Generation of spectrum compatible ground motion and its use in regulatory andperformance-based seismic analysis

Irmela Zentner1, Frederic Allain2, Nicolas Humbert3, Matthieu Caudron2

1EDF R&D, LaMSID UMR EDF-CNRS-CEA, Clamart France2EDF SEPTEN, Lyon, France3EDF CIH, Chambery Franceemail: irmela.zentner@edf.fr

ABSTRACT: This paper deals with the generation of artificial spectrum compatible ground motion complying with different goals.In particular, three distinct objectives that may emerge in performance-based seismic analysis and design are discussed:• Conservative design analysis based on only few spectrum-compatible accelerograms• Performance-based earthquake engineering approaches for risk studies, in general less than 30 accelerograms are used.• Full ”best-estimate” analysis for probabilistic and reliability analysis of structures subject to seismic random excitation.The ground motion time-histories have to be in agreement with these goals. Different methods and ground motion models areavailable in literature. Vanmarcke was among the first to implement these approaches to be used for engineering design (SIMQKE[9]), later models introduced non stationary features e.g. Preumont [1]. More recently, a module for the generation of artificialground motion has been introduced in Code Aster [23] that addresses the above objectives.The issues of practical use and code implementation are addressed and some results illustrating the different approaches arepresented.

KEY WORDS: ground motion, simulation, variability, spectral acceleration, correlation.

1 INTRODUCTION

In many international seismic codes and guidelines, seismicload is defined by design spectra. When transient structuralanalysis are carried, then it is required that the responsespectra of the accelerograms used for design envelope thegiven target spectrum. The resulting analysis are considered tobe conservative. The design spectrum is generally smoothedand broadened and does not represent any actual seismic loadbut represents an envelope of possible spectral accelerationvalues. On the other hand, performance based safety assessmentmethods and more recent guidelines such as ATC58 [2]do require more realistic description of seismic load. Inconsequence, median spectral accelerations are used and record-to-record variability and uncertainty has to be accounted for.Acknowledging that the response spectrum is itself a randomquantity, not only the median spectral acceleration values butthe whole distribution has to be matched by the syntheticaccelerograms to be in agreement with the Ground MotionPrediction equations (GMPE). The following three goals areemerging from these considerations for the generation ofsynthetic spectrum-compatible ground motion:

1. One-by-one spectrum match: generally used for conservativedesign analysis based on only few spectrum-compatibleaccelerograms2. The median (or mean) spectrum is the target to bematched: the accelerograms exhibit ”peak-to-valley variability”,performance-based earthquake engineering approaches for riskstudies, in general less than 30 accelerograms are used. This

might be the case for example for the evaluation of fragilitycurves in Probabilistic Risk Assessment (PRA) studies in thenuclear industry [21], [10]3. The whole distribution of the spectral accelerations ismatched: accelerograms featuring variability close to the onesof the database to be used for, full ”best-estimate” analysisfor probabilistic and reliability analysis of structures subject toscenario earthquakes.These three cases are illustrated in figures 1-3 where the targetresponse spectra (red) are shown together with the responsespectra of simulated accelerograms (blue) and the their statistics(magenta).

In this paper the simulation of synthetic accelerograms bymeans of stochastic ground motion models, complying with thethree goals, is presented. In the stochastic approach, seismicground motion is modelled as a stochastic, most often Gaussian,process characterized by its power spectral density. Thesimulation of time histories from such models is straightforwardand relies on rigorous mathematical background.Moreover,once the model constructed, it allows to simulate an unlimitednumber of time history at very low cost.

2 SIMULATION OF SPECTRUM COMPATIBLEACCELEROGRAMS

Seismic ground motion is modelled by a stochastic processcharacterized by a power spectral density (PSD). In order tocomply with a target response spectrum, a response-spectrumcompatible PSD has to be identified. The resulting PSD allows

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

381

2

10−1

100

101

102

10−3

10−2

10−1

100

frequency Hz

pseu

do−

acce

lera

tion

g

simulatedtarget

Fig. 1. 1. One-by-one match (EC8 design spectrum)

10−1

100

101

102

10−2

10−1

frequency Hz

pseu

do−

acce

lera

tion

g

TargetMedianSimulated

Fig. 2. 2. Median match (NGA Campbell & Bozorgnia GMPE)

