construction and application of bayesian probabilistic network for earthquake management

DISS. ETH NO. 18781

CONSTRUCTION AND APPLICATION OF BAYESIANPROBABILISTIC NETWORKS FOR EARTHQUAKE RISK

MANAGEMENT

A dissertation submitted to

ETH ZURICH

for the degree of

Doctor of Sciences

presented by

YAHYA YILMAZ BAYRAKTARLI

Dipl.-Ing., University of Karlsruhe (TH)

born 03.02.1972

citizen of Germany

accepted on the recommendation of

Professor Michael H. Faber, examinerProfessor Ton Vrouwenvelder, co-examiner

Martin Bertogg, co-examiner

2009

To Esra, Selma and Mihri

iii

Acknowledgements

I appreciate the support provided by the Swiss National Science Foundation (SNF) as part of theinterdisciplinary research project "Management of Earthquake Risks using Condition Indicators"(MERCI).

I would like to thank Professor Faber for his valuable supervision. He gave me the freedomto search my way and guidance to find it. I would also like to thank Professor Vrouwenvelderand Mr. Bertogg for acting as co-referees. The many discussions with Ufuk Yazgan, OliverKübler, Jens Ulfkjaer, Jack Baker, Kazuyoshi Nishijima and Matthias Schubert contributed tothis work, for which I am very thankful. I would also like to thank the MERCI research groupfor the many fruitful discussions. Many thanks to my friends and colleagues at the Institute ofStructural Engineering for the pleasant atmosphere, the shared activities and the Friday gamesof soccer.

My deepest gratitude goes to my wife Esra and my daughters Selma and Mihri especially forthe many working weekends towards the end of this dissertation.

Zurich, December 2009 Yahya Y. Bayraktarli

v

Abstract

Developments in earthquake engineering research during the last decades provide within ac-ceptable limits reliable design of new structures against earthquakes. The existing buildingstock however, still represents a significant risk in regard to the safety of people as well as to theeconomical assets of society.

Assessing the earthquake risk for buildings necessitates the consideration of uncertainties aris-ing from seismic hazard, site effects, structural response, and immediate/indirect consequences.Furthermore, uncertainties emerge when the earthquake risk is analyzed for portfolios of build-ings, e.g. cities. A city is more than a conglomeration of individual buildings, since interdepen-dencies exists between its various elements. These "system effects" are investigated in order toset a framework for capturing the complex consequences after natural catastrophes.

First, a system theoretic definition for a city considering functional and hierarchic dependen-cies between its elements is given. Afterwards, a framework for risk assessment is proposed.Both perspectives are jointly considered by the application of Bayesian probabilistic networks(BPN). The elements within a BPN comprise the set of parameters considered within the riskanalysis problem. The joint probability distribution of these parameters would be of highestpossible value. In very rare cases it is possible to set the joint probability distribution. BPNsconstitute a very efficient way of representing the joint probability distribution by exploitingconditional dependencies.

BPNs are constructed for modules of earthquake risk analysis; seismic hazard, structural dam-age, soil response and consequence assessment. The application of these BPN models is illus-trated by four examples considering a portfolio of 5-story buildings in a city in Turkey. The firstexample considers the decision problem of whether or not to retrofit a class of structures in acity. The second example illustrates that the framework also facilitates portfolio loss estimation.The proposed framework allows for a consistent representation of the effect of dependencies(e.g. common events or common models) in the estimation of losses for important buildings(e.g. hospitals). It is shown that inclusion of such effects may have a significant impact on port-folio loss estimation. The third example illustrates the application of BPNs for updating seismicfragility curves based on data from post-earthquake building inspections. The forth examplediscusses the assessment of consequences using a newly developed concept of robustness.

The proposed framework has several merits. By using the proposed framework consistent rep-resentation of uncertainties and the consideration of crucial effects of dependencies are possible.Furthermore, risk updating based on new information is facilitated.

vii

Kurzfassung

Neue Erkenntnisse in der Erdbebeningenieurforschung ermöglichen eine zuverlässige Bemes-sung neuer Bauwerke. Die vorhandene Bausubstanz jedoch stellt weiterhin ein signifikantesRisiko sowohl für die Sicherheit der Menschen als auch für die wirtschaftliche Wertschöpfungder Gesellschaft dar.

Die Erdbebenrisikoanalyse von Bauwerken erfordert die Berücksichtigung der Unsicherheitenin der Erdbebengefährdung, in den Standorteffekten, in den Bauwerksantworten sowie in dendirekten und in den indirekten Konsequenzen. Darüber hinaus entstehen Unsicherheiten, wenndas Erdbebenrisiko für ein Portfolio von Bauwerken analysiert wird. Eine Stadt ist mehr alseine Ansammlung individueller Bauwerke, da gegenseitige Abhängigkeiten zwischen ihren El-ementen existieren. Diese "Systemeffekte" werden ebenfalls untersucht, um einen theoretischenRahmen zur Erfassung der komplexen Konsequenzen im Falle von Naturgefahren aufzustellen.

Zu Beginn wird eine systemtheoretische Definition einer Stadt unter Berücksichtigung funk-tionaler und hierarchischer Abhängigkeiten gegeben. Daran anschiessend wird ein theoretischerRahmen für die Risikoanalyse eingeführt. Durch die Anwendung von Bayes’schen Netzen wer-den beide Perspektiven berücksichtigt. Die Elemente eines Bayes’schen Netzes stellen die ex-plizit berücksichtigten Variablen innerhalb eines Risikoanalyseproblems dar. Die gemeinsameWahrscheinlichkeitsverteilung dieser Variablen kann effizient durch Bayes’sche Netze ausgew-ertet werden. Für den Analysten ist die gemeinsame Wahrscheinlichkeitsverteilung von grossemNutzen und kann analytisch nur in den seltensten Fällen ermittelt werden.

Bayes’sche Netze werden für die Module der seismischen Gefährdung, der Bodenantwort, derBauwerksschäden und der Konsequenzen einer Erdbebenrisikoanalyse entwickelt. Die Anwen-dung dieser Module wird mit vier Beispielen anhand eines Portfolios von Stahlbetonbauwerkenin einer Stadt in der Türkei illustriert. Das erste Beispiel behandelt ein Entscheidungsproblemzur Ertüchtigung einer Bauwerksklasse gegen Erdbeben. Das zweite Beispiel illustriert, wie mitBayes’schen Netzen die Verlustüberschreitungskurve ("loss exceedance curve") eines Portfoliosberechnet wird. Es wird gezeigt, dass die Berücksichtigung der Abhängigkeiten (z.B. in denEreignissen oder in den Modellen) einen signifikanten Einfluss auf das Ergebnis hat. Im drittenBeispiel wird gezeigt, wie mit Bayes’schen Netzen eine Aktualisierung der seismischen Ver-letzbarkeitskurven ("seismic fragility curve"), basierend auf Schadensinspektionen nach einemErdbeben, durchgeführt wird. Das vierte Beispiel diskutiert die Anwendung eines neu entwick-elten Robustheitskonzeptes auf die Ermittlung der Konsequenzen.

Der hier vorgestellte theoretische Rahmen hat einige Vorteile. Mit der vorgeschlagenen Meth-ode ist eine konsistente Berücksichtigung der Unsicherheiten und der Abhängigkeiten möglich.Ein weiterer Vorteil besteht darin, dass einzelne Risiken mit eingehenden Informationen system-atisch aktualisiert werden können.

ix

Contents

1 Introduction 11.1 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Scope and outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . 4

2 Fundamentals 52.1 City as a system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Decision theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Uncertainty, probability, utility and risk . . . . . . . . . . . . . . . . . . . . . 11

3 Methodologies 153.1 Existing earthquake loss estimation methodologies . . . . . . . . . . . . . . . 153.2 Proposed framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Models 354.1 Seismic hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Soil failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Structural damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.5 Verification and validation of the models . . . . . . . . . . . . . . . . . . . . . 73

5 Examples 775.1 Example 1: Decision for retrofitting structures . . . . . . . . . . . . . . . . . . 835.2 Example 2: Assessment of seismic risk . . . . . . . . . . . . . . . . . . . . . . 915.3 Example 3: Update of fragility curves . . . . . . . . . . . . . . . . . . . . . . 1035.4 Example 4: Index of robustness . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Conclusions 111

References 117

A BPN Algorithms 129

B Soil Parameters 139

C Software Codes 147

xi

1 Introduction

1.1 Problem definition

Efficient management and consistent quantification of natural and man-made risks is increas-ingly becoming an issue of societal concern. Sustainable and consistent societal decision makingrequires that a framework for risk management is developed, which, at a fundamental level al-lows for the comparison of risks from different natural hazards such as the comparison betweenrisks due to earthquakes with the risks due to flooding or due to draughts.

