Studying near fault endurance time acceleration functionscientiairanica.sharif.edu/article_1840_bb726bad... · acceleration for periods over 0.6 sec [3], compared to far fault ones,

Scientia Iranica A (2015) 22(1), 15{29

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Studying near fault endurance time accelerationfunction

B. Ghahramanpoora;�, N. Fanaiea and H. Jahankhahb

a. Department of Civil Engineering, K.N.Toosi University of Technology, Tehran, Iran.b. International Institute of Earthquake Engineering and Seismology, Tehran, Iran.

Received 12 May 2013; received in revised form 9 February 2014; accepted 5 August 2014

KEYWORDSEndurance timemethod;Near faultearthquakes;Intensifyingacceleration function;Forward directivity.

Abstract. The Endurance Time (ET) method is a time-history based dynamic pushoverprocedure. In this method, the structures are subjected to gradually intensifyingacceleration called an acceleration function. Then, their performances are assessed fromlinear to collapse level based on the interval time through which they can satisfy the requiredobjectives. ET Acceleration Functions (ETAFs) are calibrated upon actual earthquakespectra compatible with the Iranian National Building Code (standard no. 2800). Inthis code, there is no distinction between far and near fault regions. ETAFs scale basedspectrum increases uniformly at all periods as the time passes which is why this uniformincrement is not desirable in near fault earthquakes. As the rupture propagation is orientedtowards a site, near fault earthquakes have further spectral acceleration in the period over0.6 sec compared to far fault earthquakes. The shape of the spectrum becomes richer overlong periods as its level increases. In this research, ETAFs are presented for near faulttarget spectra, provided by Abrahamson-Silva attenuation relations. These relations havebeen modi�ed by Somerville near fault correction coe�cients. According to the resultsobtained in this research, while the new acceleration function can model directivity e�ectsat higher periods, it also satis�es endurance time concepts.© 2015 Sharif University of Technology. All rights reserved.

1. Introduction

Nowadays, it is very important to properly design andretro�t structures against strong earthquakes. Physicaland economical losses, caused by earthquakes in recentdecades, has revealed the de�ciencies and shortcomingsof traditional methods in earthquake resistance de-sign. Therefore, engineers and researchers are seekingnew approaches for design and analytical methods.Performance Based Design (PBD) is a new conceptin which various performance objectives are satis�ed

*. Corresponding author. Tel.: +98 914 1601788;Fax: +98 21 88757802E-mail addresses: [email protected] (B.Ghahramanpoor); [email protected] (N. Fanaie); [email protected] (H. Jahankhah)

under di�erent ground motions from minor to severeranges. The Endurance Time (ET) method is themost recent performance based analytical procedureprovided and developed by Estekanchi et al. [1], whichis basically a simple dynamic pushover procedure. Inthis method, the damage measures of structures arepredicted at di�erent intensity measures by subjectingthem to some pre-designed intensifying dynamic exci-tations called Endurance Time Acceleration Functions(ETAFs). ETAFs are designed in such a way that theirintensities increase through time. As the demand ofETAFs increases, the structures gradually go from anelastic to a nonlinear inelastic phase, and �nally meetglobal dynamic instability [1].

Application of the ET method in linear andnonlinear analysis of structures has been proved incomparison with most known analysis methods such

16 B. Ghahramanpoor et al./Scientia Iranica, Transactions A: Civil Engineering 22 (2015) 15{29

as Incremental Dynamic Analysis (IDA), and the non-linear static procedure (pushover method) [1,2].

ETAFs are calibrated based on 7 real earthquakescompatible with the design spectrum of the IranianNational Building Code, corresponding to soil pro�letype 2. Near fault e�ects have not been considered inthe conventional ETAFs. In this research, ET conceptshave been developed for near fault earthquakes due tothe importance of such motions.

In ETAFs, each time represents one earthquakelevel which increases as the time goes up. Thesefunctions scale the spectra uniformly at all periods asthe time increases; while, in near fault earthquakesthe spectrum is relatively richer over long periods asits level increases. So the spectral characteristics ofconventional ETAFs are generally di�erent from thoseof near fault earthquakes.

Near fault earthquakes have intensi�ed spectralacceleration for periods over 0.6 sec [3], compared to farfault ones, due to the forward directivity phenomenon,and rupture propagation towards a site. The incrementrate of spectral amplitude rises at longer periods andstronger earthquake levels.

As there are almost no near fault design spectrumin seismic codes, the modi�ed attenuation relationshipsare usually used in engineering research to considerdirectivity e�ects. In this research, �rst, 5 near faulttarget spectra have been produced corresponding to 5moment magnitudes. They have been developed us-ing Abrahamson-Silva (1997) [4] attenuation relationsmodi�ed by Somerville et al. (1997) [3] near faultcoe�cients. Then, the frequency content of Gaussianwhite noise is modi�ed so that the resulting responsespectrum matches the target spectrum at prede�nedtarget times. As known, intensi�cation of near faultground motion response spectra in long period re-gions is a result of the directivity-pulse existence inrelative records. From this point of view, correctionfactors to spectra ordinates for periods more than0.6 sec re ect the probable existence of such pulsesimplicitly. However, the generated ground motionsbased on such corrected spectra may not percontaina single dominant velocity pulse explicitly. Productionof the near fault acceleration function has been gen-erally expressed in the following sections. Accordingto the results, the new acceleration functions canproperly predict the directivity e�ects of near faultearthquakes.