10−1

100

101

102

10−2

10−1

100

frequency Hz

pseu

do−

acce

lera

tion

g

Target median Target +1σMedian simulated+1σ simulated Simulated

Fig. 3. 3. Distribution match (NGA Campbell & BozorgniaGMPE)

to simulate amplitude modulated groundmotion whose responsespectrum matches the target. The amplitude modulation isintroduced by a deterministic function that is applied to the timehistories. When an evolution of the frequency content has tobe accounted for, then the model has to be parametrised and atime dependant central frequency has to be introduced. Such amodel can be expressed in a general manner by the followingevolutionary PSD:

S (ω , t) = h(t)2A (ω , t)S0(ω)A ∗(ω , t) (1)

where h(t) is the deterministic modulating function. Once thePSD constructed, the sample of ground motion time historiescan be simulated using the classical spectral representationtheorem. This is briefly described described in the followingsections.

2-1 Identification of a spectrum-compatible PSD

Vanmarcke & Gasparini [9] were among the first to model theseismic input as a modulated stationary Gaussian process, withspectrum compatible power spectral density. They evidencedthe fundamental relationship between the response spectrum andthe ground motion power spectral density via the so-called firstpassage problem and established a formula linking the responsespectrum to the moments of the power spectral density of theinput. The response spectrum Sa(ωk,ξ ) is given for a couple offrequenciesωk and for damping ratio ξ . According to Gasparini& Vanmarcke [9], a spectrum-compatible PSD G (ω) can beapproached by the formula:

G (ωk) =1

ωk(π/2ξ − 2)

[S2

a(ωk,ξ )η2

TSM

− 2∫ ωk

0G (ω)dω

], (2)

where ηTSM is the peak factor; it depends on the strong motionduration TSM . The above expression has been derived from thefollowing relation between the peak factor ηTSM and the standarddeviation σk of the (stationary) ground motion process filteredby an oscillator with central frequencyωk and reduced dampingξ :

Sa(ωk,ξ )≈ ω2kηTSMσk. (3)

The integral in expression (2) can be evaluated numerically. Theinitial value of 2ξS2

a(ωk,ξ )/(ω0πηTSM ) is chosen for initiatingthe procedure at 1

4πHz. The amplitude variation is generallyintroduced by a deterministic modulating function. Both theJenning & Housner [11] and the Gamma modulating functionhave been implemented in Code Aster [23]. These functionsare parametrised by the strong motion duration TSM .

2-2 Description of the non stationary model implemented inCode Aster

The spectrum-compatible and non stationary ground motionmodel based on evolutionary PSD [17] implemented inCode Aster uses a general formulation where ground motion isexpressed as filtered white noise, close to the classical Kanai-Tajimi PSD model [12], [20] . Time dependency of the centralfrequency is introduced by considering the central frequencyof the filter as a function of time, yielding an evolutionaryPSD model. The variation of the amplitude is introduced bya deterministic modulating function.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

382

3

Non stationary Kanai-Tajimi models, where the centralfrequency is a function of time, have been first proposed by[14] and later reported by [8] and many others [22]. Thisapproach is also adopted here. The variation of the amplitude isintroduced by a deterministic modulating function as describedpreviously. Reference [14] provides both time-domain andfrequency-domain formulations. We consider a rational PSDfunction which constitutes a generalisation of the Kanai-Tajimimodel:

SR(ω) =R2

0+R21ω2

(ω2 −ω20)

2 +2ξ 2ω20ω2

. (4)

The parameters of this model, R0,R1,ω0 and ξ are identified byminimising the distance to the spectrum compatible PSD G (ωk)(in the sense of mean least squares). The minimization problemis performed by using the simplex algorithm. The evolutionof the frequency content is introduced by considering a timedependent central frequency ω0(t) in expression (4) yieldingthe evolutionary PSD S (ω , t). Several studies, among themthe recent analysis of Rezaeian & Der Kiureghian [19], showedthat the central frequency is generally decreasing with time.According to Rezaeian & Der Kiureghian [19] a linear relationis chosen for the evolution of the frequency content:

ω0(t) = ω0 +ωp(t −Tmid), (5)

where ωp is the slope (in general we consider ωp < 0 sincecentral frequency is decreasing with time) and Tmid is the instantwhen half of the strong motion phase is reached.

The last authors also found that the damping ratio does notvary significantly with time so that ξ can be considered to beconstant. In order to obtain also the variation of the amplitudeof the ground motion process, the modulating function h(t) hasto be applied to the PSD developed in this section.

