point and fault rupture stochastic methods for generating simulated accelerograms considering soil...

Soil Dynamics and Earthquake Engineering 43 (2012) 329–341

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Soil Dynamics and Earthquake Engineering

0267-72

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Point and fault rupture stochastic methods for generating simulatedaccelerograms considering soil effects for structural analysis

Jo~ao M.C. Estev~ao a,n, Carlos Sousa Oliveira b

a Civil Engineering Department, Instituto Superior de Engenharia, University of Algarve, Campus da Penha, 8005-139 Faro, Portugalb ICIST-IST, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

a r t i c l e i n f o

Article history:

Received 25 November 2009

Received in revised form

23 December 2010

Accepted 14 July 2012Available online 1 September 2012

61/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.soildyn.2012.07.019

esponding author. Tel.: þ351 289 800 154; fa

ail addresses: [email protected] (J.M.C. Estev~ao

civil.ist.utl.pt (C.S. Oliveira).

a b s t r a c t

Many seismic codes such as the Eurocode 8 allow the use of simulated accelerograms for structural

analysis, provided that the samples used are adequately qualified with regard to the seismogenetic

features of the sources and to the soil conditions appropriate to the site. In the present work we studied

the possibility of using stochastic methods for that purpose. In that context, two computer programs for

stochastic ground motion simulation considering soil effects were developed: ACELGER based on a

point source model, and SIMULSIS based on a finite fault model. Both programs were used to simulate

the 1992 Landers earthquake for their validation. Simulation results obtained with these programs

were compared between each other to better understand the influence of source fault plane geometry

in structural response. Results seem to indicate that finite fault models are better options for structural

analysis, because only with them it is possible to reproduce directivity effects and non stationary

structural response observed with recorded accelerograms. SIMULSIS was also used to carry out the

simulation of the 1980 Azores earthquake (January 1, 1980, Portugal) in two islands, with different local

site conditions, which were compared with observed damages, to better understand the influence of

soil geology in structural response, and showed that site effects have a major importance in structural

behaviour.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Structural response history analysis demands the use of groundmotion acceleration histories compatible with the seismic codedesign response spectrum for the site. According to many codesnamely the Eurocode 8 [1], time-history representation of seismicaction can be divided in three major groups: recorded accelero-grams, artificial accelerograms (synthetic accelerograms generatedso as to match an elastic response spectra), and simulated accel-erograms (synthetic accelerograms generated through a physicalsimulation of source and travel path mechanisms).

In many regions like Portugal where the historical knowledgeof massive destruction due to the 1755 earthquake exists, but notenough strong-motion records for engineering purposes areavailable, the alternative is to use ‘‘synthetic’’ accelerograms.For a structural engineer artificial accelerograms are very attrac-tive, mainly because they do not depend on seismological knowl-edge. However, it is now widely accepted that the use ofartificial accelerograms in seismic nonlinear analysis have many

ll rights reserved.

x: þ351 289 800 183.

),

problems, such as the fact that these accelerograms tend to beparticularly unrealistic [2]. The main alternatives are the simu-lated accelerograms.

For the Structural Engineering point of view, it is importantthat the selected simulation method is easy and fast to use so thatany structural engineer can adopt it in practical dynamic struc-tural analysis, and accurate enough to guarantee the reliability ofthe results.

There are many methods available for strong ground motionssimulations, which can be classified as deterministic, empirical,semi-empirical, stochastic and hybrid, being the stochastic meth-ods an ‘‘Engineering’’ approach to the problem with successfulcomparisons of predicted and recorded data [3]. Boore proposed astochastic method [4,5], in which a white noise time series isadjusted to a seismological determined point source Fourierspectrum; however, these methods have the disadvantage ofbeing reliable only for frequencies higher than 1 Hz.

Recently, many computer programs for stochastic earthquakeground motion simulation have been developed for seismichazard analysis purposes. Boore’s point source method wasimplemented in SMSIM computer program [6]. FINSIM [7] andEXSIM [8] are other well-known programs for stochastic groundmotion simulation, which adopt a finite fault model. Anotherfinite fault model, including site effects, was developed [9] and

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J.M.C. Estev~ao, C.S. Oliveira / Soil Dynamics and Earthquake Engineering 43 (2012) 329–341330

implemented in LNECloss computer program which has beenused for Portugal seismic risk assessment [10].

To better understand the influence of seismological para-meters in structural response, two computer programs for sto-chastic ground motion simulation were developed using the samestochastic approach but with different source models: ACELGER(with a point source model) and SIMULSIS (with a finite faultmodel). Both programs were developed in Object Pascal forWindows, with DELPHI2007 [11], and consider nonlinear geolo-gical site amplification in the simulation of ground motions.

