probabilistic seismic hazard analysis using reliability...

Scientia Iranica A (2017) 24(3), 933{941

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Probabilistic seismic hazard analysis using reliabilitymethods

M. Kia and M. Banazadeh�

Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, P.O. Box 15875-4413, Iran.

Received 2 July 2015; received in revised form 12 January 2016; accepted 26 April 2016

KEYWORDSProbabilistic seismichazard analysis;Reliability methods;Probabilistic model;Epistemic andaleatory uncertainties;Limit-state function.

Abstract. By considering uncertainties in the input parameters (e.g., magnitude,location, wave path, etc.), the Probabilistic Seismic Hazard Analysis (PSHA) aims tocompute annual rate of various exceeding ground motions at a site or a map of sites of allanticipated given earthquakes. Uncertainties may be originated due to inherent randomnessof the phenomena or variability in the mean values of di�erent models parameters, mainlydue to use of �nite-sample size of observations. The �rst, in literature reviews, is commonlynamed aleatory uncertainty; the second is known as epistemic uncertainty. The totalprobability numerical integration, generally employed to calculate PSHA, only considersaleatory uncertainties, and variability in the models' parameters is neglected to simplifycalculation. In this paper, as an alternative to the total probability numerical integration,matured and standard reliability methods tailored to e�ortlessly consider both types ofuncertainties are put forward to compute site-speci�c PSHA. Then, as an applicationstudy, the peak ground acceleration hazard curve for the site, at which a historical bridgeis located, is developed and compared with those obtained from the total probabilitynumerical integration.

© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

The Probabilistic Seismic Hazard Analysis (PSHA)is an approach broadly applied to describe probablefuture ground shakings at a site. In this approach,annual rate of exceeding for a given ground motionhazard parameter �(im) at a speci�c site is estimatedby quantifying uncertainties in the characteristics ofseismic source such as magnitude, distance, and wavepathway. The mathematical description of the PSHAwas �rst set up by Cornell [1]. The formulationwas based on total probability theorem. Eq. (1)demonstrates general form of the formulation:

*. Corresponding author. Tel./Fax: +98 21 64543049E-mail addresses: [email protected] (M. Kia);[email protected] (M. Banazadeh)

�(im) =nXi=1

viP (IM > im)

=nXi=1

vix

P (IM > imjM = m;

R = r)fM (m)� fR(r)dm:dr; (1)

where vi is the mean annual rate of occurrence ofall earthquakes which are greater than or equal tothe minimum magnitude, IM denotes a ground mo-tion hazard parameter. Since, in this study, thereis more emphasis on the methodology rather thanthe hazard parameter, the well-known Peak GroundAcceleration (PGA) is chosen as the ground motionmeasure. n describes number of seismic sources andP (IM > imjM = m;R = r) is the conditional

934 M. Kia and M. Banazadeh/Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 933{941

probability of exceeding the chosen IM for a givenmagnitude and distance. In this term, dispersionaround the mean value of the IM is addressed usinglognormal distribution. Finally, fM (m) and fR(r)indicate probability density functions of magnitudeand distance, respectively. These probabilistic modelsre ect variability inherent in the characteristics ofseismic source (i.e., magnitude and location). However,probabilistic models developed based on �nite-samplesize observations pose another type of uncertainty,which is known as epistemic uncertainty. This typeof uncertainty is technically considered by employingprobabilistic model with random, rather than con-stant, parameters. It is obvious that considering bothmentioned uncertainties in the conventional frameworkmakes calculation extremely di�cult. Since impactsof epistemic uncertainties on performance evaluationand design have recently received serious attention [2],employing a framework, which is capable of consideringboth types of uncertainties, would be interesting.

