estimates of average inelastic deformation demands for...

Transaction A: Civil EngineeringVol. 16, No. 5, pp. 388{402c Sharif University of Technology, October 2009

Estimates of Average Inelastic DeformationDemands for Regular Steel Framesby the Endurance Time Method

H.T. Riahi1, H.E. Estekanchi1;� and A. Vafai1

Abstract. The Endurance Time (ET) method is a new dynamic pushover procedure in whichstructures are subjected to gradually intensifying acceleration functions and their performance is assessedbased on the length of the time interval that they can satisfy required performance objectives. In thispaper, the accuracy of the Endurance Time method in estimating average deformation demands of lowand medium rise steel frames using ETA20f series of ET acceleration functions has been investigated. Theprecision of the ET method in predicting the response of steel frames in nonlinear analysis is investigatedby considering a simple set of moment-resisting frames. An elastic-perfectly-plastic material model anda bilinear material model with a post-yield sti�ness equal to 3% of the initial elastic sti�ness have beenconsidered. For frames with an elastic-perfectly-plastic material model, which are P �� sensitive cases,the ET analysis for the maximum interstory drift ratio somewhat underestimates the nonlinear responsehistory analysis results. The di�erence between the results of the ET analysis and the nonlinear responsehistory analysis for the material model with 3% post-yield sti�ness is acceptable. The consistency of thebase shears obtained by the two methods is also satisfactory. It is shown that, although the results of theET analysis are not exactly consistent with the results of ground motions analysis, the ET method canclearly identify the structure with a better performance even in the case of structures with a relativelycomplicated nonlinear behavior.

Keywords: Nonlinear response history analysis; Dynamic pushover; Endurance time method;Performance-based seismic engineering.

INTRODUCTION

The concept of Performance Based Seismic Engineering(PBSE) is gaining increased interest among researchersand practitioners [1]. PBSE includes the conceptthat designs should be capable of satisfying variousperformance objectives under a spectrum of designground motions ranging from minor to severe. Dueto inherent randomness in ground shaking, lack ofknowledge in the precise de�nition of the structure'scharacteristics, and an inability to model the actualbehavior accurately, the estimation of seismic perfor-mance entails signi�cant uncertainty [2].

Quanti�cation of seismic demands for perfor-mance assessment implies the statistical and proba-

1. Department of Civil Engineering, Sharif University of Tech-nology, Tehran, P.O. Box 11155-9313, Iran.

*. Corresponding author. E-mail: [email protected]

Received 12 July 2008; received in revised form 12 November2008; accepted 25 April 2009

bilistic evaluation of Engineering Demand Parameters(EDPs), i.e. story drifts, oor acceleration etc. as afunction of ground motion Intensity Measures (IMs),i.e. peak ground acceleration, spectral acceleration atthe �rst-mode period etc. Sensitivity of the relation-ship between EDPs and IMs to important structuraland ground motion characteristics should also be stud-ied [3]. Several research e�orts have focused on theevaluation of demands for both Single and MultipleDegrees Of Freedom (SDOF and MDOF) systems inwhich displacement demands from nonlinear responsehistory analyses have been quanti�ed as a functionof a normalized strength or ground motion intensitylevel [4-10].

Current structural engineering practice estimatesseismic demands by the nonlinear static procedure orpushover analysis detailed in the Federal EmergencyManagement Agency (FEMA-356) or Applied Technol-ogy Council (ATC-40) guidelines [11,12]. The seismicdemands are computed by nonlinear static analysis

Estimates of Deformation Demand for Frames by ET Method 389

of the structure subjected to monotonically increasinglateral forces with an invariant or variant height-wisedistribution until a target value of roof displacementis reached. This roof displacement value is determinedfrom the earthquake-induced deformation of an inelas-tic Single-Degree-Of-Freedom system derived from thepushover curve [10].

Another promising tool for estimating inelasticdeformation demands is Incremental Dynamic Analysis(IDA). In IDA, the seismic loading is scaled anddi�erent nonlinear dynamic analyses are performedto estimate the dynamic performance of the globalstructural system [13]. By using this method, EDPsof the structures can be obtained at di�erent intensitymeasures and therefore the performance of the struc-tures can be reviewed more precisely. The large numberof nonlinear dynamic analyses needed for the IDAmethod and the di�erent performances of the structureto the di�erent records applied to it are the maindrawbacks of this method in practical applications.

The Endurance Time method is basically a simpledynamic pushover test that tries to predict the EDPsof structures at di�erent IMs by subjecting themto some predesigned intensifying dynamic excitations.These predesigned excitations in the ET method arecalled \acceleration functions" in this paper in orderto clearly identify them from ground motions andsimulated accelerograms that are usually compatiblewith ground motions. ET acceleration functions aredesigned in a manner that their intensity increases withtime. In order to practically apply the ET methodas a tool for the design and assessment of structures,ET acceleration functions should preferably representdi�erent earthquake hazard levels at di�erent times,as far as possible. For this purpose, the concept ofresponse spectra has been taken advantage of in devel-oping ET acceleration functions [14,15]. Optimizationtechniques are used in order to create a set of ETacceleration functions with the property of having aresponse spectra that proportionally intensi�es withtime, while remaining compatible to a pre-speci�edtarget response spectra curve. A detailed procedure forgenerating ET acceleration functions is described in thefollowing section. Because of the increasing demandof the ET acceleration function, structures graduallygo through elastic to yielding and nonlinear inelasticphases, �nally leading to global dynamic instability.

