Site based stochastic seismic spectra

Rita Greco a,n, Giuseppe Carlo Marano b

a DICATECH, Department of Civil Engineering, Environmental, Territory, Building and Chemical, Technical University of Bari, via Orabona 4, 70125 Bari, Italyb DICAR, Department of Civil Engineering and Architecture, Technical University of Bari, via Orabona 4, 70125 Bari, Italy

a r t i c l e i n f o

Article history:Received 27 February 2013Received in revised form25 June 2013Accepted 27 September 2013Available online 26 October 2013

Keywords:Seismic spectraEnvelope functionRandom dynamicNonstationary stochastic process

a b s t r a c t

In this paper, a method to develop stochastic response spectra on the basis of the random vibrationtheory is proposed. The main innovation of this study with respect to previous ones in literature is toderive response spectra in stochastic way, starting from a seismological consistent stochastic model ofthe seismic motion, trying to increase in this way the accuracy in response peak evaluation to futureearthquakes at a given site. In fact, the approach evaluates the maximum of structural response, namelythe response spectrum, without requiring the use of repeated time history analyses, but directly, byobtaining the peak response of a single degree of freedom system subject to a site based stochastic modelof ground motion. A nonstationary stochastic model for the ground motion is adopted, defined by atemporal function and by a set of parameters that describe, respectively, the evolution of the amplitudeduring the time and the frequency content of the record. These are selected on the basis of theinformation contained in a data bank of real seismic events by means of a regression analysis. In this way,it is possible to formulate relations between stochastic model parameters and seismological parameters.

After the stochastic model of the earthquake is constructed and the peak response of a single degreeof freedom is evaluated by means of random vibration theory, the stochastic response spectra are finallyobtained. A parametric analysis is then performed to evaluate the influence of seismological parameters,such as epicentral distance and magnitude, on stochastic spectra. A comparison with Eurocode 8spectrum is also evaluated. The possibility to construct by means of this approach site based stochasticenergy spectra adds value and novelty to the proposed method.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that the accelerogram contains the most directengineering information of an earthquake. However, the mostpractical tools to evaluate earthquake effects on a structure arespectra. An earthquake, in fact, is usually specified by responsespectrum which is the most suitable tool developed to express thestructural response in earthquake engineering and seismic design.In a linear-elastic response spectrum analysis, response spectra definethe free field ground motion for the design earthquake. In seismicengineering, its first introduction was due to Housner in 1941 for asingle past earthquake event. It simply consists in plotting of peak ofinterest responses, such as displacement, velocity or acceleration, fora series of oscillators with different natural frequencies, but all forcedby the same acceleration.

Even though this is an indirect measure of ground intensity, it givesthe maximum response in a straight line, which is one of the mostimportant concerns in structural design, and therefore it is able toprovide acceptable structural response prediction of structures underfuture seismic events. It is well known that if sets of response spectra

are generated for different ground motions recorded at differentlocations in past earthquakes, a great discrepancy would be noticedin both the response spectral values and the shape of the spectrumcurves from one set to another. This outcome depends on manyfactors: the mechanism of energy release in the region of thehypocentre, the epicentral distance and the depth of the focus, thegeology and its variation on the path of waves, etc. Nevertheless, paststudies have shown that the response spectra constructed fromearthquakes recorded on similar soil conditions are similar, both inshape and amplification. For this reason, response spectra derivedfrom accelerograms having similar characteristics can be averaged andsmoothed and, finally, used in the design. In fact, since the exhaustivefeatures of future earthquakes are not known, an earthquake designspectrum is developed by averaging a set of response spectra obtainedfrom records with similar characteristics. For these reasons, an“idealized seismic response spectrum”, based on a range of responsespectra generated for available past earthquake records was devel-oped. This was then further turned into a design response spectrum forusing in structural design, and this basic form (with some modifica-tions) is now the basis for structural design in seismic regionsthroughout the world.

It is well known that the design spectra selection is based on localsoil conditions. Nevertheless, after the design spectrum is selected,the analysis is developed deterministically, using expected spectral

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0267-7261/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.soildyn.2013.09.020

n Corresponding author. Tel./fax: þ39 0805963875.E-mail address: r.greco@poliba.it (R. Greco).

Soil Dynamics and Earthquake Engineering 55 (2013) 288–295

parameters obtained from the codes. It is evident that, althoughthese spectra are derived from many earthquake time histories, theyresult is a single smoothed line that describes the expected groundmotion, at a prescribed level of damping, with a single line, withoutany explicit consideration of the uncertainty intrinsic in the local siteconditions and other factors. Specifically, the uncertainties due tolocal site conditions, distance from the site to the source, earthquakemagnitude, focal distance, and soil–structure interaction are notconsidered explicitly in the development of the design spectrum.The spectral ordinates are linearly scaled by only the peak accelera-tion, and site effects are considered by assigning larger spectralaccelerations at longer periods for softer and deeper soil profiles.Therefore, the resulting ordinates are deterministic and limited dataavailable before the more recent earthquakes made it difficult toisolate various influencing factors.