Recently, reliability framework has been widelyapplied in literature to explicitly address aleatoryand epistemic uncertainties in engineering reliabilityanalysis and reliability-based design [3-6]. Haukaas [7]proposed the \uni�ed reliability analysis" as analternative to the seismic risk analysis based onconditional probabilities. In this approach, FirstOrder Reliability Method (FORM) is implementedto carry out design optimization for a system withlimit-state function developed in terms of monetaryloss. Du [8] quanti�ed the e�ects of the aleatory andepistemic uncertainties with belief and plausibilitymeasures in the context of the evidence theory. Heimplemented the �rst order reliability method forprobabilistic analysis and nonlinear optimizationfor interval analysis to handle black-box functions.As an application of the methodology proposed byHaukaas [7], Bohl [9] developed loss curve for anexample building and compared the result with theone obtained from Paci�c Earthquake EngineeringResearch (PEER) center approach. Der Kiureghianand Ditlevsen [10] focused on the in uence of thetwo types of uncertainties on reliability assessment,codi�ed design, performance-based engineering, andrisk-based decision-making. Two examples werestudied to demonstrate the in uence of statisticaldependence arising from epistemic uncertaintieson systems and time-variant reliability problems.Koduru and Haukaas [11] studied the viability of theFORM in the context of both static and dynamic�nite-element problems. It was concluded thatFORM is not an appropriate reliability method fortime-variant reliability problems. Xiao et al. [12]proposed an optimization model for sensitivity analysisunder both aleatory and epistemic uncertainties.The parameters with su�cient information were

modeled using probability distributions, while otherswere modeled using a pair of upper and lowercumulative distributions (the so-called P-box).Mahsuli and Haukaas [13] developed an open-sourcereliability software, named \Rt" , as a computationalplatform for evaluating reliability problems. Thissoftware, which is available on the following websitelink \http://www.inrisk.ubc.ca", is employed in thepresent study to solve reliability-based PSHA problem.

According to the above description, in this paper,reliability framework as an alternative to the conven-tional approach is put forward to address problemsin Eq. (1). Contrary to the conventional approach,this framework is able to e�ortlessly and explicitlyaddress both aleatory and epistemic uncertainties.Magnitude model with random parameters is de�nedas an example to address the ability of the frameworkto re ect both types of uncertainties within calculationof PSHA. This ability would be more notable whenattenuation relations with random model parametersare implemented in calculating PSHA. Furthermore,availability of reliability software, such as \RT", ef-fortlessly capable of carrying-out reliability analysis,makes the reliability-based PSHA more interestingand appealing. This paper starts with describingreliability analysis and limit-state function in detail.Next, magnitude and location models are developedto be applied to conventional and reliability-basedframework to develop seismic hazard curve. The resultsare presented and compared at the end to provideinsight into the accuracy and e�ciency of the suggestedreliability methods.

2. Reliability analysis

In classical structural reliability analysis, the systemreliability R and probability of failure pf are de�nedas [4-6]:

R = P [G(X) � 0] =Zg(X)�0

:::ZfX(X)dX; (2)

pf = P [G(X) < 0] =Zg(X)<0

:::ZfX(X)dX; (3)

where P [:] denotes a probability indicator, G(:) isa limit-state function which de�nes the performanceevent for which the probability is being assessed, andX = (X1; X2; :::; Xn) is the vector of random variables.fX(X) denotes the joint probability density function(pdf) of X. Reliability is graphically interpreted asthe volume underneath the surface (hyper-surface forhigher than 2-D problems) of fX(X) on the side ofsafe region [6] (Figure 1). The direct evaluation of theprobability integration is extremely di�cult. The mainreasons given in di�erent reliability texts are [3-5]:

M. Kia and M. Banazadeh/Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 933{941 935

Figure 1. Graphical interpretation of the probabilityintegration [6].

� fX(X) is practically impossible to obtain; evalu-ating the multiple integral is di�cult even if thisinformation is available.

� In many engineering applications, integrationboundary (i.e., G(X) = 0) is not de�ned explicitly.Even if this function is available, it is usually anonlinear and complicated function of X.

According to the above reasons, numerical solu-tions are the only choice available. Several reliabilitymethods are introduced to estimate the probabilityintegration numerically. Gradient-based methods, i.e.FORM and Second-Order Reliability Method (SORM)along with more sophisticated Monte Carlo Simulation(MCS), are such techniques broadly applied to solvethe probability integration in Eq. (3). MCS generatesrandom outcomes of the random Variables and eval-uates G(X) for each realization. Then, it counts thenumber of times a failure occurred and estimates theprobability of failure by [3,5,9]:

pf =Number of failure

Total number of sampling: (4)

This method is extremely robust. It can solve probabil-ity integration in Eq. (3) with any type of limit-statefunctions, even those with discontinuities. However,the main disadvantage of MCS is that it requires alarge number of evaluations of the limit-state functionto obtain the failure probability. Hence, the MCSis time-consuming, especially for random event withsmall probability of failure. This de�ciency reducesthe use of MCS in structural reliability analysis. Incontrast, gradient-based methods are highly e�cientdue to their search for only the point with the highestcontribution to the probability integration; no time is