Earlier studies have shown that in the linearseismic analysis of structures, the ET method canreproduce the results of codi�ed static and responsespectrum analysis procedures with acceptable accu-racy [14]. The compliance and level of accuracy ofthis method in the nonlinear seismic analysis of SDOFstructures has also been investigated. In this paper,the accuracy and consistency of the ET method inestimating the average inelastic deformation demands

of regular steel frames on sti� soil are examined. Thesestudies are required in order to provide a basis forpractical application of the ET method in steel framesseismic assessment and design problems. To reachthis goal, a set of ground motions is selected andtheir average response spectrum is calculated. Thisspectrum is set to be the target response spectrumfor generating a set of acceleration functions usedin this study (i.e. ETA20f series). A set of steelmoment-resisting frames with a di�erent number ofstories was used in this study. This set consistsof underdesigned, properly designed and overdesignedframes to examine the capability of the ET methodin di�erentiating dissimilar structures. An Elastic-Perfectly-Plastic (EPP) material model and a bilinearmaterial model with a post-yield sti�ness equal to 3% ofthe initial elastic sti�ness (STL) are used to study thenonlinear behavior of the frames. The results computedwith the ET method were compared to the resultsof nonlinear response history analyses. A procedureis described to �nd an equivalent time in the ETanalysis to compare its results with the results of thenonlinear response history analysis. Mean values anddispersions of the results obtained by two methods arecompared for di�erent frames. Finally, the potentialapplication of the ET method for seismic rehabilitationof structures is explained by using dampers at di�erentstories of a sample structure.

For frames with an EPP material model, whichare P�� sensitive cases, estimations of an ET analysisfor a maximum interstory drift ratio are less thannonlinear response history analysis results. A nonlinearresponse history analysis of exible structures thatare subjected to large displacements may be severelyin uenced by the P � � e�ects. For these cases,the maximum interstory drift ratio becomes very sen-sitive to ground motions that are relatively strong.Therefore, for these ground motions, P � � e�ectsdestabilize the structure and increase the maximuminterstory drift ratio drastically resulting in the averagevalue of this parameter to become unreliable andscattered. As will be shown later, average values ofdeformation demands cannot be reliably predicted bythe ET method in these cases. Unlike real earthquakes,ET acceleration functions used in this study have quitesimilar characteristics. Consequently, the dispersion ofthe results of nonlinear response history analyses forthese frames is high, but in ET analysis, the dispersionof the results is much less. However, it should be notedthat the deformations resulted from the destabilizinge�ect of P �� in EPP models are usually beyond thedrift levels that are practically important for design andare of little signi�cance. The consistency between theresults of the ET analysis and the nonlinear responsehistory analysis for the material model with 3% post-yield sti�ness is satisfactory. These frames are much

390 H.T. Riahi, H.E. Estekanchi and A. Vafai

less sensitive to P � � e�ects when subjected tostrong ground motions. The consistency of the baseshears obtained by two methods is acceptable forboth material models. As will be shown, ET can beconsidered as a useful approximate analysis procedurethat provides a practical tool in intermediate levels ofthe design process by drastically reducing the numberof required nonlinear time-history analyses at each stepof the design re�nement.

CONCEPT OF ET METHOD

The Endurance Time method is a simple dynamicpushover procedure that predicts the seismic per-formance of structures by analyzing their resiliencewhen subjected to predesigned intensifying dynamicexcitations. In this method, numerical (or experimen-tal) models of structures are subjected to intensifyingacceleration functions. Major structural responses,such as displacements, drift ratios, stresses or otherappropriate EDPs are monitored up to the desiredlimiting point where the structure collapses or failurecriteria are met. Each speci�c time in the ET analysiscan be correlated to a speci�c IM that expresses anearthquake hazard level. PBSE consists of a selectionof di�erent building performance levels for di�erentseismic hazard levels. In ET analysis, the equivalenttime for each seismic hazard level can be de�ned andthe performance of the structure until that time can becompared by prede�ned performance objectives. Basedon each di�erent structural performance level, suchas Immediate Occupancy (IO), Life Safety (LS), Col-lapse Prevention (CP) and the design or rehabilitationobjectives, the corresponding equivalent time can bede�ned and alternative designs can be compared atthese milestone times representing di�erent excitationlevels in each single analysis.

Basically, the longer that the structure can en-dure imposed excitations, it is judged to have betterperformance. In practice, the analysis or experimentneed not be continued until the real collapse of thestructure. Any convenient performance parameter,such as maximum drift, stress ratio and plastic rotationcan be considered, and the analysis or experiment canbe commenced until the desired level of excitation hasbeen covered [16].

One of the most important issues in successfulimplementation of the procedure is the determinationof a suitable acceleration function, so that the resultsfrom ET analysis (or testing) can be correlated reliablywell with the response of structures subjected to earth-quakes. The concept of response spectra can be usedin producing the intensifying ET acceleration functionsfor this purpose [14,15]. Optimization techniquesare used in order to create a set of ET accelerationfunctions with response spectrum that proportionally

intensify with time, while remaining compatible to apre-speci�ed template response spectra curve as faras possible. This means that the response spectrumof any window of these acceleration functions, fromt0 = 0 to t1 = t, resembles that of the target spectrumwith a scale factor that is proportional with time,(t) [14]. Even though other strategies can also beused to de�ne intensifying ET acceleration functions,the stated method seems to provide a well suitedacceleration function for the purposes of this study.To apply the ET method for PBSE, it is suitable togenerate acceleration functions whose response spectraare compatible with the response spectra of di�erenthazard levels. Generally, seismic hazard due to groundshaking is de�ned for any earthquake hazard levelusing spectral response acceleration. The responsespectra for di�erent hazard levels can be used forthe generation of ET acceleration functions. Theoptimization procedure for generating ET accelerationfunctions is drastically time-consuming, but once a setof acceleration functions based on earthquake hazardlevels of a seismic code is generated, it can be usedeasily for any structure.