ACELGER is based on SMSIM but with a different strategy foracceleration time series generation, which we believe is moresuitable for structural analysis because it considers a part of thestrong ground motion as being stationary, and with the capacityto reproduce local non linear site soil amplification (which isdependent not only on the local soil characteristics, but also onthe simulated earthquake characteristics itself) in response toseismic code demands. SIMULSIS adopts a finite fault model and itis an upgrade of ACELGER. As an attempt to optimized the use ofthe already existent stochastic methods for structural analysispurposes, SIMULSIS incorporates some changes, such as the timeseries generation scheme, the consideration of a variable radia-tion pattern and a non uniform rupture velocity and rake, and alsothe correction of source energy (at low and high frequencies), sothat far field results are independent of fault discretization.

2. Stochastic time series

An accelerogram with non stationary frequency content can berepresented by the sum of No sinusoidal waves with amplitudesAn, circular frequencies on (equal spaced at Do) and randomphase angles yn, given by

agðtÞ ¼XNo

n ¼ 1

An � cosðontþynÞ, ð1Þ

An ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� Gaðt,onÞ �Do

p, ð2Þ

where wave amplitudes are dependent on the non-stationarypower spectral density function

Gaðt,oÞ ¼ gðtÞ2 � GaðoÞ, ð3Þ

where g(t) is a deterministic envelope function and Ga(o) is astationary power spectral density function [12].

Eq. (1) phase angles calculation requires a pseudo randomnumber generator procedure sufficiently ‘‘random’’ for the pur-pose. Normally, this kind of computer procedures always returnsthe same sequence of values each time the program runs. To avoidthis problem a pseudo random seed number (normally dependenton the time of day) must be changed each time a sequence ofsimulations is preformed.

The definition of strong motion earthquake duration is veryimportant in time-domain simulations for structural nonlinear

Fig. 1. Envelope function adopted in (A) ACELGER and SI

analysis and may be represented as the sum of the duration ofrupture process (TF), the duration due to propagation path effects(TP) and the prolongation of the motion caused by local siteconditions (TL) [13]. In this work we adopted the concept ofeffective duration (TE) defined as the interval of time between twolimits of the accelerogram Arias intensity [14].

The total duration of the ground motion can be defined as

Tgm ¼NZ � TE ð4Þ

where NZ is a constant value.Fig. 1 presents the envelope function adopted for the devel-

oped software, which is defined so that the effective duration isbetween 2.5% and 97.5% of total Arias intensity, and is accom-plished by solving the following expression, for each simulatedaccelerogram.

Z Tgm

t2

e�c�ðt�t2Þdt ¼ 0:75t�t2

Z t2

0gðtÞdt

¼ 0:75t�t2

Z t1

0

t

t1

� �2

dtþ

Z t2

t1

1dt

" #, ð5Þ

which is equivalent to

1

c1�e�c�ðTgm�t2Þ� �

¼ 0:75t1

3þt0

� �¼ 0:3875� TE: ð6Þ

3. Site effects

The variations on local site conditions (for example, geologicalformations, thickness and properties of soil and rock layers anddepth of bedrock) have significant effects on the characteristics ofearthquake motions on the ground surface [15], so it is importantto account such effects in any simulation method.

For stochastic methods implementation, local site effects L(o)can be expressed as the result of two functions:

LðoÞ ¼HRðoÞ � HSðoÞ: ð7Þ

HR(o) is a filter that accounts for the diminution of the high-frequency motions in a rock outcropping reference site, and canbe given by the combination of two filters [5]:

HRðoÞ ¼ 1þo

omax

� �8" #�1

2

� exp �k0 �o

2

� �, ð8Þ

where omax (¼2pfmax) is the cut-off frequency and k0 is adiminution parameter.

SHAKE91 [16], a version of SHAKE [17], is a commonly usedand referenced computer program for geotechnical earthquakeengineering. In this type of programs, the nonlinear soil transferfunction (HS) for S waves is obtained from an equivalent linearanalysis of a one-dimensional ‘‘soil column’’. This procedure(Fig. 2) was implemented in both programs ACELGER and

MULSIS compared with implemented in (B) SMSIM.

Fig. 3. Equivalent linear analysis iterations procedure in terms of shear modulus

and damping ratio nonlinear variation (logarithmic scale).

Fig. 4. Results of stochastic approach implemented and from program EERA [20]

for the same data.

Fig. 2. Site ‘‘soil column’’ response analysis scheme. Am is the amplitude of the incident

S wave travelling upwards at layer m and Bm is the amplitude of the reflected S wave

travelling downwards at the same layer. ANN and BNN are the wave amplitudes at the

bedrock. At the surface, the incident S wave is equal to the reflected S wave.