The recent series of earthquakes, in Turkey (1999), Taiwan (1999), India (2001) and Sumatra(2006) not only led to a greater awareness of seismic hazards, but also highlighted the difficultiesinvolved in an efficient decision making process in such situations, especially in less developedcountries. Consistent and quantitative risk assessment tools for buildings and infrastructure inseismically active areas are urgently needed to ensure an efficient decision making process thatfacilitates the optimal allocation of available economical resources for the management of risksbefore and after an earthquake.

The developments in earthquake engineering research during recent decades provide withinacceptable limits reliable design of new structures against earthquakes. In most countries suchmethods are implemented in the best practice design of new structures. The existing buildingstock however, still represents a significant risk in regard to the safety of people as well as to theeconomical assets of society.

Assessing the earthquake risk for existing buildings necessitates the consideration of uncer-tainties from the earthquake source mechanism, the site effects, the structural response, andthe immediate consequences to the indirect consequences. Furthermore, additional uncertaintiesemerge when the earthquake risk is analyzed for groups of buildings and infrastructure elements,e.g. cities or portfolios of buildings. A city is more than a conglomeration of individual build-ings, since interdependencies between its elements exist. These "system effects" need to beinvestigated in order to set a framework for capturing the complex consequences after devastat-ing events such as natural catastrophes.

Significant efforts have been devoted in the past to assessing the seismic risk related to ex-isting structures subject to earthquake hazards. One of the first major projects concerning theassessment of the seismic risk is the ATC-13 (1985) project. The ATC-13 report provides a set ofvulnerability functions in the form of damage probability matrices to be used in the assessmentof the seismic vulnerability of a stock of structures located within the same region (Whitmanet al., 1973).

In the 1990’s, the period designated as the International Decade for Natural Disaster Reduc-tion (IDNDR) by the UN (1987), a number of seismic risk assessment studies were carried out

1

1 Introduction

around the world. One of the major projects within this period is the RADIUS (Risk Assess-ment Tools for Diagnosis of Urban Areas against Seismic Disasters) project GHI (2004). In thisproject, practical tools for a preliminary estimation of the possible damage scenarios and for thepreparation of a risk management plan for nine case study cities are developed.

Another important earthquake loss estimation methodology is developed by the Federal Emer-gency Management Agency through the National Institute of Building Sciences, the HAZUSmethodology (Whitman et al., 1997). Details of the HAZUS methodology are summarized inKircher et al. (1997b) and Kircher et al. (1997a). The HAZUS methodology is implemented inthe software package HAZUS99-SR2. HAZUS originally contained very general methods forestimating earthquake losses on a regional scale. With the addition of the Advanced Engineer-ing Building Module to HAZUS99-SR2, building-specific damage estimation studies are madepossible (HAZUS, 2001). An overview of other existing loss estimation methodologies is givenin Section 3.1.

The proposed framework in this dissertation together with Bayesian probabilistic networks(BPNs) for earthquake risk management has, unlike the aforementioned methodologies, the ad-vantage of forming a basis for consistently integrating all aspects affecting the damage on astock of structures located within a region subjected to the same earthquake exposure. The un-certainties which influence the functional chain of an earthquake from the source mechanism, thesite effects, the structural response, the immediate consequences (damage) to the indirect con-sequences can be handled consistently and with new information a consistent actualization orupdating of the results can be performed. The latter facilitates the extension of the methodologyto decision problems related to risk management during and after earthquakes.

BPNs are constructed for modules of earthquake risk analysis; seismic hazard, structural dam-age, soil response and consequence assessment. The application of these BPN models is illus-trated by four examples considering a portfolio of reinforced concrete structures in a city locatedclose to the western part of the North Anatolian Fault in Turkey. The first example considers thedecision problem of whether or not to retrofit a specific class of structures in a city. The sec-ond example illustrates that the framework also facilitates the assessment of the portfolio lossexceedance probability distribution function. The proposed methodology allows for a consistentrepresentation of the effect of dependencies in the estimation of losses for important buildings(e.g. hospitals). It is shown that the inclusion of such effects may have a very significant impacton portfolio loss estimates. Based on data of post-earthquake building inspections in Adapazariafter the 17th August, 1999 Kocaeli Mw7.4 earthquake the application of BPNs for updatingone of the main models, e.g. building fragility curves, is illustrated in the third example. Thefourth example discusses the assessment of consequences using a newly developed concept ofrobustness.

The proposed framework has several merits compared to traditional schemes for risk assess-ment in the context of large scale natural hazards. Traditional schemes generally assess risksfor individual hazards on an object by object basis. By using the proposed framework consistentrepresentation of uncertainties and the crucial effects of dependencies are possible. Furthermore,risk updating through new information is facilitated.

2

1.2 Objectives

1.2 Objectives

The dissertation aims to develop a indicator based generic risk assessment framework for theconsistent quantitative and rational management of earthquake risks. The proposed frameworkis designed for decision-makers responsible for the safety of personnel, the environment andassets of a larger area such as e.g. a city. The framework is generic in the sense that it isformulated in terms of observable characteristic descriptors (indicators). It can thus easily beadapted to the characteristics of a specific region or city. The main emphasis is on the risks dueto potential failures and collapse of buildings as well as infrastructure systems such as hospitals.An important feature of the proposed framework is that it provides decision support on how tooptimize investments into risk reducing measures before and after an earthquake.

A key element in the proposed framework is the quantification of the effect of various types ofindicators on the risks. These indicators may have very different characteristics and necessitatethat different types of expertise are integrated. An example of such an indicator is the informa-tion about the structural design codes applied for the design of a group of structures. Anotherindicator could be the characteristics of the soil implicitly describing earthquake related fail-ures, e.g. liquefaction. In addition, more traditional indicators of damage in structures, such asinterstory drift ratios, are utilized.

The decision-making process with regard to the efficient and targeted allocation of resourcesfor the purpose of reducing and/or mitigating earthquake risks for cities or regions is complexdue to the broad representation of structures, possible damage states and the numerous uncer-tainties prevailing within the problem area. This dissertation attempts to establish a frameworkfor risk assessment in such situations. Risks are thereby quantified, and efficient risk-reducingor mitigating activities are identified in two distinct situations; before and after an earthquake.The proposed framework is generic and facilitates the assessment of risks at different levels ofaccuracy as appropriate under the particular circumstances. A further benefit is achieved by thefact that the proposed framework is adaptable to other types of risk assessment and manage-ment problems which allow for integral risk analysis considering all relevant risks in a givengeographical region.

The main objectives of the dissertation includes:

• Construction and application of BPN models in individual disciplines

– BPNs for seismic hazard assessment.– BPNs for spatial modeling of soil failures.– BPNs for structural response and damage assessment.– BPNs for consequence assessment.

• Systematic application of BPNs to cities

– Establishment of a risk assessment framework and a loss estimation methodology.– Construction of a GIS-based tool for the integration of individual BPN models.– Discussion of "system effects" when applying the methodology for cities.

3

1 Introduction

1.3 Scope and outline of the dissertation

Scope

The dissertation proposes a framework for risk assessment based on a system theoretic defin-ition for a city considering functional and hierarchic dependencies between its elements. Theapplication of Bayesian probabilistic networks (BPN) within this framework is discussed. Theelements within a BPN comprise the set of parameters considered within the risk assessmentand management problem. The joint probability distribution of these parameters would be ofthe highest possible value for the analyst. In very rare cases it is possible to set the joint prob-ability distribution. BPNs constitute a very efficient way of representing the joint probabilitydistribution by taking into account conditional dependencies.

Earthquake risk assessment and management requires the integration of several related dis-ciplines. The dissertation is hence written in the framework of the multidisciplinary projectManagement of Earthquake Risks using Condition Indicators (MERCI, www.merci.ethz.ch),which comprised of research groups from five institutes in two departments at ETH Zurich.Furthermore, the application to building portfolios or cities impose additional difficulties. Thedissertation focuses on the "system effects". It does not aim to provide novel methods in definingseismic hazard, treating soil behavior or assessing structural damage. At these process orientedlevels state-of-the-art methods are applied in this dissertation. The applied state-of-the-art meth-ods are introduced with the main focus on their "translation" into BPNs and on the therefornecessary additional modeling (see Sections 4.1 to 4.4). The dissertation also does not aim toprovide new algorithms or computer codes for construction and evaluation of BPNs. The per-formance of several existing commercial, free and open-source software packages was also notthe focus. Therefore, one quality assured software package was chosen and the main focus ofthe dissertation is pursued with this software package.

The dissertation should provide the reader the advantages and disadvantages of constructionand application of BPNs in a structured way for earthquake related problems in cities. Decision-makers, risk analysts and risk managers as well as the specialist in the relevant disciplines arethe main readership of the dissertation.