Although current sets of ET acceleration func-tions are generated based on linear response spectra,their performance in estimating the nonlinear responseof SDOF systems has been satisfactory. In thisresearch, the accuracy of the ET analysis in estimatingaverage inelastic deformation demands is examined bycomparing its results with nonlinear response historyanalysis results. It should be noted that it is not theaim of the ET method to estimate the response ofstructures to each individual earthquake. The responsespectra used for the generation of ET accelerationfunctions are representative of the average response ofthe structures subjected to a set of earthquakes. ETanalysis is aimed to predict this average response. Fordesign purposes, the response spectrum function canbe adjusted to properly re ect the level of dispersionby applying statistical procedures. ET accelerationfunctions can then be generated based on these designspectrum. In this study, however, the average responsespectrum from seven earthquakes is directly used inorder to investigate the accuracy and consistency of ETanalyses and directly compare them with the results oftraditional time-history analyses.

ET acceleration functions that have been used inthis research are designed in such a way that theirresponse spectrum remains proportional to that ofthe average of seven strong motions recorded undera sti� soil condition. These acceleration functionsare generated to be compatible with some groundmotions to facilitate the comparison of the results ofET analysis and nonlinear response history analysis.Dynamic properties of these acceleration functions willbe discussed in the next sections.


SPECIFICATIONS OF STEEL FRAMES,GROUND MOTIONS ANDACCELERATION FUNCTIONS

A set of steel moment-resisting frames with a di�erentnumber of stories were studied. This set consists oftwo-dimensional regular generic frames with 3, 7 and12 stories. These frames have either a single bay orthree bays. The generic frames of this study are basedon the models developed by Estekanchi et al. [14].These frames are designed according to the AISC-ASD design code [17]. To facilitate the comparativestudies, frames are designed for di�erent levels oflateral load, named \Standard", \Underdesigned" and\Overdesigned". Standard frames have been designed,according to the recommendations of the INBC for ahigh seismicity area [18]. Frames are designed withthe aid of an equivalent static procedure. The nameof these frames ends with the letter S. Underdesignedframes have been designed assuming one half of thecodi�ed base shear as the design lateral load. The nameof these frames ends with the letter W. Overdesignedframes have been designed for twice the standardlateral load. The name of these frames ends with theletter O. Frame masses are considered to be the samefor these three kinds of frame. A response modi�cationfactor of R = 6 is used for the design of the frames andthe interstory drift ratio is limited to 0.005 for all ofthem per code requirements. Usually, in standard andoverdesigned frames, the story drift turns out to be thecontrolling design criteria. In underdesigned frames,

element forces control the design. The geometry andsection properties of the frames with seven storiesand one bay are depicted in Figure 1. Some of thespeci�cations of all frames are shown in Table 1.

The Elastic-Perfectly-Plastic material model andthe bilinear material model with a post-yield sti�nessequal to 3% of the initial elastic sti�ness are used tostudy the nonlinear behavior of the frames. The EPPmodel has been used widely in previous investigationsand, therefore, represents a benchmark to study thee�ect of hysteretic behavior. Furthermore, recentstudies have shown that this is a reasonable hystereticmodel for steel beams that do not experience lateralor local buckling or connection failure [19]. For morerealistic nonlinear behavior, the STL material model isalso used in this study. To apply these material modelsin the analysis, the OPENSEES beam-column elementwith nonlinear distributed plasticity is utilized [20].This element is used for the beams and columns ofthe frames to account for the nonlinearity for bothof them. Only one (horizontal) component of theground motion has been considered, while dynamicsoil-structure interaction is neglected. P � � e�ectshave been included in the analysis. A viscous dampingof 5%, as customary for these types of frame, has beenapplied in the analyses. This value of damping isconsistent with the value used for generating codi�edresponse spectra and ET acceleration functions.

To investigate the accuracy of the ET methodin estimating the nonlinear response of ground mo-tions, a set of ET acceleration functions (ETA20f)

Table 1. Speci�cations of the frames.

Frames Numberof Stories

Numberof Bays

MassParticipation

Mode 1

FundamentalPeriod (sec)

Design BaseShear (KN)

FM03B1RGW 3 1 90.98% 1.20 59.7FM03B1RGS 3 1 88.03% 0.89 116.32FM03B1RGO 3 1 85.15% 0.60 244.92






Figure 1. Schematics of the frames with seven storiesand one bay.

are generated using the average response spectrum ofground motions. To reach this goal, 20 accelerogramsthat are recorded on site class C, as de�ned by theNEHRP and used in FEMA 440, are selected [21].From these ground motions, 7 records whose responsespectra shapes were more compatible with the responsespectrum of soil type II of INBC standard 2800 are se-lected (Table 2) [22]. These 7 accelerograms are scaledto produce a response spectrum that is compatiblewith the INBC standard 2800 spectrum. Finally, theaverage of the pseudo acceleration spectrum of thesescaled accelerograms is obtained and smoothed. Thesmoothed spectrum is used as the target spectrum ingenerating new ET acceleration functions. As can be

Figure 2. Total acceleration response spectra of ETA20fseries acceleration functions for � = 5% at di�erent times.

seen in Figure 2, the response spectrum of a window ofET acceleration functions from t0 = 0 to t1 = 10, i.e.t 2 [0; 10], matches reasonably well with the averageresponse spectrum of the seven strong motion records.It is important to note that the ET response spectraremain proportional to the target spectra from sevenground motions at all times, e.g. it is 0.5 and 1.5times the target spectra at t = 5 sec and t = 15 sec,respectively. A sample acceleration function generatedin this way is shown in Figure 3. To compare theresults of ET analysis with earthquakes, a set of groundmotions (GM1) are used. The set consists of 7 recordsthat are used for the generation of the ETA20f set ofacceleration functions.