J.M.C. Estev ~ao, C.S. Oliveira / Soil Dynamics and Earthquake Engineering 43 (2012) 329–341 331

SIMULSIS, in which:

HSðoÞ ¼9u199uNN9

¼A1

ANN, ð9Þ

um ¼ umðz,tÞ ¼ ðAm � eiln

mzþBm � e�iln

mzÞ � eiot , ð10Þ

with complex wave number given by

ln

m ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirm �o2

Gn

m

s, ð11Þ

with

Gn

m ¼ Gm � ð1þ2ixmÞ, ð12Þ

and where rm is the unit mass, Gm is the shear modulus and xm isthe damping ratio of the soil layer m.

Am and Bm can be determinate through the following recursiveexpressions.

Amþ1 ¼1

2� Am � ð1þan

mÞ � eiln

mhmþ1

2� Bm � ð1�an

mÞ � e�iln

mhm

ð13Þ

Bmþ1 ¼1

2� Am � ð1�an

mÞ � eiln

mhmþ1

2� Bm � ð1þan

mÞ � e�iln

mhm ,

ð14Þ

with complex impedance ratio equal to

an

m ¼ln

m � Gn

m

ln

mþ1 � Gn

mþ1

ð15Þ

Gm can be obtained through the nonlinear relation betweenGm/Gmax and soil distortion (g). Several relations have beenproposed as the result of experimental studies [18,19]. The valueof Gmax is dependent on the soil unit mass (r) and shear wavevelocity (VS), as

Gmax ¼ r� V2S : ð16Þ

The equivalent linear analysis procedure implemented inprograms ACELGER and SIMULSIS has the follow main steps:

– Determination of Gm and xm for each soil layer (or sub-layer).– Quantification of Am and Bm.– Quantification of maximum distortion (gmax) of each soillayer or sub-layer.– Determination of the effective maximum distortion, which isdependent on the earthquake magnitude (M) [16].

geff ¼M�1

10� gmax ð17Þ

The previous steps are repeated until convergence criteria arereached (Fig. 3), and then soil transfer function is computed.

The maximum soil distortion can be computed in time domain orin frequency domain using stochastic methods [9]. Both approacheswere implemented in the programs developed, but the stochasticmethod is faster (especially for long duration accelerograms) withsimilar mean results. Fig. 4 presents the comparison between theresults of stochastic approach implemented and the results of anotherequivalent linear program called EERA [20].

4. Point source model

The spectrum of the ground motion is an essential part of thestochastic method. It is convenient to divide the total spectrum ofthe motion at a site A(o,M0,R) into contributions from earthquakesource F(o,M0), path P(o,R) and local site effects L(o).

Aðo,M0,RÞ ¼ Fðo,M0Þ � Pðo,RÞ � LðoÞ: ð18Þ

Fig. 5. Peak acceleration obtained with (A) ACELGER and with two attenuation

laws (B) Ambraseys et al. [21] and (C) Tromans and Bommer [22] for several

magnitudes (5, 6, 7 and 8) and stress drops (respectively 130, 100, 80 and 60 bars).


The point source spectrum of shear waves can be modelled by

Fðo,M0Þ ¼ C0 �M0 � A0ðoÞ, ð19Þ

where M0 is the seismic moment (in dyne cm) of the earthquake,and C0 is a constant given below

C0 ¼Ryf � CV � CF

4p� r� b2� R0

� 10�20, ð20Þ

where Ryf is the radiation pattern of the source, CV represents thepartition of total shear-wave energy into horizontal components,CF is the effect of the free surface (equal to 2 for SH waves),r (g/cm3) and b (km/s) are the density and shear-wave velocity inthe vicinity of the source, R0 (¼1 km) is a reference distance,respectively, and A0(o) is an acceleration spectrum

A0ðoÞ ¼o2

1þ ooC

� �2, ð21Þ

which is related to the corner frequency

oC ¼ 2p� 4:9� 106� b

DsM0

� �1=3

, ð22Þ

where Ds is the stress drop (in bars) [4,5].The path effects are represented by the combination of two

functions, given by

P o,Rð Þ ¼ PG Rð Þ � PA o,Rð Þ, ð23Þ

where PG(R) accounts for geometrical spreading of seismic energy,which can be defined as

PGðRÞ ¼

ðR0=RÞp0 ,RrR1

PGðR1Þ � ðR1=RÞp1 ,R1oRrR2

PGðRnÞ � ðRn=RÞpn ,RnoR

8><>: ð24Þ

and PA(o,R) for anelastic attenuation, being

PAðo,RÞ ¼ exp �o� R

2� Q ðoÞ � cQ

ð25Þ

The anelastic attenuation is a function of the quality factorQ(o), in which cQ is the velocity of the seismic waves used in thedetermination of Q(o).