Outline

Chapter 2 gives a thorough definition of cities from a system theoretic perspective, and the basicsof decision theory. Existing methodologies dealing with large-scale earthquake loss estimationare discussed in Chapter 3. The strengths and weaknesses of these existing methodologies arediscussed and the main characteristics required for an efficient integral methodology are derived.Different methods are considered and a framework is proposed.

The application of the proposed framework is prepared in Chapter 4 by adapting state-of-the-art earthquake-related modules to the proposed methodology. Using the four example applica-tions outlined in Chapter 5 the pros and cons are discussed in Chapter 6.

4

2 Fundamentals

2.1 City as a system

According to predictions of demographers the fraction of urban dwellers will rise from the cur-rent level of 50 percent to 80 percent of the world’s total population in 2030. Urbanization is notsimply a process of accumulation of people in towns and cities; it also involves distinctive waysof life and subcultures and patterns of individual and social interaction. Many of the world’seconomical, social and political processes are connected between the world’s towns and cities(Knox and Marston, 2006).

The very first known impulse leading to urbanization was the first agricultural revolution.Cities in Mesopotamia and the Nile valley grew up around 350 B.C. Later, cities appeared in theIndus Valley by 2500 B.C., in Northern China by 1800 B.C., in central and south-western NorthAmerica by 600 B.C. and Andean America around 800 A.D. The original Middle Eastern hearthcontinued successive generations of city-empires, e.g. Athens, Rome and Byzantium. TheseEuropean cities almost collapsed during the early Middle Ages. Interestingly elaborate systemsof cities developed from them into what have today become centers of global economy.

Through colonization the Europeans became the leaders of the rest of the world’s economiesand societies. Colonial city systems were established in Latin America by the Spanish andPortuguese conquerors. The Spaniards founded their cities mostly in strategically importantdefensible sites such as hilltops and channeled their growth in relation to the population theycontrolled. The Portuguese founded their colonies with commercial rather than administrativeconsiderations. They chose coastal sites with proper natural harbors or inland areas along navi-gable rivers. From late Middle Ages onwards a centralization of political power resulted in theformation of nation-states, the beginning of industrialization. Port cities and Atlantic coast citiestook advantage of their location and grew considerably (e.g. London).

The geographical planning of ancient cities were based on a grid system. Key buildingsand neighborhood relations were carefully considered in the planning of settlements. In China,Taoist ideas played a great role in city planning. The major streets and the interior layout ofbuildings were designed to be in perfect harmony with the cosmic energy. In Europe betweenthe 15th and 17th centuries the new wealth and power were reflected in new urban design.Additionally, the advance in military ordnance lead to a surge of planned urban redevelopmentfeaturing impressive fortifications. These new constructions resulted in a need for a new designfor city centers (e.g. Copenhagen, Karlsruhe and Nancy).

The establishment of the important gateway cities also coincided with this period. Thesecities serve as a link between regions and countries (e.g. Boston, Rio de Janeiro and CapeTown). Some of these cities grew rapidly because of their function in colonial expansion. Rio de

5

2 Fundamentals

Janeiro grew due to gold mines, Sao Paulo on the basis of coffee or Accra on the basis of cacao,to mention just a few.

In the 19th century industrialization led to blossoming cities. These industrial economiesneeded a large pool of labor, transportation network, physical infrastructure of factories andconsumer markets. This period is the fastest growing in urbanization. Migrants from rural areaswere attracted by higher wages and greater opportunities in the urban labor market.

Geographers conceptualize urbanization through attributes and functions of city systems. Acity system is an interdependent set of urban settlements within a region. It is possible to talkof a German urban system, a European urban system or a global urban system. Every cityis part of the interlocking urban system connecting local-regional, national and internationalscale human geography in a complex network of socio-economical interdependence. Space isorganized through hierarchies of cities of different size and functions. Functional differencescan be observed within urban system hierarchies. Some cities evolve as general purpose urbancenters providing an evenly balanced range of functions.

Based on the network for goods and services cities may be classified as leading world cities(e.g. New York), world cities (e.g. Zurich), major regional world cities (e.g. Bangkok), cities ofnational influence (e.g. Ankara) and regional cities (e.g. Adapazari/Turkey). This classificationis made considering not only the primary nature of the cities (the internal structure), but alsothe secondary nature, namely the external global network relationships (Taylor, 2004). Theinfluence of this characteristic on the consequences of possible damage, in particular due tonatural hazards, should be considered in risk studies.

The city of Adapazari/Turkey, which is the subject of the illustrative examples in the dis-sertation, may be considered to be a regional city. The consequences of possible earthquakesare assumed to be restricted to the region the city belongs, i.e. the potential consequences ofan earthquake in Adapazari are not assumed to have any significance outside the region. Thecity as a system is considered to be comprised of buildings and infrastructural elements. Thedamage-inducing behavior of these system constituents is assumed to be constructed by the sameprinciples and models for different construction periods. Hence a dependency of the individualelements is always present. On the other hand, the damage to the individual elements will alsohave an influence on consequences for other elements in the system, and this will be more pro-nounced when dealing with lifeline elements. Keeping these in mind we can define a city as asystem comprising elements (i.e. buildings and infrastructure elements) which have "commonmodels" and which lead to so-called indirect consequences, i.e. consequences beyond the simplesum of all elements of the city. The "common models" may be applied to subclasses of buildingsdepending on the system representation. The aggregation of losses may be performed assuminga purely linear consequence model (i.e. for residential buildings the damage to one building hasno influence on the consequence of the damage to another building) or any kind of nonlinearconsequence models (i.e. the collapse of one hospital has a significant influence on the totalconsequence of the damage of another hospital, as the lack of treatment capacity may lead toadditional indirect losses for the considered society).

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2.1 City as a system

Characteristics of a system

Cities can be regarded as living systems. Their elements grow over time in such a complicatedway, that they could not be seen as a conglomerate. The rise and decline of cities are mostlyrelated to war, natural catastrophes, disease and fire. The influence of these devastating effectson cities has been the topic of much research work. Recent catastrophes indicated in almostall cases an underestimation of the consequences. Not only well before the events, but alsoafter the event took place, the consequence estimation fell well short of being realistic. For thisreason, the dissertation adopts a two-way approach: a holistic definition of the subject (here city)considering system theoretical perspective, and an analytical perspective by breaking down theproblem set considering a risk assessment framework is formulated. After a short excursion intothe fundamentals of general systems theory, the "system" city is formulated in its broadest senseusing systems theory.

In science theory the "systematic way" is often understood as a counter-current to a splittingof the sciences in highly specialized areas, which may be the main cause for the lack of commu-nication between experts. A system is in its broadest sense a functional relationship between itsconstituent parts, with which the whole acquires existential attributes independent of its parts.

The very first ideas in system theory are found in the works of van Bertalanffy (1968). Heformulated the integral ideas, when the scientific atmosphere was dominated by the philosoph-ical schools of rationalism. The general system theory can be understood to be a by-product ofthe philosophy of Kant, who formulated a strict epistemology in his philosophy of sciences.

Natural science uses mathematical language in formulating general laws, which can be ex-perimentally verified. A theory in natural science is hence primarily mathematical and shouldfollow logical principles. The general systems theory on the other hand tries to formulate aneven broader aspect, especially when it considers "living" systems. Organization, goals andaims gain focus.

The definition of systems and their classification in literature are divergent. Rapoport (1986)uses a two-dimensional perspective in explaining this divergence: The analytical-holistic di-mension and the descriptive-normative dimension. The analytical method aims to understand anobject, phenomenon or process by understanding its constituents. The other side of this perspec-tive is the holistic approach. Here, the "identification" is not based on the piecewise investigationof the constituents, but on the cognitive understanding of the "whole". This divergence can befound in many disciplines of science and philosophy. For example, Euclidean geometry is basedon recognizing configurations of fundamental shapes and hence can be seen as an holistic ap-proach. On the other hand, the analytical geometry of Descartes starts from the "point" to inves-tigate geometrical figures. The descriptive approach is more interested in understanding "how"something works, whereas the normative approach deals with the more value-oriented question"for what". For example, the descriptive decision theory explains how individuals would makedecisions in real situations. On the other hand the normative decision theory defines optimal andrational decisions for certain situations. Following Rapoport (1986) an integral system theoreticconception for cities considering these two perspectives will be attempted. In the following,several applications of system theoretic conceptions will be discussed.

Many system conceptions were proposed, e.g. Fuchs (1972), Klir (1972) and Ropohl (1979).