To be consistent with the seismic codes, the GM1set of ground motions should be scaled. All framesare analyzed as planar structures subjected to a singlehorizontal component of ground motion. Therefore,records are scaled individually rather than scaling themas pairs. Most codes stipulate that the ground motionsbe scaled such that the average of the ordinates of the5 percent-damped linear response spectra does not fallbelow the design spectrum for the period range 0:2 Ti to

Table 2. Description of GM1 set of ground motions used in this study.

Date EarthquakeName

Magnitude(Ms)

StationNumber

Component(deg)

PGA(cm/s2)

Abbreviation

06/28/92 Landers 7.5 12149 0 167.8 LADSP000

10/17/89 Loma Prieta 7.1 58065 0 494.5 LPSTG000

10/17/89 Loma Prieta 7.1 47006 67 349.1 LPGIL067

10/17/89 Loma Prieta 7.1 58135 360 433.1 LPLOB000

10/17/89 Loma Prieta 7.1 1652 270 239.4 LPAND270

04/24/84 Morgan Hill 6.1 57383 90 280.4 MHG06090

01/17/94 Northridge 6.8 24278 360 504.2 NRORR360


1:5 Ti where Ti is the fundamental period of vibrationof each frame modeled as a linear system. Here,scale factors are obtained in a way that the groundmotion spectrum matches the ASCE-7 spectrum in thementioned range [23]. Scale factors obtained by thismethod for the GM1 set are shown in Table 3 for eachframe.

COMPARISON BETWEEN THE RESULTSOF ET ANALYSIS AND NONLINEARRESPONSE HISTORY ANALYSIS

For each set of ground motions and acceleration func-tions, the mean value and standard deviation of thespeci�ed EDP can be calculated. For example, for aset of ground motions, the mean value and standarddeviation of the EDP can be calculated by the followingequation:

EDPex =1n

nXi=1

EDPex;i; (1)

�ex =

vuuut nPi=1

(EDPex;i � EDPex)2

n� 1: (2)

In this equation, EDPex;i is the value of EDP for aground motion, n is the number of ground motions inthe set, EDPex is the mean value of EDP for the setand �ex is the standard deviation of it. Similar valuescan be calculated for the set of acceleration functions.

An important question is how the results of twomethods can be compared. Results of ET analysis areobtained through time and as mentioned before in thismethod, the time is correlated with IM. Therefore,di�erent values of EDP are calculated for di�erentvalues of IM in an ET analysis. To establish a relationbetween the results of the ET method and any othermethod, the IM value of the other method shouldbe found in the ET analysis. Therefore, a procedureshould be de�ned to �nd an equivalent time in the ETanalysis in which the IM values of the two methods areequal.

Many quantities have been proposed to charac-terize the intensity of a ground motion record. Inthe ET method, intensity increases through time and,therefore, scalable IMs can be conveniently used for thismethod. Common examples of scalable IMs are PeakGround Acceleration (PGA), Peak Ground Velocity(PGV) and Spectral Acceleration at the structure's�rst-mode period (Sa(T1)). In this research, �rst-modespectral acceleration (Sa(T1)) is used as IM to obtainthe equivalent time. Most frames used in this studyare �rst-mode dominated structures that are sensitiveto the strength of the frequency content near their�rst-mode frequency, which is well characterized bySa(T1), but not by PGA. Moreover, researchers showthat Sa(T1) produces a lower dispersion over the fullrange of EDP values in IDA analysis [13]. Certainly,other IMs can be used easily to calculate the equivalenttime. The equivalent time can be calculated for a singlerecord or a set of records. To compare the results ofa set of records with the results of ET analysis, the

Table 3. Scale factors of GM1 set for di�erent frames.

Frames Scale Factors EquivalentLPAND270 LADSP000 MHG06090 LPGIL067 LPLOB000 NRORR360 LPSTG000 Time (sec)

FM03B1RGW 2.89 3.97 1.74 2.35 2.63 1.11 1.61 10.79FM03B1RGS 2.55 3.66 1.61 2.12 1.87 1.15 1.76 10.26FM03B1RGO 2.36 3.64 1.79 1.91 1.60 1.31 1.87 9.63






Figure 3. a) ETA20f02 acceleration function; b)ETA20f02 velocity function; and c) ETA20f02displacement function.

average of the �rst-mode spectral acceleration of therecords (Sa;Ave) should be calculated. Furthermore,the value of the smooth response spectrum used forthe generation of ET acceleration functions at the �rst-mode period (T1) is calculated (Sa;ET). Finally, theequivalent time is obtained by the following equation:

teq =Sa;Ave

Sa;ET� 10: (3)

Constant 10 is used in this equation because theresponse spectrum of the ET acceleration function att = 10 seconds matches the target smooth responsespectrum. Equivalent times of each frame for the GM1set of records are shown in Table 3.

