application of endurance time method in performance …

Transaction A: Civil EngineeringVol. 17, No. 6, pp. 482{492c Sharif University of Technology, December 2010

Research Note

Application of Endurance TimeMethod in Performance-Based

Design of Steel Moment Frames

A. Mirzaee1, H.E. Estekanchi1;� and A. Vafai1

Abstract. In this paper, application of the Endurance Time (ET) method in the performance-baseddesign of steel moment frames is explained from a conceptual viewpoint. ET is a new dynamic pushoverprocedure that predicts the seismic performance of structures by subjecting them to a gradually intensifyingdynamic action and monitoring their performance at various excitation levels. Structural responses atdi�erent excitation levels are obtained in a single time-history analysis, thus signi�cantly reducing thecomputational demand. Results from three analyses are averaged to reduce the random scattering of theresults at each time step. A target performance curve is presented based on the required performancecriteria, as a continuous function of an increasing intensity measure. The actual performance is thenplotted against this target performance based on the results of ET analysis. The overall performance of thestructure can be anticipated by comparing the target to actual performance at various intensity levels andthe design can be improved based on the observed performance. Results are indicative of a good potentialfor application of the ET method in the performance-based design of steel moment frames.

Keywords: Dynamic pushover analysis; Performance-based design; Endurance time method; Intensifyingaccelerograms.

INTRODUCTION

Seismic performance of structures during strong earth-quakes is one of the most sensitive considerations,regarding their safety and economic requirements, tobe set as their design criteria. Nowadays, usually theowners like to know the performance of their structuresduring an earthquake in order to make relevant eco-nomic decisions with a reasonable level of con�dence.This interest has led engineers to develop methodsfor designing structures that are capable of deliver-ing a predictable performance during an earthquake.Performance-based earthquake engineering essentiallyconsists of various procedures, whereby the structureis ensured for an acceptable seismic performance. Theprocedure involves identi�cation of the hazard levelfor the site, development of conceptual, preliminary

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

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

Received 31 January 2010; received in revised form 24 May 2010;accepted 2 October 2010

and �nal structural designs, construction, and themaintenance of a building during its lifetime [1].

As per FEMA-302 NEHRP Recommended Pro-visions for the Seismic Regulations for New Buildingsand Other Structures (BSSC, 1997) [2] and FEMA 273NEHRP Guidelines for the Seismic Rehabilitation ofBuildings (BSSC, 1997) [3], three Performance Levels(PLs) are considered. These are termed Immediate Oc-cupancy (IO), Life Safety (LS) and Collapse Prevention(CP). In the �rst damage state Immediate Occupancy(IO), only minor structural damages are visible andno substantial reduction in building gravity or lateralresistance has occurred. In the Life Safety (LS)level, although signi�cant damage to the structure hasoccurred, structural elements have enough capability toprevent collapse. The Collapse Prevention (CP) level isde�ned as the post-earthquake damage state in whichcritical damages are occurred and the structure is onthe verge of experiencing collapse [3]. Another practicalnotion is \Performance Objective", which consists ofthe speci�cation of a structural performance level (e.g.collapse prevention, life safety, or immediate occu-

Application of ET Method in P-BD of Steel Moment Frames 483

pancy) for a given level of seismic hazard. For example,in accordance with SAC 2000, ordinary buildings areexpected to provide less than a 2% chance over 50 yearsof damage exceeding CP performance [1].

Evaluating the performance of existing struc-tures during an earthquake is another important taskthrough which the operational situation of a structureduring and after the event can be predicted. The per-formance evaluation consists of structural analysis withcomputed demands on structural elements comparedagainst speci�c acceptance criteria provided for eachof the various performance levels [4]. These acceptancecriteria are really some limitations that are speci�ed forvarious structural parameters (such as interstory driftand plastic rotation at joints) at di�erent performancelevels.

The performance evaluation might sound astraightforward process, but in reality it is not asimple undertaking. The erratic nature of earthquakes,uncertainties in the existent analysis methods and lackof enough information about the current strength of thestructures are some factors that make the procedureintricate.

In this paper, a new methodology for extendingthe application of Endurance Time method into thearea of performance-based design is introduced [5]. Inthe ET method, the structural responses at di�erentexcitation levels are obtained in a single time-historyanalysis, thus signi�cantly reducing the computationaldemand. So using the ET method and regarding theconcepts of performance-based design, the performanceof a structure at various levels of seismic hazard canbe predicted in a single time-history analysis. Inother words, one can estimate if the structure satis�esits performance objectives by only one time-historyanalysis. This characteristic of the ET method can bestbe utilized by extending its concepts to incorporate thebasic concepts from performance-based design, which isthe main purpose of this paper.