Although today response spectra are commonly used for theestimation of the largest peak response of linear structural systems ina seismic environment, one should point out that very stronglimitations exist on their applications. First, these methods do notconsider uncertainty in response due to variability of seismic actionand are not able to incorporate other intensity measures parametersthat account for the energy released by the event. In addition, theyhave natural limitation of being applied accurately only for few typesof structural systems. In fact, for analyzing some kinds of earthquakeengineering problems, or for the design of special or very importantstructures, it may be necessary to account for other relevantcharacteristics of the ground motion time histories.

Sure of limits associated to use of response spectrum, a crucialpoint in further researches is going in trying to increase the accuracyin response peak evaluation to future earthquakes at a given site.Therefore, the problem of the evaluation of the response of astructure having known dynamic characteristics to an assignedcharacterization of the ground motion represents one of the mostimportant topics in seismic safety assessment, and the responseanalysis must necessarily account for the uncertainty associated withthe description of ground motion process. As consequence of randomnature of earthquakes, stochastic procedures are recommended toobtain the needed information for the design and for the safetyassessment of structures subject to seismic action. For this purposeearthquake ground motion must be necessarily characterized as arandom process dependent on a number of factors which can berelated to the recorded strong motion characteristics, and should bedescribed by a transient stochastic function. This assumption is thestart point of the methodology here proposed to construct stochasticresponse spectra.

The construction of response spectra from a probabilistic pointof view has followed from the pioneering work of Udwadia andTrifunac [1]. The authors showed how a probabilistic responsespectrum can be computed from the Fourier transform of a givenground motion by using the statistics of the largest peak ofoscillator response. Amini and Trifunac [2,3] were the first toapply this idea and then proposed a response spectrum super-position technique for probabilistic response analysis of simple,fixed-base, shear buildings to translational component of groundmotion. After these works, a lot of suggestions have been for-mulated by various scholars, intended to develop a stochasticapproach to earthquake response spectrum.

Estimation of time domain and spectral parameters of groundmotion can be obtained either by empirical relations that connectthese to earthquake magnitude, distance, and local soil conditions(source scaling and attenuation relations) or by means of mathe-matical modeling. At present, there is no doubt that these relationsare different for different seismic regions, and “region and site-specific” models should be developed on the basis of availablestrong ground motion records. Different methods have beenformulated, some of these starting from seismological models for

earthquake source mechanisms and waves propagation model,where deterministic source-to site characteristics were convertedto stochastic ones, by regarding various filter parameters, such asthe apparent duration of fault rupture, the dislocation rise time thedistance to source, the rate of wave amplitude decay, etc asrandom variables. Examples include the construction of empiricalGreen's function that approximate the impulse response of theground motion between earthquake source and observation sitethat can be used for the generation of synthetic accelerograms [4],derivation of a stochastic model for strong ground motions basedon the normal mode theory [5], etc. The earthquake sourcemechanism can be then extended a step further through thegeneration of response or design spectra that are compatible withthe statistical description of the ground motion [6]. The core of theprocedure is to relate the power spectrum of the stochasticmodeling of the ground motion to design spectrum. This latter isdefined by the peck response of a linear oscillator subject to thepower spectrum input. By using the extreme value theory, thedistribution of the maximum response can be finally predicted.

In general, methods for generating stochastic response spec-trum belong to two categories in relation with two distinct groupsof method for describing the ground motion. Concerning this lasttopic, essentially two main distinct approaches can be distin-guished for predicting earthquake ground motion: the mathema-tical approach and the experimental one. In the mathematicalapproach a model is derived analytically, based on physicalprinciples. Seismological source models, like those before men-tioned, belong to this class. More precisely, the model involvessome parameters (source parameters) which are identified fromrecords of previous earthquakes. Hanks and McGuire [7] and Boore[8] were the first to adopt this kind of seismological model for thestochastic simulation of earthquakes.

Laboire and Orosco [9] developed a methodology to simulateartificial acceleration records based on the theory of nonstationaryrandom processes with evolutive power spectral density function.The random process model made use of the definition of the Boorepower spectral density function for taking into account the seismiccharacteristics of the zone.