Figure 2. Comparison of the �rst order and secondreliability methods [6].

wasted in calculations and they do not signi�cantlycontribute to the probability of failure. This charac-teristic of the gradient-based methods making themtailored to compute probability of rare event would begreatly interesting for practical purposes. Since theapproximation of the limit-state function in SORMis better than that in FORM (Figure 2), SORM isgenerally more accurate than FORM, especially forhighly nonlinear limit-state function. However, SORMis not as e�cient as FORM due to computing thesecond-order derivatives. Therefore, a great numberof solutions are proposed to improve e�ciency of theSORM using various quadratic approximations, oneof which is the Breitung's improved formula usingthe theory of asymptotic approximations [14]. Thissimple closed-form solution, applied to the presentstudy as the second-order analysis, modi�es failureprobability of the FORM using the �rst principlecurvature. Generally, gradient-based methods takethe three following steps to estimate probability offailure [3,6,12].

1. The vector of random variables, X = (X1; X2; :::;Xn), in their original random space is trans-ferred into independent standard normal vector,U = (U1; U2; :::; Un), in standard normal space (U -space). After transformation, limit-state functionG(X) is changed to ~G(U) in U -space. Note thatmathematical formulation of limit-state functionafter transformation is di�erent from that in theoriginal space.

2. Next, a point located on ~G(U) = 0 and has thehighest contribution to the probability integrationmust be sought. In reliability text, this point isnamed Maximum Probable Point (MPP), or designpoint, and computed by an iterative optimizationprocess as:


Figure 3. Probability integration in the U -space [6].(Max ~f(U) = �n

i=11p2�

Exp��u2

i2

�Subjected to ~GL(U) = 0

(5)

where ~f(U) demonstrates the mathematical de-scription of the integrand in U -space. Since�ni=1

1p2�

Exp��u2

i2

�= 1p

2�Exp

��Pni=1 u

2i

2

�, max-

imizing �ni=1

1p2�

Exp��u2

i2

�is equivalent to min-

imizingPni=1 u

2i . Accordingly, MPP is the point

on the integral boundary, i.e. ~G(U) = 0, whichhas the shortest distance to the origin in U -space(Figure 3). In technical texts, this minimumdistance is called reliability index and is shown by�.

3. Finally, based on the obtained reliability index, theprobability of failure is estimated by the followingequations:

pf =

8>>><>>>:�(��)! If FORM selected�(��)�n�1

i=1 (1 + ��i)12 ! If SORM

(Breitung's improved formulation)selected

(6)

where �i denotes the principal curvatures of thelimit-state function, i.e. ~G(U), at the minimumdistance point.

3. Limit-state function

According to the previous section, limit-state functionde�ning desired event is the core of the reliabilitymethods. A limit-state function consists of two mainparts: threshold and probabilistic models. Threshold

is the value assigned by analyst, but probabilisticmodel is an equation (or algorithm) that simulates allpossible scenarios of physical phenomena. In addition,probabilistic models implemented in a reliability anal-ysis should have other characteristics that are fullydescribed in [7,9,13]. A few of these characteristicsinclude: accounting for both aleatory and epistemicuncertainties in terms of random variables and beingcontinuously di�erentiable. Since the foundation ofall reliability algorithms are based on several trialrealizations of random variables, probabilistic modelsmust be able to yield deterministic output(s) for a givenrealization of input variables. The exact number ofrealizations depends on the e�ciency of the algorithmemployed to solve a reliability problem.

Depending on the objective performance, thenumber of probabilistic models required to de�nedesired limit-state function would be di�erent. Forexample, in a seismic risk analysis which computesexceedance probability of monetary loss, three intensi-ties, demand, and consequence models are required todevelop limit-state function. But, for seismic hazardanalysis which is the subject of this paper, only proba-bilistic intensity model would be adequate. Therefore,following limit-state function is extended in this paperto address the problem of calculating Eq. (1):