As seen in Table 3, for frames that have longperiods, the equivalent time is larger than 10 seconds,which is the target time in the generation of ETacceleration functions. The main reason for obtainingsuch a large equivalent time is the scaling procedure

used for the records. For example, the equivalenttime for a FM12B3RGW frame is 15.36 sec. Theperiod of this frame is 2.89 sec and, therefore, thescaling procedure is done for the range of 0.578 to4.335 sec. In this range, the smooth spectrum usedfor the generation of ET acceleration functions is lessthan the ASCE-7 spectrum (Figure 4). Therefore, largescale factors are used for this frame (Table 3). Itcan be concluded that for a better comparison of theresults of ET analysis with the results of other methods,the response spectrum used for the generation ofacceleration functions should be consistent with thespectrum used in other methods in a wide range ofperiods (e.g. 0 to 5 sec).

All the frames have been subjected to the ETA20fset of acceleration functions and GM1 set of groundmotions. The response data summarized in thisresearch are part of a comprehensive database on EDPsacquired for the previously de�ned generic frames.The discussion presented here focuses on maximuminterstory drift ratios for frames. This EDP is relevantto structural damage, if the damage is dominated bythe maximum story deformation over the height andis a measure of damage to nonstructural components.For P�� sensitive structures, the maximum interstorydrift ratio is the most relevant EDP for global collapseassessment, because dynamic instability is controlledby the story in which the story drift grows mostrapidly [3]. To examine the consistency of the IM ofnonlinear response history analysis and IM obtainedfrom ET analysis, the base shears of the framesobtained by two methods are compared.

The average of maximum interstory drift ratios forET acceleration functions through time and values ofequivalent time are presented for FM03B1RG framesin Figure 5. These results are obtained consideringthe EPP material model for the frames. It should benoted that ET analysis results are usually presented by

Figure 4. Comparison of smooth spectrum and ASCE-7spectrum with the average spectrum of the scaled recordsfor FM12B3RGW frame.


Figure 5. ET maximum interstory drift ratio curves forFM03B1RG frames with EPP material model and valuesof teq of nonlinear response history analysis.

increasing curves, where the y coordinate at each timevalue, t, corresponds to the maximum absolute valueof the required parameter in the time interval, [0; t], asgiven in Equation 4:

(f(t)) � max(Abs(f(�) : � 2 [0; t]); (4)

where is the Max-Abs operator as de�ned aboveand f(t) is the response history, such as base shear,interstory drift, damage index or other parameters ofinterest.

Because of the statistical characteristics and dis-persion of ET analysis results in a nonlinear range,the resulted curves are serrated. Sometimes, the valueof the response does not pass the maximum valueexperienced before in a long time interval. Therefore,the resulted ET curve has a constant value in thatinterval. In this research, a moving average procedure

is used to reduce the serrated nature of the ET curvesin a nonlinear range.

Looking through a typical ET curve can revealsigni�cant information about a structure. During theinitial phase of the excitation, the structure behaveslinearly until it reaches a certain point where a plastichinge is created, i.e. plastic behavior. By increasing theintensity of the acceleration function through time, thestructure experiences more plastic deformations until itreaches the collapse limit. Occurrence of the collapsefor a structure through an ET analysis is dependenton its lateral sti�ness. For example, a FM03B1RGWframe experiences signi�cantly high displacements after12 second. But, the two other frames do not collapsebefore the 20th second, as can be seen in Figure 5.

Maximum interstory drift ratios of FM03B1RGframes with an EPP material model obtained froma nonlinear response history analysis are presented inFigure 6. As can be seen in Figure 6, all the recordsrank the structures based on their lateral sti�ness.This result can be seen in ET curves too. Maximuminterstory drift ratios of a FM03B1RGW frame forNRORR360 and LADSP000 records are far beyondthe average values of other records. In a nonlinearresponse history analysis, such exceptions can signi�-cantly change the average value of the response. It willbe discussed later that the main cause of the di�erencesbetween the results is the P �� e�ect. As can be seenin Table 4, the ET method underestimates the resultsof this frame. It means that the ET method could notestimate these exceptions well.

Figures 7 and 8 show the interstory drift ratioresponse history of the FM03B3RGW frame for theLPAND270 record and the ETA20f02 accelerationfunction, respectively. Most of the records and all ETacceleration functions anticipate that the maximum

Figure 6. Maximum interstory drift ratios of the accelerograms and their average for FM03B1RG frames with EPPmaterial model.


Table 4. Comparison between the results of nonlinear response history analysis and ET analysis for di�erent frames withEPP material model.

Maximum Interstory Drift Ratio Base Shear (KN)Frames Average Standard Deviation Average

Nonlinear TimeHistory Analysis

ETAnalysis


ETAnalysis


ETAnalysis

FM03B1RGW 0.043 0.034 0.021 0.004 218.1 217.8FM03B1RGS 0.024 0.021 0.009 0.003 323.0 335.6FM03B1RGO 0.015 0.015 0.005 0.001 505.3 558.3

FM03B3RGW N.A. 0.057 N.A. 0.022 N.A. 543.6FM03B3RGS 0.045 0.025 0.037 0.003 912.5 933.7FM03B3RGO 0.017 0.014 0.007 0.002 1424.7 1513.3


FM07B3RGW N.A. 0.037 N.A. 0.007 N.A. 738.1FM07B3RGS 0.025 0.023 0.003 0.005 1256.4 1274.4FM07B3RGO 0.016 0.015 0.003 0.003 2098.2 2109.0


interstory drift ratio is obtained in the second storyfor this frame. The di�erence between the results ofthis story and the others is signi�cant in high intensitymeasures.