Rezaeian and Der Kiureghian [10] developed a fully nonsta-tionary stochastic model for strong earthquake ground motion,which makes use of filtering of a discretized white-noise process.Nonstationarity was obtained by modulating the intensity andchancing filter properties in the time. More precisely, the modelwas fitted to target ground motions by matching a set of statisticalcharacteristics, including the mean-square intensity, the cumula-tive mean number of zero-level up-crossings and a measure of thebandwidth, all expressed as functions of the time.

Nelson et al. [11] described the development of a methodologyto construct design response spectra utilizing a generic attenua-tion relationship provided by the component attenuation model(CAM). In addition, they adopted limited ad hoc observations oflocal isolated earthquake events to assist in the crustal classifica-tion of the region and to confirm the generic attenuation relation-ships. In opposition to seismological models, other approaches,like experimental ones, have been proposed for simulating groundmotion. In these approaches, a mathematical model is formulatedbut without any physical meaning. When a model is based only onavailable measured data, it is named Black-box modeling, whichsimply consists in fitting low order discrete time series (ARMA)models to the data.

This paper is inspired by a previous work of a co-author [24],where a consistent stochastic model of the seismic action isdeveloped. Starting from the cited paper, this study is focused onthe construction of stochastic response spectra, starting from astochastic simulation of earthquake record consistent with theaccelerograms recorded at the site. More precisely, a correlation

R. Greco, G.C. Marano / Soil Dynamics and Earthquake Engineering 55 (2013) 288–295 289

law between the earthquake model and the earthquake recordedat the site is constructed. After the consistent stochastic model isobtained, the stochastic response spectrum is evaluated as the plotof the maximum response of a linear system to the stochasticprocess versus its natural period. A parametric analysis is thenperformed to evaluate the influence of seismological parameters,such as epicentral distance and magnitude, on stochastic spectra.Energy spectra are also developed and comparisons with Euro-code8 spectrum are made. Some conclusions on the influence ofseismological parameters on stochastic spectra are given; inaddition, interesting suggestions are provided, useful from anengineering point of view, which add practical engineering valueto the present work.

The novelty of present study with respect to previous ones inliterature is to develop response spectrum in stochastic way, startingfrom a seismological consistent stochastic model of seismic motion.The advantage of stochastic approach to develop seismic responsespectra with respect to classical one is to consider in an effective waythe intrinsic nature of seismic phenomenon. As a consequence, alsosystem response and reliability take into account the intrinsicuncertainty of earthquakes, trying to increase in this way theaccuracy in response peak evaluation to future earthquakes at agiven site.

2. Construction of stochastic response spectra

In this section the procedurei for constructing stochasticresponse spectra is presented. The steps are described below:

Simulation of earthquake records consistent with the acceler-ograms recorded at the site. More precisely, the earthquakesignal is described by a uniformly nonstationary stochasticprocess, obtained by multiplying a filtered stationary stochasticprocess for a temporal function. The accelerograms adopted inthis study consist of records from the strong-motion databaseof the NGA project [12]. Not all of the available records havebeen considered in this study; the particular subset that hasbeen adopted is exactly the same as that which has previouslybeen used by Stafford et al. [13].In addition, by performing such analyses for a large number ofrecorded ground motions, a correlation law between the modelparameters and the earthquake and site characteristics isconstructed. The Chough and Penzien (CP) model is adoptedto represent the stationary part of the process. CP parameterswill be assumed as given quantities, depending on soil type.After the stochastic model is obtained, the stochastic responsespectrum is evaluated as the plot of the maximum response ofa linear system to the stochastic process input, versus theperiod of the system. For this aim, the theory of peaks ofstochastic process is used.

In next sections, the phases before briefly described, will beexplained in detail.

2.1. Stochastic simulation of site based earthquake records

Many models have been proposed to simulate earthquakeground motion. A method based on the seismological meaningof the problem will be adopted in this study: in this way, it ispossible to generate acceleration records compatible with localseismic characteristics, therefore incorporating local seismic char-acteristics in the evaluation of structural response. The approachhere proposed is then a site based stochastic model, which is ableto generate artificial ground motions, with statistical characteris-tics similar to those of the target ground motion.

In the following, as before pointed out, an earthquake accelero-gram will be considered as a realization of a nonstationary Gaussianstochastic process. Several stochastic models for the ground motionhave been formulated in the past; their mean characteristics areprincipally related to the possibility to represent the seismic motionas a stationary or a nonstationary process, with different frequencycontent. A complete review of site based stochastic models forground acceleration modeling has been done by [14–18].

The simplest approach to simulate an earthquake signal is bypassing a stationary with noise process wðtÞ by a linear filter. Moreprecisely, the filtering effect of a soil layer is simulated by means ofa damped single degree of freedom linear system, with dampingratio ξf and circular frequency ωf .