This point source model was implemented in the computerprogram ACELGER, in combination with Eq. (1), and with thefollowing source power spectral density function

GaðoÞ ¼Fðo,M0Þ

2

p� TF, ð26Þ

so that the amplitudes of sinusoidal waves at the site are

An ¼ gðtÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�Dop� TF

s� Fðo,M0Þ � Pðo,RÞ � LðoÞ: ð27Þ

Acceleration is evaluated at an even number (Nt) of points witha time interval of Dt. The number of equally distant harmonicwaves will be dependent on Nyquist frequency, so

Do¼ pNo � Dt

¼2p

Nt � tD: ð28Þ

For program validation purposes, the mean stochastic peakacceleration results of ACELGER were compared with the valuesobtained with two different Europe attenuation laws [21,22] forrock outcropping. The simulations were carried out consideringthe same values adopted in North American stochastic studies[23], with exception of stress drop.

Results for several magnitudes are presented in Fig. 5 andseem to be similar when mean stress drop is adjusted (lowerstress drop values for higher magnitudes and higher stress dropvalues for lower magnitudes).

5. Fault rupture model

The methodology implemented in SIMULSIS was based onseveral concepts that support the program EXSIM [8] but withsome modifications.

The simulated accelerogram results from the contribution of anumber of small earthquakes as subfaults that comprise a bigfault (Fig. 6). A large fault is divided in NF subfaults and eachsubfault is considered as a point source event. In SIMULSIS therupture spreads radially from the hypocenter, with a constant or avariable rupture velocity Vri on each subfault i, so time series

Fig. 6. Global SIMULSIS ground motion generation procedure.


results from a superposition of sinusoidal waves that are summedwith a proper delay

agðtÞ ¼XNF

i ¼ 1

XNo

n ¼ 1

Ai,nðDtiÞ� cosðonDtiþyn,iÞ: ð29Þ

The wave amplitude Ai,n(Dti ) is the contribution of the pointsource i to the frequency n.

Point source events are simulated with the proceduredescribed earlier for program ACELGER, but with a modifiedsource spectrum for each subfault i, which is

Fiðo,M0iÞ ¼Hi � HðoÞ � C0i �M0i � A0iðoÞ: ð30Þ

The concepts of dynamic corner frequency [8] and activepushing area (DLr) [24] were adopted, but with the functionNR(DtRi) representing the cumulative percentage of rupturedsubfaults, in terms of seismic moment, between the most distant(less or equal to DLr) subfault N0 and the subfault i, given by

oci ¼ 2p� 4:9� 106� b�

DsNRðtRiÞ �M0

� �1=3

, ð31Þ

NRðDtRiÞ ¼1

M0�Xi

n ¼ N0

Mon, ð32Þ

and

Mon ¼DunPNf

i ¼ 1 Dui

�M0, ð33Þ

to account for the existence of fault asperities and a non uniformslip distribution (Dui).

If N0¼1 and i¼NF then Eq. (31) gives the same result asEq. (22) adopted for the point source model.

Hi and H(o) are scaling factors which guarantee that the sourceradiated energy obtained with a finite fault model equals thevalue obtained with a global point source model, independentlyof the number and size of subfaults considered.

The scaling factor Hi guaranties that the subfault energy isconserved at high frequencies. The SIMULSIS have the option of

using two different expressions for the determination of Hi,which are:

Hi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM0

M0i�

P½A0ðoÞ�2P½A0iðoÞ�2

vuut ; ð34Þ

similar to the expression that is implemented in EXSIM program,or the following proposed expression

Hi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

NF�½A0ð1Þ�

2

½A0ið1Þ�2

s: ð35Þ

The scaling factor H(o) that we proposed, which is notpresented in earlier methods, corrects the total energy contentat all frequencies, and is given by

HðoÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM0 � A0ðoÞ½ �

2PNF

i ¼ 1 Hi �M0i � A0iðoÞ½ �2

vuut : ð36Þ

The radiation pattern (Ryfi) for the S waves is calculated usingand Eq. (3) proposed by Boore and Boatwright [25], as follows

Ryfi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR y2i

y1i

R 2p0 ðF

2SVþF2

SHÞ � sinðyÞdfdyR y2i

y1i

R 2p0 sinðyÞdfdy

vuuut , ð37Þ

where the FSV and FSH radiation pattern are obtained according toEq. (4.90–91) proposed by Aki and Richards [26]. Ryfi is a functionof the strike (fS), the dip (d), the rake (l) and the azimuthal angle(f) of subfault i, and is also a function dependent on the limitsbetween two takeoff angles (y1 and y2) of seismic ray trajectories(Fig. 7).