As another example, the average of maximuminterstory drift ratios for ET acceleration functionsthrough time are presented for FM07B1RG frames withan EPP material model in Figure 9. As can be seen,the general trend of the results is like the FM03B1RGframes results, but there are some exceptions too. ETresults show that between t = 9 to 13 seconds, the

Figure 7. Interstory drift ratio response history ofFM03B3RGW frame for LPAND270.

maximum interstory drift ratio of the FM07B1RGWframe is less than that obtained for the FM07B1RGSframe. This estimation of the ET method can bechecked by a nonlinear response history analysis. Todo so, a reduction scale factor is applied to the GM1set to change the equivalent time of the FM07B1RGWframe to the equivalent time of the FM07B1RGS frame.Finally, the FM07B1RGW frame is analyzed for theGM1 set with this reduction scale factor. Now, theresults of FM07B1RGW and FM07B1RGS frames canbe compared at teq = 11:47 seconds. At this time,

Figure 8. Interstory drift ratio response history ofFM03B3RGW frame for ETA20f02.



the maximum interstory drift ratio of the FM07B1RGSframe obtained from the ET analysis is larger than thecorresponding value of the FM07B1RGW frame. But,the results of the nonlinear response history analysisare vice versa. The maximum interstory drift ratio ofthe FM07B1RGS frame is 0.0226 and the correspond-ing value for the FM07B1RGW frame at teq = 11:47seconds is 0.024. It shows that when the EDPs ofthe frames are very close together, small di�erencesin ET analysis curves might be a randomness e�ectand should not be interpreted as an indication thatone structure has better performance over the other.In these cases, performance di�erences may actuallybe insigni�cant. A more re�ned analysis, using moreground motions and ET acceleration functions, arerequired if a de�nitive conclusion is to be made in suchcases.

Figure 10 shows the average of the maximuminterstory drift ratios of FM07B3RG frames with an


EPP material model for ET acceleration functionsthrough time. Like the previous examples, ET curvesdi�erentiate between three frames. Between t = 18 to20 seconds, the maximum interstory drift ratios of theFM07B3RGO frame are larger than the correspondingvalue for the FM07B3RGS frame. This can be checkedby doing a nonlinear response history analysis forthe equivalent IM for this range of time. To do so,increasing scale factors are applied to the GM1 setfor FM07B3RGO and FM07B3RGS frames. Thesescale factors change the equivalent time of these framesto 20 seconds. Maximum interstory drift ratios ofFM07B3RGS and FM07B3RGO frames at teq = 20seconds are 0.0444 and 0.0663, respectively. Thistime, the results of the nonlinear response historyanalysis are consistent with the results of the ETanalysis.

The average and standard deviation of maximuminterstory drift ratios and the average of base shearsobtained from a nonlinear response history analysisand the ET analysis of the frames with the EPPmaterial model are compared in Table 4. It shouldbe noted that for two cases of underdesigned frames,the nonlinear response history analyses of some groundmotions did not converge and, therefore, no results arepresented for them. In most frames, estimations of ETanalysis for the maximum interstory drift ratio are lessthan nonlinear response history analysis results. Thedi�erence between the results is more in underdesignedframes which experience more nonlinearity in theiranalyses. Usually, in these frames, the dispersion of theresults of nonlinear response history analyses is high.But, in frames that behave more linearly than others,the di�erence between the results of ground motions isless and the results match the results of the ET analysisbetter.

Table 4 shows that the consistency of the baseshears obtained by two methods is acceptable. It meansthat the procedure to �nd the equivalent time in theET analysis to match the IMs of the two methodsworks well. It should be noted that although theequivalent time tries to show consistency between theIMs of the two methods, this is done just for oneperiod. Therefore, the average response spectrum ofET acceleration functions at t = teq and the averageresponse spectrum of scaled accelerograms have someminor di�erences, which cause the inconsistency of thebase shears in some frames like the FM03B1RGO andFM03B3RGO. The di�erence between the base shearsobtained by two methods is more in overdesignedframes, and in overdesigned and properly designedframes, the ET method always overestimates the baseshear.

Figure 11 shows the mean and mean plus andminus standard deviation (STDEV) of maximum in-terstory drift ratios of frames with the EPP material


Figure 11. Maximum interstory drift ratios of the frameswith EPP material model obtained by nonlinear responsehistory analysis and ET analysis.

model obtained by nonlinear response history analyses.The average of maximum interstory drift ratios ofthe frames obtained by ET analysis is also shown inthis �gure. If this �gure is compared with Table 4,it can be concluded that when the dispersion of theresults is high, the di�erence between the resultsof the two methods is high either. For example,Figure 11 shows that the dispersion of the results ofunderdesigned frames is larger than properly designedand overdesigned frames. For these frames, Table 4and Figure 11 show the maximum di�erence betweeninterstory drift ratios obtained by two methods. Forthe FM03B3RGS frame where the dispersion is high,the di�erence between the results of the two methodsis high either.

The P � � e�ect increases the dispersion of theresults of nonlinear response history analyses. The

nonlinear seismic response of steel moment-resistingframe structures, which are usually quite exible,may be severely in uenced by the structure P � �e�ect. This occurs especially when these structuresare subjected to large displacements under severeground motions. For structures in which this ef-fect induces negative post-yield story sti�ness, theresponses become very scattered under severe groundmotions [7].