2. Endurance time method

The endurance time method is a new dynamic pro-cedure that can predict the seismic performance ofstructures. In this method, the structures are subjectedto a gradually intensifying dynamic excitation andtheir performances are monitored at di�erent excita-

Figure 1. Demand/capacity of frames under accelerationfunction [1].

tion levels. The structures that endure the imposedintensifying acceleration function for a longer time areexpected to sustain stronger seismic excitation. Infact, in ET analysis, the time axis can be correlated toexcitation intensity because of intensifying accelerationfunctions [2].

The concept of the ET method can be physicallypresented by a hypothetical shaking table experiment.For this purpose, three di�erent frames with unknownseismic properties are �xed on the table and subjectedto an intensifying shaking. One of the frames (frame 1in Figure 1) fails after a short time and the third onewill also collapse as the amplitude of the vibration in-creases. It is assumed that the same will happen to thesecond frame also. These three frames can be rankedaccording to their seismic resistance times, i.e. the timeat which the models failed. It should be mentionedthat the lateral loads, induced by the shaking table,are somehow corresponded with the earthquake loads.Therefore, the endurance time of each structure canbe considered the seismic resistance criterion againstintensifying shaking. In this hypothetical analysis, thesecond frame shows the best performance because of itslonger endurance time and, therefore, is the strongeststructure. The �rst one is the worst and the third issomewhere in between the �rst and second frames [1,5].The demand/capacity ratios of these frames presentedin Figure 1 have been calculated as the maximumabsolute values of the endurance index during the timeinterval of \0.0" to \t". Since a structure collapses,when its demand/capacity ratio exceeds unity, theendurance time for each frame can be easily derivedfrom this �gure [5].

In order to start an ET time history analysis, therepresentative model should be constructed and theproper damage measure and ET accelerogram shouldbe set (Figure 2).

The analysis results are usually presented by acurve in which the maximum absolute value of thedamage measured in the time interval [0; t] (as given

B. Ghahramanpoor et al./Scientia Iranica, Transactions A: Civil Engineering 22 (2015) 15{29 17

Figure 2. Typical ET accelerogram [5].

in Eq. (1)) corresponds to time [5].

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

where, is maximum absolute operator; and f(�) isthe desired structural response history, such as inter-story drift ratio and base shear or other damage indices.To apply the ET method, the intensifying accelero-grams are generated in such a way as to producedynamic responses equal to those values induced bythe desired response spectrum (such as the code designspectrum) at a prede�ned time (tTarget) [1,5]. Also,the results obtained by ET time history analysis canbe compared with those of other analysis methods and,the performances of structures with di�erent periods offree vibration can be compared. The �rst intensifyingaccelerograms suggested for ET had a linear intensi�ca-tion scheme. It means that the response spectrum of anET accelerogram should be intensi�ed proportionallywith time. Hence, the target acceleration responseof an ET accelerogram can be related to the codi�eddesign acceleration spectrum [5], as follows:

SaT(T; t) =t

tTarget� SaC(T ); (2)

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

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

minF (ag) =Z Tmax

0

Z tmax

0fAbs[Sa(T; t)� SaT(T; t)]2

+ �Abs[Su(T; t)� SuT(T; t)]2gdt:dT;(3)

where, ag is the ET accelerogram; SaT(T; t) is thetarget acceleration response; SuT(T; t) is the targetdisplacement response at time t; Sa(T; t) and Su(T; t)are the acceleration and displacement responses ofthe acceleration function at time t, respectively; andT is the period of free vibration. � is a relativeweight parameter that can be used to adjust the

e�ective penalty attributed to displacement deviations,as compared to acceleration deviations from targetvalues. High values of � make the target function moresensitive to displacement response di�erences, and lowvalues of � make the target function more sensitiveto acceleration di�erences. Considering the theoreticalcorrelation between the acceleration response and therelative displacement response, a high value of � shouldresult in a better �t in the high period range, and a lowvalue of � results in a better �t in the low period range.For di�erent generations of the ET accelerograms, �may vary. For most ET generations, � has been setto 1.0, forcing a balanced penalty on either responseparameter [1,2,5].

Based on the mentioned linear scheme, di�erentsets of ET accelerograms can be generated in di�erentranges (linear or nonlinear), according to their compat-ibilities with di�erent spectra. Each set consists of agroup of intensifying acceleration functions (usually 3).

For example, three acceleration functions, named\ETA20f01-03" or, brie y, \f series", are created insuch a way that the template response spectrum att = 10 second matches the average response spectrumof major components of 7 real accelerograms (listed inFEMA440 for soil type C). In Figure 3, the responsespectra of the f series are compared at di�erent times.The spectral acceleration values corresponding to thetarget times of 5, 10 and 15 sec have been plotted in this�gure. According to this �gure, and, as expected fromthe ET concept, the values corresponding to 10 and 15sec are twice and triple that of 5 sec, respectively [6].