Boore and Boatwright [25] considered three different ranges oftakeoff angles: y1¼171 and y2¼251 for use at teleseismic dis-tances; y1¼601 and y2¼1201 for use at regional distances (tens tohundreds of km); y1¼1201 and y2¼1801 for use at close distances(within a source depth).

SIMULSIS also allows the use of a constant pair of y1 and y2

values for the determination of the radiation pattern, or differenty1 and y2 values for each subfault, namely by considering the


mean value between two ray trajectories, a direct path and anindirect path that is tangent to the vertical direction at the site,limited by a maximum depth (Zmax).

If a non-uniform rupture velocity is considered, the modelimplemented in SIMULSIS for the determination of the subfaultrupture trigger instant (Dtri) is based on the hypothesis that therupture is propagated radially from each subfault. First a networkof all possible links between each two adjacent subfaults i and j

are created. Then the distances (dij) and rupture duration (tij) ofeach network link i–j are computed assuming a linear velocityvariation, which leads to

dij ¼

Z tij

0VrðtÞdt ð38Þ

tij ¼2� dij

VriþVrjð39Þ

Fig. 8. Studied sites position and 1992 Landers earthquake fault considered (Silent

Fig. 7. Definition of the subfault-orientation parameters (strike fSi and dip di),

slip-direction parameter (rake li) and takeoff angles of seismic rays (between y1i

and y2i) used in the radiation pattern determination procedure.

Finally subfault trigger instants are computed so that the totalrupture duration between the focus and any subfault is minimized.

6. Simulation of the 28 June 1992 Landers earthquake

For the purpose of SIMULSIS validation, the response spectra ofthe simulated accelerograms were compared with the responsespectra of the records obtained after the 28 June 1992 Landersearthquake, in California.

The Landers earthquake resulted from lateral shear of fivemajor faults, with different directions, within an 80 km wide belt.Its seismic moment of l027 dyne cm is equivalent to a magnitudeMw of 7.3 [27]. Because of SIMULSIS limitations (does not considermultiple sources), it was considered only one equivalent verticalfault plane with L¼80 km and W¼15 km, divided into23�9¼207 subfaults (Figs. 8 and 9).

Time-domain simulations were carried out for three sitesaround the rupture (Fig. 8), as presented in Table 1.

Stress drop of 110 bars was adopted [28], with b¼3.7 km/s,r¼2.8 g/cm3, and DLr¼20 km. The path attenuation expressionsadopted are the same that were adopted in North Americanstudies [23].

Valley/Poppet Flat, Lake Cahuilla/County Park and Pearblossom/Pallet Creek).

Fig. 9. (A) Fault slip distribution (mean fault azimuth of 1571) and (B) variable

rupture velocity (km/s) with schematic representation of rupture progression,

adopted for 1992 Landers earthquake simulations (the dot indicates de hypocen-

ter location).

Table 1CGS-CSMIP stations information.

Site identification N %o Latitude (1) Longitude (1) Altitude (m) Geology description

Silent Valley/Poppet Flat 12206 33.851N 116.852W 1098 Weathered granite

Lake Cahuilla/County Park 12624 33.628N 116.28W 35 Hard granodiorite bedrock

Pearblossom/Pallet Creek 23584 34.458N 117.909W 1206 Granitic rock

Fig. 10. Recorded and simulated accelerograms (cm/s2) and corresponding

response spectra at Silent Valley/Poppet Flat (56 km from epicentre).


response spectra at Lake Cahuilla/County Park (67 km from epicentre).


Fault slip distribution adopted (Fig. 9A) was adjusted fromother simulation results for the Landers earthquake, with differ-ent models [29,30]. An adaptation of the rupture progressionpresented by other authors [31] was considered, with a variablerupture velocity between 1 and 4 km/s (Fig. 9B), and a meanvalue of 2.7 km/s. For each simulation and subfault, it wasintroduced a random variation on slip (710%), on rake(7201), on rupture velocity (710%), on dip (751) and onazimuthal angle (7101), which leads to a variable radiationpattern for each subfault.

No soil effects were considered, k0¼0 and cut-off frequencyfmax equal to 8 Hz (Pearblossom/Pallet Creek), 13 Hz (LakeCahuilla/County Park) and 14 Hz (Silent Valley/Poppet Flat).

The effective duration prediction was obtained with an empiricalexpression, which considers the three components of duration [32],as described earlier.