Figure 12 compares the results of the nonlinearresponse history analysis of the FM03B1RGW framewith the EPP material model by considering and elim-inating the P �� e�ect. In most of the accelerograms,the maximum interstory drift ratio obtained for themodel, eliminating the P � � e�ect, is less than themodel considering this e�ect. The largest di�erencebetween these results is obtained for LADSP000 andNRORR360 records. Also, the responses of the frameto these records are the largest in the GM1 set. Byeliminating the P � � e�ect, the results of theserecords approach the average response for the GM1set and the dispersion of the results reduces signi�-cantly.

The same trend can be seen in the results ofthe ET analysis. Figure 13 compares the results ofthe ET analysis of the FM03B1RGW frame with theEPP material model by considering and eliminatingthe P � � e�ect. It can be seen that the curves areseparated from each other at about t = 11 seconds.After this time, the P�� e�ect increases the maximuminterstory drift ratio of the frame. As can be seen inthe �gure, the P �� e�ect changes the EDP after theequivalent time computed for the nonlinear responsehistory analysis. In other words, e�ects of P � � onthe results of the ET analysis and nonlinear responsehistory analysis are not seen at the same IM. The

Figure 12. Maximum interstory drift ratios of the accelerograms and their average for FM03B1RGW frame with EPPmaterial model with and without considering P �� e�ects.


Figure 13. ET maximum interstory drift ratio curves forFM03B1RGW frame with EPP material model with andwithout considering P �� e�ects.

reason for this phenomenon is the di�erence in natureof acceleration functions and ground motions. Forsome of the ground motions, P � � e�ects increasethe maximum interstory drift ratio drastically and theaverage value of this parameter is changed a lot. Thisis due to the fact that the characteristics of the groundmotions are really di�erent from each other. Unlike realearthquakes, ET acceleration functions have similarcharacteristics. Consequently, the dispersion of theresults of nonlinear response history analyses for theseframes is high but, in ET analysis, the dispersion of theresults is not signi�cant.

P � � e�ects are not critical if the e�ectivesti�ness at maximum displacement remains positive [7].As mentioned before, the EPP material model wasused for previous models. Because this material modelhas no strain hardening, the frames tend to reach thenegative post-yield sti�ness especially in underdesignedframes and, therefore, P � � e�ects can change theirdrifts a lot. If the STL material model that has 3%post-yield sti�ness is used instead of the EPP materialmodel, it can be guessed that the results of the ETanalysis approach nonlinear response history analysisresults.

Figure 14 compares ET maximum interstory driftratio curves for FM03B1RG frames with the EPP andSTL material model. As can be seen, by increasingthe strain hardening of the material model, the P ��e�ects are decreased. If this �gure is compared withFigure 13, it can be judged that ET results of theFM03B1RGW frame with the STL material model,considering P �� e�ects, are very similar to the resultof this frame with the EPP material model eliminatingP � � e�ects. The average and standard deviationof maximum interstory drift ratios, and the averageof base shears obtained from the nonlinear responsehistory analysis, and ET analysis of the frames withthe STL material model are compared in Table 5. If

Figure 14. ET maximum interstory drift ratio curves forFM03B1RG frames with EPP and STL material model.

this table is compared with Table 4, it can be seenthat by considering 3% strain hardening, the di�erencesbetween the results of the two methods are totallydecreased. It can be concluded that for structures thatare extremely sensitive to P �� e�ects, the results ofET analysis should be used with special care. To avoidunderestimated values for EDPs, such as a maximuminterstory drift ratio, it is better to de�ne a limit forthe lateral deformation of the structure. This limitshould specify the onset of reaching negative post-yieldstory sti�ness. The interstory drift ratio obtained atthe maximum base shear of a pushover curve can be agood value for this limit.

Although the maximum interstory drift ratios andcorresponding base shears are calculated at teq, thesevalues do not necessarily happen at the same time. ETcurves are obtained by the Max-Abs operator describedin Equation 4 and perhaps the maximum value for thebase shear is not obtained at the same time that themaximum value of the maximum interstory drift ratiois gained.

The results of the ET analysis reported in thisresearch can be used in order to estimate the meandemand structures due to ground motions. However,a scatter of the results from earthquakes is not antici-pated by the ET method. Therefore, a safety marginfor expecting seismic demands should be de�ned. Inspectral analysis, this issue is taken into account byusing the mean spectrum plus the standard deviation ofthe ground motions. The same method can be used inET analysis. In addition to generating ET accelerationfunctions based on a mean spectrum, another set ofacceleration functions can be generated by assumingthe mean spectrum plus the standard deviation asthe target spectrum. The results of the ET analysisobtained for this set can be used as an upper estimationof the results of ground motions and it can be addressedas a safety margin in the ET analysis.


Table 5. Comparison between the results of nonlinear response history analysis and ET analysis for di�erent frames withSTL material model.

Maximum Interstory Drift Ratio Base Shear (KN)Frames Average Standard Deviation Average


ETAnalysis


ETAnalysis


ETAnalysis






APPLICATION OF ET METHOD INSEISMIC REHABILITATION OFBUILDINGS

The analysis of underdesigned, properly designed andoverdesigned frames discussed in the previous sectionwere for explanatory purposes and their relative per-formances could be guessed without advanced analysis.In order to further demonstrate the signi�cance of ETanalysis, the capability of the ET method in a moreinvolved situation is demonstrated in this section.