G() = pga� PGA(�;M;R); (7)

where pga indicates peak ground acceleration thresh-old, pga(�;M;R) represents an intensity model pre-dicting peak ground acceleration at a designated site.In this model, �, M , and R indicate model param-eters, earthquake magnitude, and wave propagatione�ects, respectively. Fortunately, most of attenuationrelations proposed in the literature comply with thementioned characteristics of a probabilistic model andcan be directly amenable to a reliability analysis.One example is the following ground-shaking intensitymodel predicting peak ground acceleration at a site [15]and implemented in this paper to describe reliability-based PSHA. It is noteworthy that analyst wouldbe autonomous in the use of his developed or otherprobabilistic intensity model:

Log(PGA) =aMw � bp(R2 +H2)

� d:Log �pR2+H2�

+cisi+�:": (8)

In the above equation, a; b; ci; and si are the parametersof the intensity model; Mw; R; and H represent earth-quake magnitude, focal depth, and epicentral distance,respectively; " is the standard normal random variablerepresenting the model error; and � is the standarddeviation of the model error. Furthermore, the modelyields a unique deterministic value of PGA for a given


realization of input variables and is continuously di�er-entiable. Hence, Eq. (8) can be directly implementedin reliability analysis. It is worthwhile to mention thatthe above model is not capable of addressing epistemicuncertainty due to use of constant parameters. In otherwords, variability in the model parameters, i.e. a; b; ci;and si, is ignored when the model is developed. Byutilizing the above relationship, the limit-state functionG() is rewritten to address the problem of calculatingthe probability that PGA at a site exceeds a speci�cthreshold value:

G() =

pga�10haMw�b

p(R2+H2)�d:Logp(R2+H2)+cisi+�:"

i: (9)

4. Case study example

In this section, both traditional and reliability-basedmethodologies are implemented to develop peak groundacceleration hazard curve at a site of a historical bridge.This bridge, dating back to about 250 years, is locatedat 52:665� east and 36:525� north. Seismicity catalogof the site was �rst developed based on all historicaland systematic earthquakes occurred at a distance upto 150 km from the site. For the sake of simplicity,three circular area sources with di�erent radii werede�ned. Only major excitations were considered,and near-source e�ects are limited by de�ning lowerbound radius being equal to 10 km (Figure 4). Next,both magnitude and location probability functions aredeveloped for each seismic zone. These probabilitydensity functions are then utilized in the intensitymodel proposed in the previous section to formulatelimit-state function, Eq. (9), for each seismic zone. Af-ter extending limit-state function, reliability methodsare employed to calculate the probability that peakground acceleration exceeds certain ground motionlevels.

Figure 4. Concentric circular seismic sources.

4.1. Magnitude modelThe relative frequency of various-sized earthquakeshas been usually speci�ed by a truncated exponentialdistribution function developed based on Gutenbergand Richter's recurrence law:

Ln(N(m)) = a� b:m! N(m) = ea:e�b:m; (10)

where N(m) indicates the number of earthquake withmagnitude equal to or greater than \m"; \a" and\b" are the recurrence model parameters. Accord-ing to Eq. (10), cumulative distribution function ofearthquake magnitude and consequently its probabilitydistribution function can be computed as:

Fm(m) = P (M � mjmmin < M < mmax)

=(n(mmin)� n(m))n(mmin)� n(mmax)

=1� e�b(m�mmin)

1� e�b(mmax�mmin) ; (11)

fm(m) =@Fm(m)@m

=b:e�b(m�mmin)

1� e�b(mmax�mmin)

mmin < m < mmax; (12)

where mmin indicates lower bound value of earthquakemagnitude selected based on engineering judgmentand seismicity catalog of the site; mmax is the upperbound value of magnitude used to demonstrate thatmaximum magnitude may be generated by seismicsource. Magnitude model parameters, i.e. mmax; b; andmmin, for each of the above three seismic zones arepresented in Table 1. Except for the latter, the otherparameters are computed by a sophisticated procedurewhich considers uncertainty in the input magnitudedata [16,17].

4.2. Location modelTo predict ground motion intensity at a site location, itis also necessary to describe probability distribution ofsite to source distance as a more common parameterapplied to re ect wave propagation e�ects. Sincerupture can be extended over ten kilometers, variousdistance measures have been developed to estimateground motion parameters at a site (Figure 5). InEq. (9), the epicentral distance R associated with focaldepth H is used to re ect wave propagation e�ects.Probabilistic description of epicentral distance can beeasily computed by assuming that earthquakes occurwith equal probability anywhere at a seismic zone.Considering the above assumption, the probability oflocating an epicenter at a distance of less than \r" andgreater than rmin can be calculated by:


Table 1. Parameters of magnitude model for each seismic zone.