Simulations were carried out considering a constant pair ofvalues for takeoff angles (y1¼1201 and y2¼1801 for the SilentValley/Poppet Flat and the Lake Cahuilla/County Park site, whichcorresponds to a mean value Rfyi¼0.55, y1¼601 and y2¼1201 forthe Pearblossom/Pallet Creek site, which corresponds to a meanvalue Rfyi¼0.70) and Eq. (35), as described earlier. The accelera-tion waveforms and corresponding response spectra of some ofthe best simulation results are presented in Figs. 10–12.

7. Point source model versus finite fault model

Studies have pointed some discrepancies between the resultsof SMSIM (a point source model) and EXSIM (a finite fault model)for the same stress drop [33].

In this work we compared the mean peak acceleration resultsof ACELGER and SIMULSIS obtained for the same earthquakecharacteristics and hypocenter distances, but with different faultorientations and focus locations (Fig. 13). In the first source casefor finite fault model, the focus is at the middle of the fault andthe site direction is perpendicular to the fault plane (case 1 ofFig. 13). In the second case, the earthquake focus is at the end ofthe fault and the rupture is towards the site, and the direction isaligned with the fault plane (case 2 of Fig. 13). The last case issimilar to the second one, but the rupture is in the oppositedirection of the site location (case 3 of Fig. 13).

Simulations were carried out for two different magnitudes(M¼6 and M¼8), considering a vertical fault with no asperitiesand constant rupture velocity, and showed (Fig. 14) that for anepicentre distance (D) much greater than the fault length, peakacceleration results are similar in both ACELGER (point sourcemodel) and SIMULSIS (finite fault model).

It is also interesting to notice the peculiar shape of the secondsource case results for M¼8 (Fig. 14), and to point that the

Fig. 13. Different sources cases (arrows indicate rupture direction) considered in

finite fault model simulation results comparison (with SIMULSIS).

Fig. 14. Simulation mean peak acceleration results for different shape sources,

magnitudes (6 and 8) and corresponding fault length (L), width (W) and stress

drops (Ds), as a function of epicentre distances (D).


response spectra at Pearblossom/Pallet Creek (138 km from epicentre).


maximum peak acceleration value does not correspond to thenearest epicentre distance.

To evaluate the influence of those models in the simulatedaccelerograms characteristics, several dynamic linear structuralanalyses were performed using simulated and recorded 1992Landers earthquake ground motions at Silent Valley/Poppet Flat.This site was selected because, according to the previous results(Figs. 13 and 14), this is the site where the differences betweenACELGER and SIMULSIS results could be minimal, because the sitedirection is almost perpendicular to the fault plane.

Structural time-history analysis of a six storey structure wereperformed with computer program SAP2000 [34] to observe thedifferences obtained with a simulated earthquake excitationversus a real one. To better compare the structural behaviour,

AVI videos were created for those analysis results, and showedthat, for the recorded accelerograms, resonant vibration shapewas not always the same, changing essentially between the firstthree modes through time in a chaotic manner.

The structural response in result of SIMULSIS simulated accel-erograms excitation (finite fault model) exhibits the same patternof behaviour as for the recorded ground motions. However, theresults with the point source model (ACELGER) do not reveal thatkind of behaviour, showing a structural response dependentbasically on one vibration mode (normally the first mode). Thisevidence can be related to the fact that in this kind of models thewave amplitude is non stationary, but not the frequency content,which is not the case of the finite fault model where both waveamplitude and frequency content are non stationary. Moreover,ACELGER simulations at Silent Valley/Poppet Flat presented muchhigher peak acceleration and lower effective durations (Fig. 15).

Several linear time-history analyses of a single-degree-of-freedom structure was also carried out for 20 different naturalfrequencies equally spaced between 1.4 and 3.3 Hz. As presentedin Fig. 15, the results obtained with ACELGER (point sourcemodel) exhibits almost the same pattern for all studied frequen-cies and are much different from those obtained with one record.Only SIMULSIS (finite fault model) simulates earthquake groundmotions with non stationary frequency content and seems to besimilar to recorded accelerograms in terms of duration andamplitude, although it does not match the earthquake records.

8. Simulation of the 1 January 1980 Azores earthquake

After SIMULSIS validation, the software was used to simulatethe 1980 Azores earthquake. This seismic event (epicentre coor-dinates were 38.811N, 27.781W with depth focus about 10 km)affected a large percentage of the building stock in the Portuguese

Fig. 15. Results from a linear time-history analysis of a single-degree-of-freedom structure subject to recorded and simulated 1992 Landers earthquake ground motions at

Silent Valley/Poppet Flat. Natural frequencies equally spaced between 1.4 (top) and 3.3 Hz (bottom).