As will be explained in detail, in this proposedmethodology, two new concepts of continuous \Per-formance Curves" and generalized \Damage Levels"are incorporated. Utilizing some equations, such asGutenberg-Richter equations, the equivalent endurancetimes corresponding to each PL are obtained. A newlyintroduced \Target Performance Curve" or \TargetCurve" representing the required performance as acontinuous function of the excitation level is drawn.The de�nition of the mentioned curve is accomplished,assuming that the PLs are continuous, and, theoreti-cally, it is possible to consider an unlimited number ofPLs between these three previously said levels in theform of a continuous curve. Then in order to have amore versatile numerical presentation of the PLs, anindex is introduced called \Damage Level" (or \DL"),which is de�ned as a numerical index in the arbitrary

range of (0, 4) for the purpose of this study. Integernumbers in this interval are representatives of codi�edPLs, thus a convenient numerical equivalent is createdfor each PL. As will be explained, this index is used inorder to draw a continuous target curve as the indicatorof PLs.

The actual performance is then plotted againstthis target performance based on the results of ETanalysis. The overall performance of the structure canbe anticipated by comparing the target to the actualperformance and the design can be improved based onthe observed performance.

Finally, the performance of three steel momentframes is evaluated using the target curve, and theadvantages and limitations of this procedure are ex-plained. As will be explained, by providing a good es-timate of structural performance at di�erent excitationlevels in each response-history analysis, the ET methodcan considerably reduce the huge computational e�ortrequired for the practical performance design of struc-tures. Also the concepts of \Performance Curve" and\Damage Level" provide a simpli�ed presentation ofperformance analysis results that can be used as a toolin practical design cases.

Endurance Time Concept

Endurance Time (ET) is a new dynamic pushoverprocedure that predicts the seismic performance ofstructures by subjecting them to a gradually intensify-ing dynamic action and monitoring their performanceat di�erent excitation levels. Structures that canendure the imposed intensifying acceleration functionfor longer are expected to be capable of sustainingstronger seismic excitation. In fact, since intensifyingacceleration functions are used, the time axis in ETanalysis can be correlated to the intensity of excita-tion [6].

The concept of the ET method can be physicallypresented by a hypothetical shaking table experiment.Three di�erent frames with unknown seismic propertiesare �xed on the table and the table is subjected to anintensifying shaking. After a short time, one of theframes (assume frame 1) fails. The second model willalso fail as the amplitude of the vibration is increased.Assume that this happens to frame number three.Considering the times at which the models have beenfailed, and regarding that the lateral loads induced bythe shaking table somehow correspond with earthquakeloads, one can rank the three frames according to theirseismic resistance. Hence, here, the endurance timeof each structure against intensifying shaking can beconsidered as the seismic resistance criterion. In thishypothetical analysis, the second frame is the strongestor the best performer, the �rst frame is the worst andthe third's performance is somewhere in between [7].

484 A. Mirzaee, H.E. Estekanchi and A. Vafai

Figure 1. demand/capacity of frames under accelerationfunction action [6].

In Figure 1, the schematic result of the above hy-pothetical analysis is presented. The demand/capacityratio has been calculated for these frames as the maxi-mum absolute value of the endurance index during thetime interval from 0.0 to t. Since a structure collapseswhen its demand/capacity ratio exceeds unity, theendurance time for each frame can be easily derivedfrom this �gure.

To start an ET time history analysis, after arepresentative model has been constructed, one shouldset a suitable damage measure and an appropriateET accelerogram (Figure 2). The analysis results areusually presented by a curve in which the maximumabsolute value of the damage measured in the timeinterval [0; t] (as given in Equation 1) corresponds totime.

(f(t)) � max(Abs(f(�) : � 2 [0; t]): (1)

In the above equation, is a max�Abs operatorand f(t) is the desired structural response history,such as interstory drift ratio, base shear or damageindex. For application of the ET method, intensifyingaccelerograms are generated in such a way as to pro-duce dynamic responses equal to the desired responsespectrum (such as code's design spectrum) at a pre-de�ned time, tTarget [7]. If such an accelerogram weredesigned and used, it would be possible to compare

Figure 2. Typical ET accelerogram.

the results of the ET time history analysis with thoseobtained from other analysis methods and, moreover,to compare the performance of di�erent structures withdi�erent periods of free vibration. The �rst suggestedintensifying accelerograms for ET had a linear inten-si�cation scheme, i.e. the response spectrum of anET accelerogram should intensify proportionally withtime. Hence, the target acceleration response of anET accelerogram can be related to the codi�ed designacceleration spectrum as:

SaT (T; t) � SaC(T )� ttTarget

; (2)

where SaT (T; t) is the target acceleration response attime t, T is the period of free vibration and SaC(T ) isthe codi�ed design acceleration spectrum.