The ground acceleration ast is then obtained as the absoluteacceleration of the filter:

€xþ2ξfωf _xþω2f x¼ �w ð1Þ

ast ¼ €xþw¼ �ð2ξfωf _xþω2f xÞ ð2Þ

The power spectral density function of the filtered white noise is:

SðωÞ ¼ S0jHðiωÞ2j ð3Þwhere HðiωÞ is the complex frequency response function of thefilter and S0is the power spectral density function of the whitenoise excitation. For a second order linear filter, the power spectralof the acceleration process is:

SðωÞ ¼ S0½1þξ2f ðω=ωf Þ2�

½1�ðω=ωf Þ2�2þ4ξ2f ðω=ωf Þ2ð4Þ

Eq. (4) expresses the power spectral density function of the Kanaiand Tajimi stochastic process [19,20], whose major attractivenessconsists in the possibility of representing the ground motionresonance property.

Unfortunately, by means of this approach, the seismic motion isregarded like a stationary process. To overcome this limit thestationary white noise process w is first passed through a linearfilter in order to obtain the desired power spectral density functionand then is multiplied by a deterministic envelope function E tð Þ, sothat the stationary filtered white noise ast gets the desired time-dependent amplitude. In this way, the uniformly modulated nonsta-tionary stochastic model of ground motion in obtained.

Despite the Kanai-Tajimi process is widely adopted [21], it hasbeen demonstrated (see for example [22]) that this model is toopoor to represent real seismic motions because of the presence ofonly two filter parameters and it is not so able to represent groundmotions from medium to large structural periods. To overcomethis limit, the Clough and Penzien (CP) model [23] can beconveniently adopted. This considers a linear fourth order filterforced by a white noise. Therefore, by using the above CP model,ground acceleration a is:

a¼ €xg ¼ �ω2pxp�2ξpωp _xpþω2

f xf þ2ξfωf _xf€xpþω2

pxpþ2ξpωp _xp ¼ω2f xf þ2ξfωf _xf

€xf þ2ξfωf _xf þω2f xf ¼ �EðtÞw

8>><>>: ð5Þ

where xf is the response of the first filter, having frequency ωf anddamping coefficient ξf , xp is the response of the second filtercharacterized by ωp and coefficient ξp.

When earthquakes are modeled by means of a uniformlymodulated nonstationary stochastic process, one needs to selectin a suitable way the modulation function EðtÞ and the filterparameters. In this study, the definition of the envelope will beperformed by a procedure carried out by a co-author of this studyin a previous research [24], and it will be summarized in the nextsection. With regard to filter parameters that define the spectraldensity function, in this study some literature parameters will be

R. Greco, G.C. Marano / Soil Dynamics and Earthquake Engineering 55 (2013) 288–295290

assumed, even if these can be obtained from the same datautilized for defining the modulation function.

2.2. Envelope evaluation

Many different envelope functions have been proposed over thepast 40 years or so; the most common of which are discussed byJangid [25]. In this study, the envelope function is related directly tothe seismological and ground motion parameters that are routinelyspecified during a design situation. The envelope function is tieddirectly to the expected Arias intensity of the accelerogram, givensome seismological scenarios as well as to two shape parameters thatare in turn related to seismological parameters.

The input required to derive the envelope consists of dataregarding the scenario, i.e., magnitude, distance, average shear-wave velocity, etc., as well as an estimate of the Arias intensityassociated with this particular combination of seismological para-meters. This information may be specified directly during ascenario-based analysis, where one simply defines the seismolo-gical parameters from the outset, or via a probabilistic seismichazard analysis (PSHA), where the most relevant seismologicalparameters are identified following disaggregation of the ground-motion level corresponding to a particular return period [26].

The form of the envelope function derives by the considerationof the manner in which the energy is temporally distributedthroughout an accelerogram. For a more exhaustive deepening ofthe method to derive the envelope function one can examine thecited research; in this section a brief description of main features iscarried out. A key objective of this approach is to ensure that theenvelope is consistent with energy characteristics of real accel-erograms. This consistency is monitored through the use of Ariasintensity, both in terms of the total Arias intensity of a record, aswell as the manner in which this total Arias intensity is achieved.

It is well known the Arias intensity is proportional to theintegral of the squared acceleration time-history and is defined as:

Ia ¼ π

2g

Z 1

0a2ðtÞ dt ð6Þ

where aðtÞ is the acceleration time-history (see [24] for a morethorough definition). According to [13], the temporal accumulationof Arias intensity can be viewed using a Husid plot (either in anabsolute or normalized form), which may be defined:

HðtÞ ¼R t0 a

2ðtÞ dtR10 a2ðtÞ dt ¼

π

2gIa

Z t

0a2ðtÞ dt ð7Þ

The form of the envelope function derives from the realization thatthe function HðtÞ satisfies all of the required conditions to allow itto be regarded as a cumulative probability distribution functionand that, furthermore, the shape of HðtÞ for real accelerograms isstrikingly similar to that of the cumulative distribution function(CDF) of the lognormal distribution (see Fig. 1).