7

2 Fundamentals

These can mainly be classified into three categories: the functional system concept, the struc-tural system concept and the hierarchical system concept. In the first, system is understood asa whole, which performs functionality. Hereby, neither the internal parts nor their relation is ofimportance. The boundaries of the system and the input-output characteristics of its function-ality are of central concern. Within the structural system conception the internal parts and theirrelationships are considered. Here, the parts of the whole have an intrinsic character of being anelement of the whole. Hence, they cannot be fully understood independent of the whole, whichalso imposes complex interdependencies among the elements. Within the system some elementsmay form another whole, which shows completeness in its structure and functionality. The char-acteristic of this system is then related to the original system within certain functionality. Thesesystems can be defined using hierarchical system conceptions.

Depending on the problem at hand, one of these three system conceptions or a combinationof these is favored. Examples for the application of the functional system concept can be foundin cybernetics (Mesarovic and Takahara, 1975), of the structural system concept in Klir (1972)and of the hierarchical system concept in combination with structural system concepts in Lin(1999). A combination of all three system concepts was given in Ropohl (1979).

Rapoport (1986) assigns three attributes to systems: identity, organization and goal-directed-ness. Preservation of identity is a fundamental property of a system for two reasons. Changes inthe elements or relations among the elements should not prevent something from being recog-nized as a system. Additionally, if something cannot develop means to preserve its internalorganization in regard to distractions, it ceases to be a system. Organization is the most funda-mental property of a system. The structural and functional relations form a complete definitionof an organization. Goal-directedness is naturally based on the decision theory, since rationaldecision is formulated with a set of predefined actions regarding the preferences for reachinggoals. An overview of decision theory is given in the next section.

These theoretical considerations will be used in defining a city as a system in Chapter 3. Aftera comprehensive definition, the application to a city will be given. The problem complex willbe simplified to such extent that the illustration in the examples in Chapter 5 is easy to follow,without disregarding the fundamental issues pointed out in this chapter.

2.2 Decision theory

The descriptive decision theory deals with the way people make decisions, without strictly con-sidering the efficiency of their choices. The goal of the actors can be derived from the decisions,as long as they follow certain patterns. The prospect theory of Kahneman and Tversky (1979) de-scribes how people make choices in situations where they have to decide between alternatives.The normative decision theory on the other hand attempts to identify the way people shouldmake decisions, if they follow certain goals. Given that human beings are entirely rational, thenormative decision theory predicts their behavior.

A different classification can be made when the main items of the decision theory are con-sidered: actors and goals. Problems with a single actor and a single goal are the subject of thenormative decision theory (Luce and Raiffa, 1957). Problems with a single actor dealing with

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2.2 Decision theory

multiple goals or criteria belong to operations research (Churchman et al., 1957). Decision prob-lems where multiple actors follow one single goal are investigated in the team theory (Marschakand Radner, 1972). Problems with multiple actors, each following an own goal are the subjectsof the game theory (Neumann and Morgenstern, 1944).

In this dissertation, decision problems related to earthquake hazards are discussed. Irrespec-tive of whether decisions are to be made by building owners regarding the retrofit of their build-ing or decisions to be made by a city governor regarding resource allocation, the decision prob-lems belong to the first kind. A single actor has to make decisions following one single goal ofhaving the least cost or highest benefit. The main point is to identify, what the decisions of arational decision-maker "should" be rather than to understand how decisions are made. Hence,later on the normative decision theory for a single actor with a single goal is investigated.

Decision situations can also be divided into decisions under certainty, decisions under uncer-tainty and decisions under risk. Decisions under certainty lead to predefined results. Decisionscan be defined, if the actor is able to prioritize his preferences. This is not always simple. Ex-periments have shown that test persons make contradicting preference prioritization. A typicalexample is the intransivity of preferences. When preferences were compared in pairs, an alterna-tive x may be preferred over y and alternative y over z. But when in addition alternatives x and zis to be prioritized, x is not always prioritized over z as expected. In decisions under uncertainty,the decisions cannot be related one-to-one to the true state. A decision may lead to differentresults depending on the true state, which is independent of the actor. The main characteristicsof decisions under uncertainty are that the true state is not known.

Several decision principles were suggested as being applicable for solving decision problemsunder uncertainty. According to the maximin principle, regardless of which decision is made, theworst case of the unknown true state is assumed to result. The decision alternative with the "best"worst case is to be chosen. The minimax-regret principle represents an optimistic pendent. Toeach unknown true state a numerical value is assigned reflecting the degree of preference for eachdecision alternative. Each of these values is replaced by the algebraic difference between itselfand the maximum value related to that unknown true state given the decision alternatives. Thesedifferences are a measure for the regret of the decision maker not to have chosen that alternative.Finally, for each decision alternative, the maximum is chosen (Savage, 1954). The Hurwitz-αprinciple combines the above two extremes by assigning a factored linear combination of them.For each decision alternative, the minimum and maximum of the assigned preference numbersare weight by α and the 1−α . A linear combination of these two factors is assigned to eachdecision alternative. The alternative with the maximum is the optimal decision. α is a measure ofthe pessimism of the decision-maker. Setting α to zero leads to the maximin principle, setting α

to 1 leads to minimax-regret principle. The fourth decision principle goes back to Laplace. Sincethe true state is unknown, no preferences were made and equal probabilities are assigned to thetrue states. A rational decision is defined as the one leading to the maximum expected utility. Insome cases the pessimistic maximin has to be used, as the other principles require a preferenceprioritization. These assignments were done independent of information and knowledge.

When preferences and probabilities are assigned to true states based on experience, informa-tion or convictions, the situation becomes one of decision-making under risk. Decision-makingunder risk is based on the utility theory. The utility theory provides a formalization of the pref-

9

2 Fundamentals

erences of the decision-maker. Von Neumann and Morgenstern (1944) states a set of axioms,whose fulfillment allows the assignment of numerical values (utility) to the preferences of thedecision maker. The algebraic order of utilities reflects the order of preferences. The decisionmaker has to choose from alternatives, which leads to a certain utility with an assigned probabil-ity. The multiplication of the probability with the utility is called expected utility. Von Neumannand Morgenstern (1944) have shown that the optimal decision is the one which leads to thehighest expected utility.

Assigning utilities and probabilities to preferences and unknown true states respectively isstrongly associated with subjectivity. The Bayesian decision analysis is based on the utilitytheory, but additionally provides a formal basis for taking subjectivity into account (Raiffa andSchlaifer, 1961). The fundamental assumption of Bayesian decision analysis is the ability of thedecision-maker to assign subjective probabilities to all uncertain variables and to assign utilitiesto all combinations of outcomes of a decision problem. Although the subject is of controversialdebate because of the formalization of subjectivity, the theory has been applied to many engi-neering problems. In Faber (2003) and Kübler (2006) the applicability of Bayesian decisionanalysis to engineering problems is discussed. A fundamental reference for civil engineeringapplications of Bayesian decision analysis is Benjamin and Cornell (1970).

Bayesian decision analysis has its strength in its ability to combine different aspects whendealing with probabilities. The uncertainty about the true state can be reflected either by afrequentistic view based on data or by a subjective degree of belief. Furthermore, the inclusionof new information in the decision process is systematized. Bayesian decision analysis can beperformed in three different situations depending on the state of information processing. Inthe prior decision analysis the probabilities to uncertain states are assigned based on presentknowledge. After assigning probabilities and utilities to each possible outcome the expectedutilities are evaluated. Prior decision analysis evaluates the expected utility of each alternativeand selects as the optimum the one with the highest expected utility.

New information related to the uncertain states can be considered systematically using BayesTheorem. Here, observations, tests and new rationales can be regarded as new information. Theprobabilities assigned a priori are updated to posteriori probabilities using the new informationand the expected utilities are recalculated using these posteriori probabilities. Decision problemsconsidering posteriori probabilities are called posterior decision analysis.

In the third kind of Bayesian decision analysis the "potential" of new information is assessed,not the new information itself. Prior to evaluating the process of acquiring new informationthe possible outcomes are assessed. Systematically all possible outcomes are evaluated eachforming in itself a posterior decision analysis. The maximum expected utility for each posteriordecision analysis is calculated. As the analysis is done prior to the new information acquiringprocess these kinds of problems are referred to as pre-posteriori analysis. The value of infor-mation is related to pre-posterior decision analysis. It is assessed as the difference between theexpected utility of the optimal action with the new information and the expected utility with-out the new information minus the cost of acquiring the information. This so-called Value ofInformation analysis is extensively described in Raiffa and Schlaifer (1961) and Benjamin andCornell (1970).

10

2.3 Uncertainty, probability, utility and risk

So far, two basic conceptions concerning risk have been considered; probability and utility.In the following a short overview is given of these conceptions.