Using unconstrained optimization in the timedomain, the problem was formulated as follows:

Find ag(t)j 8 T 2 [0;1);

t 2 [0;1)! (�u(t)) = SaT (T; t): (3)

Or:

Minimize F (ag(t))=TmaxZ0

tmaxZ0

fAbs[Sa(T; t)�SaT (T; t)]

+ ��Abs[Sd(T; t)� SdT (T; t)]gdt dT; (4)

where ag is the ET accelerogram being sought,SaT (T; t) and SdT (T; t) are the target accelerationresponse and displacement response at time t, re-spectively, Sa(T; t) and Sd(T; t) are the accelerationresponse and displacement response of the accelerationfunction at time t, respectively, � is a weight parameterset to 1 and T is the period of free vibration [6].

It should be noted that based on the mentionedlinear scheme, di�erent sets of ET accelerograms canbe generated according to their compatibility withdi�erent spectrums and in di�erent ranges (linearrange or nonlinear range). Each set consists of agroup of intensifying acceleration functions (usually3). For example, three acceleration functions named\ETA20d01-03" or brie y \d series" in this study, arecreated in such a way that the response spectrum att = 10 sec would be compatible with the INBC 2800design spectrum and support nonlinear ranges.

Sample response spectra generated using varioustime windows of ET accelerograms are shown in Fig-ure 3. In this �gure, the curves are taken as averagevalues between the results of three accelerograms of thed series. As can be seen in this �gure, the responsespectra produced by the ET acceleration functionproportionally grow with time.


Figure 3. Acceleration response generated by ETaccelerograms at various times.

Performance Levels

Performance levels are structural damage states thatmust be clearly de�ned as one of the �rst steps in aperformance-based design procedure. These levels areusually expressed as some distinct bands in the damagespectrum of a structure, and divide the damage statusof structures according to the amount of damage tostructural and nonstructural components. Moreover,some other concepts, such as cost, repair time andinjury can also be related to performance levels [8].

Noting that the Performance Levels (PLs) areusually investigated at some speci�c levels of designearthquake motion, they can be thought of as a criteriafor limiting values of measurable structural responseparameters (such as interstory drift and absolute accel-eration and displacement) under each mentioned levelof earthquake motion.

There are three performance levels consideredby FEMA-273 [3]: Immediate Occupancy (IO), LifeSafety (LS) and Collapse Prevention (CP), which weredetermined according to structural damages observedin earthquakes. For example, at the Immediate Occu-pancy Level, the building has experienced limited dam-age, since at the Collapse Prevention Level, damage isrelatively extreme.

On the other hand, in FEMA-356 Prestandardand Commentary for the Seismic Rehabilitation ofBuildings, four levels of probabilistic earthquake haz-ard are de�ned [9]. Combining these levels with thePLs, a table of Performance Objectives (POs) can becreated. These objectives are di�erent according tothe type of structure that is to be built. For example,if a hospital is planned to be built, an appropriatePO might be that it is capable of meeting the LSperformance level in an earthquake with a mean returnperiod of 2475 years, and the IO performance level inan earthquake with a mean return period of 475 years.So if one wants to evaluate the performance of a speci�c

Table 1. Selected performance objective for a residentialbuilding.

EarthquakeHaving

Probabilityof Exceedance

MeanReturnPeriods(years)

PerformanceLevel

50% per 50 year 75 IO

10% per 50 year 475 LS

2% per 50 year 2475 CP

model of a hospital, he or she should �rst analyze themodel under two sets of considered earthquakes withthe de�ned mean return periods separately and, then,see if the model satis�es the related code criteria foreach PL.

Since the descriptions of the performance objec-tives are mostly qualitative, some performance criteriahave been de�ned to bind these descriptions to engi-neering demand parameters, so that the performanceobjectives can be predicted in the analysis and designprocess [1]. In fact, these criteria are the rules andguidelines that must be met to ensure that the designedstructure satis�es the performance objectives. In thisresearch, the performance level shown in Table 1 isconsidered for a residential building.

TARGET AND PERFORMANCE CURVE

As previously mentioned, the target curve is a conceptby means of which the speci�c properties of the ETmethod can be readily put into use in the performance-based design procedure. Using the target curve isan appropriate way to evaluate the performance ofstructures in the ET method. In fact, the target curvewill be used as the criteria curve for the ET responsecurve and the performance of a structure at di�erentexcitation levels can be evaluated by comparing the ETresponse curve with the target curve.