As was shown in Table 4, the maximum interstorydrift ratio of the FM03B3RGW frame with the EPPmaterial model obtained by ET analysis is 0.057. Thenonlinear response history analysis of this frame doesnot converge for two ground motions and it can bejudged that the performance of this frame is notacceptable. Now let us assume that the maximumacceptable interstory drift ratio for this frame hasbeen set to 0.04. In order to improve the frame'sperformance, a viscoelastic damper is to be used. Thequestion to be answered is, at which story level shoulda damper with a damping constant of 2000 KN.sec/mbe installed in order to result in the best performance?i.e. the least overall maximum interstory drift ratio.

ET analysis and nonlinear response history anal-ysis are done for this frame by installing the damperin di�erent stories. Results of the analysis are shownin Figure 15 and Table 6. Figure 15 shows that byinstalling the damper in the 3rd story, no signi�cant

Figure 15. ET maximum interstory drift ratio curves forFM03B3RGW frames with di�erent locations for dampers.

performance improvement is achieved. The same resultis also obtained by comparing the values of maximuminterstory drift ratios for di�erent records obtainedfrom nonlinear response history analyses. Analyses donot converge for LADSP000 and NRORR360 recordsfor both cases. Another point that can be concludedfrom Figure 15 is that, by installing the damper inthe 1st or 2nd story, the performance of the frameimproves. The best performance is obtained when thedamper is installed in the 2nd story. The performanceof this case is really better in high IMs. Maximuminterstory drift ratios of the frames having dampersin the 1st or 2nd story at the equivalent time are


Table 6. Comparison between the results of nonlinear response history analysis for FM03B3RGW frames with di�erentlocations for dampers.

Maximum Interstory Drift Ratio

Records No Damper Damper in1st Story

Damper in2nd Story

Damper in3rd Story

LPAND270 0.0699 0.0401 0.0246 0.0534

LADSP000 N.A. 0.0219 0.0497 N.A.

MHG06090 0.0434 0.0534 0.0468 0.0456

LPGIL067 0.0244 0.0199 0.0250 0.0231

LPLOB000 0.0290 0.0241 0.0221 0.0287

NRORR360 N.A. 0.2797 0.0504 N.A.

LPSTG000 0.0444 0.0314 0.0335 0.0362

0.0344 and 0.0272, respectively. Again, the results ofnonlinear response history analyses con�rm the resultsof the ET analysis. Nonlinear response history analysisshows that the averages of maximum interstory driftratios of the frames having dampers in the 1st or 2ndstory are 0.0672 and 0.0360, respectively. The resultsof ET analysis somewhat underestimate the results ofearthquakes again because of the P �� e�ects.

As mentioned before, one of the bene�cial ad-vantages of the ET method is its capability in di�er-entiating between di�erent structural systems with aminimum number of analyses. The previous exampleshows that although the results of the ET analysisare not exactly consistent with the results of groundmotions analysis, the ET method can pinpoint thestructure with a better performance, even in the case ofstructures with relatively complicated behavior. Thiscan be put into good use in performance-based seismicdesign where a lot of trial designs should be checked inorder to �nd the structure that meets the performanceobjectives in an optimal manner.

SUMMARY AND CONCLUSIONS

In most frames with an EPP material model, esti-mations of ET analysis for the maximum interstorydrift ratio are less than nonlinear response historyanalysis results. The di�erence between the resultsis more in underdesigned frames, which experiencemore nonlinearity in their analysis. But, in framesthat behave more linearly than others, the di�erencebetween the results of ground motions is less and theresults match the results of ET analysis more closely.The consistency of the base shears obtained by twomethods is reasonable. The procedure to �nd theequivalent time in the ET analysis to match the IMsof two methods is acceptable. The ET method issuccessful in locating the story with the maximuminterstory drift ratio.

The dispersion of the results of the nonlinear

response history analysis for the frames with the EPPmaterial model that experience more nonlinearity isrelatively high. The dispersion of results cannotbe estimated using current sets of ET accelerationfunctions. When the dispersion of the results of thenonlinear response history analysis is high, ET analysisunderestimates the maximum interstory drift ratiomore signi�cantly.

The main reason for the dispersion of the resultsof the nonlinear response history analysis is P � �e�ects. In cases where P �� e�ects are excluded, theresults of the two methods closely match. The samephenomenon can be seen in the results of ET analysis,but e�ects of P �� on the results of ET analysis showup at a higher IM as compared to a nonlinear responsehistory analysis. Frames with an EPP material modelare more sensitive to P � � e�ects, because theydevelop a negative post-yield sti�ness under severeground motions. Therefore, for some of the groundmotions, P�� e�ects increase the maximum interstorydrift ratio drastically and the average value of thisparameter is a�ected considerably making it unusable.

In STL material models that have 3% post-yieldsti�ness, the results of the ET analysis match thenonlinear response history analysis results with goodprecision.

It is shown that, although the results of the ETanalysis are not exactly consistent with the results ofthe ground motions analysis in all cases of materialproperties and IMs, in most cases the ET method isquite successful in di�erentiating between structures(or design alternatives) with better performance, evenin the case of relatively complicated structures. Itshould be noted that when the ET analysis estimationfor the EDPs of the frames shows closely identicalresults, some reservations should be considered beforedrawing a conclusion. The di�erences observed in thesecases may have been resulted from pure randomnesse�ects. A detailed nonlinear response history analysis,using a relatively large set of relevant earthquakes, is


needed for the �nal veri�cation in cases where twosystems show a very close response.

ACKNOWLEDGMENTS

The authors would like to thank Sharif University ofTechnology Research Council and the Structures andEarthquake Engineering Center of Excellence for theirsupport of this Research.

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