The shape for envelope function has been related to theprobability density function equivalent of the normalized Husidfunction, which is defined in Eq. (8). Details can be found in [26].

hðtÞ ¼ dHðtÞdt

¼ π

2gIaa2ðtÞ ð8Þ

To derive the shape of the envelope function, it is simply assumedthat the function h tð Þ may be modeled using the probabilitydensity function (pdf) of the lognormal distribution. This lognor-mal pdf is directly related to the shape of the envelope andrequires the specification of two parameters that correspond tothe mean and standard deviation of the distribution. These shapeparameters are defined as functions of common seismologicalparameters [24].

The general expression for the envelope function E tð Þ is:

EðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4gIaπ

hðtÞr

ð9Þ

The Husid pdf, given by hðtÞ, is assumed to be modeled by the pdfof a lognormal distribution which leads to the final expression forthe envelope function:

EðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4gIa

tsπffiffiffiffiffiffi2π

p exp �ðlnðtÞ�μÞ22s2

" #vuut ð10Þ

Hence, the Husid Envelope Function (HEF) is completely definedfor each instant, provided that one has the estimates of the Ariasintensity corresponding to the scenario, as well as the estimates ofthe shape parameters μ and s that correspond to the mean andstandard deviation of the lognormal distribution. The estimates ofthe Arias intensity have been made using existing empiricalmodels such as that of Travasarou et al. [27]. The shape parametersmust be related to the seismological characteristics of the scenariothrough new empirical predictive models by nonlinear random-effect regression analyses.

The final functional forms are given in Eqs. (11) and (12), whilethe coefficients of these models are presented in Appendix.

μ¼ b1þb2Mwþðb3þb4MwÞlnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2rupþb25

qþb6 ln VS;30þb7Ztor ð11Þ

ln s¼ c1þc2Rrupþc3 ln VS;30þc4Ztor ð12ÞThe variance of the logarithmic envelope values has been found bypropagating the variances associated with the Arias intensity andthe shape parameters [24]. It is expressed as follow:

s2ln EðtÞffi14

s2ln IaþKCs2μþðKC�1Þ2s2ln sþ

ffiffiffiffiffiKC

ps ρln Ia ;μsln Iasμ

þðKC�1Þρln Ia ; lnssln Iaslns

þðKC�1ÞffiffiffiffiffiKC

ps ρμ;lns

sμslns

264

375

ð13Þ(see [24] for a more detailed explanation). The standard deviationterms in Appendix are defined as follows: sE is the inter-event

Fig. 1. Example of the typical fit that may be achieved by modeling the observed Husid pdf as a lognormal distribution.

R. Greco, G.C. Marano / Soil Dynamics and Earthquake Engineering 55 (2013) 288–295 291

standard deviation, sA is the intra-event standard deviation, includ-ing the component-to-component variability, sC is the component-to-component standard deviation, s1 is the intra-event standarddeviation, excluding the component-to-component variability, sT,ARBis the total standard deviation for an arbitrary component, and sT,GMis the total standard deviation corresponding to the geometric meanof two horizontal components.

Finally, KC is equal to ðlnðtÞ�μÞ2=s2.

3. Stochastic seismic spectra

The response spectrum is given by the plot of the maximumresponse of a linear one degree of freedom system to the recordedearthquake versus its natural period. In stochastic meaning, in ananalog way, the response spectrum is the plot of the maximumresponse of the system to a stochastic model of the ground motionversus the natural period, for any assigned value of the structuraldamping. By adopting the formulation explained in previoussection, one can provide the response characteristics of the systemto the modeled stochastic process, whose envelope and spectralproperties reflect those of the recorded ground accelerations.

Therefore, concerning the stochastic modeling of the groundmotion, the seismological consistent envelope is used as reas-sumed in previous section. With regard the shape of the powerspectral density function, the CP model is used, whose parametersare selected according with some literature data and derived inrelation of soil condition.

3.1. Stochastic response to nonstationary input

To develop probabilistic spectra, the response of a simple linearsingle degree of freedom system subject to the previous formu-lated stochastic characterization of the ground motion must beevaluated.