3. Spectral characteristics of near faultearthquake

An earthquake is a shear dislocation that begins froma point on the fault and spreads with almost the samevelocity of shear wave. In forward directivity, the faultrupture is propagated towards a site with very highvelocity, producing high seismic energy. Propagation ofthe fault ruptures towards a site (forward directivity)

Figure 3. Response spectra of ETA20f01-03 at di�erenttimes [6].


at very high velocity causes most of the seismic energyfrom the rupture to arrive in a single large long periodpulse of motion, which occurs at the beginning of therecord. It is oriented in the direction perpendicularto the fault, causing the strike-normal peak velocityto be larger than the strike-parallel peak velocity.Forward directivity results from the increasing of theresponse spectrum of the horizontal component normalto the fault strike at periods longer than 0.5 sec. Thiscauses the peak response spectral acceleration of thestrike-normal component to shift to longer periods; forexample, from 0.25 seconds to as much as 0.75 seconds.Near fault e�ects cannot be adequately described byuniform scaling of a �xed response spectral shape.The increasing of spectral values is higher in thelong periods as the earthquake magnitude increases.Figure 4 shows the e�ects of rupture directivity onthe response spectrum of the Rinaldi record of theNorthridge earthquake, 1994 [3].

Forward directivity e�ects are based on two con-ditions: 1) The rupture front is propagated towardsthe site; 2) The direction of slip on the fault isaligned with the site. The conditions for generatingforward directivity e�ects are met mainly in the strike-slip faulting. Accordingly, the fault slip directionis oriented horizontally along the strike of the faultand the rupture is propagated horizontally along thestrike, unilaterally or bilaterally. Backward directivitye�ects occur when the rupture is propagated away fromthe site. This fact results in the opposite e�ects offorward directivity, like long duration motions with lowamplitudes at long periods. Forward directivity occursin dip-slip faulting, including both reverse and normalfaults as well [3].

Somerville et al. (1997) used a large set of nearfault strong motion records to develop a quantitativemodel of rupture directivity e�ects. This model can be

Figure 4. 5% damped response spectra for strike-normaland strike-parallel horizontal components of Rinaldistation [3].

used to modify existing ground motion attenuation re-lations for incorporating directivity e�ects. The groundmotion parameters, modi�ed for directivity e�ects, arethe average horizontal spectral acceleration and theratio of strike-normal acceleration to strike-parallelacceleration. Strike-normal refers to the horizontalcomponent of the motion normal to the fault strike.Strike-parallel refers to the horizontal component of themotion parallel to the fault strike [3].

In the Somerville's model, the amplitude variationdue to rupture directivity depends on two geometricalparameters: 1) The smaller the angle between thedirections of rupture propagation and the waves travel-ling from the fault to the site, the larger the amplitude;2) The larger the fault rupture surface between thehypocenter and site, the larger the amplitude. Theazimuth angle (�) and length ratio (X) are consideredfor strike-slip, and the zenith angle (') and widthratio (Y ) for dip-slip faults in the directivity model arepresented in Figure 5. The e�ects of rupture directivityon ground motion amplitudes are modeled using thedirectivity functions, X cos(�), for strike-slip faults andY cos(') for dip-slip faults. The dependency of thespectral ampli�cation factor on X cos(�) and Y cos(')has been illustrated in Figure 6. These e�ects arestarted at the period of 0.6 sec and increase as theperiod increases [3].

Following the above model, the ratio of strike-normal to average horizontal of spectral accelerationdepends on the magnitude, fault distance and direc-tivity functions. The strike-normal motion is obtainedthrough multiplying the attenuation relation value �rstby the average response spectral ratio, and, second, bythe ratio of strike-normal to average horizontal. Strike-parallel spectral acceleration can be estimated usingaverage horizontal and strike-normal components aswell.

Figure 5. Rupture directivity parameters: � and X forstrike-slip faults; ' and Y for dip-slip faults [3].


Figure 6. The dependency of response spectral factor on period and directivity function: X cos(�) for strike-slip faulting;Y cos(') for dip-slip [3].

In this research, the Abrahamson-Silva [4] atten-uation relation has been used to generate the targetspectra, modi�ed by the Somerville directivity modeland discussed in the following.

3.1. Abrahamson-Silva spectral attenuationrelation

The general functional form of the relation is given asfollows:

ln (Sa(g)) =f1(Mw; rrup) + Ff3(Mw)

+ HWf4(Mw; rrup) + Sf5(PGArock);(4)

where, Sa(g) is spectral acceleration in g; Mw ismoment magnitude; rrup is the closest distance to therupture plane in km; F is fault type; HW is a dummyvariable for hanging wall sites; and S is a dummyvariable for the site class [4].

3.2. Average horizontal response spectraAverage horizontal response spectra are obtained bymultiplying Sa(g) in Eq. (4) by Eq. (5) in the corre-sponding periods:FAH = ey; (5)

where:y = C1 + C2X cos(�) for strike-slip Mw > 6;

y = C1 + C2Y cos(') for dip-slip Mw > 6:

X, Y , � and ' are directivity parameters, shown inFigure 5; C1 and C2 are period dependent coe�cients,clari�ed in Table 1, for both strike-slip and dip-slipfaulting [3].