Fig. 16. Slip distribution (mean fault azimuth of 1491) considered for the

simulations of 1980 Azores earthquake (the dot indicates the hypocenter

location).


islands of S. Jorge, Graciosa and Terceira with evidence ofdirectivity and local site effects [35–37].

The only strong motion record available was obtained at‘‘Observatorio Prıncipe do Monaco’’ in Horta, and it is notcomplete [35]. In the past, to establish the upper and the lowerbounds for the strong motion in other Azores islands that suffermajor damages, several structural analysis of concrete damagedstructures were made [38]. At that time, several laboratorialtests were carried out to characterise the mechanical propertiesof the buildings materials, and the Portuguese code responsespectra [39] were used in the 3D dynamic structural analysis as abase to estimate the peak ground acceleration that leads to thedamage observed, namely for the ‘‘Semaforo do Monte Brasil’’ (itwas used the type 1 response spectrum, which is a near fieldmoderate magnitude earthquake, for soil I which is a rock or avery stiff soil, with a peak acceleration of 177 cm/s2) and for theHospital of Angra do Heroısmo (it was used the type 2 responsespectrum, which is a far field higher magnitude earthquake, forsoil III which is a soft soil, with a peak acceleration of 108 cm/s2),both in Terceira. Based on the observed damage and on theprevious mentioned Portuguese response spectra, the estimatedvalues for the peak acceleration were, respectively, 27 and144 cm/s2.

Several simulations were carried out with a constant pair ofvalues for takeoff angles (y1¼1201 and y2¼1801 which corre-sponds to a mean value Rfyi¼0.55) and with Eq. (35), for‘‘Observatorio Prıncipe do Monaco’’ in Horta, for ‘‘Semaforo doMonte Brasil’’ and for the Hospital of Angra do Heroısmo, inTerceira, to compare to the recorded and the estimated seismicaction values, as describe earlier.

In these simulations we considered a fault plane withL¼30 km and W¼10 km, divided into 15�5¼75 subfaults(Fig. 16). Two asperities were also considered, representing 22%


of total fault area, which is the mean value proposed by severalauthors [40] for an earthquake source event of this type.

The magnitude (Mw¼6.8), the strike (1491) and dip (851)considered were the proposed in other studies [41], as well therupture velocity of 1.8 km/s. The seismic moment was obtainedfrom an empirical equation [42].

It was adopted a stress drop of 30 bars, with b¼3.7 km/s,r¼2.8 g/cm3, and DLr¼8 km. In the absence of regional pathattenuation expressions, we used the same expressions that wereconsidered in North American studies [23].

For each site twenty simulations were carried out to computethe mean response spectrum which was compared with therecorded strong motion response spectra or with the Portugueseresponse spectra adjusted to the peak acceleration estimatedvalues as described earlier. A random variation on slip (725%),on rake (751), on rupture velocity (710%), on dip (751) and onazimuthal angle (751) was considered for each simulation, whichleads to a variable radiation pattern. Local site conditions wereadjusted so that the simulations fit the results of structuralbehaviour observed during the earthquake.

Fig. 17. Soil column shear velocity profile considered and mean transfer function ob

Heroısmo.

Fig. 18. Response spectra comparison between earthquake records and the mean value

epicentre).

�

tain

s of

‘‘Observatorio Prıncipe do Monaco’’ in Horta, Faial—Previousstudies [43] have shown the evidence of local amplification forthis hill. The simulations for this site (fmax¼7 Hz) were carriedout considering a 35 m soil column composed by multiplelayers of pumitic pyroclastic and clinker deposits and lavaflows, with the same properties as used in other studies forHorta [44], with some minor adjustments on shear velocity,and using Ishibashi and Zhang [19] and Schnabel et al. (forrock) [17] dynamic shear moduli and damping ratios. Themean transfer function obtained (Fig. 17) is consistent within situ measured results [45], and mean simulation responsespectra seems to be well related to the recorded responsespectra values (Fig. 18).
� ‘‘Semaforo do Monte Brasil’’ in Angra do Heroısmo,
Terceira—The simulations for this place consider basaltic rockoutcropping (fmax¼15 Hz). The mean results were comparedwith type 1 (soil I) response spectrum of Portuguese code [39]multiplied by the ratio of 27/177 to adjust to the peak value of27 cm/s2 as proposed by Oliveira et al. [38] according withobserved structural behaviour (Fig. 19).

ed for (A) ‘‘Observatorio Prıncipe do Monaco’’ and (B) Hospital of Angra do

20 SIMULSIS simulations at ‘‘Observatorio Prıncipe do Monaco’’ (80.2 km from

Fig. 19. Response spectra comparison between the mean values of 20 SIMULSIS simulations at ‘‘Semaforo do Monte Brasil’’ in Angra do Heroısmo (51.3 km from epicentre)

and the Portuguese Code earthquake type 1 and soil I adjusted to the estimated peak ground acceleration.