Since in the target response curve (or simplytarget curve), the target and the ET response curveswill be compared in a single chart, the horizontal andvertical axes of the target curve should be de�ned,such as to match with the corresponded axes in theET response curve, i.e. the time in seconds on thehorizontal axis and a damage index on the vertical axis.On the other hand, the target curve should be inclusiveof a relation between the performance levels and thecorresponding performance criteria. The performancecriteria are conceptually similar to the damage indices.Thus the challenge is to correlate the performancelevels to the endurance times. In other words, the�rst step in the process of creating the target curveis to identify the respective endurance time of eachPL. To do so, the procedure shown in Figure 4 should


Figure 4. Target curve construction procedure.

be followed step by step. It is noticeable that thisprocedure is not an exact one. To have an exactcalculation, it is needed to obtain the hazard curvefor a special site and the PGA should be acquiredaccording to that curve, but since the main purposeof this research is to illustrate the basic concepts ofthe proposed procedure, the following approximateprocedure can be considered as appropriate. Thecorresponding magnitude of each earthquake hazardhas been obtained, �rst, using the Gutenberg-Richterrelation as follows [10]:

logN = a� bM; (5)

where:

N is the return period of earthquake (years);M is the magnitude of earthquake in Richter;b is called the \b-value", and is typically in the

range of 0.8-1.2a is called the \productivity".

`a' and `b' are some parameters that severely dependon the properties of the region in which the earthquakehad occurred. For Iran, the following form is recom-mended by Kaila [11]:

logN = 6:02� 1:18M: (6)

After that, the Peak Ground Acceleration (PGA) mustbe acquired from magnitude. The following formula issuggested for Iran [12]:

ln(PGA) = 3:65 + 0:678M � 0:95 ln(R); (7)

where `PGA' is peak ground acceleration in cm/sec2,`M ' is magnitude in Richter and `R' is the distance tothe fault in km, which is to be considered as 18 in thisresearch.

This acceleration can be easily related to theendurance time. For this purpose, it is needed tospecify which series of accelerograms are to be used.According to its conformability with the Iranian code2800 standard [13], the `d' series of accelerograms(i.e. ETA20d01-03) has been chosen. Consideringthis series, the equivalent endurance times in ETrecords, corresponding to the 3 mentioned PGA, canbe identi�ed. To do so, it is enough to trace the timesin the ET acceleration function at which the PGAsexceed the values of the PGAs corresponding to eachPL.

The results of the above procedure are shown inTable 2. In this table, the relevant interstory drift foreach PL is indicated. These quantities are for steelmoment frames and based on the FEMA-356 standard.Although these values are not intended in FEMA-356to be used as acceptance criteria for evaluating theperformance of structures, and are just some quantitieswhich qualitatively indicate the behavior of structuresat each level of performance, in this research, thesevalues will be used as an index to show the limitsof each performance level. Plastic rotation in beamsis the other index which is used here to evaluate theperformance of structures. To accurately evaluate theperformance, one should obtain the values of plastic de-formations in all elements (including beams, columns,panel zones, braces etc.), and compare them to theacceptance criteria given in the FEMA-356 standard.

This means that the design of a typical structureshould be such that if the `d' series of accelerogramswere applied to the structure, it would be capableof meeting the IO performance level up to 4.14 sec,the LS performance level up to 10.21 sec and the CPperformance level up to 19.11 sec.

Table 2. Endurance times related to each PL.

PerformanceLevel

Mean ReturnPeriods (years)

Magnitude(Richter)

PGA(g)

EnduranceTime (sec)

InterstoryDrift (%)

IO 75 6.7 0.22 5.16 0.7

LS 475 7.3 0.35 10.16 3.5

CP 2475 7.9 0.53 15.46 5


Figure 5. Target and existing performance curves.

Based on the above discussion, the target curvehas been drawn and compared with a 3-story frameperformance curve in Figure 5. In this �gure, the limitof each performance level has been speci�ed on thetarget curve. This frame is subjected to the `d' seriesof ET accelerograms and its interstory drift responseis considered as the damage index. It should be notedthat, while there are no criteria for damages below theIO level, this area is restricted by a horizontal line inthe target curve.

Damage Level

The above target curve has some ambiguities andincompetency; one is that in order to evaluate theperformance of structures, the performance of all el-ements should be checked by observing their plasticdeformations and comparing the values with the ac-ceptance criteria. Since di�erent limits are set onthese parameters for various elements in each PL, itis di�cult to compare the performance of di�erentelements and clearly de�ne the critical one. Thus toaccurately evaluate the performance of a structure, oneshould create a target curve for each mentioned indexand element, and compare the related response curvewith that target curve. In this way, even though theperformance can be identi�ed, the speci�cation of thecritical index is not a simple matter. A combineddamage indicator has been de�ned for the purpose ofthis study that simpli�es the compilation of damagelevels indicated by various indexes into one normalizednumerical value. This index is named \Damage Level"or DL in this study. Another property of the damagelevel is that this dimensionless index creates a numer-ical presentation for performance levels, i.e. one coulddistinguish and compare the performance of di�erentstructures with only one number. Moreover, if twostructures lie within the same PL, their performancecan be still comparable with this index.

