Spatial Distribution of Simulated Response for Earthquakes, Part I Ground Motion

Fenves,c c M.EERI, Jaesung Park,d d and Bozidar Stojadinovice ,,e M.EERI The objective of this study is to examine, by computer computationalsimulation, the spatial and temporal distr

Spatial Distribution of Simulated Response for Earthquakes, Part I: Ground Motion

Jacobo Bielak, Antonio Fernández, Gregory L Fenves, Jaesung Park, and Bozidar Stojadinovic

Corresponding author: Gregory L Fenves

Mailing address: Department of Civil and Environmental Engineering

University of California, BerkeleyBerkeley, CA 94720-1710

Phone: 510-643-8543

Fax: 510-643-5264

Email: fenves@ce.berkeley.edu

Submission date for review copies: February July 301, 2004

Submission date for camera-ready copies:

<<<Greg, even though I agree with the change you made on page 3 regarding finitedifferences and finite elements, I prefer to put finite differences and finite elements on anequal footing to avoid controversy Otherwise we would have to add a more detailedexplanation since a number of people have developed efficient FD codes for forward wavepropagation in basins>>>

<<<Jaesung: Could you please add the authors’ affiliations on the next page? Please copy from Paper 2 Also, please reformat the abstract to conform to Earthquake Spectra style Thanks.>>>

Spatial Distribution of Simulated Response for Earthquakes, Part I: Ground Motion

Jacobo Bielak,a ) a) M.EERI, Antonio Fernández, b ) b) M.EERI, Gregory L

Fenves,c ) c) M.EERI, Jaesung Park,d ) d) and Bozidar Stojadinovice ) ,,e) M.EERI

The objective of this study is to examine, by computer computationalsimulation, the spatial and temporal distribution of the earthquake response ofidealized structures to near-source ground motionground motion near a causativefault In this paper we model theAnground motion of idealized 20 km by 20 kmregion is modeled as a layer on an elastic halfspace with dense spatial samplingfor frequencies up to 5 Hz Two scenario events are considered, a a strike-slipfault and a thrust fault, in a layer on a halfspace, by the use of finite dislocationmodels These idealized models are representative of two common types ofearthquakes, in which the directivity of the rupture generates large, short-duration,velocity pulses in the forward direction, and large spatial variation of the free-surface motion throughout the epicentral region We obtain a dense spatialsampling of ground motion over a large region for frequencies up to 5 Hz in order

to elucidate the effects of the source and path on the near-field ground motion.For the strike-slip fault event, the fault normal component exhibits a very strongforward directivity effect with a strong pulse-type motion approximately twice theamplitude of the motion in the fault parallel direction The dynamic effect in thefault parallel direction produces a pulse-type motion near the epicenter For thethrust fault event the greatest concentration of ground displacement occurs nearthe corners of the fault opposite the hypocenter, in the rake direction In contrastwith the strike-slip fault, the ground displacement in the direction of the slip isgreater by a factor of two than in the direction normal to the slip In a companionpaper, we examine how the free-field ground motion from the two scenarioearthquakes influences the spatial and temporal distribution of structural response

of a family of simple single -degree-of-freedom elastoplastic systems

A companion paper then uses the synthetic records to examine how the field ground motion for the two scenario earthquakes influences the spatialdistribution of structural response of a family of SDF elastoplastic systems in aregion close to the causative fault

free-a ) Professor, Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213.

b Formerly, Graduate Student Researcher, Department of Civil and Environmental Engineering, Carnegie )

Mellon University, Pittsburgh, PA, 15213; currently, Manager of International Projects, Paul C Rizzo Assoc.,

520 Exposition Mall, Monroeville, PA, 15245.

INTRODUCTION

Large earthquakes in urban regions cause a highly variable spatial distribution of damage

to the built-infrastructure because of source effects, path effects, large-scale geologicalstructures such as sedimentary basins, site response effects, and the structural characteristics

of the buildings and other infrastructure components The distribution of damage in the 1994

Northridge M w=6.5 earthquake (Somerville, 1995) and the 1995 Hyogoken-Nanbu, Japan,

Mw=6.9 earthquake (Akai et al., 1995) raised profound questions about these effects, withparticular concern about the large velocity pulses at sites near the faults Additional data

obtained from the 1999 Kocaeli, Turkey, M w=7.4 earthquake (Rathje et al., 2000) and the