The obtained evolutionary PSD model will yield timehistories whose median spectrum is close to the target. However,since the PSD is a parametrical model, exact matching is ingeneral not possible. The equal energy criterion [1] allows thento conciliate the spectrum-compatible model h(t)2G (ω) withthe evolutionary PSD h(t)2S (ω , t)S0(ω), where S0(ω) is thecorrective term:

∫ T

0h(t)2S (ω , t)S0(ω)dt =

∫ T

0h(t)2G (ω)dt. (6)

Moreover, a high pass filter allows to delete the low frequencycomponents of synthetic ground motion (ω2 decrease), yieldingnon diverging behaviour of the integrated velocity anddisplacement time histories. The filtered PSD reads (Clough &Penzien [6]) :

S (ω) = |ω2

ω2f −ω2+2iξ fω fω

|2SR(ω). (7)

The filter frequency ω f is also known as the corner frequencyin the literature [18]. The power spectral density of equation (7)will be considered as the median value in section 2-5.

2-3 Simulation of accelerograms matching a given targetdesign response spectrum

If it is desired that each of the accelerograms best matchesthe design spectrum, then formula 2 can be directly applied

to determine a spectrum-compatible PSD. Iterations on thefrequency content of the accelerogram are performed to adjustthe spectral match ”one-by-one” that is for each of the syntheticaccelerograms [9]. An example of ”one-by-one” match isprovided by figure 1. ”One-by-one” match is used when thetarget or design spectrum is an envelope of possible spectralaccelerations. According to the recommendations of seismiccodes such as EC8 [7], ASCE [15] and ASN [3], only a fewtransient analysis are carried out. At least 3 time history analysisare required by those codes : permutation of the directionsallows then to obtain 3 different 3D load cases (triplets ofhorizontal and vertical load). Peak-to-valley variability isnot desired and has to be excluded since it might lead tonon conservative results. For example, EC8 requires that theresponse spectra should not take values less than 90% of thetarget spectrum. This criterion can be applied to each of theaccelerograms or to the mean of the suite of accelerograms.The last case is treated in the next issue (section 2-4) where thesimulation of accelerograms whose median response spectrummatches the target is addressed. Note that the median rather thanthe mean has been chosen here since common GMPE considerthe spectral acceleration to be lognormally distributed. Thismeans that our criterion corresponds to matching the mean logspectral accelerations.

2-4 Simulation of accelerograms whose median responsespectrum matches the target

The ”one-by-one” matching does not guarantee that theobtained acelerograms are not correlated from a statistical pointof view (ASN [3] requires a correlation coefficient less than0.3). This problem does not arise when the mean or medianis compared to the target. The match of the target spectrumcan be improved by iterations on the spectral conent [9]. Incontrast to the ”one–by-one” match of the preceding section2-3, the median of the response spectra of accelerograms iscompared to the target.This is illustrated in figure 2. However,iterations on the spectral content are in general not necessary ifthe target spectrum is physical (that is the median of observedaccelerograms) and more than a dozen accelerograms aregenerated. An evolution of the central frequency of groundmotion can be accounted for by using the formulation of section2-2.

2-5 Simulation of accelerograms featuring record-to-recordvariability close to natural time histories

In the framework of performance-based transient seismicanalysis, the ground motion has to be as ”realistic” as possibleand the simulation of adequate accelerograms requires specialcare. Ideally, the accelerograms have to be spectrum-compatibleand non-stationary featuring variability close to the one ofrecorded accelerograms. The variability observed in the datais can be expressed by the disp Thus, in terms of spectral shape,not only the median spectrum but also its quantiles (and thusthe whole distribution) have to be yielded. This is achievedby not only considering a target response spectrum, but byspecifying the response spectrum as a random vector with givendistribution. This is illustrated in figure 2 where the mediantogether with the ±oneσ spectra are represented: the red curves

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

383

4

are the target while the magenta curves represented the valuesestimated from the simulated time histories (blue).

More details can be found in [23] and [24]. The spectrumcompatible power spectral density becomes in turn a randomvector whose distribution can be linked to the one of the spectralacceleration values. Indeed, current ground motion predictionequations do not only give median or mean response spectralvalues but also provide information on dispersion and onspectral correlations (Baker et al. [4]). According to the NGAground motion prediction equations (GMPE), the log responsespectrum can be considered as a Gaussian random vectorwith given marginal (normal) distributions and a covariancematrix. It can be shown that, for small damping ξ , the PSDis proportional to the squared median spectral acceleration:

Sa(ω)2 ≈ σ2(ω)ω4η2TSM

≈ S (ω)πωη2

TSM

2ξ, (8)

(the approximation in the relation between the first two termslies in the approximate expression of the peak factor). Relation(8) is written with respect to the median spectrum Sa(ω) andthe median PSD S (ω). Then, in agreement with the responsespectra a lognormal distribution can be assumed for the powerspectral density [24]. The correlation between different spectralaccelerations and thus between the values assumed by the PSDat different frequencies is accounted for through the correlationcoefficients proposed by Baker and co-workers.