To specify such an index, �ve performance levelsare considered as OP (Operational), IO (ImmediatelyOccupancy), LS (Life Safety), CP (Collapse Preven-

tion) and CC (Complete Collapse, an arbitrary pointto extend the target performance curve beyond CP),which is a rather arbitrary level to simplify formulation.It should be noticed that OP and CC levels are justused as the limits of the performance and also the limitsof the damage level. Until more research is available tode�ne the CC point, based on more rational criteria,it will be considered arbitrarily in such a way that theslope of the target performance curve before the CPlevel is maintained. This additional point is requiredin order to theoretically extend the performance curvebeyond CP, and has no practical signi�cance in thisstudy. The formulation proposed for the DL hasbeen arranged is such a way as to assign an explicitnumber (preferably an integer) to each PL and usethe determining parameters (such as interstory driftand plastic rotation) to compute the DL in a clear andunderstandable way.

An appropriate formulation, which satis�es theabovementioned considerations, can be expressed asfollows:

DL =nXi=1

max[�i�1;min(�; �i)]� �i�1

�i � �i�1; (8)

where � is the related parameter-like drift, whichshould be computed from analysis and �i is the FEMA-356 Prestandard boundary of that parameter for eachperformance level. The values of �i for interstory driftand plastic rotation for each PL and the correspondingDLs are given in Table 3.

As can be seen in Table 3, the obtained DLwill satisfy its purpose satisfactorily. Because, �rstly,it denotes the performance level of the structure,secondly, since it is a number, it will satisfy the need tocreate a numerical presentation for performance levels,thirdly, it can present all parameters in a normalizedform and this will ease future computations.

In light of the above discussion, the determiningparameter like interstory drift (or plastic rotations)can be replaced by DL in the target curve. Likewise,the structure performance or response curve should bedrawn according to this new index to comply with thetarget curve.

MODEL DESIGN

In order to demonstrate how the target curve can easethe visualization of the performance of structures, a setof 2D steel moment-resisting frames with a di�erentnumber of stories and spans were selected and usedin this research. These models consist of three-storyone-bay, and seven-story three-bay frames that aredesigned in three alternatives (standard, weak andstrong frames), and a twelve-story three-bay frame thatis designed in two standard and strong alternatives.


Table 3. Assigning damage levels to performance levels.

Performance Damage �i ��i (Plastic Rotation)

Level Level (Drift) Case (a) Case (b) Case (c)

IO 1 0.7 1 0.25 Interpolation

LS 2 3.5 6 2 is

CP 3 5 8 3 required

CC�� 4 7 11 6

* Case (a): bf=2tf < 52=pFye and h=tw < 418=

pFye,

Case (b): bf=2tf > 65=pFye or h=tw > 640=

pFye,

Case (c): Other.

** CC is an auxiliary point included so that the performance curve can be theoretically

extended beyond CP.

These frames are designed according to the AISC-ASD89 design code. To compare the performanceof the frames with varying seismic resistance, thestandard, weak and strong frames have been designedusing base shears equal to 1, 0.5 and 1.5 times thecodi�ed base shear, respectively. As an example, thegeometry and section properties of the seven-storythree bay frame are shown in Figure 6. In this �gure,the black circles stand for the plastic hinges and showthat the failure mechanism follows the strong column-weak beam concept. As can be seen in Figure 7, thesehinges have been modeled as a rotational spring witha moment-rotation curve shown in this �gure. Thecapital letters in this �gure (A to E) determine theboundaries of various behaviors of the hinge model.

Figure 6. Standard seven-story frame geometry andsections.

Figure 7. Plastic hinge model and its generalizedforce-deformation curve.

ANALYSIS RESULTS

The modeling and nonlinear analysis were done withOpensees software [14] Nonlinear models of the framesare prepared by using the beam element with nonlineardistributed plasticity. In this research, the dampingratio is assumed to be 0.05 of the critical value andP � � e�ects have been included in the nonlinearanalysis. Applicability of the ET method in nonlinearanalysis and the acceptability of its approximation inestimating various damage indexes have already beenstudied [6,15,16]. A similar level of approximation isconsidered to be adequate for the purpose of this study.