In decision problems, uncertainty is expressed quantitatively in terms of probabilities. A care-ful and consistent modeling of these uncertainties is essential for the identification of optimaldecisions. Uncertainties can be classified into two categories; aleatory uncertainties and epis-temic uncertainties. Aleatory uncertainty, also referred to as randomness, represents the naturalvariability in the considered phenomenon. On the other hand, epistemic uncertainty results frominsufficient knowledge of the considered phenomenon. The insufficient knowledge may lie inthe description of models representing the real phenomena or it may lie in the modeling of therandom variables. The former is known as model uncertainty, the latter as statistical uncer-tainty. Epistemic uncertainty can be reduced by improved models and better data basis, whereasaleatory uncertainty remains unaffected.

The different interpretations of probability can be classified as classical, frequentistic and sub-jective. The classical interpretation defines probability of an event as the ratio of the number ofcases favorable to it, to the number of all cases possible. This conception requires the completeknowledge of the considered phenomenon as all possible cases need to be identified. The fre-quentistic interpretation is based on counting the favorable cases relative to a large number ofobservations. It does not require a complete knowledge of the considered cases. The subjectiveinterpretation expresses a degree of belief that a considered event will occur.

In complex engineering decision problems the application of only one of these conceptions israrely possible. Either data for a frequentistic interpretation is missing, or experience in somespecific cases or a complete physical understanding is lacking. A combination of these threeinterpretations is required in such cases in particular. The Bayesian perspective constitutes aconsistent modeling basis for such complex engineering problems, see JCSS (2001).

Considering the consequences of earthquakes for a city, the utility can be assumed linearlywith respect to monetary units for the considered range of events. Therefore the risks are as-sessed based on the expected cost criterion. By analogy, expected benefit can also be used ifbenefits of activities were also included in the analysis (Benjamin and Cornell, 1970). The ex-pected cost criterion requires that all consequences of an event are expressed in monetary terms.Hence the loss of human life during a catastrophic event needs also to be quantified. Althoughstill the subject of controversial debate among structural engineers, decision analysts point outthat disregarding this would lead to inconsistent decisions (Benjamin and Cornell, 1970). TheLife Quality Index (LQI) provides a theoretical basis for the quantification of life saving costs,see Nathwani et al. (1997) and Rackwitz (2001). An outline of the LQI is given in Section 4.4.

To date, discussions have been about making decisions by comparing expected costs or ben-efits of the different decision alternatives. But when the different decision alternatives havedifferent time horizons a simple comparison would be misleading. For example, when decidingwhether to retrofit a building with regard to a possible damaging seismic event, the expenditure

11

2 Fundamentals

for retrofit at present is compared to a future cost due to a possible damaging seismic event.Since the value of money changes over time, it has to be considered explicitly.

Discounting is the process of finding the present value of cash at some future date. The dis-counted value of a cost or benefit is determined by reducing its value by an appropriate discountrate for each unit of time between the time when the cash flow is to be valued to the time ofthe cash flow. The discount rate is usually expressed as an annual rate. The discount rate is theinterest rate at which a central bank lends to commercial banks.

Discounting became a major issue in the years after the 1929 market crash. The first formallyexpressed discounting was done by Williams (1938). The nominal costs or benefits C(t) arereduced by a discount factor δ (t) to obtain the corresponding present value C(t = 0).

C(t = 0) = δ (t)C(t) (2.1)

The discount factor is given by

δ (t) =1

(1+λ )t (2.2)

where λ is the discount rate.As discussed above, all consequences of an event need to be expressed in monetary terms,

including life saving costs. This makes the quantification of the interest rate difficult. Corotis(2005) differentiates between private and public investments and argues that the rate of interestcan be different for public and private sectors. In Rackwitz et al. (2005) an intergenerationaldiscounting model is suggested, which takes the economic growth per capita δ into account.This is also known as natural interest rate. Additionally, the rate of pure time preference ρ

and the elasticity of marginal consumption ε is considered. The former takes into account thatindividuals prefer to consume earlier than later. The interest rate γ is given as:

γ = ρ +δε (2.3)

This model considers a discounting with γ for all present generations and a discounting withδε for future unborn generations. On the same basis, Nishijima et al. (2007) determine inflation-free public interest rates of about 2% annually, which is significantly lower than interest ratesfor private investments.

Closely related to discounting is the notion of net present value (NPV ). The sum of all costsand revenues discounted to their present value represents the NPV . It takes into account allconsequences throughout the projects lifetime. A project is evaluated as positive and hence eco-nomically feasible when the NPV is positive. Different alternatives are compared by referringto their NPV , and the one which yields the highest NPV is generally selected. In principle, thepoint in time at which it is discounted is arbitrary. In decision analysis the point in time is chosenat which the decision is to be made.

In this dissertation "decision problems under risk" are considered from a normative perspec-tive. In the following, the second concept "risk" is considered in more detail. After the decisionproblem is formulated, the governing risks need to be assessed. The management of the risk isthen based on this assessment. Risk assessment and management form the basis for designing

12


new structures and assessing existing structures as well as for the planning of inspection and themaintenance of structures.

Risk assessment is the first step in a risk management problem. It is the estimation of quantita-tive or qualitative risk related to a specific situation and a recognized hazard. When quantifyingrisk, two components need to be calculated: the magnitude of the potential loss and the proba-bility that the loss will occur.

Expressed in mathematical terms, the risk RE associated with the particular event E is:

RE = pECE (2.4)

where pE is the probability that the event will occur and CE is the consequence associatedwith the event. The consequence can be quantified in a common metric as currency or somenumerical measure of a locations quality of life. In civil engineering it is quantified as theexpected consequences which are associated with activities throughout the service life of thestructures, JCSS (2001).

Nathwani et al. (1997) formulated four principles for managing public risk: accountability,maximum net benefit, compensation and life measure. According to accountability, decisionsmust be open, quantified, defensible, consistent and applied across the full range of hazards.When risks are assessed openly and transparently, the consideration of public preferences isassured. When the risks of the full range of hazards are quantified, they become comparable.Transparency allows unpopular decisions taken by the decision-maker to become defensible.The principle of maximum net benefit is in line with the normative decision theory. Only thequantification of immaterial consequences such as environmental impacts and human life stillpresent difficulties. As discussed above, the life quality index represents a mean to overcome thisdifficulty, see Section 4.4. Some people in society take greater risks through hazardous activitiesand must therefore be compensated. The benefit of an activity is given if these people are fullycompensated and there is still benefit. The length of life is an indicator of maximum net benefitto society. A reliable and accepted measure for assessing this is life expectancy. When dealingwith the maximum net benefit of society, the effect of a decision on life expectancy needs to beconsidered.

A structured approach to managing the uncertainties related to a hazard is referred to as riskmanagement. Several strategies are possible, including transfer of the risk to another party, inorder to avoid risk, to mitigate the effect of risk, and to accept partly or all the consequences ofthe risk. The main objective is to reduce risks related to a defined domain to a level which isaccepted by society.

Risk management can be structured into a generic format as illustrated in Figure 2.1, (AS/NZS,1999). First the context must be established. Important questions such as who the decision makeris, which other parties may be affected and what the acceptable level is, need to be answered.In other words, the system has to be identified at this first step. All relevant risks are then iden-tified, and irrelevant hazards and opportunities are eliminated through a risk screening process.In the next step the risks are analyzed. The analyzed risks are then evaluated for their accept-ability. Risks identified as not acceptable have to be treated by mitigation, reduction or transfer.

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2 Fundamentals

In the whole risk management process a review and monitoring as well as communication andconsultation with the involved parties are required.

Figure 2.1: Risk Management process (AS/NZS, 1999).

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3 Methodologies

Loss estimation studies can be categorized as regional loss estimation and building-specific lossestimation studies. This dissertation concentrates mainly on the former. Regional loss estimationdeals with the quantification of economic losses for a portfolio of buildings within a geographicalregion such as a city, county, state or country. Describing the advantages, disadvantages, andlimitations of existing earthquake loss estimation methodologies requires considerable attentionto detail which is beyond the intent of the dissertation. Thus, the existing methodologies willonly be briefly discussed here. For details, please refer to items in the selected bibliography.

After the short overview, the methodology proposed in this dissertation is introduced andits capabilities with regard to the limitations of the existing loss estimation methodologies arediscussed. The remaining part of the dissertation deals with the development of the modelsrequired for the application of the proposed methodology and the illustration through examples.Here, the limitations of the proposed methodology are also discussed.

3.1 Existing earthquake loss estimation methodologies

In loss estimation, mathematical models for the contributory elements of the loss-inducingprocess are used to arrive at estimates for the potential losses that might result from an earth-quake. Loss estimation is a quantitative science which uses tools developed in seismology,geology, geotechnical engineering, structural engineering and economics linked by probabilitytheory and statistics (Khater et al., 2003).