1999 Chi-Chi, Taiwan, M w=7.6 earthquake (Li and Shin, 2001) provided further evidence thatthe spatial distribution of ground motion in a region is related to the fault mechanism andpath effects The large number of ground motion data recorded in these earthquakesconfirmed that near-fault pulse-type ground motion can be very damaging to buildings, asfirst identified in the 1971 San Fernando earthquake (Bertero et al., 1978)

Researchers have examined the spatial distribution of ground motion based on theextensive records obtained in the 1994 Northridge earthquake and its aftershocks.Aftershock records obtained by Meremonte et al (1996) and Hartzell et al (1997) showedthat the amplitude of peak ground velocity varied by a factor of two or greater at differentsites with the same general geological structure but separated by distances of only 200 m.Boatwright et al (2001) developed correlations between ground motion parameters andeffects on structural response, as measured by an intensity measure based on building tags.The correlations show that peak ground acceleration (PGA) is a poor predictor of damage,peak ground velocity (PGV) is much better, and that an averaged pseudo-velocity (between0.3 and 3.0 seconds period), which is linearly related to PGV, is a good predictor Contours

of PGV and averaged pseudo-velocity correspond to the spatial distribution of damage based

on a tagging intensity In another study using 1994 Northridge earthquake strong motiondata, Bozorgnia and Bertero (2002) examined various responsethe spatial distribution ofstructural response parameters associated with damage quantities associated with structuraldamage as the spatial distribution of damage parameters The spatial distribution of ductilitydemand for single- degree-of-freedom (SDF) systems of 1- and 3-sec vibration periods using

a code-specified strength are qualitatively similar to the PGV contours of peak groundvelocity Their analysis shows that 1-sec and 3-sec period SDF systems reach a ductilitydemand of 9 and 5, respectively The large ductility demands are concentrated in the updipregion direction of slip of the buried thrust fault for the 1-sec case and the updip direction andepicentral area for the 3-sec case Additional results are presented for spatial distributions ofstructural damage measures that include the effect of ductility and hysteretic energy demandsand capacities

An important feature of large-magnitude earthquakes in the near field is the large amount

of seismic energy from the rupture that is concentrated in the forward directivity zone of near

a the fault This energy is generally manifested as a single large pulse of ground motion(e.g., Somerville et al., 1997) Somerville (1998) defines the forward directivity effectsoccurring at sites where the fault rupture propagates towards the site and the direction of faultslip is aligned in the direction of the site These conditions are met with strike-slip faults anddip-slip rupture in thrust faults Forward directivity effects in the fault normal directionoccur in all locations along a strike-slip fault For thrust faults, however, forward directivityeffects occur mostly in the surface projection of the fault, updip from the hypocenter

Backward directivity and neutral directivity effects generally produce longer period, butlower amplitude, motion in the fault-normalfault normal and fault-parallelfault paralleldirections Somerville et al (1997) have developed a procedure for taking intoconsiderationconsidering the effects of near-fault motion on the ground motion from strike-slip faults by modifying the elastic response spectra The In this approach, the spatial effect

of directivity depends on geometric parameters related toon the direction of fault rupture,andthe location of the site with respect to the epicenter, and also the fraction of fault rupturebetween the epicenter and site

To account for near-fault effects in seismic design of buildings, the 1997 UniformBuilding Code (ICBO, 1997) incorporated near-fault amplification factors for theacceleration-sensitive and velocity-sensitive period region bands of the elastic designresponse spectrum The factors depend on the distance to the fault and the fault mechanismand magnitude, and also the The near-fault factors depend on the bands of distance to thefault (less thanwithin 15 km) and the type and magnitude of the anticipated fault rupture.The 1997 UBC provisions near-fault factors are compatible comparable with the mean valuesfor spectrum fault-normalmodification factors and fault-parallel values from Somerville et

al (1997), but they do not distinguish between fault-normalfault normal and parallelfault parallel ground motion, for which the former is substantially larger greater inthe forward directivity zone

fault-Although recent large earthquakes have produced extensive ground motion records as aresult of increased instrumentation in urban areas, their coverage is still far from complete,and theirthe spatial resolution is not high enough to describe fully the distribution of groundmotion with sufficient fidelity Thus, there is an important need for developingfor high-resolution, realistic simulations of ground motions in a region to study the seismological,geotechnical, and structural effects of urban earthquakes in detail A number of studies havebeen devoted recently to modeling earthquake ground motion in realistic basins (e.g.,Frankel, 1993; Graves, 1993; Olsen et al, 1995; Olsen and Archuleta, 1996; Pitarka et al,1998; Hisada et al., 1998; and Stidham et al., 1999) These simulations, however,are wereusually generally limited to low frequencies ( ≤ 0.5 Hz) Motivated by the 1994 Northridgeearthquake, Hall et al (1995) and Hall (1998) simulated the ground motion produced by anearthquake on a thrust fault to examine the effect of near-fault ground motions on 20-storysteel frame buildings and long-period based isolated buildings The fault rupture parameters