The drift and plastic rotation responses of frameswere obtained and converted to the DL index, ac-cording to Equation 8. Then, the ET response curve(performance curve) of each frame is plotted for eachaforementioned parameter's related DL, separately. InFigure 8, the response curve for the plastic rotationand drift in a 3-story standard frame is depicted.As illustrated in this �gure, using the concepts ofendurance time and DL, it is possible to compare


Figure 8. Performance curves for plastic rotation anddrift in frame F3s1b.

the situation of various parameters, such as plasticrotation and drift, and identify the critical parameterin any seismic intensity. For example, in a 3-storystandard frame, as can be seen in Figure 8, drift isthe critical parameter in low-intensity ground motions,but plastic rotation is critical at high-intensities. Notethat the �nal performance curve for each frame shouldbe created considering the maximum value of DL andconsidering both the drift and rotation (or any otherparameters that need to be considered based on thedesign criteria). The �nal DL response curve (orperformance curve) for three alternatives of a 3-storyframe is shown in Figure 9. Using this �gure, onecan easily study and compare the performance of thesethree alternatives in various seismic intensities. Forexample, according to Figure 9, it can be concludedthat all three alternatives fail the criteria of the IO per-formance level, but remain in the safe region of LS andCP performance levels; the standard frame performssimilar to the weak frame at low-intensities, but itsperformance resembles the strong frame performanceas seismic intensity increases.

Figure 10 shows the performance curves of three

Figure 9. Performance curves for 3-story frames.

Figure 10. Performance curves for 7-story frames.

types of 7-story frame. A good performance of thestrong frame in comparison with two other frames canbe easily observed in this �gure. Moreover, it can beseen that the weak frame lies above the target curvealmost at all times. The behavior of 12-story framescan be evaluated with a similar procedure.

Using the target curve, the performances of strongframes with various numbers of stories and bays canalso be compared with each other. Such a comparisonhas been done for three strong frames with 3, 7 and12 stories (Figure 11). Although all these three frameswere designed on the basis of 1.5 times the codi�ed baseshear, their performance is not similar. The twelve-story frame is the best performer and the three-storyframe is the worst one in this study.

To show and verify the versatility of the targetcurve, the 3-story standard frame has been subjectedto some earthquake records and its performance is eval-uated by the target curve. To have a good evaluation, 7earthquake records have been selected from the FEMA-440 recommended records and scaled in such a way asto create 7 modi�ed records at each performance level(i.e. 21 records). To do so, some correction factors are

Figure 11. Performance curves for strong frames.


Table 4. Properties of used records and the correction factors.

No Record HP LP DT Name CF for CP CF for LS CF for IO PGA

1 Northridge 24278 0.12 23 0.02 NRORR360 4.035 2.222 1.053 0.171 g

2 Landers 12149 0.07 23 0.02 LADSP000 2.664 1.467 0.695 0.259 g

3 Loma Preita 1652 0.2 41 0.005 LPAND270 1.575 0.868 0.411 0.438 g

4 Loma Preita 47006 0.2 45 0.005 LPGIL067 2.255 1.242 0.588 0.306 g

5 Morgan Hill 57383 0.1 27 0.005 MHG06090 1.933 0.504 0.504 0.357 g

6 Loma Preita 58065 0.1 38 0.005 LPSTG000 2.828 1.577 0.615 0.244 g

7 Loma Preita 58135 0.2 40 0.005 LPLOB000 1.342 0.739 0.35 0.514 g

required to be computed and applied to each record.The correction factors for each PL and each record arethe ratio of PL related PGA (see Table 2) to record'sPGA. In Table 4, the properties of used records and thementioned correction factors for each PL are indicated.

Thus, 21 time history analyses have been per-formed and the responses of the frame are calculatedin the form of the DL index. To provide a logicaloverview of the performance of the standard frame, itis recommended that the maximum and average of theDL indices in each PL be considered and comparedto those related to ET analysis results. Figure 12shows the aforementioned comparison. In this �gure,the crosses stand for the results of 21 time historyanalyses. As can be seen in this �gure, the averagevalues are compatible with ET analysis results, i.e. theperformance estimated by the ET method is consis-tent with that of time history analysis using groundmotions. Now, it seems that if a line connects theaverage (or maximum) values, the performance of theframe will be coincident with this line at various seismicintensities. In fact, this line is similar to the familiarperformance curve. However, it should be mentionedthat selecting the optimal form of the connecting curverequires further research in this area.

Figure 12. Comparison of ET and earthquakes results atequal PGA for 3-story standard frames.