The motion equations of a linear system subject to a groundacceleration modeled by means of the uniformly modulated CPstochastic process are:

€xs ¼ �2ξsωs _xs�ωs2xsþω2

pxpþ2ξpωp _xp�ω2f xf �2ξfωf _xf

€xpþω2pxpþ2ξpωp _xp ¼ω2

f xf þ2ξfωf _xf€xf þ2ξfωf _xf þω2

f xf ¼ �EðtÞw

8>><>>: ð14Þ

where ξs and ωs are, respectively, the damping coefficient and thenatural frequency of the system.

The stochastic response is achieved by solving the Lyapunovmatrix differential equation.

_RZZðtÞ ¼ARZZðtÞþRZZðtÞATþBðtÞ ð15Þ

In Eq. (15) RðtÞ ¼ ⟨ZZT ⟩ is the covariance, BðtÞ is the forcing vectorhaving all elements equal to zero except the last one that holds2πS0EðtÞ2, and A is the state matrix.

To construct the stochastic spectrum, the problem of evaluatingthe peak response of a system should be developed. This issummarized in the next section.

3.2. Peak response and stochastic response spectrum evaluation

In structural dynamic a very important issue is the evaluationof the response under a fixed probability of failure that is theprobability that a certain limit state will not be exceeded duringthe system lifetime.

For a zero-mean, non-stationary, Gaussian stochastic process,the reliability during the total duration Tof the process, under the

Poisson hypothesis, is [28]:

r¼ 1�Pf ¼ exp �Z T

0νðτÞdτ

� �ð16Þ

where ν is the total average rate of exceeding a given barrier.In this study, stochastic displacement and acceleration spectra are

obtained. The way of constructing these spectra is similar to thedeterministic approach, and it is based on the assumption that:

SdðTS; ξsÞ ¼ pnðTS; Pf Þ max0r tr td

jsXs ðtÞj ¼ pnsXmax ð17Þ

In Eq. (17) TS and ξs are, respectively, the structural period and thedamping coefficient of the system; pnsXmax is peak response for aspecific value of probability of failure Pf (being pn the so called peakfactor) and td is the duration of seismic action.

The spectral values of the relevant quantity are determined as:

(1) Assume a given level of reliability r .(2) Solve Lyapunov matrix differential equation for every TS (linearly

spaced from TS¼0.05 s to TS¼2.5 s).(3) Determine pnðTSÞ by Eq. (17), in which the expression of νðtÞ

can be restated as:

log r ¼ �Z T

0

1π

s _XsðτÞ

sXs ðτÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ρX _XðτÞ

qexp �1

2pnsXmax

sXs ðτÞ

� �2" #

χ½dXðτÞ�dτ

ð18Þassuming the initial value pnðTSÞ ¼ 3.

The quantities in Eq. (18) are specified in [30].The procedure above described cannot be applied to develop

the acceleration spectrum SaðTS; ξsÞ, due to the intrinsic difficultyin evaluating expression (18). However, in this case a conventionalapproach will be considered by assuming a constant value pnðTSÞ ¼ 3

3.3. Results analysis

Fig. 2 shows the displacement spectra obtained by assumingthree values of Pf (0.10, 0.05 and 0.01). In order to constructresponse spectra only seismological parameters are assigned: infact, these are sufficient to completely define the shape of themodulation function.

Input data are magnitude M¼5.2, epicentral distance Rrup¼11 km, Ztor¼5.28; in addition a rock soil type (Vs30¼874 m/s) isconsidered. Concerning the CP power spectral density function, theparameters of Table 1 have been adopted corresponding to a rocksoil condition [22].

A structural damping equal to ξ0 ¼ 5% has been assumed. Theresults achieved by the peak theory are compared with the more“conventional” solution, assuming as the maximum system displace-ment Xmax ¼ 3 maxðsX ðtÞÞ.

Fig. 2. Displacement stochastic spectra evaluated with different criteria andcompared with the standard Eurocode8 spectrum.

R. Greco, G.C. Marano / Soil Dynamics and Earthquake Engineering 55 (2013) 288–295292

In addition, in Fig. 2 the stochastic displacement spectraevaluated for different Pf are compared with Eurocode8 spectrum[29] on firm soil. This spectrum has been evaluated considering aPGA corresponding to the peak value of the envelope function hereassumed, being the power spectral density function scaled in orderto give a unit maximum acceleration. From the analysis of thisfigure it is possible to deduce that by using the “conventional”peak factor (equal to 3), the displacement spectrum obtained is ingood agreement with those obtained by adopting Pf in the range of0.05 and 0.01. However, it is quite evident that the stochasticspectra overestimate the Eurocode8 spectrum.