Fig. 20. Response spectra comparison between the mean values of 20 SIMULSIS simulations at Hospital of Angra do Heroısmo (51.9 km from epicentre) and the Portuguese

Code earthquake type 2 and soil III adjusted to the estimated peak ground acceleration.


�
Hospital of Angra do Heroısmo, Terceira—This reinforcedconcrete structure was built with piles foundation (20 mdepth) on a soft soil formation, and had suffered somemoderate structural damage on the ground floor columns.Simulations were carried out considering soil effects (20 m ofsoil layers composed by multiple layers of pumitic pyroclasticand clinker deposits with similar characteristics as adopted inHorta simulations and with fmax¼15 Hz). The shear wave soilprofile and the mean transfer function obtained are presentedin Fig. 17. Mean results were compared with the type 2 (soilIII) response spectrum of Portuguese code [39] multiplied bythe ratio of 144/108 to adjust to the peak value of 144 cm/s2 asproposed by Oliveira et al. [38] based on the observedstructural behaviour (first 13 vibration modes between 1.36and 6.1 Hz), as it is presented in Fig. 20.
9. Conclusions

All simulations carried out in this work seem to indicate thatstochastic methods are able to reproduce peak acceleration andresponse spectra, with good approximations of earthquake strongmotions records, if adequate source and site characteristics areconsidered. However, as each simulation gives a different result,which is dependent on the generated random phase angles, forstructural analysis purposes it is advisable to select the acceler-ograms which are closer to the mean simulated values. We alsonoticed that the simulation mean values results are very sensitiveto the random phase angles generation procedure.

Mean peak acceleration obtained with point source modelseems to be well related to attenuation law curves if properregional parameters are adopted.

When comparing finite fault models with point source models,mean peak acceleration results seem to be comparable only if thedistance to the source is much higher than the fault dimension.Moreover, the comparison between ACELGER and SIMULSISsimulations indicate that directivity effects can only be repro-duced by finite fault models.

The simulations carried out for the 1992 Landers earth-quake showed an overall good approximation to the realitywhen comparing the response spectra and the accelerationwaveforms to the recorded values. To adjust the simulatedacceleration waveforms to the observed ones it is very impor-tant the consideration of the rupture velocity variationthrough the fault and also the adoption of a correct slipdistribution. The poor result obtained for the simulated accel-eration waveforms was for the Lake Cahuilla/County Park site.Moreover, Pearblossom/Pallet Creek site presented the great-est differences between the recorded and simulated meanresponse spectrum. Those differences can be related to thepossibility that the earthquake fault cannot be represented bya unique fault plane. Instead, several planes with differentfault azimuths should be considered. For periods higher thanone second the simulated mean response spectrum forthe Cahuilla/County Park site is almost coincident with therecorded ones, but for the Silent Valley/Poppet Flat and thePearblossom/Pallet Creek sites the simulations mean resultspresented higher spectral values.


Linear time-history analysis of structures subject to recordedearthquake accelerograms that were performed in this workshowed a chaotic change of structural resonance vibration modesthrough time, which is probably an important ground motioncharacteristic for the nonlinear structural behaviour. Only simu-lated accelerograms obtained with a finite fault model seemed tobe able to reproduce this observed structural behaviour, becausethe studied point source models do not have the capability toreproduce the non stationary frequency content of recordedaccelerograms.

Simulations for the 1980 Azores earthquake indicate that localsite effects have a major influence in structural damage so it isvery important to survey site soil profiles to be able to reproducethe earthquake effects. Moreover, response spectra of simulatedground motions for soft soils presented much higher maximumspectral acceleration amplification when comparing with peakacceleration amplification.

Overall, it seems that computer programs using stochasticmethods for ground motion simulation combined with equivalentlinear methods to reproduce site amplification, like SIMULSIS(with a fault rupture model), can be used to generate simulatedaccelerograms for nonlinear structural analysis purposes and fulfilseismic code demands for its use.

In future work, for seismic hazard analysis purposes, multiplefault planes will be considered in SIMULSIS. For the samepurposes the hypothesis of the simulation of east–west andnorth–south components of ground motion will be separated asproposed by other authors [46], and the frequency dependentradiation pattern will be investigated.

Acknowledgments

We thank to an anonymous referee for the thorough reviewwhich contributed to improve the method implemented inSIMULSIS as well as the quality of the manuscript. The secondauthor acknowledges the partial financial support given byFundac- ~ao para a Ciencia e a Tecnologia, Portugal (FCT) throughits Pluri-Annual Programme.

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