were calibrated to correspond to a scenario M w=7.0 event on the Elysian Park buried thrustfault in Los Angeles Ground motions were simulated on an 11 by 11 grid with 5-kmspacing The peak ground velocity was 1.8 m/s and large areas had velocities greater than 1m/s The simulations of the 20-story buildings located at the grid points showed that peakstory drift ratios reached 0.064 in the forward directivity zone because of the large velocitypulses Standard design procedures for a base-isolated three-story building resulted in largebearing displacements and story drifts Hall (1998) then investigated the earthquake response

of six-story and two-story steel frame buildings designed according to U.S and Japanesecodes using Elysian Park thrust fault simulation and a simulation based on the 1994Northridge earthquake The near-source velocity pulses produced large drifts in buildingslocated at grid points in the forward-directivity zones, and fracture of the moment-resistingconnections has a detrimental effect on drifts

new means to obtain qualitative and quantitative descriptions of thesimulate ground motion

in a region, including near-field fault ground motioneffects This was followed bycomprehensive parametric studies by Boore et al (1971), Archuleta and Hartzell (1981),Anderson and Luco (1983a, 1983b), and Luco and Anderson (1983), among others, In thepast 10 ten years numerical modeling methods for anelastic wave propagation that takes intoconsideration the earthquake source, propagation path, and local site effects in realisticmodels of basins have become increasingly available There are using several types of suchmethods Boundary element and discrete wavenumber methods have been popular formoderate-sized problems with simple geometry and geological conditions (e.g., Mossessianand Dravinski, 1987; Kawase and Aki, 1990; Bielak et al., 1991; Hisada et al., 1993;Sánchez-Sesma and Luzón, 1995; Bouchon and Barker, 1996) Finite differences methodsare common (e.g., Frankel and Vidale, 1992; Frankel, 1993; Graves, 1993, 1996; Olsen et al.1995; Pitarka, 1999; Stidham et al., 1999; Sato et al., 1999), and) and fFfinite elements (e.g.,Lysmer and Drake, 1971; Toshinawa and Ohmachi, 1992; Bao et al., 1998; Aagard et al.2001) are better suited for largerlarge-sized problems that involve realistic basin models withhighly heterogeneous materials, because of their flexibility and simplicity

Bao et al (1998) developed one of the first scalable, parallel, finite element tools forlarge-scale ground motion simulation in sedimentary basins with heterogeneous materials.Aagaard et al (2001) used parallel finite element simulations to investigate the sensitivity oflong-period ground motion to different seismic source parameters, on strike-slip fault andthrust fault mechanisms, with particular emphasis on the large velocity pulses in the forwarddirectivity zones Their simulations show that the peak ground motion occurs within thebands zones for the near-fault factors in thewhich the 1997 UBC for faults with surfacerupturespecifies near-fault factors A buried thrust fault simulation, however, shows peakground motions occurring in the up-dip direction over a larger area than defined for the near-fault factor

The previous researchPrevious research using recorded ground motion data and earlysimulations has examined on the spatial distribution of ground motion and its effects onstructural response has s <<<the subject is “research”>>>veidentified many important trends.Due to the limited relatively few number of ground motion records and earlier limitations incomputational capabilitescapabilities, there remain additional questions concerning thecharacterization of the spatial effects, the systematic interpretation of such effects usingstructural dynamic concepts, and the evaluation of building code approaches for accountingfor the effects Thus, the objective of this paper is to use the ground motion simulation oftwo representative scenario earthquake events with frequencies up to 5 Hz to obtain a densespatial sampling of ground motion over a large region in order to elucidate the effects of thesource and path on the near-fieldnear-fault ground motions AA companion paper (Fenves

et al., 2004) then uses the dense sampling of ground motion to examine the distribution ofstructural response in a region close tonear the causative faults in the two scenarios The use

of an idealizedsimplified two-layer model of a region and single degree-of-freedom inelasticsystems to represent structural response allows the papers to focus on fundamental trends