Furthermore, an evolution of the central frequency of groundmotion can be accounted for by using the relations establishedin section 2-2.

3 APPLICATION TO NGA SPECTRA

The Campbell & Bozorgnia relations [5] are used. This modelis valid for 5% damping and frequencies ranging from 0.1Hz−100Hz. The spectral acceleration values have been retrievedfrom the NGA database [16]. The scenario considered is amagnitude M = 7 and distance D = 20km event for a strike slipfault with dip angle 90 deg. The site properties are mediumsoil with Vs30 = 500m/s. Moreover, according to Kempton &Steward [13], the strong motion duration was assumed to beTs = 17.5s.

3-1 Median match

The match of the NGA target spectrum after 20 iterations isillustrated on figure 2: the median spectrum (target) is the redcurve, while the spectra of the 10 simulated time histories aregiven by the blue curves together with the estimated medianvalues (magenta). Three of the simulated accelerograms areshown on 4. The respective displacement time histories aredisplayed on figure 5. None of the integrated time historiesshowed diverging behaviour but amplitudes decreasing to zeroat the end of shaking.

3-2 Match of the median spectrum and ±1σ values

The match of the NGA target spectrum considering variabilityis illustrated on figure 3: the median and +1 σ target spectra arein red, the spectra of the 30 simulated time histories are givenby the blue curves together with the estimated median and +1σ

0 5 10 15 20 25 30 35 40−2

0

2

0 5 10 15 20 25 30 35 40−2

0

2

acce

lera

tion

m/s

2

0 5 10 15 20 25 30 35 40−2

0

2

time s

Fig. 4. An example of three simulated accelerograms.

0 5 10 15 20 25 30 35 40

−0.1

0

0.1

0 5 10 15 20 25 30 35 40

−0.1

0

0.1

disp

lace

men

t m

0 5 10 15 20 25 30 35 40

−0.1

0

0.1

time s

Fig. 5. Displacement time histories corresponding to the 3accelerograms of figure 4.

values in magenta. No iterations on the spectral content areperformed in this configuration.

Three of the simulated accelerograms are shown on 6.The respective velocity and displacement time histories aredisplayed on figures 7 and 8. None of the integrated timehistories showed diverging behaviour but amplitudes decreasingto zero at the end of shaking.

ACKNOWLEDGMENTS

The financial support from ANR via the MODNAT project isgreatfully acknowledged.

REFERENCES

[1] Preumont A. The generation of nonseparable artificial earthquakeaccelerograms for the design of nuclear power plants. NuclearEngineering Design, 88:59–67, 1985.

[2] ATC-58. Seismic perfomance assessment of buildings.[3] ASN French Nuclear Safety Authority. Prise en compte du risque

sismique a la conception des ouvrages de GC dinstallations nucleairesde base a lexception des stockages a long terme des dechets radioactifs.ASN/GUIDE/2/01, septembre 2006.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

384

5

TABLE ICOMPARISON OF GROUND MOTION PARAMETERS OF THE 10 SYNTHETIC ACCELEROGRAMS TO THOSE OF THE C&B GMPE.

Parameter PGA PGV CAVmedian median median

Target 0.159 0.153 0.620Simulated 0.161 0.151 0.716

TABLE IICOMPARISON OF GROUND MOTION PARAMETERS OF THE 30 SYNTHETIC ACCELEROGRAMS FEATURING ”NATURAL”

VARIABILITY TO THOSE OF THE C&B GMPE.

Parameter PGA PGV CAVmedian std median std median std

Target 0.159 0.510 0.153 0.525 0.620 0.420Simulated 0.158 0.481 0.164 0.465 0.767 0.476

0 5 10 15 20 25 30 35 40

−2

0

2

0 5 10 15 20 25 30 35 40

−2

0

2

accé

lera

tion

m/s

2

0 5 10 15 20 25 30 35 40

−2

0

2

time s

Fig. 6. An example of three simulated accelerograms.