3.3. Ratio of strike-normal to averagehorizontal motions

The ratio of strike-normal to the average horizontalmotion is expressed as follows:

SNAH

= ey2 ; (6)

Table 1. The model's coe�cients for spatial variation ofaverage horizontal spectral acceleration [3].

For strike-slip For dip-slipPeriod C1 C2 C1 C2

0.6 0 0 0 00.75 -0.084 0.185 -0.045 0.008

1 -0.192 0.423 -0.104 0.1781.5 -0.344 0.759 -0.186 0.3182 -0.452 0.998 -0.245 0.4183 -0.605 1.333 -0.327 0.5594 -0.713 1.571 -0.386 0.6595 -0.797 1.757 -0.431 0.737

where:

y = C1 + C2 ln(rrup + 1) + C3(Mw � 6) Mw > 6;

where, rrup is the lowest distance from the ruptureplane (here, assumed as 5 km); Mw is moment mag-nitude; and C1, C2 and C3 are period dependentcoe�cients, listed in Table 2 [3].

The strike-normal spectral value is obtained bymultiplying Eq. (6) to the modi�ed average value ofhorizontal components achieved in Section 3.2. Thevalues of horizontal components and strike-normalspectral values are manipulated in order to obtain thestrike-parallel motion.

Figure 7 shows the procedure described for strike-slip faulting, with respect to Mw = 7 and deep soil.

4. Near fault acceleration function

For generating acceleration functions, the target timesand spectra should primarily be speci�ed. In thisresearch, the acceleration function has been generatedfor both strike-slip and dip-slip faulting consideringdeep soil and shallow crustal soil, according to theAbramson-Silva assumption. Strike-normal, averagehorizontal and strike-parallel components have been


Table 2. The coe�cients of the ratio of strike-normal toaverage horizontal spectrum [3].

Period C1 C2 C3

0.5 0 0 0

0.6 0.048 -0.012 0

0.7 0.079 -0.019 0

0.75 0.093 -0.022 0

0.8 0.108 -0.026 0

0.9 0.124 -0.03 0

1 0.139 -0.033 0

1.5 0.192 -0.046 0

2 0.231 -0.055 0.005

2.5 0.27 -0.065 0.011

3 0.304 -0.075 0.025

3.5 0.331 -0.084 0.046

4 0.351 -0.096 0.063

4.5 0.362 -0.098 0.081

5 0.369 -0.103 0.093

Figure 7. Near fault acceleration spectra consideringstrike-slip faulting, deep soil, Mw = 7 and X cos(�) = 1.

considered in both cases of mentioned soils and focalmechanisms (dip-slip and strike-slip).

4.1. Target spectraHere, target spectra are produced according to Sec-tion 3 for �ve moment magnitudes of 4, 5, 6, 7 and 8 inthe 5 km distance from the rupture plane. Regardingthe average horizontal component, �ve directivity con-ditions have been considered. Accordingly, X cos(�)and Y cos(') are set as 0, 0.25, 0.5, 0.75 and 1. Toconsider directivity e�ects on strike-normal and strike-parallel components, these values are set just as 0.5and 1 to induce average and highest forward directivityconditions, respectively.

Figure 8. Spectral acceleration of strike-normalcomponent in deep soil considering forward directivity fordi�erent magnitudes (rrup = 5 km).

Figure 8 shows the target spectra of strike-normalcomponents for deep soil and di�erent magnitudes.

4.2. Target timesThe most important spectral characteristics of ETAFsare target times and spectra. If the target time is setas t = t0, then, ETAFs are calibrated in such a waythat their response spectra in a window from t = 0to t = t0 match the design or codi�ed spectrum witha unity scale. When the window of an accelerationfunction is from t = 0 to t = K � t0, its responsespectrum is corresponded to K multiplied by thetemplate spectrum at all periods. This characteristicis used to determine the target times for near faultacceleration functions.

t = 10 sec is assumed as the time at which theearthquake has the moment magnitude of Mw= 6.Therefore, this time and its corresponding target spec-trum are considered as the bases. Target time forother moment magnitudes is approximately obtainedas follows:

tp(Mw) =Sa(Mw)T=1

Sa(6)T=1� 10; (7)

where, Mw is moment magnitude; tp(Mw) is targettime corresponding to Mw; Sa(Mw)T=1 is spectralamplitude corresponding to Mw at T = 1 sec; andSa(6)T=1 is spectral amplitude corresponding to Mw =6 at T = 1 sec. Final target times are achieved basedon engineering judgment of the above values. Targettimes and their corresponding moment magnitudes arelisted in Table 3.

4.3. Generating the acceleration functionNear fault ETAFs has been generated based on amethod used in the �rst generation of ET accelerationfunctions. Some deviations, generally minor for all


Table 3. Target times and their corresponding momentmagnitudes.