DESCRIPTION OF MODEL AND SCENARIO EARTHQUAKE EVENTS

The model considered in this study consists of an elastic layer on an elastic halfspace, asshown in Fig 1 The density,ρ, shear-wave velocity, β, and P-wave velocity, α, of the layerare 2.6 g/cm3, 2.0 km/s, 4.0 km/s, respectively, and the corresponding values for the

halfspace are 2.7 g/cm3, 3.464 km/s, 6.0 km/s These values were chosen to model the region

as being made up of rock material and material attenuation is not considered.The modelconsidered in this study consists of an elastic layer on an elastic halfspace, as shown inFigure 1, to represent a 20 km by 20 km region of rock material We consider a buriedstrike-slip fault (Figure 1a) and a buried thrust fault (Figure 1b) in order to model representtwo of the most common types of earthquakes The top edge of the vertical fault is 1 kmbeneath the interface between the layer and the halfspace, whereas the top edge of theinclined fault touches the interface Red circles in Figure 1 denote The the hypocenters of thetwo earthquakes are denoted by red circles on the respective faultsearthquakes The dipangle (40º) of the thrust fault is be similar to that of the fault that caused the 1994 Northridgeearthquake source The density, , shear-wave velocity, v s, and P-wave velocity, v p, of the

layer are 2.6 g/cm3 , 2.0 km/s, 4.0 km/s, respectively, and the corresponding values for thehalfspace are 2.7 g/cm , 3.46 km/s, 6.0 km/s Material attenuation is not considered in the3simulations

We assume kinematic rupture of the fault by imposing a, that is, a dislocation is imposedacross the fault (jump in the tangential displacement and continuous normal displacement).The rupture propagates radiallradiallyy from the hypocenter at a constant speed of 0.8 km/suntil it reaches the edges of the fault The slip direction of rake, is 0º and 90º for the verticaland the inclined faults, respectively;, that is, the particle motion across the fault is along thelength of the fault in both cases For the vertical fault the motion is right lateral strike- slip,and for the inclined fault the motion of the hanging wall is upward and that of the footwall isdownward (thrust slip) The variation of the dislocation with time is defined by the slipfunction u(t) Df (t), u(t) = ū f(t), where Dū is the (uniform) amplitude of thedislocationuniform slip, and,

f(t) = [1 – (1 +t/T0)exp(-t/ T0)]H(t) where T0T0 definesis the slip rise time There is a delay

time at each location with respect to the onset time at the hypocenter of r / v ruptr/Vrupt, where

r is the hypocentral distance from the particular location, and v rupt Vrupt is the rupture velocity.

In the simulations, we assumeWe take v rupt=Vrupt = 3.0 km/s and , T0 T0= 0.1 sec, andmodelwhich allows modeling seismic waves up to 5.0 Hz The model for the strike-slipevent is the same that has been used by a group of modelers, including some of this paper’sauthors, in verifying several finite difference and finite element codes for simulatingearthquake ground motion in large regions (Day et al, 2002)

The seismic moment, M0M0, of the an earthquake can be expressed as (Aki, 1966),:

M0  AD,

M0 = μAū,Aū,

to the seismic moment in dyne-cm, are related by (Kanamori, 1977):

M w 0.67 log 10 M 0 10.7 <<<Previous formula needs a ⅔ in front of the log(10 )>>>

Mw = ⅔log(10M0, in dyne-cm) – 10.7

For our the scenario simulationsevents, we set Mw = 6.0 and 5.8 is selected for the strike-slipfault and the thrust fault earthquakes, respectively These magnitudes correspond to slip D

ofū =0.112.5 cm m and 0.05.42 cmm, respectively, for the two events

Despite their simplicity, models of this type have been shown to capture the essentialnature of earthquake ground motion in the near-fault region For example, in a recentsimulation of the 1992 Landers earthquake in southern California, Hisada and Bielak (2004)modeled the crustal region in the vicinity of the epicenter as a single layer on a halfspace andthe causative fault by piecewise strike-slip planar surfaces, and calculated the ground motion

at the Lucerne Valley station using an integral representation technique (Hisada and Bielak,2003), with satisfactory results Similarly, results for the spatial distribution of groundmotion to be presented later for the thrust-fault event are qualitatively similar to thoseobserved during the 1994 Northridge earthquake

COMPUTATIONAL METHOD FOR GROUND MOTION SIMULATION

To simulate slip on the fault and the resulting ground motion within the domain, we use

an elastic wave propagation, finite element code developed for modeling earthquake groundmotion in large sedimentary basins (Bao et al., 1998) The wave propagation code is builtusing Archimedes, a software environment for solving unstructured-mesh finite elementproblems on parallel computers (Bao et al., 1998) Archimedes includes two- and three-dimensional mesh generators, a mesh partitioner, a parceler, and a parallel code generator