Loss estimation quantifies earthquake losses, providing a first step to a better understandingof the loss contributors, of the alternatives for risk reduction and the level of the acceptable loss.Loss estimation methodologies are needed since the relatively infrequent earthquakes in any spe-cial location do not allow precise knowledge of the precise impact of future earthquakes. Theestimation of the impacts of future events are however necessary for the design of new construc-tions, planning for emergencies and the management of insurance commitments (Scawthorn,1995).

The topic was first treated by Freeman in the 1930’s. In Freeman (1932) the earthquake dam-ages of past events and their relation to insurance were reviewed. In the 1960’s, Steinbrügge andcoworkers developed scenario loss estimates for major U.S. cities (NOAA, 1972). Algermissenet al. (1972) estimated damage and losses that would result from major earthquakes in the SanFrancisco Bay Area in an interdisciplinary group of experts in seismology, geology and struc-tural engineering. In the 1970’s and 1980’s loss estimation was first treated probabilistically inWhitman et al. (1973) and Scawthorn (1981). In the late 1980’s loss estimation methodologiesbegan to be used by insurance and reinsurance companies. The Mw6.7 Northridge earthquake

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3 Methodologies

in 1994 accelerated the research in loss estimation methodologies. By that time it had becomeevident that extrapolating loss results from the past as a basis for the expectation of losses in thefuture leads to an underestimation of the losses. In the U.S., governmental institutions startedto support the development of comprehensive earthquake loss estimation methodologies. Themost important of these methodologies is HAZUS.

The earthquake loss estimation methodology HAZUS has been developed by the U.S. FederalEmergency Management Agency (FEMA) to assess the physical, economical and human conse-quences of earthquakes, hurricanes and floods throughout the U.S. (HAZUS, 2001). HAZUS isa GIS-based tool for use as a U.S. nation-wide decision support tool for policy analysis, emer-gency response planning and disaster response preparedness at all levels - federal, regional andlocal. It generates an estimate of the consequences to a city or region of a "scenario earthquake",i.e. an earthquake with a specified magnitude and location. After initial release of HAZUS97,the model was released three times; HAZUS99, HAZUS99-SR1 and HAZUS99-SR2. In 2004a multihazard version HAZUS-MH was released including flood and hurricane hazards besidesearthquakes.

HAZUS enables analysis at several levels depending on the data set available. At Level 1 na-tional level data sets are used. At Level 2 local data may be substituted for national data. At Level3 besides local data, specific analysis tools for studying special conditions such as liquefactionmay be implemented by the user. Capabilities in HAZUS include a hazard characterization toolfor earthquakes, floods and hurricanes, damage analysis tool for buildings and lifelines, casualtyand shelter estimation tool and economic analysis tool. In the hazard characterization tool forscenario earthquakes with a specified magnitude and location, the probabilistic ground motiondata is provided by the U.S. Geological Survey. Historical earthquakes are supplied by severalcatalogues and databases. HAZUS provides a general building stock database comprising 36structural types (e.g. steel braced frame, concrete frame with masonry infill) and 28 occupancytypes (e.g. single-family dwelling, heavy industry). Each structural type is further subdividedbased on the number of stories and construction period indicating the foreseen earthquake re-sistance for that structural type. For each structural type, capacity curves and fragility curvesare provided. Capacity curves are used in combination with damping-modified response spec-tra to determine the peak structural response of the structure according to the ATC-40 (1996)procedure.

Fragility curves describe the exceedance probability of different damage states given the peakstructural response. The damage to structural and nonstructural systems is described in HAZUSby five damage states: None, Slight, Moderate, Extensive, and Complete. The damage probabil-ities are then calculated using the damage analysis tool. The casualty estimation tool estimatesinjuries and deaths caused by structural damage. Casualties are estimated in four classes: mi-nor injuries, more severe injuries not requiring hospitalization, injuries requiring hospitalizationand deaths. Casualty estimations are produced for three scenario times: day time, night timeand commute time. The economic analysis tool calculates the building, content and inventorycosts, business, personal and rental income and disruption costs and lifeline valuations. HAZUSalso include models for estimating shelter needs, lifeline damages, debris generation and firefollowing damage.

The RADIUS (Risk Assessment Tools for Diagnosis of Urban Areas against Seismic Dis-

16


asters) initiative was launched in 1996 by the secretariat of the International Decade for Nat-ural Disaster Reduction (IDNDR 1990-2000) of the United Nations with financial assistancefrom the Japanese government. The main motivation was the inadequacy of the existing seis-mic risk assessment and management tools with reference to developing countries. Nine casestudy cities were selected, Addis Ababa (Ethiopia), Antofagasta (Chile), Bandung (Indonesia),Guayaquil (Ecuador), Izmir (Turkey), Skopje (The former Yugoslav Republic of Macedonia),Tashkent (Uzbekistan), Tijuana (Mexico), and Zigong (China) from 58 applicant cities. Tech-nical guidance was provided by three international institutes, namely, GeoHazards International(GHI, USA), International Center for Disaster-Mitigation Engineering (INCEDE)/OYO Group(Japan), and Bureau de Recherches Géologiques et Minières (BRGM, France). In an 18-monthperiod earthquake damage scenarios and seismic risk mitigation action plans were developedbased on a methodology developed by GHI for risk management projects in Quito (Equador)and Kathmandu (Nepal).

The case studies in RADIUS were carried out in two phases: the evaluation phase and theplanning phase. In the evaluation phase, seismic risk assessment for the city was performed bycollecting existing data and estimating the potential damage due to a hypothetical earthquake.The potential damage was estimated by a theoretical and non-theoretical step. In the theoreticalestimation seismic intensity distributions for the hypothetical event were combined with thebuilding stock inventory and infrastructure using vulnerability functions. In the non-theoreticalestimation, opinions from the local experts were collected through a series of interviews allowingadopting the special characteristics of the city system to be included in the damage estimation.In the planning phase, the results of the first phase was used to develop an action plan that wouldreduce the earthquake risk to the city. Using these case studies, practical tools for earthquakedamage estimation were developed, thus providing a basis for similar efforts as a first step ofearthquake risk management for other cities in developing countries. The OYO Group developeda computer programme for simplified earthquake damage estimation, aiding users to understandthe seismic vulnerability of their cities. As input data, population, structure types, soil typesand lifeline facilities are required. The tool provides outputs of seismic intensity, structure andlifeline damage and casualties (GHI, 2004).

The RISK-UE initiative supported by the European Commision developed a methodologyfor creating earthquake risk scenarios with special focus on the distinctive features of Europeancities, including existing buildings and monuments. The project duration was about 42 monthsand ended with a final conference in Nice, France at the end of March 2004. The aim was toproduce a standard manual for assessing earthquake risk in urban areas, for use not only by allEuropean countries, but also by countries subject to conditions similar to those found in Europe,e.g. the Mediterranean region. The initiative comprises two parts: a methodology enabling theassessment of seismic risk of European cities and an application of this methodology to sevenEU and Eastern European cities, which are Barcelona (Spain), Bitola (Former Yugoslavian Re-public of Macedonia), Bucharest (Romania), Catania (Italy), Nice (France), Sofia (Bulgaria)and Thessaloniki (Greece). The methodology comprises a state-of-the-art seismic hazard as-sessment, a systematic inventory typology of the structures at risk, with emphasis on distinctivefeatures of European cities. Distinctive European urban features particularly concern complexbuilding aggregates in old city centers and monuments and historical buildings. The classifica-

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3 Methodologies

tion of European buildings resulted in a European Building Typology Matrix (BTM), where 23building types were identified leading to a total of 65 typologies. The vulnerability of the build-ing typologies helping to identify the weak points of the infrastructure systems are estimatedeither by assigning a score or vulnerability index or by the capacity spectrum method (ATC-40, 1996). The economic loss scenarios are based on the reposition cost of the buildings: theexpected number of casualties and injuries was obtained using the model proposed by Coburnand Spence (2002). The established earthquake scenarios, structured within a GIS, could thenbe put forward to town councils as a basis for discussion and drawing up plans of action forsystematically reducing earthquake risk (Risk-UE, 2004).

The recently completed LESSLOSS project funded by the European Community is anotherinterdisciplinary approach for earthquake loss estimation. For selected European cities the vul-nerability of the building stock, the estimation of human casualties and the direct economicallosses were performed. The main goal of the project was the assessment of earthquake risks,the impacts of an earthquake on the environment and on the cities as well as emergency andmitigation measures (LESSLOSS, 2005).

In 1995 the Russian Ministry of Emergency Situations was actively engaged in the devel-opment of a GIS-based automated system for the estimation of consequences of severe earth-quakes. A global GIS System EXTREMUM was developed for forecasting the consequencesof destructive earthquakes. This system allows performance round-the-clock and forecasts (esti-mate) consequences of earthquakes all over the world. Since August 2000 EXTREMUM is usedfor the common benefit of the world community (Shakhramanjyan et al., 2001).