Targettime (sec)

Moment magnitude(Mw)

2 44 510 632 745 8

Figure 9. Gaussian white noise with time duration oft = 45 sec.

magnitudes, except for M=8, from the target spectraare accepted, while no optimization procedure fora perfect match has been used. First, a series ofrandom numbers are produced having the Gaussiandistribution of zero mean and the variance of unitywith the time duration of the last target time (45 seccorresponded to magnitude 8), shown in Figure 9. Thetime step is set as 0.01 sec.

The frequency content of the above random num-bers is then modi�ed in order to resemble the accelero-grams of real earthquakes. For this purpose, the �lterfunctions, explained by Clough and Penzien [7], areapplied to the random accelerograms as follows:

H1(i!) =

�1 + 2i�1( !!1

)�

(1� !2

!21) + 2i�1( !!1

); (8)

H2(i!) =

�1 + 2i�2( !!2

)�

(1� !2

!22) + 2i�2( !!2

); (9)

where, !1 = 2�0:5 , !2 = 2�

0:1 and �1 = �2 = 0:2 are used,such as the �rst generation of ETAFs [8]. It shouldbe mentioned that frequencies higher than 25 Hz are�ltered and neglected in this study. A sample frequencycontent resulted from the accelerogram of Figure 9 isplotted in Figure 10.

The acceleration whose frequency content hasbeen modi�ed is returned to the time domain with

Figure 10. Modi�ed frequency content resulted from theaccelerogram shown in Figure 9.

Figure 11. Intensifying acceleration.

inverse fast Fourier transformation. The resultedacceleration is multiplied by a pro�le function, f(t) =t

45 , to take intensifying form, as shown in Figure 11.Like conventional ETAFs, the resulted accelera-

tion is modi�ed in such a way that at some target times,the related response spectra are matched with theircorresponding target spectra in the time window of 0- target time. As mentioned before, �ve target timesand spectra have been considered here; therefore, theprocess has 5 steps numbered by j. The target times(Tp) are de�ned as follows:

Tp = [2; 4; 10; 32; 45] sec:

The acceleration presented in Figure 11 is multipliedby the window shown in Figure 12 to meet step j.

Tp(j) is the jth target time, shown in Figure 12.The result will be 5 acceleration functions, whosevalues follow the initial acceleration function, shown inFigure 11, in the consecutive target times; their valuesare zero at other times. Fourier spectrum amplitude ismodi�ed for the resulted accelerations as follows:

Anew(i!)j = Aold(i!)j � SaT (!)jSa(!)j

; (10)

where, Aold(i!)j is the Fourier amplitude of the accel-


Figure 12. shape window for the jth step.

eration function of the jth step; SaT (!)j is the jthtarget spectrum corresponding to 5% damping andTp(j), described in Sections 4.1 and 4.2; Sa(!)j isthe 5% damped response spectrum of the accelerationfunction of the jth step; and Anew(i!)j is the modi�edFourier amplitude of the acceleration function.

This process is repeated up to meet proper con-vergence between the response spectrum and the targetspectrum. The above mentioned procedure is presentedin Figure 13 for the 4th step as an example.

The �nal acceleration function is obtained bysumming the accelerations resulted from the abovementioned method and shown in Figure 14. The recordand its relevant spectra at di�erent target times aredepicted in this Figure.

Like conventional ETAFs, three acceleration func-tions have been generated for each case (di�erent focalmechanisms, soil type and directivity condition) andthe average value of results is used for analysis. Thegenerated near fault acceleration functions and theirabbreviations have been summarized in Table 4.

4.4. Comparing new ETAFs with realaccelerograms

Every synthesized accelerogram should be able tosatisfy some characteristics of real earthquakes. Al-though near fault earthquakes, which have furthercharacteristics in addition to the directivity e�ect,rationally cannot be compared with new ETAFs, which

Figure 14. a) Generated ETAF. b) Response spectra atdi�erent target times.

model directivity e�ects only, some similarities can stillbe seen in several parameters such as the responsespectrum, frequency content, power spectral densityand Arias intensity. In this study, new accelerationfunctions are compared with real records selected onlybased on moment magnitude and distance from therupture plane, as listed in Table 5, for both strike-slip and dip-slip faulting, and no directivity conditionwas considered in this selection. The records havemagnitudes around 6 and are compared at t = 10 sec.

The acceleration response spectrum is one of

Figure 13. The procedure of new acceleration function generated for 4th step.


Table 4. The abbreviations of near fault acceleration functions.

Acceleration function type Abbreviation

Strike-slip ET average horizontal component SSETAH01-03Strike-slip ET strike normal component SSETSN01-03Strike-slip ET strike parallel component SSETSP01-03Dip-slip ET average horizontal component DSETAH01-03Dip-slip ET strike normal component DSETSN01-03Dip-slip ET strike parallel component DSETSP01-03

Table 5. Real earthquakes used in this study.