Seismic-zone Model parameter Median Standard deviation fm(m) with medianvalues of mmax and b

Zone 1b 0.63 0.11

0:6844:e�0:63(m�3)mmax 7.02 0.16mmin 3 |

Zone 2b 1.16 0.07

1:167:e�1:16(m�3)mmax 7.37 0.12mmin 3 |

Zone 3b 1.13 0.12

1:173:e�1:13(m�3)mmax 5.92 0.28mmin 3 |

Figure 5. Graphical interpretation of the epicentral distance.

FR(r) = P (rmin < R < r)

=

8><>:0 r < rmin�:r2��r2

min�r2

max��r2min

rmin � r � rmax

1 r � rmax

(13)

According to Eq. (13), the probabilistic description ofepicentral distance can be computed by:

fR(r) =@FR(r)@r

=

(2�r

�r2max��r2

minrmin � r � rmax

0 Otherwise (14)

Based on the above equation, epicentral probabilityfunction of the seismic zones is developed as follows:

f1(r) =r

175010 � r < 60;

f2(r) =r

540060 � r < 120;

f3(r) =r

4050120 � r < 150: (15)

It is strongly emphasized that the above description isnot only limited to circular area source, but also canbe extended to line source or area source with a moregeneral shape.

According to Eq. (9), to complete location model,focal depth also requires to be speci�ed. Statisticalanalysis of focal depth values presented in publishedglobal earthquake catalogues shows that lognormaldistribution is a suitable distribution function to de-�ne focal depth variability. However, focal depthspublished in seismic catalogues are inaccurate by upto 60 km for events in Iran and the surrounding re-gions [18]. Therefore, it is recommended that only focaldepth values determined using accurate methods, suchas local network recordings or teleseismic wave forminversion, be used instead of seismic catalogues valuesin a PSHA [18]. Therefore, based on a study conductedby Magge et al. [19], focal depth value, in this paper,is considered constant and set equal to 12 km.

4.3. Comparison of analysis optionsIn this section, seismic hazard curve is derived us-ing di�erent reliability methods, and the results are


then compared with those obtained from conventionalmethod. The objective followed in this comparison isto �nd the appropriate reliability model for computingPSHA with respect to e�ciency and accuracy criteria.To this end, for a given threshold, the exceedanceprobability is �rst calculated for each zone using dif-ferent reliability methods. Next, the probability of ex-ceedance is converted to the annual rate of exceedanceby a simple modi�cation. To this end, it is enough tomultiply the probability by annual rate of occurrenceof earthquake at the site. Finally, hazard curve isdeveloped using direct summation of the annual rate ofexceedance over all seismic zones. The reliability of theobtained results is investigated by its comparison withthe results of conventional approach. To demonstratedi�erences between methods, vertical axes in Figure 6are plotted in logarithmic space. According to Figure 6,FORM, especially at the tail of the curves, tends tooverestimate the probability of failure compared withconventional and other reliability methods. This isquite reasonable and mainly due to the linearizationof the limit-state function in the FORM. To examinethe accuracy of the results better, PGA is computedand compared with those obtained from conventionalmethod in terms of percentage errors by applyingPoisson occurrence model and reliability-based hazardcurves, according to di�erent occurrence probabilitiesin 50 years (Table 2). According to the results ofTable 2, it is concluded that MCS is the most accuratemethod. Figure 6 graphically con�rms this conclusion.However, it is too time-consuming to be used forpractical purposes. To appreciate this fact, the numberof simulations and CPU time required to computethe probability of exceeding a speci�c threshold valueis shown in Table 3 for di�erent reliability methods.According to the results of Table 3, the computa-tional cost of MCS increases dramatically when theprobability of rare event is sought. For example, toachieve 2% coe�cient of variation of the probabilityresults, MCS requires 750616 realizations for thresholdequal to \0.75g" at the �rst seismic zone. The CPUtime corresponding to these numbers of simulations is5665 seconds, namely 1.57 hours. In return, the runtime required to compute the probability of exceeding\1.1g" is equal to 32849.02 seconds which is more than6 times of the run time required for the threshold equals\0.75g". It is noteworthy that the aforementionedcomputational cost is relevant to only one thresholdvalue. By considering computational e�orts of di�erentthreshold values at all seismic zones, it is concludedthat MCS is too time-consuming to be applied forpractical purposes. In contrast, to compute the prob-ability of exceeding \0.75g", gradient-based methods,i.e. FORM and SORM (Breitung's improved formula),only 11 realizations are required which are equivalentto 0.102 and 0.149 seconds CPU time for FORM and

Figure 6. Comparison of the results obtained fromdi�erent methods.