The displacement-based earthquake loss assessment (DBELA) method is a fully probabilisticframework that incorporates variability both in demand and capacity parameters. It was devel-oped by Pinho et al. (2002) and Crowley et al. (2004). The main focus of this methodology is theevaluation of the displacement capacity of classes of buildings at various damage limit states.

Pointing to the ever growing research efforts in natural sciences, engineering and social sci-ences regarding catastrophes, the Alliance for Global Open Risk analysis (AGORA) started theinitiative for open source software code development. Current end-to-end risk models have notbeen designed to respond to emerging knowledge and data. The paradigm of having an openplatform promises shorter implementation periods for new findings. The open source seismicrisk related software (OpenRisk) is being developed by AGORA. OpenRisk estimates the perfor-mance of assets such as buildings subjected to earthquakes in terms of economic costs, humansafety and loss of use. Besides OpenRisk, AGORA supports the development of several otheropen source softwares; OpenSHA for seismic hazard, OSRE for risk assessment, MIRISK fordecision making (AGORA, 2009).

Beside those listed above, many other earthquake loss estimation methodologies have beendeveloped and successfully applied. No claim is made to provide a complete list of existingearthquake loss estimation methodologies worldwide, nor is it possible. As can be seen from theshort descriptions of the methodologies, the following properties can be identified.

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Integrality and generality

Risk-based decision making requires loss estimation which represents an integral approach. Thisrequires that the interaction between all relevant agents, i.e. technical systems, nature, humanbeings and organizations are explicitly considered. An integral approach to risk assessment en-sures that significant risk contributors originating from the interactions between the differentagents are taken into account (Faber, 2008). All of the aforementioned loss estimation method-ologies can be regarded as integral approaches to some degree.

Generality, i.e. context independence, is especially important when the loss estimation method-ology is to be applied to other regions as well as for other hazard types. The lack of generalityshould not be seen categorically as a shortcoming of a loss estimation methodology. Some of theaforementioned earthquake loss estimation methodologies have been developed to be applied todifferent regions like HAZUS, even though this is only in the US. Others are designed as projectsof finite duration with application in chosen pilot cities, e.g. Radius and Risk-UE, and do notclaim to be by implication generic.

Modularity

Earthquake loss estimation requires an interdisciplinary approach. Research in individual dis-ciplines such as seismology, soil dynamics and earthquake engineering provides scientific andtechnological improvements which may result in the existing loss estimation methodologiesbecoming obsolete, if these improvements are not considered. A modular structure enablesthe implementation of new models in all the disciplines without the need to reset the over-all methodology. All of the aforementioned earthquake loss estimation methodologies have amodular structure. They comprise modules for seismic hazard, for soil response, for structuralresponse, for damage and losses. The interfaces of these modules is defined and calculationsare executed from end-to-end. They classify the buildings in the city or region considered intoclasses and predict the portion of the building class falling within a predefined damage classfor specified earthquake demand by vulnerability modeling. The vulnerability methods rangefrom damage probability matrices such as that developed by the Applied Technology council ofthe U.S. (ATC-13, 1985) based on expert opinion, over analytical derived fragility curves as inHAZUS to more mechanically-derived formulae that describe the displacement capacity of theclasses of buildings at different damage limit states in DBELA.

The following shortcomings of the aforementioned earthquake loss estimation methodologiescan be stated.

Inference

The earthquake risk is evaluated in the aforementioned methodologies in forward direction. Theanalysis of the seismic hazard, soil response, structural response, damage and loss assessmentare performed in a sequence leading to a quantification of the risk or to a damage scenario.Queries of the kind "What would be the total loss due to an earthquake of magnitude MW = 7?"can be answered. For a decision-maker it is also valuable to perform diagnostic analysis. Typical

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3 Methodologies

queries are of the kind "Which magnitude of earthquakes would lead to complete unavailabilityof important structures such as hospitals?" The performance of sensitivity analysis is also valu-able. Sensitivity analysis refers to analyzing how sensitive the condition (e.g. the probability ofcollapse of a hospital) is to minor changes. The changes may be variations of the parameters ofthe model or may be changes in the diagnosis of evidence.

None of the existing loss estimation methodologies considers the prevailing parameters ex-plicitly by considering their uncertainties. Explicit consideration of the parameters would enablethe inference in both directions.

Updateability

The main engineering task is to represent reality by means of models. These models providea means for understanding the problem. The continuous adaptation of the models to realityis a major challenge. In Bayesian understanding the constructed models represent the currentknowledge and with all incoming information which could be for the present case a damag-ing or non-damaging earthquake, the models could be updated. The existing loss estimationmethodologies in some ways consider the knowledge from new earthquake by remodeling, butnone of them provides a framework for systematic update even though the underlain modulesare continuously improved and replaced as in the case of HAZUS.

The knowledge and information basis is not very broad when assessing earthquake risks.Damaging earthquakes are not very frequent; data on built environment is usually very scarceand research on the occurrence of earthquake and its effects on soil and structures is ongoing.Hence, the processing of knowledge and data of any source, from scientifically verified knowl-edge over statistically representative data to experience to reflecting expert opinions, is neces-sary. Especially when the problem complex is modeled explicitly by observable characteristicdescriptors called indicators, an update is facilitated. A Bayesian perspective in loss estimationmodeling provides a sound and thorough means for this. None of the aforementioned earthquakeloss estimation methodologies considers such an integral approach to knowledge.

Dependencies

The lack of explicitly considering the dependencies of the prevailing parameters is another pitfallof the existing methodologies. Disregarding the dependencies leads to suppressing importantsystem effects when considering loss estimation to portfolios of buildings. Statistical depen-dency may be appropriately represented through correlation. Functional dependency or commoncause dependency is appropriately represented through hierarchical probabilistic models.

Multi-detailing

A decision is an allocation of resources. Considering risks due to earthquakes, decision-makersare to some degree all individuals within the earthquake prone region. The decision-maker isan authority or person who decides on the allocation of the available resources and takes re-sponsibility for the consequences of the decisions on others. Hence, the formulation of the

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3.2 Proposed framework

decision problem depends on the decision-maker. Different decision-makers will have differentpreferences and objectives. The following decision making levels can be distinguished: Privateowners, e.g. individuals, local authorities, e.g. municipalities, national authorities, e.g. gov-ernmental agencies, international private companies, e.g. insurance and reinsurance companies,Supranational authorities, e.g. United Nations.

For individuals and private owners information about the earthquake risk is important as it al-lows the adaptation of behavior and activities in order to minimize risks. Information on risk alsomakes it possible to consider risk transfer to third parties at affordable costs. Local and nationalauthorities deal with earthquake risks from the perspective of resource allocation. Here, neitherthe risks for one group of persons should be considered, nor the risks associated with individ-ual hazard processes. In a holistic perspective risks due to all prevailing hazard types must beconsidered before decisions can be made. At a national level, codes and regulations concerningthe design of structures and the use of land can be based on the assessed overall risks. Inter-national private companies accept risk for other stakeholders and help in reducing the impactof earthquakes at a cost. Knowledge of portfolio risks, including the statistical characteristicsof their magnitude, dispersion and influencing factors is important for setting the premiums andfor diversifying or uncoupling potential losses. Supranational authorities provide and allocateeconomic and knowledge resources for the purpose of reducing risks especially in countries withinsufficient capacity to do so themselves (Faber et al., 2007).

None of the aforementioned loss estimation methodologies can be applied to different decisionmakers with different levels of detail.

The proposed framework in this dissertation aims to represent the problem complex earth-quake risk by explicitly modeling the prevailing parameters, to explicitly model the dependen-cies among the parameters to consider system effects and to provide a framework for systematicupdate. Comparable to the aforementioned methodologies, it will use a modular structure inmodeling seismic hazard, soil response, structural response and damage and loss assessment.The main difference is the explicit modeling of the prevailing parameters, especially thosewhich can be observed, and hence denoted as indicators, using so-called Bayesian probabilis-tic networks. The following section describes the proposed methodology of "Indicator-basedlarge-scale risk assessment framework" along with the main tool for evaluation, the Bayesianprobabilistic networks.


Risk assessment is the first step in quantifying risks. In this step, risks are already analyzed andthrough a screening process limited to their governing constituents. Managing the risk within adecision-making process requires a sound and reliable risk assessment.