Station name Mw Mechanism Component Rrup (km)

Morganhill, Anderson dam 6.19 Strike-slip Strike-normal 3.3Morganhill, Hallsvalley 6.19 Strike-slip Strike-normal 3.6

Morganhill, Coyotolake dam 6.19 Strike-slip Strike-normal 0.5ParkField, Cholame#8 6.19 Strike-slip Strike-normal 12

NPS-cabazon 6.06 Dip-slip Strike-normal 4Desert hot springs 6.06 Dip-slip Strike-normal 6.8

Palm spring airport 6.06 Dip-slip Strike-normal 10

the most important parameters from an earthquakeengineering point of view. The spectral accelerationof real records and ETAFs for both strike-slip and dip-slip faulting are plotted at t = 10 sec and shown in Fig-ure 15(a) and (b). According to this �gure, althoughthe spectral values of near fault ETAFs and real recordsare di�erent in some periods, they are signi�cantlyclose to each other. The displacement and velocityof new ETAFs for the strike normal component ofstrike-slip faults are presented in Figure 15(c) and (d).The velocity and displacement values generally increasethrough time duration and do not reach zero at theend of the records. This phenomenon has also beenobserved for the ETAFs generated for far fault [9].

Dynamic responses of structures are very sensitiveto the frequencies at which they are loaded. Earth-quakes produce complicated loading, with componentsof motion that span a wide range of frequencies.The amplitudes of ground motions are distributed indi�erent frequencies according to their contents [9].The frequency content of ETAFs, related to the strike-normal component of the strike-slip fault, is comparedwith that of a real earthquake and presented in Fig-ure 15(e). Based on the �gure, the results of nearfault ETAFs and real earthquakes are similar at almostthe whole domain. The Intensity Arias (Ia) is closelyrelated to the root mean square of acceleration, whichis e�ective in characterizing the frequency content andpower spectral density of accelerograms. According toFigure 15(f), the intensity Arias parameter matches theaverage ground motions for near fault ETAFs at t = 10sec. Its value increases with a hyperbolic trend as thetime increases, similar to conventional ETAFs [9].

5. The responses of SDOF systems

Here, three linear single degree of freedom systemswith natural periods of 0.5, 1 and 1.5 sec have beenconsidered to study the e�ects of directivity on theresults of near fault ETAFs. Acceleration functionscorresponding to the average horizontal componentof three directivity conditions have been provided bysetting three values of 0, 0.5 and 1 for the directivityparameter, X cos(�), and applied to the mentionedsystems. In the ET method, 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 Eq.(1)) corresponds to time.Figure 16 shows the ET displacement curves. Therate of directivity e�ect on the responses increasesas the periods of the systems and, also, the time,increase. According to Somerville, for producing nearfault ETAFs, directivity a�ects responses beyond thestructural period of 0.6 sec. This fact can be followedbest in Figure 16. Regarding the system with a periodof 0.5 sec, all site-source geometry conditions lead tothe same results. It means that directivity has noe�ect on structural responses in this system. However,for the two other systems with periods of 1 and 1.5sec, three directivity conditions a�ect the results after10 sec. Moreover, the results for various directivityconditions in the system with period of 1.5 sec havemore di�erences with each other compared to othersystems, as expected.

Like conventional ETAFs, the equivalent time isused to compare analysis results of ET records withthose of real records [10]. In this regard, the average


Figure 15. Comparing di�erent parameters of near fault ETAFs and real earthquakes: (a) Comparison of new ETAFsand real earthquake response spectra for strike-slip faults; (b) comparison of new ETAFs and real earthquake responsespectra for Dip-slip faults; (c) displacement of new ETAFs; (d) velocity of new ETAFs; (e) comparison of frequencycontent between new ETAFs (at 10 sec) and ground motions; and (f) Arias intensity for SSETSNs acceleration functionsand ground motions.

value of the �rst-mode spectral acceleration (Sa;Ave) iscalculated for real earthquake records.

Accordingly, the spectral acceleration of gener-ated near fault ETAFs is calculated (Sa;ET) at the �rst-mode period of MDOF systems or the natural periodof SDOF systems (T1) for the same level of those realearthquakes. Finally, the equivalent time is obtainedby the following equation:

teq =Sa;Ave

Sa;ET� TP ; (11)

where, Tp is the target time corresponding to certaintarget spectrum. Target time is assumed as 10 secto calculate the equivalent time, because the momentmagnitude of real earthquakes is about 6. The resultsof the SSETSN set, introduced in Table 4, and theaverage of real earthquakes, listed in Table 5, have beencalculated and compared for the SDOF system with aperiod of 1.5 sec and shown in Figure 17.

The displacement values of ET at 10 sec and thoseof real earthquakes are not the same, because: 1) Realearthquake spectrum and ETAF target spectrum have


Figure 16. linear displacement responses underSSETAHs for di�erent periods of SDOF systems.

Figure 17. Comparing the linear displacement responseof SSETSNs set and average of real earthquake for SDOFsystem with the period of 1.5 sec.

discrepancies, shown in Figure 15(a) and (b); 2) TheETAF spectrum is matched with the target spectrumat 5 points and has not enough accuracy to predictother spectral levels. Therefore, the ET result shouldbe compared at the equivalent time. According to the�gure, the results of these two methods become closerby using equivalent time.