With regard the PGA, as it has been pointed out in [24], themaximum of the envelope function is not able to encapsulate thePGA. However, the authors have demonstrated that the mean ofthe ratio between the peak acceleration and the maximum of theenvelope is very close to 2.5 and the logarithmic standard devia-tion is 0.2095. This simple approach for PGA prediction has alsobeen demonstrated to be generally very consistent with otherwidely diffused NGA empirical ground-motion models. On thebasis of this consideration, in this study the PGA estimated as themean of the HEF plus its standard deviation is used. Indeed, it hasfound a good agreement between this value and that proposed in[24]. Even if the PGA has a simple and intuitive engineeringmeaning, it is not the best index for earthquake severity quanti-fication. For this reason in the proposed approach, the stochasticspectra do not depend directly from this peak parameter, but overa more wide and indicative set of seismologic parameters whichwill produce a more realistic representation of maximum actionson the structure.

A comparison between the Eurocode8 spectrum and thestochastic one, obtained with the three sigma rule, has been madealso for acceleration (Fig. 3). Also in this case one can notice a goodaccordance with Eurocode8 pseudo acceleration spectrum, even if,for medium and long natural periods, the stochastic spectrumtends to overestimate the deterministic one, especially in theresonance range. On the contrary, in the range of small periods,the Eurocode8 spectrum shows accelerations greater than thoseobtained from stochastic approach.

Fig. 4 shows the mean value of energy for unit mass. This hasbeen obtained by applying the mathematical average operator tothe power balance, which is expressed through the “relative energyequation”, obtained by integrating the “relative power equation” of

the system respect to the time t:Z t

0⟨ €XS

_XS⟩ dtþZ t

02ξSωS⟨ _X

2S⟩ dtþ

Z t

0ω2S⟨XS

_XS⟩ dt ¼ �Z t

0⟨ €Xg

_XS⟩ dt

ð19ÞHence, the “relative energy balance” in mean at time t can bewritten as:

⟨ekðtÞ⟩þ⟨eξðtÞ⟩þ⟨eeðtÞ⟩¼ ⟨eiðtÞ⟩ ð20Þwhere

⟨ekðtÞ⟩¼12s2_XS

ðtÞ ð21Þ

⟨eξðtÞ⟩¼ 2ξSωS

Z t

0s2_XS

ðτÞ dτ ð22Þ

⟨eeðtÞ⟩¼ ω2S

Z t

0⟨XSðτÞ _XSðτÞ⟩ dτ ð23Þ

The meaning of the energy balance is very crucial in seismicdesign strategy. In fact, it expresses the circumstance that duringthe seismic event, part of the absorbed energy is stored into thestructure in the form both of kinetic and strain energy; theremainder part is dissipated through the damping and by struc-tural viscous behavior. The energy evaluation is then very usefulfor the choice of a seismic strategy design, through the require-ment that the structure is able to dissipate an amount of energygreater than the one introduced into it by the seismic motion. InFig. 4 the spectra of mean values of different energetic rates areplotted. From these plots one can clearly deduce how sensitive arethe energy terms whit natural periods.

Figs. 5–9 show the results of a parametric analysis on displace-ment and acceleration spectra versus epicentral distance andmagnitude. More precisely, in Figs. 5 and 6, the displacementspectra have been obtained considering a Magnitude in the range5.5–7.5 and considering two epicentral distances: 5 km and 20 km.

As obvious, the spectra obtained grow up with the magnitudeand decrease with the epicentral distance. However one can pointout that the increase of the spectrumwith the magnitude has not alinear law; in addition, the shape of the spectrum is not simplyscaled as this parameter varies. At this purpose, one shouldremember that in Technical Codes, which consider as seismicdistinctive parameter, besides the soil, also the PGA, design spectraare obtained by means of a linear dependence from this latterparameter. One should remember that in scientific literature oftenit has been recalled the necessity to free oneself from thissimplification, which sometimes can discredit the precision ofthe method, especially in some spectral ranges.

The dependence of the spectrum from the magnitude isremarkable also in Figs. 7–9, where the acceleration spectra,

Table 1Clough and Penzien model parameters.

Soil type ωg ðrad=sÞ ξg ωf ðrad=sÞ ξf

Rock 23.00 0.43 2.80 0.97

Fig. 3. Acceleration stochastic spectrum in comparison with the standard Euro-code8 spectrum.

Fig. 4. Energy stochastic spectra.

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normalized with respect the PGA, are referred to the range ofmagnitude 5–7.5. Three values of the epicentral distance areassumed, 5 km, 20 km and 50 km.

From these plots it is possible to observe that the variability ofthe spectra with the magnitude is noteworthy for Rrup¼5 km,whereas becomes insignificant gradually when Rrup increases andis practically inappreciable for Rrup¼50 km.