We use standard, Galerkin linear tetrahedral elements for the spatial discretization of thegoverning Navier equations of elastodynamics over the tetrahedral mesh The element sizesare tailored to the local wavelengths of the propagating waves, and the mesh generationstrictly controls the aspect ratio of every element such that it does not exceed a prescribedvalue The spatial discretization leads to a standard system of second-order ordinarydifferential equations with constant coefficients These equations are solved using the centraldifference method, an explicit, conditionally stable, step-by-step algorithm in the timedomain To avoid the need of solving a system of algebraic equations at each time step, alumped mass matrix is used As a result, the only significant operation at a time step is amatrix-vector multiplication

Two important issues must be considered for solving earthquake wave propagationproblems in infinite domains by the finite element method One is the requirement for afinite domain of computation and the need to limit spurious reflections at the artificialabsorbing boundaries This is accomplished by a sponge layer of viscous material near theartificial boundaries (Israeli and Orszag, 1981) and by placing a set of nodal viscous dampersdirectly on the artificial boundaries, along the lines described in Lysmer and Kuhlemeyer(1969) The second issue is the need to represent the slip on the fault in the finite elementformulation We do this, following Aki and Richards (1980), by expressing the slip in terms

of a set of double couples, which in turn are expressed as body forces The body forces arereduced to equivalent nodal forces through the Galerkin process for spatial discretization(Bao, 1988) To determine the amplitude of each double couple, the fault is divided into anumber of subfaults For the case of uniform slip, the double couple within each elementintersected by the fault is proportional to the seismic moment, based on the area of the fault

contained within the element divided by the total area of the fault The finite element codehas been verified with several finite differences codes for idealized and realistic earthquakes(Day, 2002), including the strike-slip event considered here, with satisfactory results

(a) Strike-slip fault scenario

(b) Thrust fault scenario

Figure 1 Model of faults in a homogeneous elastic layer on a homogeneous elastic halfspace

Despite their simplicity, models of this type have been shown to capture the essentialnature of the earthquake ground motion in the near-fieldnear- region For example, in arecent simulation of the 1992 Landers earthquake in southern California (Fig.Figure 2a),Hisada and Bielak (2004) modeled the crustal region in the vicinity of the epicenter as asingle layer on a halfspace and the causative fault by piecewise strike-slip planar surfaces,and calculated the ground motion at the Lucerne Valley station using an integralrepresentation technique (Hisada and Bielak, 2003), with satisfactory results Figures 2b and2c show the corresponding velocities and displacements in the fault -normal (FN) and fault -parallel (FP) directions The agreement between the synthetic motion (solid lines) and therecorded motion (dashed lines) is satisfactory Notice that at theThe Lucerne Valley station,the peak amplitudes of the FN and FP velocities are comparable, but the permanent offsetdisplacement is much greater in the FP than in the FN directions As an example of a thrust-

faultthrust fault, we examined the 1994 Northridge earthquake in the San Fernando Valleyarea Figure 3a shows the distribution of peak ground velocity near the fault, derived fromseismograms recorded in the epicentral region The causative fault is shown in the lower part

of the figure, together with the hypocenter and the distribution of the amplitude of peak slip

in the fault surface The rectangle defined by the solid black lines is the projection of thefault on the free surface Large values of peak ground velocity are observed at the updipcorners of the fault projection and in the region outside the fault opposite the hypocenter Inaddition, a concentrated area of moderately high ground velocity occurs directly above thefault in the portion of the San Fernando Valley near the foot of the Santa Susana Mountains

A similar qualitative distribution can be observed in Fig.Figure 3b, which shows the peakground velocity in the direction perpendicular to the strike of the thrust-faultthrust faultmodel in Fig 1b No concentration of large motion is observed in the area directly above thefault because our crustal model is uniform in the lateral direction.Similarly, results for thespatial distribution of ground motion to be presented later for the thrust-fault event arequalitatively similar to those observed during the 1994 Northridge earthquake

FIGURE 2 1992 LANDERS EARTHQUAKE (A) FAULT MODEL CONSISTS OF THREE LEFT-LATERAL STRIKE-SLIP FAULTS; (B) AND (C) COMPARISONS BETWEEN SIMULATIONS (SOLID LINES) AND OBSERVATIONS (DASHED LINES) AT THE LUCERNE VALLEY STATION THE OBSERVED

DISPLACEMENT RECORDS WERE CORRECTED BY IWAN (1992); (B) FAULT NORMAL (FN) AND FAULT PARALLEL (FP) COMPONENTS OF VELOCITY AT LUCERNE VALLEY STATION; (C) CORRESPONDING COMPONENTS OF

GROUND DISPLACEMENT.