CONCLUSION REMARKS

In this paper, a methodology is proposed in order toextend the application of the Endurance Time method(ET) to the performance-based design of structures.Application of the Endurance Time method in theperformance-based design of steel frames is investigatedfrom a conceptual viewpoint. In the ET method,structures are subjected to an intensifying accelerationfunction, thus an estimate of the structural responseat di�erent levels of excitation is obtained in a singleresponse history analysis, thus considerably reducingthe required computational e�ort. The concept ofperformance levels has been extended from discretepresentation to a continuous target performance curve.This target performance curve, while theoreticallymore attractive, turns out to be quite versatile wheninvestigating the ET analysis results, as shown inthe paper. In order to be able to combine severaldi�erent performance criteria into a single numericalperformance index, a generalized Damage Level (DL)index has been proposed. The DL index proposed inthis paper creates a versatile numerical representationof performance levels and, also provides a uniformindex to express a performance of structures thatincorporates various parameters (such as drift, plas-tic rotation etc.). Furthermore, the target curve isan e�ective tool for estimating the performance ofstructures under various seismic intensities by the ETresponse curve. This curve can be used to anticipatethe seismic performance of structures subjected toearthquakes. The target performance curve has a goodpotential to be used in the performance-based designof structures. The concepts of Performance Curveand Damage Level introduced in this paper lay thenecessary foundation for a more versatile applicationof the ET method in the practical performance-baseddesign of structures. The analysis results are shownto be consistent with those obtained using groundmotions scaled to represent particular excitation levels.It should be noted that extensive research in this areais required in order to assess the precision and level


of con�dence that can be expected from the proposedmethodology.

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.

NOMENCLATURE

Abs absolute value functionag(t) ET acceleration functionbf ange widthCC complete collapseCP collapse prevention performance levelF (ag) optimization target functionFye yield strengthIO immediate occupancy performance

levelLS life safety performance levelM earthquake magnitudemax maximum of the valuesN earthquake mean return periodPGA peak ground accelerationR distance to the faultSa spectral accelerationSa(T; t) acceleration response for period T at

time tSa(T ) acceleration response as a function of

period TSaC(T ) codi�ed design acceleration spectrumSaT (T; t) target acceleration response for period

T at time tSd(T; t) displacement response value for period

T at time tSdT (T; t) target displacement response value for

period T at time tT free vibration period (sec)t timetf ange thicknesstTarget target time (= 10 sec in this paper)Tmax maximum free vibration period (sec)

to be considered in the optimizationtmax time corresponding to the end of

acceleration function� weighting factor in optimization target

function (= 1.0 in this study)�i determinant parameter (drift ratio or

plastic rotation)

(f(t)) endurance time analysis result equal tomax(Abs(f(t1))) : t1 2 [0; t]

j such that8 for all values

REFERENCES

1. Krawinkler, H. and Miranda, E. \Performance-basedearthquake engineering", in Earthquake Engineering:From Engineering Seismology to Performance-BasedEngineering, Y. Bozorgnia and V.V. Bertero, Eds.,CRC Press, USA (2004).

2. FEMA-302 \NEHRP recommended provisions for seis-mic regulations for new buildings and other struc-tures", Part 1 - Provisions, prepared by the BuildingSeismic Safety Council for the Federal EmergencyManagement Agency, Washington, D.C. (1997).

3. FEMA \NEHRP guidelines for the seismic rehabili-tation of buildings", FEMA-273, Federal EmergencyManagement Agency, Washington, D.C. (1997).

4. Humberger, R.O. \A framework for performance-basedearthquake resistive design", EERC-CUREe Sympo-sium in Honor of Vitelmo V. Bertero, Berkeley, Cali-fornia (1997).

5. Estekanchi, H.E., Vafai, A. and Sadeghazar, M. \En-durance time method for seismic analysis and designof structures", Scientia Iranica, 11(4), pp. 361-370(2004).

6. Riahi, H.T. and Estekanchi, H.E. \Seismic assessmentof steel frames with endurance time method", Journalof Constructional Steel Research, 66(6), pp. 780-792(2010).

7. Estekanchi, H.E., Valamanesh, V. and Vafai, A. \Ap-plication of endurance time method in linear seismicanalysis", Engineering Structures, 29(10), pp. 2551-2562 (2007).

8. Grecea, D., Dinu, F. and Dubina, D. \Performancecriteria for MR steel frames in seismic zones", Journalof Constructional Steel Research, 60, pp. 739-749(2004).

9. FEMA-356 \Prestandard and commentary for the seis-mic rehabilitation of buildings", Federal EmergencyManagement Agency, Washington, D.C. (2000).

10. Mohraz, B. and Sadek, F. \Earthquake ground motionand response spectra", in Earthquake EngineeringHandbook, W.F. Chen and C. Scawthorn, Eds., CRCPress, Boca Raton, Florida (2003).

