenty-story 8.5 16.1 One mode 12.6 18.6Two modes 16.6 17.5Three modes 17.1 17.4

Table 5Percent errors in peak roof displacement demands under LA21–LA40

Structure Proposed procedure Modal Pushover Analysis (MPA) procedureMean (%) St. dev. (%) No. of modes Mean (%) St. dev. (%)






(c) Twenty-story.

Fig. 6. Errors in peak roof displacement demand estimates under LA21–LA40.

lateral displacement vector from the application of [w]{1} on thestructure as the mode shape of the structure.Tables 4 and 5 show that the Mass Proportional Pushover

procedure provides better estimates for the mean peak roofdisplacement demands than the estimates from the ModalPushover Analysis procedure for all of the cases considered inthe investigation (i.e., considering the three-story, nine-story,and twenty-story structures; the DBE and MCE ground motionensembles; and the MPA results including one, two, and threemodes in the solution). It can be observed that the improvementprovided by the proposedprocedure is larger for the nine-story andtwenty-story structures than the improvement for the three-storystructure and is also larger for the DBE ground motion set thanthe MCE set. Based on these results, it is concluded that the MassProportional Pushover procedure provides better estimates for theMDOF peak roof displacement demands than the Modal PushoverAnalysis procedure. These improved estimates, combined withrelative ease of use, make the proposed procedure well-suited forthe seismic design of regular building frame structures in practice.Note that a large amount of scatter is observed in Figs. 5

and 6 indicating a large variation in the errors for the peak roofdisplacement demand estimates from the different groundmotionrecords used in the investigation (the error in the estimatedpeak roof displacement demand is more than 50% under someof the ground motions). For both the Mass Proportional Pushoverprocedure and the Modal Pushover Analysis procedure, the errorsin the demand estimates vary significantly with the number of

stories (i.e., the three SAC structures) and the ground motionrecords. No significant correlation is observed between the errorsand the maximum incremental velocity of the ground motionrecords; however, the mean errors under the MCE ground motionset are in general larger than the errors under the DBE groundmotion set. The increased mean errors under the MCE set ispossibly because of the increased nonlinearity in the structuralresponse, and the corresponding increase in the mode shapevariation and coupling between the modes, which are ignored inboth approximate procedures.

4.2. Peak floor lateral displacement demand estimates

The peak floor lateral displacement demands for a MDOFsystem can be estimated by multiplying the estimated peak roofdisplacement demand with the roof-normalized mode shape. Inthe case of theMass Proportional Pushover procedure, this processresults in Eq. (13), where ue (i.e., the floor/roof lateral displacementvector obtained from the linear-elastic response range of thepushover analysis under [m]g{1} = [w]{1}, normalized withrespect to the roof displacement) is used as the normalized modeshape.Note that a different peak floor displacement vector would

result from each ground motion record using this proceduresince the peak roof displacement is different from each groundmotion. For design purposes, a more reasonable approach wouldbe to estimate a single peak floor displacement vector for the



(c) Twenty-story.

Fig. 7. Mean peak floor displacement demand estimates under LA01–LA20.

Table 6Percent errors in mean peak floor displacement demands

Structure Proposed procedure Modal Pushover Analysis (MPA) procedureLA01–LA20 (%) LA21–LA40 (%) No. of modes LA01–LA20 (%) LA21–LA40 (%)




structure based on the estimated mean peak roof displacementdemand from the entire ground motion ensemble. The resulting‘‘mean’’ peak floor displacement demands for the SAC buildings,ui,max, estimated using the Mass Proportional Pushover procedure(N markers) are compared in Figs. 7 and 8 with the ‘‘exact’’mean peak floor displacement demands determined from theMDOF nonlinear dynamic time-history analyses of the structures(�markers).The errors between the estimated mean peak floor displace-

ment demands, (ui,max)estimated, and the exact mean demands,(ui,max)exact, for the SAC buildings are given in Table 6. The errorvalues were calculated as:

error in floor displacements

=

K+1∑i=2ABS

[(ui,max)estimated − (ui,max)exact

]K+1∑i=2(ui,max)exact

(15)

where, K = 3, 9, and 20 for the three-story, nine-story, andtwenty-story structures, respectively, i = 2 represents the secondfloor level (see Fig. 2), and i = K + 1 represents the roof level.To provide a benchmark for the evaluation of the Mass Pro-

portional Pushover procedure, the mean peak floor displacementdemands for the SAC structures were also estimated using the



(c) Twenty-story.

Fig. 8. Mean peak floor displacement demand estimates under LA21–LA40.