6. The responses of MDOF systems

In this research, the responses of multi degree offreedom systems have also been studied under nearfault ETAFs. For this purpose, 3 and 6 story steelbraced frames with three bays have been used toverify the concept of the endurance time method. TheSSETSNs series have been applied for time historyanalysis. These frames are designed according tothe Iranian National Building Code (INBC), Section10 [11], which is almost identical to AISC-ASD designrecommendations [12]. The frames with various seismicresistance values have been categorized in 3 groups inorder to compare their behavior, regarding the inten-sity levels of design lateral loads. These groups are:Standard, Under-designed and Over-designed frames;their names are ended with the letters S, W and O,respectively, listed in Table 6. As an example, an over-designed frame with 6 stories and 3 bays is abbreviatedas \BF06B3O" in the mentioned table. Standardframes have been designed according to the base shearrecommended by INBC for high seismicity areas in anequivalent static procedure [13]. Under-designed andover-designed frames have been designed based on onehalf and twice the codi�ed base shear, respectively. Theframes' masses are assumed equal in order to simplifythe comparison. The geometry and section propertiesof the frames are presented in Figure 18, and some oftheir major characteristics in Table 6.

The nonlinear behaviors of the frames have beenstudied using a bilinear material model with a post-yield sti�ness equal to 1% of the initial elastic sti�ness.This material model is applied in the analysis usingOPENSEES for beam, column and truss elements, withnonlinear distributed plasticity [14]. The dynamic soil-

Table 6. The speci�cations of frames.

FramesNumber

ofstories

Numberof

bays

Massparticipation

of mode 1

Period(sec)

BF03B3W 3 3 83.12% 0.46BF03B3S 3 3 81.75% 0.4BF03B3O 3 3 80.30% 0.33BF06B3W 6 3 79.00% 0.96BF06B3S 6 3 71.28% 0.73BF06B3O 6 3 62.27% 0.64


structure interaction has been neglected, but the e�ectsof P �� have been considered in the analysis.

6.1. ET analysisET analysis has been conducted on the frames sub-jected to the SSETSNs sets of acceleration functions.

Figure 18. Schematics con�gurations of the frames usedin this study.

ET analysis is like time-history analysis, except thatin the former, the ground motions are replaced withacceleration functions. In this research, several damageindices, such as base shear, roof displacement andmaximum story drift ratio, have been investigated.Figure 19 shows the base shear and roof displacementET curves of the BF03B3S frames and their historiesfor near fault acceleration functions.

The rate of near fault e�ects has been studied onthe responses of three and six story standard frames,and shown in Figure 20. Also, threshold times de-scribing di�erent earth quake levels have been marked.According to this �gure, unlike the SDOF system, roofdisplacement ET curves, related to average and forwarddirectivity conditions, do not diverge at the same timein the mentioned frames. The speci�cations of theBF03B3S frame with the natural period of 0.4 sec aresummarized in Table 6. The directivity has no e�ecton this system at t = 10 sec, which is correspondedto Mw = 6. The curves' divergence points have beenshifted to 22 sec, due to the nonlinear behavior. Astime increases, the amplitude of excitation increases aswell, and the frame experiences nonlinear status. Thelateral sti�ness value decreases, so the frame's periodexceeds 0.6 sec. After this point, the responses area�ected by directivity, according to the assumptions ofthis research, and shown in Figure 20(a). Regardingthe BF06B3S frame, as the system's period is 0.73 sec(Table 6), directivity will a�ect the responses at 10 sec.The di�erence between the ET curves increases as thetime increases; this is also seen in SDOF systems. Ac-cording to the results obtained in this research, ETAFscan properly model the directivity e�ects. It shouldbe mentioned that the forward and average directivityconditions are obtained in ETAFs production. In thisregard, the values of 1 and 0.5 have been used forX cos(�) or Y cos('), respectively.

Figure 21 shows the di�erent performances ofa structure at di�erent earthquake levels using thenew ETAFs. Lower time is needed for the under-designed frame to meet the prede�ned damage index,compared to others, as it has a worthy performance.

Figure 19. Generating ET curves from roof displacement and base shear histories of the BF03B3S frame consideringforward directivity.


Figure 20. ET non-linear roof displacement curves of SSETSNs set: (a) BF03B3S; and (b) BF06B3S.

Figure 21. ET non-linear roof displacement curves ofSSETSNs set for di�erent alternatives of BF06B3 frames.

The over-designed frame has a higher endurance timeand the best performance, regarding the same damageindex. Therefore, the new ETAFs follow endurancetime concepts well.

The maximum drift ratios, estimated by ETanalysis under the SSETSNs set, and those of realrecords, at equivalent time, have been studied for both3 and 6 story frames, and are shown in Figure 22.According to this �gure, the results obtained by thetwo methods are not the same. However, analysis usingthe new ETAFs can predict the distribution patternof drift ratio at di�erent story levels. In general, the

frames behave similarly under near-fault ETAFs andconventional ETAFs.