11. Kaila, K.L. and Narian, H. \A new approach for thepreparation of quantitative seismicity maps", Bull.Seis. Soc. Amer., 61, No. 1275-91 (1971).

12. Ghodrati Amiri, G., Mahdavian, A. and ManouchehriDana, F. \Attenuation relationships for Iran", Journalof Earthquake Engineering, 11, pp. 469-492 (2007).

13. BHRC. Standard No. 2800 \Iranian code of practice forseismic resistance design of buildings", In: Permanentcommittee for revising the Iranian code of practice for


seismic resistant design of buildings. Tehran (Iran):Building and Housing Research Center, 3rd Ed. (2005).

14. Paci�c Earthquake Engineering Research Center(PEERC), 2004 \Open system for earthquake en-gineering simulation", (OpenSees), Berkeley, CA:Paci�c Earthquake Engineering Research Center(http://opensees.berkeley.edu/).

15. Riahi, H.T., Estekanchi, H.E. and Vafai, A. \Appli-cation of endurance time method in nonlinear seismicanalysis of SDOF systems", Journal of Applied Sci-ences, 9(10), pp. 1817-1832 (2009).

16. Estekanchi, H.E., Arjomandi, K. and Vafai, A. \Esti-mating structural damage of steel moment frames byendurance time method", Journal of ConstructionalSteel Research, 64(2), pp. 145-155 (2008).

APPENDIX

Equations

(f(t)) � max(Abs(f(�) : � 2 [0; t]); (A1)

SaT (T; t) � SaC(T )� ttTarget

; (A2)

Find ag(t)j8T 2 [0;1); t 2 [0;1)! (�u(t))

= SaT (T; t); (A3)

Minimize F (ag(t)) =TmaxZ0

tmaxZ0

fAbs[Sa(T; t)

� SaT (T; t)] + �

�Abs[Sd(T; t)� SdT (T; t)]gdt dT; (A4)

logN = a� bM; (A5)

logN = 6:02� 1:18M; (A6)

ln(PGA) = 3:65 + 0:678M � 0:95 ln(R); (A7)

DL =nXi=1

max[�i�1;min(�; �i)]� �i�1

�i � �i�1: (A8)

BIOGRAPHIES

Amin Mirzaei was born in 1983. He was a studentat the Allameh Helli Secondary and High School(a�liated to NODET; the National Organization for

Developing Exceptional Talents) in Kerman, Iran, from1995 to 2001. He received a B.S. degree in Civil En-gineering and a M.S. degree in Structural Engineeringfrom Sharif University of Technology (SUT), Tehran,Iran, in 2005 and 2007, respectively. His M.S. degreethesis was entitled \Application of Endurance TimeMethod in Performance-Based Design of Steel MomentFrames", which he successfully defended in September2007. He is currently a Ph.D. candidate in Structuraland Earthquake Engineering in the Department of CivilEngineering at SUT.

His research interests include: Dynamic Anal-ysis of Structures using Endurance Time Method,Performance-Based Design, Seismic Behavior of SteelMoment Frames, Rehabilitation of Steel Structures andEvaluating the Seismic Performance of Structures.

He was ranked 6th in the 12th National CivilEngineering Olympiad, 2007, in Iran and also inthe same year authored `Fluid Mechanics' for M.S.Candidates; a Dibagaran Publication, in Tehran.

Homayoon Estekanchi is the Associate Professor ofCivil Engineering at Sharif University of Technology(SUT), Tehran, Iran. He received his Ph.D. in CivilEngineering from SUT in 1997, where he is now afaculty member. He is a member of the IranianConstruction Engineers Organization, ASCE, IranianInventors Association and several other professionalorganizations. His research interests include a broadarea of topics in Structural and Earthquake Engi-neering with a special focus on the design of TallBuildings and Industrial Structures. He regularlyteaches the design of `Steel Structures', `Tall Buildings'and `Industrial Structures' at SUT. In recent years, hisresearch interests mainly focus around the developmentof a new seismic analysis and design procedure calledthe `Endurance Time Method', which he proposedin 2000, the area in which he has published variousresearch articles in collaboration with other colleaguesand students.

Abolhassan Vafai, Ph.D., is a Professor of CivilEngineering at Sharif University of Technology (SUT),Tehran, Iran. He has authored/co-authored numerouspapers in di�erent �elds of engineering, including:Applied Mechanics, Biomechanics and Structural En-gineering (steel, concrete, timber and o�shore struc-tures). He is also active in the area of higher educationand has delivered lectures and published papers onthe `Challenges of Higher Education', the `Futureof Science and Technology', and `Human ResourcesDevelopment'.

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