Modal Pushover Analysis procedure. Mean peak demand esti-mates considering the first (i.e., fundamental) mode, first twomodes, and first three modes (only for the twenty-story structure)were determined by combining modal mean peak floor displace-ment demands using the SRSS rule. The modal mean peak floordisplacement demands were determined bymultiplying each nor-malized mode shape with the corresponding estimated modalmean peak roof displacement demand. Following Chopra andGoel [6], each mode shape was determined as the floor/roof lat-eral displacement vector obtained from the pushover analysis ofthe structure under {sn} when the roof displacement reached theestimated modal mean peak roof displacement demand from theground motion ensemble. This process was repeated for as manytimes as the number of modes considered.The resulting mean peak floor displacement demand estimates

from the Modal Pushover Analysis procedure are shown usingthe © markers in Figs. 7 and 8, with the errors between theestimated and exact mean demands [calculated using Eq. (15)]given in Table 6. Looking at the results, it is concluded that theMass Proportional Pushover procedure provides, on average, betterpeak floor displacement demand estimates for the SAC buildingsthan theModal Pushover Analysis procedure, with the exception ofthe three-story building under the MCE-level LA21–LA40 groundmotion set. The use of nonlinear mode shapes following Chopraand Goel [6] may have played a role in the increased errorsin the mean peak floor displacement demand estimates fromthe Modal Pushover Analysis procedure; however, this was not

investigated. Note that the Modal Pushover Analysis (MPA) resultsin Figs. 7 and 8 are for demand estimates obtained using thefirst mode only. Similar to the peak roof displacement demandsdescribed previously, the errors in the estimated mean peakfloor displacement demands from the Modal Pushover Analysisprocedure tend to increase as more modes are included.

4.3. Peak story drift angle demand estimates

The story (i.e., inter-story) drift angle is defined as therelative lateral displacement between two adjacent floor/rooflevels divided by the story height. In the case of the proposedMass Proportional Pushover procedure, the peak story driftangle demands are calculated from the peak floor/roof lateraldisplacement demands described previously. Once again, theestimated mean peak roof displacement demand from the entireground motion ensemble is used to determine the ‘‘mean’’ peakstory drift angle demands for each structure.The resulting mean peak story drift angle demands for the SAC

buildings,1ui,max, estimatedusing theMass Proportional Pushoverprocedure (N markers) are compared in Figs. 9 and 10 with theexact mean peak story drift angle demands determined from theMDOF nonlinear dynamic time-history analyses of the structures(�markers).The errors between the estimated mean peak story drift

angle demands, (1ui,max)estimated, and the exact mean demands,



(c) Twenty-story.

Fig. 9. Mean peak story drift angle demand estimates under LA01–LA20.

(1ui,max)exact, for the SAC buildings are given in Tables 7 and 8.The ‘‘area’’ errors investigate the total error in the story drift angledemands considering all of the stories in the structure, and arecalculated as:

area error in story drift angle

=

K+1∑i=2ABS

[(1ui,max)estimated − (1ui,max)exact

]K+1∑i=2(1ui,max)exact

(16)

where, K = 3, 9, and 20 for the three-story, nine-story, andtwenty-story structures, respectively, i = 2 represents the secondfloor level, and i = K + 1 represents the roof level.The ‘‘peak’’ errors investigate the error in the largest mean peak

story drift angle demand occurring in any story over the height ofthe structure, and are calculated as:

peak error in story drift angle

=largest(1ui,max)estimated − largest(1ui,max)exact

largest(1ui,max)exact. (17)

Note that while the peak error can be positive or negative(indicating an over-estimation or an under-estimation in theresults, respectively), the area error is always positive.To provide a benchmark for the evaluation of the proposedMass

Proportional Pushover procedure, the mean peak story drift angle

demands for the SAC structures were also estimated using theModal Pushover Analysis procedure.Mean peak demand estimatesconsidering the first (i.e., fundamental) mode, first two modes,and first three modes (only for the twenty-story structure) weredetermined by combining modal mean peak story drift angledemands using the SRSS rule.The resulting mean peak story drift angle demand estimates

from the Modal Pushover Analysis procedure are shown usingthe © markers in Figs. 9 and 10, with the errors between theestimated and exactmean demands [calculated using Eqs. (16) and(17)] given in Tables 7 and 8. Note that, different from Figs. 5–8,the Modal Pushover Analysis (MPA) results in Figs. 9 and 10 arefor demand estimates obtained using the first two modes for thethree-story and nine-story structures and the first three modes forthe twenty-story structure since the consideration of highermodesin the Modal Pushover Analysis procedure can result in betterestimates for the peak story drift angle demands (see Tables 7 and8). This is because, the story drift angle demands are sensitive tothe derivative (i.e., slope) of the mode shape vector(s); and thus,the consideration of a larger number of mode shapes can improvethe estimation.It is observed from the peak errors in Tables 7 and 8 that

the Mass Proportional Pushover procedure underestimates thelargest mean peak story drift angle demands, except for the three-story structure under the MCE-level ground motion ensemble. Ingeneral, the errors in the mean peak story drift angle demandestimates are quite large, ranging between 14.1% and 22.7% for



(c) Twenty-story.