N130W

(a) Velocity

(b) Displacement Fault Normal Fault Parallel

Fault Normal Fault Parallel

FROM THE TWO COMPARISONS BETWEEN THE IDEALIZED MODELS AND THE REALISTIC SITUATIONS, IT SEEMS REASONABLEAPPARENT THAT A DETAILED ANALYSIS OF THE GROUND MOTION OF THE IDEALIZED MODELS PROVIDE ADDITIONAL INSIGHT INTO THE SOURCE AND PATH EFFECTS FOR STRIKE-SLIP AND THRUST FAULT EARTHQUAKES THE ; RESULTS FROM THIS ANALYSIS JUSTIFY THE NEED FOR THE DENSE SAMPLING OF GROUND MOTION THAT WE USE IN THE COMPANION PAPER

TO STUDY THE SPATIAL DISTRIBUTION OF STRUCTURAL RESPONSE FOR

THESE TWO TYPES OF EARTHQUAKES.

(A)

(B)

FIGURE 3 DISTRIBUTION OF GROUND VELOCITY DUE TO THRUST FAULT EARTHQUAKES: (A) PEAK GROUND VELOCITY DERIVED FROM

SEISMOGRAMS RECORDED WITHIN THE EPICENTRAL REGION DURING THE 1994 NORTHRIDGE EARTHQUAKE; (B) PEAK GROUND VELOCITY IN THE DIRECTION PERPENDICULAR TO THE STRIKE OF THE THRUST FAULT EARTHQUAKE MODEL THE CAUSATIVE FAULT IS SHOWN IN THE LOWER PART OF THE FIGURE, TOGETHER WITH THE HYPOCENTER AND THE

DISTRIBUTION OF THE SLIP IN THE FAULT SURFACE.

COMPUTATIONAL METHOD FOR GROUND MOTION SIMULATION

COMPUTATIONAL METHOD FOR GROUND MOTION SIMULATION

To simulate slip on the fault and the resulting ground motion within the domain, we use

an elastic wave propagation, finite element code developed for modeling earthquake groundmotion in large sedimentary basins (Bao et al, 1998) The wave propagation code is built ontop of Archimedes, a software environment for solving unstructured-mesh finite elementproblems on parallel computers (Bao et al, 1998) Archimedes includes two- and three-dimensional mesh generators, a mesh partitioner, a parceler, and a parallel code generator

We use linear tetrahedral elements and standard Galerkin ideas for the spatial discretization

of the governing Navier equations of elastodynamics over the tetrahedral mesh The elementsizes are tailored to the local wavelengths of the propagating waves, and the mesh generationstrictly controls the aspect ratio of every element such that it does not exceed a prescribedvalue The spatial discretization leads to a standard system of second-order ordinarydifferential equations with constant coefficients These equations are solved using the centraldifference method, an explicit, conditionally stable, step-by-step algorithm in the timedomain To avoid the need of solving a system of algebraic equations at each time step, alumped mass matrix is used As a result, the only significant algebraic operation at a timestep is a matrix-vector multiplication

Two important issues must be considered for solving earthquake wave propagationproblems in infinite domains by the finite element method One is the the need to render thedomain of computation finite and the need to limit the occurrence of spuriousreflectionsartificial absorbing This is accomplished here by introducing a sponge layer ofviscous material near the artificial boundaries (Israeli and Orszag, 1981) and by placing a set

of nodal viscous dampers directly on the artificial boundaries, along the lines described inLysmer and Kuhlemeyer (1969) The second point that requires attention is the need toincorporate the slip on the fault into the finite element formulation We do this, followingAki and Richards (1980), by expressing the slip in terms of a set of double couples, which inturn are expressed as body forces These body forces are reduced to applied nodal forcesthrough the spatial discretization Galerkin process (Bao, 1988) To determine the amplitude

of each double couple, the fault is divided into a number of subfaults For the case ofuniform slip considered in this study, the double couple within each element that isintersected by the fault is proportional to the total seismic moment, based on the area of thefault contained within the element divided by the total area of the fault The finite elementcode has been verified with several finite differences codes for a number of idealized andrealistic earthquakes (Day, 2002), including the strike-slip event considered here, withsatisfactory results

RESULTS OF GROUND MOTION SIMULATIONS

The large-scale finite element solution of the governing equations described in the previoussection is applied to the two scenario earthquake events with the idealized domain