7. Conclusion

In this research, the ET method has been developedfor near-fault regions. For this purpose, near faulttarget spectra have been used instead of conventionalET template spectrum. Such spectra have been pro-vided through the Abramson-Silva attenuation relationand modi�ed by Somerville coe�cients. New sets ofacceleration functions have been produced consideringdi�erent components, soils and directivity conditions.SDOF and MDOF systems have been subjected to thenew ETAFs and the obtained results are expressed asfollows:

1. Unlike conventional ETAFs in which the spectra,corresponding to various earthquakes levels, havelinear increasing relations for all periods uniformly,in the new ETAFs, establishing such relations is notreasonable due to the e�ects of directivity for higherperiods on the near-fault regions.

2. Near-fault acceleration functions are produced witha progressed procedure compared to conventionalETAFs. The results show good convergence ofthe new ETAFs response spectrum with the targetspectrum at target time. However, as the timeincreases, the rate of the mentioned convergencedecreases. Moreover, new ETAFs have not enough

Figure 22. Distribution of maximum interstory drift ratios of BF03B3S and BF06B3S at di�erent story levels obtainedby ET at equivalent time and real earthquakes time history analysis.


accuracy to predict other spectral levels at thetarget time intervals.

3. Higher time intervals are needed in the appliedmethod for matching the ETAF spectrum with atarget spectrum related to high moment magnitude,such as 7 or 8, unless other optimization proceduresare to be used.

4. In the new ETAFs, more parameters a�ect near-fault acceleration functions because of using atten-uation relations. These parameters are the distancefrom the fault, focal mechanism, soil type anddirectivity conditions; so, new ETAFs may nothave a widespread use for design purposes, unlikeconventional ETAFs.

5. Like conventional ETAFs, the responses valuesobtained by the new ETAFs and real earthquakesare not the same; therefore, the equivalent timehas been used for comparison. The results of thetwo methods become closer to each other at theequivalent time.

6. ETAFs response spectra are only matched withtarget spectrum at the target times de�ned inthis study. However, the responses keep theirintensifying forms in ET curves. Therefore, theresults will be valid for other moment magnitudesas well.

7. Studying SDOF systems shows that the directiv-ity e�ects can properly be modeled by the newacceleration function. As expected, the directivityconditions have no e�ect on the linear systems withperiods of less than 0.6 sec. However, high pe-riod systems have di�erent responses under variousstrike components and directivity conditions. Thisvariety increases as the time increases.

8. Studying nonlinear MDOF systems proves how newETAFs can follow its basic concepts. Regardingnonlinear systems with a period of less than 0.6 sec,the frame's lateral sti�ness decreases as the timeincreases, due to its nonlinear behavior. Then, theresponses are a�ected by directivity, as the frame'speriod exceeds 0.6 sec Concerning under-designed,standard and over-designed frames, new ETAFsfollow the endurance time concept properly.

Acknowledgments

The authors would like to thank Khajeh Nasir ToosiUniversity of Technology for supporting this researche�ectively.

References

1. Estekanchi, H.E., Valamanesh, V. and Vafai, A. \Ap-plication of endurance time method in linear seismic

analysis", Engineering Structures, 29(10), pp. 2551-2562 (2007).

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3. Somerville, P., Smith, N., Graves, R. and Abrahamson,N. \Modi�cation of empirical strong ground motion at-tenuation relations to include amplitude and duratione�ects of rupture directivity", Seismological ResearchLetters, 68(1), pp. 180-203 (1997).

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Biographies

Behzad Ghahramanpoor Soomaie obtained his BSin degree Structural Engineering from the Department


of Civil Engineering at Tabriz University, Iran, in2010, and his MS degree from Khajeh Nasir ToosiUniversity of Technology, Iran, in 2012. His MSthesis was conducted under the supervision of Dr.Nader Fanaie. His �eld of research includes arti�cialaccelerograms, seismic behavior of steel structure andnear fault e�ects. He has published 3 conference papersup to now.

Nader Fanaie obtained his BS, MS and PhD de-grees in Civil Engineering from the Department ofCivil Engineering at Sharif University of Technology,Tehran, Iran. He graduated in 2008 and, at present,is a faculty member of Khajeh Nasir Toosi Universityof Technology, Tehran, Iran. He has supervised 7MS theses up to now. His �eld of research includesseismic hazard analysis, earthquake simulation, seismicdesign and IDA. He has published 15 journal and

conference papers, and also 10 books. He received 3rdplace in the �rst mathematical competition, held atSharif University of Technology, in 1996, and a GoldMedal in \The 4th Iranian Civil Engineering Scienti�cOlympiad" in 1999. In 2001, he achieved the �rst rankin the exam and obtained a PhD scholarship abroad,and was acknowledged as an innovative engineer on`Engineering Day', in 2008.

Hussein Jahankhah obtained his BS, MS and PhDdegrees in Civil Engineering from the Department ofCivil Engineering at Sharif University of Technology,Tehran, Iran. He graduated in 2010 and, at present, isa faculty member of the the International Institute ofEarthquake Engineering and Seismology in Iran. His�eld of research includes soil-structure interaction andnear fault earthquakes, in which areas he has published6 journal papers.

Documents

Studying near fault endurance time acceleration functionscientiairanica.sharif.edu/article_1840_bb726bad... · acceleration for periods over 0.6 sec [3], compared to far fault ones,