Fig. 10. Mean peak story drift angle demand estimates under LA21–LA40.

Table 7Percent errors in mean peak story drift angle demands under LA01–LA20

Structure Proposed procedure Modal Pushover Analysis (MPA) procedureArea error (%) Peak error (%) No. of modes Area error (%) Peak error (%)

Three-story 19.4 −7.2 One mode 5.7 −6.7Two modes 3.9 −4.1

Nine-story 16.1 −13.8 One mode 26.8 5.3Two modes 17.3 6.2

Twenty-story 22.7 −14.1 One mode 24.0 9.8Two modes 11.3 18.3Three modes 10.3 19.0

Table 8Percent errors in mean peak story drift angle demands under LA21–LA40

Structure Proposed procedure Modal Pushover Analysis (MPA) procedureArea error (%) Peak error (%) No. of modes Area error (%) Peak error (%)


Nine-story 14.1 −13.2 One mode 18.3 6.2Two modes 10.8 6.6

Twenty-story 18.6 −25.3 One mode 14.4 −2.8Two modes 12.1 2.9Three modes 14.0 3.3


the area errors and between −25.3% and +11.8% for the peakerrors. Furthermore, the estimates from the proposed procedureare worse than those provided by the Modal Pushover Analysisprocedure. The poor performance of the proposed procedure toestimate the peak story drift angle demands is because of theuse of a single mode shape to represent the displaced shape ofthe structure during an earthquake and the inability of this modeshape to capture the changes in the lateral displacements betweenadjacent floor and roof levels.

5. Conclusions

This paper proposes an alternative pushover analysis proce-dure, referred to as the Mass Proportional Pushover (MPP) proce-dure, to estimate the peak seismic lateral displacement demandsfor building structures responding in the nonlinear range. Themainadvantage of theMass Proportional Pushover procedure over otherapproximate procedures (e.g., the Modal Pushover Analysis pro-cedure) is the use of a single pushover analysis for the struc-ture with no need to conduct a modal analysis to capture theeffects of higher modes on the lateral displacement demands.Observations made through comparisons between the estimateddisplacement demands from the Mass Proportional Pushoverprocedure, the Modal Pushover Analysis procedure, and the‘‘exact’’ demands from multi-degree-of-freedom nonlinear dy-namic time-history analysis results are given below. Note thatthese observations are based on themean seismic demands from alimited number of analyses considering three different steel build-ing moment frame structures and two ground motion ensembles,and thus, generalizations drawn from the investigation should beused with caution.(1) Both the Mass Proportional Pushover procedure and the

Modal Pushover Analysis procedure provide reasonable estimatesfor the mean peak roof and floor lateral displacement demandsfrom the MDOF analyses.(2) The Mass Proportional Pushover procedure provides better

estimates for the mean peak roof and floor displacement demandsthan the estimates from the Modal Pushover Analysis procedure.(3) The improvement from the Mass Proportional Pushover

procedure is larger for the nine-story and twenty-story structuresthan the improvement for the three-story structure and is alsolarger for the Design Basis Earthquake (DBE) ground motion setthan the Maximum Considered Earthquake (MCE) set.(4) The improved lateral displacement demand estimates,

combined with relative ease of use, make the Mass ProportionalPushover procedure well-suited for the seismic design of regularbuilding frame structures in practice.(5) As a result of modal combination, the mean errors in

the peak roof and floor displacement estimates from the ModalPushover Analysis procedure increase as more modes are includedin the solution. This situation does not occur in the MassProportional Pushover procedure since the effects of higher modesare lumped into a singlemode, and thus, there is no need formodalcombination.(6) There is a large variation in the errors for the peak lateral

displacement demand estimates from the different groundmotionrecords used in the investigation. No significant correlation isobserved between the errors and the intensity of the groundmotions; however, the mean errors under the MCE groundmotionset tend to be larger than the errors under the DBE ground motionset. The increased mean errors under the MCE set is possiblybecause of the increased nonlinearity in the structural response,and the corresponding increase in the mode shape variationand coupling between the modes, which are ignored in bothapproximate procedures.

(7) The Mass Proportional Pushover procedure provides rela-tively poor estimates for the peak inter-story drift angle demands,and the peak inter-story drift angle estimates from the ModalPushover Analysis procedure are on average closer to the MDOFanalysis results. The poor performance of the Mass ProportionalPushover procedure to estimate the peak inter-story drift angle de-mands is because of the inability of the selected mode shape tocapture the changes in the lateral displacements of the structurebetween adjacent floor and roof levels.

Acknowledgements

This work was supported by the Korea Research Foundation(KRF-2005-214-D00168) with additional funding provided by theDepartment of Civil Engineering and Geological Sciences at theUniversity of Notre Dame, Notre Dame, Indiana, USA. Their supportis greatly appreciated. The opinions, findings, and conclusionspresented in the paper are those of the authors and do notnecessarily reflect the views of the organizations acknowledgedabove.

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