In this section we examine in detail the distribution of ground motion in the epicentralregion for the two scenario earthquakes using the finite element methodology In an earlierstudiesy, Bao (1998) and Bao et al (1998) found that numerical dispersion can could be kept

to below less than 5 five percent over distances of 50 km by using 8 to 10 linear tetrahedralelements per wavelength Thus, for the present application the mesh size is 40 m for the toplayer and 80 m for the halfspace to represent ground motion up to 5 Hz The correspondingdiscretized equations of motion (N=1042 million) <<<need to confirm>>>were solved on

128 processors of the Cray T3E at the Pittsburgh Supercomputing Center

STRIKE-SLIP FAULT

Tracking the ground motion from its inception, we first examine the displacement field in

a buried horizontal plane that passes exactly through the hypocenter of the vertical strike-slipfault (Fig.Figure 11a) Figure 4 2 shows the spatial distribution of the peak horizontaldisplacement in the hypocentral plane in the FP fault parallel (FP) (Fig 4a) and FN faultnormal (FN) (Fig 4b) directions For the right-lateral slip considered in this earthquakescenario, the FP FP motion north of the fault plane is to the right, while thatwhereas themotion south of it is to the left The finite element discretization requires a continuousdisplacement field throughout the domain; yet, slip is nonetheless allowed to occur across thefault, spread over a one-element wide band The amplitude of the slip across the fault plane

is uniform along the length of the dislocation and is equal to the prescribed value of

antisymmetry considerations, there is no motion in the FP FP direction on the fault planebeyond the ruptured zone Thus, the displacement across the fault conforms exactly to theprescribed dislocationslip, including the requirement that the FN FN component of thedisplacement remain isbe continuous across the fault (Fig.Figure 4b2 bottom) This pattern

of behavior confirms that representing the prescribed dislocation slip by body forces isequivalent to imposing directly the dislocation slip on the fault Away from the fault, the FP

FP component of displacement in the hypocentral plane is concentrated primarily in theregions north and south of the fault Some asymmetry is observed with respect to a planeperpendicular to the fault plane passing through the middle of the dislocation, but it is minor

Figure 24 Strike-slip earthquake Spatial distribution of the peak horizontal displacement in the hypocentral plane in ( atop ) fault parallel and ( bbottom ) fault normal directions

On the other hand, the motion in the FN direction is dramatically different from that inthe FP direction The FN motion arises primarily as a consequence of the couples that aregenerated in the north-south direction along the fault to balance the east-west couples thatproduce the dislocationslip As a result of the fault rupture propagation, a nodal linedevelops east of the hypocenter;: points east of this line move north while those to the westmove south The nodal line is curved and the FN motion is not symmetric with respect to thenodal line because of the propagating nature of the rupture The FN displacement is muchlarger in the direction of the rupture propagation of the rupture than in the backwarddirection, and it affects a wide region on and away from the fault The FN motion reaches itsmaximum value precisely at the east end of the dislocation, where there is a stressconcentration This is evidence of the forward directivity effect, which, as will be seen later,occurs even more prominently on the free surface That theThe forward directivity effectsare dynamic in nature and a direct consequence of the constructive interference of thepropagating waves that travel in the direction of the rupture, asof the dislocation can be seen

by comparing Fig.Figure 4 2 with Fig.Figure 53, which shows the residual, permanent FP and

FN components of displacement in the horizontal hypocentral plane In contrast with thedynamic components evident especially at the bottom of in Fig.Figure 4b2, the FN residualdisplacement is antisymmetric with respect to the midpoint of the dislocation Also, whereasthe peak amplitude of the FP residual component across the fault is quite similar to thatobserved during the passage of the seismic wave, but there is a significant reduction in theamplitude of the FN permanent component with respect to that of the total displacement,denoting indicating significant dynamic action The dynamic effect is significantly muchsmaller in the FP direction

(a)

Figure 53 Strike-slip earthquake Residual (permanent) ( atop ) fault parallel and (b ottom ) fault normal components of displacement in the hypocentral plane.