SORM, respectively. This brief comparison demon-strates how much faster the gradient-based methodsare than MCS, which would be greatly interesting forpractical purposes. However, according to Table 2,SORM provides more accurate results than FORM,especially at low probabilities. This is quite reasonableand mainly due to considering limit-state curvaturewhen exceedance probability is sought. Therefore, thesecond-order reliability analysis based on Breitung's


Table 2. PGA according to di�erent occurrence probabilities in 50 years.

Occurrence probabilityduring 50 year

PGAconventional method PGAFORM Errora

90 0.0572 0.0549 -4.021%75 0.0859 0.0852 -0.815%50 0.1285 0.1325 3.113%25 0.2161 0.2305 6.664%10 0.3668 0.4137 12.786%2 0.7416 0.8967 20.914%


PGAconventional method PGABreitung's formula Errorb

90 0.0572 0.0569 -0.527%75 0.0859 0.0858 -0.117%50 0.1285 0.1291 0.465%25 0.2161 0.2194 1.504%10 0.3668 0.3735 1.794%2 0.7416 0.781 5.045%


PGAconventional method PGAMCS Errorc

90 0.0572 0.05723 0.052%75 0.0859 0.0861 0.233%50 0.1285 0.1288 0.233%25 0.2161 0.2177 0.740%10 0.3668 0.3699 0.845%2 0.7416 0.7516 1.33%

a Error = PGAFORM�PGAconventional methodPGAconventional method

� 100; bError = PGABreitung's formula�PGA conventional methodPGAconventional method

� 100;

cError = PGAMCS�PGAconventional methodPGAconventional method

� 100.

Table 3. A comparison of computational e�orts of the di�erent reliability methods for computing probability of exceeding.

Threshold MCS FORM/SORMNumber ofevaluations

CPU time(Second)

Number ofevaluations

CPU time(Second)

Zone 1

0.25 71363 62.918 5 0.077 / 0.10.5 286548 783.961 8 0.086 / 0.1160.75 750616 5664.16 11 0.096 / 0.1121.1 2080833 32849.02 14 0.119 / 0.16

improved formula would be the authors' suggestedreliability method for computing PSHA.

5. Conclusion

In this paper, an alternative approach based on theapplication of Reliability Methods is put forward to cal-culate Probabilistic Seismic Hazard Analysis (PSHA).The pioneering work (i.e., Eq. (1)) in PSHA is consid-ered as a reference to exhibit e�ciency and accuracy ofthe reliability methods. In both approaches, the results

are peak ground acceleration hazard curves for the siteat which a historical bridge in Iran is located. It isconcluded that the subsequent use of the SORM (Bre-itung's improved formula) provides accurate resultswith respect to conventional method with a fractionof the computational e�orts involved in MCS. That is,SORM can be adopted as an alternative approach tocalculate PSHA because of good balance between accu-racy and e�ciency. The ability to explicitly account forepistemic uncertainties associated with the capabilityof e�ortlessly addressing correlation between random


variables is another advantage, making the reliability-based approach powerful for practical purposes com-pared to conventional method. These advantagesare particularly signi�cant when attenuation relationswith random model parameters are implemented incalculating PSHA.

References

1. Cornell, C.A. \Engineering seismic risk analysis", Bul-letin of the Seismological Society of America, 58(5),pp. 1583-1606 (1968).

2. Wen, Y. \Probabilistic aspects of earthquake engi-neering", Earthquake Engineering from EngineeringSeismology to Performance-Based Engineering, Ed. byBozorgnia, Y. and V.V. Bertero, CRS Press LLC(2004).

3. Ditlevsen, O. and Madsen, H.O., Structural ReliabilityMethods, Wiley New York (1996).

4. Melchers, R.E., Structural Reliability Analysis andPrediction, John Wiley & Sin Ltd (1999).

5. Haldar, A. and Mahadevan, S., Probability, Reliability,and Statistical Methods in Engineering Design, JohnWiley & Sons (2000).