0 5 10 15 20 25 30 35 40

−0.2

0

0.2

0 5 10 15 20 25 30 35 40

−0.2

0

0.2

velo

city

m/s

0 5 10 15 20 25 30 35 40

−0.2

0

0.2

time s

Fig. 7. Velocity time histories obtained by integration from the3 accelerograms of figure 6.

0 5 10 15 20 25 30 35 40−0.1

0

0.1

0 5 10 15 20 25 30 35 40−0.1

0

0.1

disp

lace

men

t m

0 5 10 15 20 25 30 35 40−0.1

0

0.1

time s

Fig. 8. Displacement time histories corresponding to the 3accelerograms of figure 6.

[4] J.W. Baker and N. Jayaram. Correlation of spectral acceleration valuesfrom NGA ground motion models. Earthquake Spectra, 24(1):299–317,2008.

[5] K.W. Campbell and Y. Bozorgnia. NGA ground motion model forthe geometric mean horizontal component of PGA, PGV, PGD and 5%damped linear elastic response spectra for periods ranging from 0.01s to10s. Earthquake Spectra, 24(1):139–171, 2008.

[6] R.W. Clough and J. Penzien. Dynamics of Structures. McGraw-Hill, 1975.[7] EUROCODE 8. Calcul des structures pour leur resistance aux seismes

partie 1 : regles generales, actions sismiques et regles pour les batiments.European norm Eurocode 8 part 1, ref. NF EN 1998-1, septembre 2005.

[8] F.G. Fan and G. Ahmadi. Nonstationary Kanai-Tajimi models for ElCentro 1940 and New Mexico City 1985 earthquake. ProbabilisticEngineering Mechanics, 5:171 – 181, 1990.

[9] D.A Gasparini and E.H. Vanmarcke. SIMQKE : Users manualand Documentation. Dept. of Civil Eng. Massachusetts Institute ofTechnology, 1976.

[10] Zentner I. Numerical computation of fragility curves for NPP equipment.Nuclear Engineering Design, 240(6):1614–1621, 2010.

[11] P.C. Jennings, G.W. Housner, and C. Tsai. Simulated earthquake motionsfor design purpose. In Proc. 4th World Conference Earth. Eng., volumeA-1, pages 145–160, Santiago, 1969.

[12] K. Kanai. Semi-empirical formula for the seismic characteristics ofthe ground motion. In Bulletin of the Earthquake Research Institute,volume 35, pages 308–325. University of Tokyo, 1957.

[13] J.J. Kempton and J.P. Stewart. Prediction equations for significant durationof earthquake ground motions considering site and near-source effects.Earthquake Spectra, 22(4):985 –1013, 2006.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

385

6

[14] Y.K. Lin and Y. Yong. Evolutionary Kanai-Tajimi earthquake models.ASCE J Eng Mech, 113(8):1119 – 1137, 1987.

[15] American Society of Civil Engineers. Seismic analysis of safety relatednuclear structures ans commentary. ASCE/SEI 4-98, september 2000.

[16] PEER. US NGA strong-motion database. http://peer.berkeley.edu/nga/,september 2012. last accessed.

[17] M.B. Priestley. Evolutionary spectra and nonstationary processes. J. RoyalStatistical Society, 27(2):204 – 237, 1965.

[18] Sanaz Rezaeian and Armen Der Kiureghian. A stochastic ground motionmodel with separable temporal and spectral nonstationarities. EarthquakeEngineering & Structural Dynamics, 37(13):1565–1584, 2008.

[19] Sanaz Rezaeian and Armen Der Kiureghian. Simulation of syntheticground motions for specified earthquake and site characteristics. Earth-quake Engineering & Structural Dynamics, 39(10):1155–1180, 2010.

[20] H. Tajimi. A statistical method of determining the maximum response ofa building structure during an earthquake. In 2nd WCEE, pages 781–798,Tokyo, 1960.

[21] D. Wakefield, M. Ravindra, K. Merz, and G.. Hardy. Seismic ProbabilisticRisk Assessment Implementation Guide. Final report 1002989. EPRI, PaloAlto, 2003.

[22] I. Zentner. Simulation of non stationary conditional ground motion fieldsin the time domain. Georisk, 7(1):37–48, 2013.

[23] I. Zentner. Generation de signaux sismiques. Code Aster documentationhttp://www.code-aster.org/V2/doc/default/fr/man r/r4/r4.05.05.pdf, 2014.last accessed.

[24] I. Zentner, L. D’Amico, and P. Cacciola. Simulation of non stationaryground motion compatible with NGA-spectra. In Proceedings ofICOSSAR, New York, 2013.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

386