To examine the ground motion generated by the propagating waves on the free surface ofthe layered system, we next consider the spatial distribution of the absolute values of the FPand FN components of the free surface peak displacements and velocities, as well as themaximum amplitudes of the corresponding resultant displacement and velocity vector fields,

as shown in Fig.Figure 64 Velocity quantities are shown in Figures 6a, b, c and theircorresponding displacements counterparts in Fig.Figure 6d, e, f Figure 6a The FP velocityclearly shows that most of the dynamic effect in the FP direction occurs in the neighborhoodofnear the epicenter As in at the hypocentral plane, the FN component of velocity(Fig.Figure 6b) exhibits a very strong forward dynamic directivity effect The peak value of

FN component is twice that of the FP component This difference in amplitudes is obvious inFig.Figure 6c for the total resultant velocity, in which the contributions of both the FP and

FN components are apparent, but where the effect of the FN component is much stronger.The distribution of the acceleration (not shown) is very similar to that of the velocity Thepeak values of the FP and FN components of displacement (FigsFigure 6d, e) also differ by

a factor of about two, as for the velocity components In contrast with the velocity, however,both the FP and FN components of displacement show a strong forward directivity effect onthe free surface Interestingly, the decay decrease of in the the velocity with distance fromthe fault plane is quite rapid, compared with that attenuation of the displacement Theimportant implications that these differences in ground motion have for the response of long-and short-period structures will beare examined in detail in the companion paper (Fenves etal., 2004)

To explore further the effects of directivity on the ground motion, the syntheticseismoscope records in Figure 5 show the displacement path at a number of locations on aregular grid on the ground surface Red dotsSolid circles indicate the position of each point

at the end of the earthquake These records show that the direction and amplitude of motionvariesy widely over the region Due to symmetry, points directly north of the faultexperience motion only in the FN direction This motion reaches its maximum in the forwarddirectivity direction, some distance away from the edge of the dislocation DirectlyFurther

location Figure 6 shows seismograms of ground velocity and displacement in twoorthogonal directions at selected locations These have been low-pass filtered up to 5 Hz.Open circles indicate the locations of the observation points, and the direction of particlemotion at each location is perpendicular to the time axis of the corresponding seismogram.Thus, for example, the point at station S3 experiences a peak velocity of 1.92 m/s in the north(FN) direction and a peak value of 0.03 m/s in the west (FP) direction, while the peak FPvelocity at station S5 is 0.93 m/s, and the FN component is 0.09 m/s It is noteworthy thatwhereas at stations in the forward directivity direction (e.g., S3, S4, S7, S8) the FNcomponent of velocity that exhibits a strong pulse-like ground motion, at stations locateddirectly north of the epicenter (e.g., S5) it is the FP component of velocity that exhibits astrong pulse-like behavior This behavior is evident also for the other stations located at shortdistances north (and south) of the fault (e.g., S17, S6) Farther away, the seismogramsbecome more complex in shape and their duration becomes longer; also, the dominantperiods appear to become longer with distance from the fault (e.g., S15) These effects aremainly due to wave dispersion and contributions from the surface waves In the direction offorward directivity only the FN component is significant, whereas at S6, north and middle ofthe fault, the amplitude of both the FP and FN components is significant The same is true atthe location S17 at the back of the fault As expected, the displacement seismograms aresmoother than those for the velocity In addition, displacement seismograms exhibitpermanent offsets This fling is related to the permanent deformation at the site.

To examine differences in frequency content between the seismograms it is useful to useresponse spectra, andas these also provide insight into the spatial distribution of the dynamicresponse of elastic SDF systems to the ground motion Figures 7 and 8 show the responsespectra for five percent critical damping in the FN and FP directions, respectively, calculatedfrom the corresponding free-field synthetic seismograms The spectra are plotted in the usualtripartite logarithmic representation for the pseudo-velocity, pseudo-acceleration anddisplacement to study the period ranges for which the spectra are most sensitive to the groundvelocity, ground acceleration, and ground displacement All the spectra are drawn for

periods, T, in the range of 0.5 sec 8.0 sec Also shown in the spectra plots are the peak

ground acceleration (PGA), peak ground velocity (PGV), and peak ground displacement(PGD) at each location, drawn as piecewise straight lines

fault normal components of the free surface peak velocity and displacement, and the maximum amplitudes of the corresponding resultant velocities and displacements.

To explore further the effects of directivity on the ground motion, the synthetic seismoscoperecords in Fig 7 5 show the displacement path at a number of locations on a regular grid onthe free surface These records show that the direction and amplitude of motion varieswidely throughout the free surface Red dots indicate the position of each point at the end ofthe earthquake Due to symmetry, points directly above the strike of the fault experiencemotion only in the FN direction This motion reaches its maximum in the forward directivitydirection, some distance away from the edge of the dislocation Directly north and south ofthe fault the motion is predominantly in the FP direction, while at other locations the groundexperiences FP and FN motion of comparable size