Concerning the case Rrup¼5 km, the increase of the accelera-tion spectrum with the magnitude is particularly sensible in theresonance range and tends to diminish for high periods. It isevident that this aspect, which is not considered in most TechnicalCodes is of primary importance to correctly predict the seismicaction on structures. At this purpose one can observe that theresonance peak increases of the 15% when the magnitude variesfrom M¼5.5 to M¼7.5.

4. Conclusion

In this paper a stochastic approach to develop design spectra ispresented. The novelty of present study with respect to previousones in literature is to construct response spectrum in a stochasticway, starting from a seismological consistent stochastic model ofthe seismic motion. Therefore, a site-dependant random model ofseismic events has been used, so that earthquakes are representedby consistent stochastic processes instead of natural seismicground motion records. After the stochastic model of the earth-quake has been obtained, the stochastic response spectra arefinally obtained by means of the peak theory of stochastic pro-cesses. A parametric analysis has been developed to assess theinfluence of seismological parameters, such as epicentral distanceand magnitude, on stochastic spectra. In addition, a comparisonwith Eurocode8 spectrum has been performed. Analysis of theresults allows tio conclude that by using the conventional peakfactor (equal to 3), the displacement spectrum obtained well agreeswith those obtained by adopting a failure probability Pf in the rangeof 0.05–0.01. However, stochastic spectra overestimate the Euro-code8 spectrum. Moreover, with regard to acceleration spectrumone can notice a good accordance with Eurocode8 pseudo accelerationspectra, even if, for medium and long natural periods, the stochasticspectrum overestimates the classic one, especially in the resonancerange. On the contrary, for small periods, the EC8 spectrum showsaccelerations greater than those obtained by stochastic way.

The parametric analysis developed by varying seismologicalparameters, points out that the increase of the spectrum with themagnitude has not a linear law, and that the shape of the spectrumis not simply scaled as this parameter varies. This result adds animportant engineering value to the study, if one considers that inTechnical Codes, which assume as seismic distinctive parameter, as

Fig. 5. Displacement spectra for different magnitudes and epicentral distanceof 5 km.

Fig. 6. Displacement spectra for different magnitudes and epicentral distanceof 20 km.

Fig. 7. Acceleration spectra for different magnitudes and epicentral distanceof 5 km.

Fig. 8. Acceleration spectra for different magnitudes and epicentral distanceof 20 km.

Fig. 9. Acceleration spectra for different magnitudes and epicentral distance of 50 km.

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well the soil, also the PGA, design spectra are obtained by means ofa linear dependence from this latter parameter. At this purpose,one should remember that in scientific literature, often, it has beenrecalled the necessity to free oneself from this simplification,which sometimes can discredit the precision of the method,especially in some spectral ranges.

The advantage of stochastic approach to construct responsespectra with respect to classical spectra, is the capability to takeinto account in a suitable way, the intrinsic character of earth-quakes, trying to increase in this way the accuracy in responsepeak evaluation to future earthquakes at a given site. This aspect isvery important if one considers that classical spectra have thenatural limitation of being applied accurately only for few types ofstructural systems. In fact, for analyzing some kinds of earthquakeengineering problems, or for the design of special or very impor-tant structures, it may be necessary to account for other relevantcharacteristics of the ground motion time histories.

The limit of the applicability of this approach is in an enoughaccurate model of seismic input as stochastic process. Increaseaccuracy in representing classes of real seismic events willenhance the effectiveness of seismic spectra evaluation.

The possibility to construct by means of this approach also sitebased stochastic energy spectra, adds value and novelty to theproposed method. Finally, the main advantage of this procedure isa direct extension of the methodology also to nonlinear systems,without a dramatic increase of computational cost. This approachis, in fact, the preliminary to construct site based consistent non-linear stochastic spectra.

Appendix A

See Table A1.

References

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Table A1Parameter values and associated 95% confidence intervals (defined as7values) forthe empirical models for the shape parameters of Eq. (11) (left two columns) andEq. (12) (right two columns).

Shape parameter μ Shape parameter s

Parameter Estimate Parameter Estimate

b1 �6.039170.7769 c1 0.315170.1139b2 1.089570.1188 c2 �0.00370.0003b3 1.841570.1316 c3 �0.095770.0165b4 �0.206570.0193 c4 �0.010470.0096b5 5.757571.356b6 �0.138370.0217b7 �0.023970.0198sE, μ 0.326970.0491 sE, lns

0.177470.0286sA, μ 0.293670.0059 sA, lns

0.224470.0045sC, μ 0.107870.003 sC, lns

0.082170.0023s1, μ 0.273170.0065 s1, lns

0.208970.005sT,ARB, μ 0.439470.0367 sT,ARB, lns

0.286170.0181sT,GM, μ 0.425970.0379 sT,GM, lns

0.27470.0189

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