6. Du, X. \ME/AE probabilistic engineering design",Class notes, Engineering Uncertainty Repository,http://web.mst.edu/ due/repository, 2016, unpub-lished.

7. Haukaas, T. \Uni�ed reliability and design optimiza-tion for earthquake engineering", Probabilistic Engi-neering Mechanics, 23(4), pp. 471-481 (2008).

8. Du, X. \Uni�ed uncertainty analysis by the �rst orderreliability method", Journal of Mechanical Design,130(9), pp. 091401-10 (2008).

9. Bohl, A. \Comparison of performance based engi-neering approaches", A Thesis Submitted in PartialFul�llment of the Requirements for the Degree ofMaster of Applied Science, the University of BritishColumbia (2009).

10. Kiureghian, A.D. and Ditlevsen, O. \Aleatory orepistemic? Does it matter?", Structural Safety, 31(2),pp. 105-112 (2009).

11. Koduru, S. and Haukaas, T. \Probabilistic seismic lossassessment of a Vancouver high-rise building", Journalof Structural Engineering, 136(3), pp. 235-245 (2009).

12. Xiao, N.-C., Huang, H.-Z., Wang, Z. Pang, Y. andHe, L. \Reliability sensitivity analysis for structuralsystems in interval probability form", Structural andMultidisciplinary Optimization, 44(5), pp. 691-705(2011).

13. Mahsuli, M. and Haukaas, T. \Seismic risk analysiswith reliability methods. Part II: Analysis", StructuralSafety, 42, pp. 63-74 (2013).

14. Breitung, K. \Asymptotic approximations for multi-normal integrals", Journal of Engineering Mechanics,110(3), pp. 357-366 (1984).

15. Zare, M., Ghafory-Ashtiany, M. and Bard, P.Y. \At-tenuation law for the strong motions in Iran", InProceedings of the Third International Conference onSeismology and Earthquake Engineering, 1, pp. 345-354 (1999).

16. Kijko, A. and Sellevoll, M.A. \Estimation of earth-quake hazard parameters from incomplete data �les.Part II. Incorporation of magnitude heterogeneity",Bulletin of the Seismological Society of America, 82(1),pp. 120-134 (1992).

17. Kijko, A. \Estimation of the maximum earthquakemagnitude, m max", Pure and Applied Geophysics,161(8), pp. 1655-1681 (2004).

18. Maggi, A., Priestley, K. and Jackson, J. \Focal depthsof moderate and large size earthquakes in Iran",Journal of Seismology and Earthquake Engineering,4(2-3), pp. 1-10 (2002).

19. Maggi, A., Jackson, J., Priestley, K. and Baker,C. \A re-assessment of focal depth distributions insouthern Iran, the Tien Shan and northern India: Doearthquakes really occur in the continental mantle?",Geophysical Journal International, 143(3), pp. 629-661 (2000).

Biographies

Mehdi Kia started his university studies in BabolNoshirvani University of Technology, Babol, Iran, inthe �eld of Civil Engineering, in September 2004.Having �nished his undergraduate study in 2008, hecontinued his studies as a graduate student in SharifUniversity of Technology in Structural Engineering.He successfully defended his thesis titled \ModelingLocal Joint Flexibility using a new element addedto the library of OpenSees software" to obtain hisMS degree in December 2010. He was acceptedas a PhD student in the University of Amir-Kabir,Iran, in 2012, where he worked on reliability analysisand its application on seismic risk evaluations. Heis currently a Structural Engineer at Pars Gas OilCompany (P.O.G.C).

Mehdi Banazadeh obtained his BS (Civil Engineer-ing) and MS (Earthquake Engineering) degrees fromthe University of Tehran with honor. He then pursuedhis PhD degree in the University of the Ryukyus inJapan and accomplished it in 2004. He has manyyears of experience in the next generation performance-based engineering and reliability analysis. He hasbeen a faculty member in Amir-Kabir University ofTechnology in Tehran, Iran. He has authored andcoauthored over 30 technical papers. His research areais mainly focused on seismic risk evaluation using re-liability framework and nonlinear structural modelingand analysis.

probabilistic seismic hazard analysis using reliability...

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