Assessing and managing earthquake risks should be seen relative to the occurrence of theseismic event, i.e. before and after the earthquake event occurs. This is necessary as the decisionalternatives and problem context or boundary conditions change over the corresponding timeframe. Before an earthquake occurs the issue of concern is the allocation of resources for optimalpreventive measures such as adequate strengthening or renewal of the built environment. The

21

3 Methodologies

BEFORE AFTER

Optimal allocation of available ressources for risk reduction

Loss estimation

Condition assessment and updating of reliability and risk

Loss assessment

Figure 3.1: Decision situations for management of earthquake risks.

estimation of expected losses due to a potential earthquake is another major interest. After anearthquake the issue is to estimate the losses occurred and to assess the structural conditionof the buildings and lifelines. The knowledge gained from the effects of the event which hasoccurred is then used in updating the underlying models used to assess the losses, especially themodels for assessing the behavior of soils and structures. After an earthquake, the situation iscomparable to the situation before an earthquake. Figure 3.1 illustrates the different decisionsituations.

The identification of optimal decisions could be performed by means of traditional cost/benefitanalysis, if all constituents of the decision problem would are known with certainty. As thepresent understanding of the physical phenomena "earthquake", its spatio-temporal occurrence,its effect on soil and structure as well as its financial effects are far from perfect, the decisionproblems are subject to significant uncertainties. However, the risks associated with the differentdecision alternatives can be assessed. The assessed risks can then be used to rank the decisionalternatives consistently.

This dissertation aims to establish a framework for assessing earthquake risks in a region. Theframework attempts to meet the identified modeling basis discussed in Section 3.1:

1. Integrality and generality

2. Modularity

3. Inference

4. Updateability

5. Dependencies

22


6. Multi-detailing

In the following the risk assessment framework, Bayesian probabilistic networks and the de-tails of the modeling basis are outlined.

Risk assessment framework for systems of cities

A system can be considered as an ensemble of interrelated constituents (or assets); buildings,components, lifelines, human beings and environment. The individual constituents and theirinterrelation define the characteristics of the system. In describing a system, spatial and temporalrepresentation of its constituents are required; including the interrelation between all relevantexposures (hazards), the assets and possible consequences. Direct consequences are related todamages to the individual constituents of the system. Indirect consequences are any consequenceoccurring beyond the direct consequences, i.e. associated with losses in regard to system effects.

Modeling a system always requires a choice of an appropriate level of detail or scale. Thechoice depends on the characteristics of the system and the spatio-temporal characteristics ofthe consequences. A decision-maker responsible for one or a couple of buildings in a city mayprobably choose a more refined model for evaluating the structural behavior of the buildings dueto earthquakes than a decision-maker responsible for hundreds of buildings.

The representation of a system should accommodate information collection about the indi-vidual constituents. A performance update of the system in regard to the response to externaleffects such as earthquakes is thus possible. The system is modeled based on the availableknowledge of the individual constituents. The mapping of reality which is nothing but the mod-eling process, lacks perfect knowledge. The physical phenomenon earthquake, its effect on thebuilt environment and the generation of losses are all subject to significant uncertainty. Hencethe consistent treatment of knowledge and uncertainty play a key role. The consistent repre-sentation of knowledge and uncertainty justify the integration and aggregation of risk estimatesobtained for different assets and for individual hazards (Faber, 2008).

Vulnerability

Robustness

Exposure

Ris

k re

duct

ion

mea

sure

s

Ris

k in

dica

tors

Figure 3.2: Illustration of risk assessment framework.

23

3 Methodologies

Risk assessments may be facilitated by considering the generic representation illustrated inFigure 3.2. In this framework three levels are distinguished: exposure, vulnerability and robust-ness. The risk assessment framework allows the use of any type of risk indicators in regard tothe exposure, vulnerability and robustness of the considered system. Risk indicators are anyobservable or measurable characteristics of the system or its constituents containing informationabout risk. The possibilities for collecting additional information in regard to the uncertaintiesassociated with the risk indicators can be considered as comprising the total set of measures forrisk reduction. The risk reduction measures may be considered as decision alternatives. Riskmeasure can be implemented at different levels in the system representation, in regard to expo-sure, vulnerability and robustness (Figure 3.2). Exposure can be considered to be an indicator ofthe hazard potential for a given object or system of consideration. Considering earthquakes, theexposure EX is an inherently uncertain phenomenon with probabilistic characteristics usuallyprovided in terms of earthquake intensities and corresponding return periods. The vulnerabil-ity of a system, assessed through the term P(D|EX)C̄D, can be considered as the ratio betweenthe risks due to direct consequences and the total value of the considered asset considering allevents in a specified time frame. Considering an earthquake event, vulnerability is associatedwith significant uncertainty and is appropriately described by a probability distribution of differ-ent damage states of structures conditional on the exposure event, e.g. the earthquake intensity.Robustness, assessed through the complement of the term P(CID|D,EX) ¯CID, can be consideredas the ratio between the risks due to direct consequences and the sum of the risks due to directand the indirect consequences. Considering again the event of an earthquake, robustness is asso-ciated with the conditional probability of losses of various degrees conditional on the exposureand a given damage state.

For a system corresponding to a city or region, societal losses including loss of lives as wellas economical losses may depend strongly on the specific time of the year, week and day whenan earthquake occurs. In this way the robustness of the system exposed to an earthquake willalso be dependent on the specific time when an earthquake takes place. Whereas the uncertaintymodeling associated with the assessment of exposure and vulnerability may be based on wellestablished frameworks such as the Joint Committee on Structural Safety Probabilistic ModelJCSS (2001), the uncertainties involved in the assessment of the robustness are generally subjectto significant epistemic uncertainty. One of the reasons for the significant subjective elementof uncertainty in consequence modeling is the uncertainty on the decision maker in regard toappropriateness and completeness of the applied consequence assessment models. However, incorrespondence with the risk assessment framework outlined above and consistent with Faberand Maes (2003) it is proposed that risk assessment be performed on the basis of the follow-ing expression for the risk R or the equivalent expected direct consequences CD and indirectconsequences CID:

R = E[CD +CID] (3.1)

E[CD] =∫ ∫

C̄D p(D|EX)p(EX)dDdEX (3.2)

E[CID] =∫ ∫ ∫

¯CID p(CID|D,EX)p(D|EX)p(EX)dCIDdDdEX (3.3)

24


Earthquake risk assessment framework

Based on Equation 3.1 and consistent with the PEER performance-based earthquake engineer-ing framework (Cornell and Krawinkler, 2000), the following expression is elaborated in theproposed methodology:

E[CD] =∫ ∫ ∫ ∫

C̄DdP(D|EDP)dP(EDP|IM)dP(IM|SE)p(SE)dSE (3.4)

E[CID] =∫ ∫ ∫ ∫ ∫

¯CIDdP(CID|D)dP(D|EDP)dP(EDP|IM) (3.5)

dP(IM|SE)p(SE)dSE

where CD, CID are the direct and indirect consequences, D is the damage state, EDP is theengineering demand parameter (i.e. interstory drift ratio, dissipated energy, etc.), IM is theintensity measurement (i.e. the spectral displacement, peak ground response, magnitude, etc.)and SE is the seismic event. In the following sections the components of equations 3.4 and 3.5are further explained.

Exposure is represented by the probability that a specific ground motion intensity measure-ment, IM, exceeds a specific value at a given site in a specific time interval. This probability isalso denoted as the seismic hazard at the site. In Equation 3.4, the terms dP(IM|SE) and p(SE)are related to the assessment of the seismic hazard. The main sources of uncertainties involvedin evaluating the seismic hazard at a given site arise from the definition of seismic sources, recur-rence relations, attenuation relationships, local site effects and soil-structure interaction effects.Seismic sources are points, lines or areas with a uniform level of seismic activity and are definedon the basis of geological, geophysical and seismological data. Recurrence relationships aredefined for each source zone according to the assessed recurrence frequencies of earthquakeswith different magnitudes. Attenuation relationships provide estimates of major characteristicparameters of a strong ground motion (i.e. peak ground acceleration, acceleration response spec-tra) at a given site due to a seismic event with given major characteristics, such as magnitude,distance and faulting mechanism. A review of the major developments in the field of seismichazard analysis is available in Atkinson (2004). In particular, the probabilistic approaches forestimating the seismic hazard provide a structured framework for explicit quantification of theuncertainties involved. The Probabilistic Seismic Hazard Analyses is one of the major stud-ies in this field (Cornell, 1968). It should be noted that explicit quantification of uncertaintiesinvolved in the estimation of seismic hazards in turn provides the possibility of evaluating thesensitivity of results to various uncertain parameters. Such information forms a valuable basisfor allocating the available resources in order to gain more information for the improvementof hazard estimates. As a result of the modular approach followed in this study, a number ofseismic hazard models with varying levels of detail can be utilized.

The vulnerability of a structural system is expressed through the probability that a specificlevel of damage, D, occurs when the structure is subjected to a specific loading intensity, IM.In equations 3.4 and 3.5, the terms dP(D|EDP) and dP(EDP|IM) are directly related to the as-sessment of the seismic vulnerability of a structure. The methods for obtaining the vulnerability

25

3 Methodologies

functions for structures can mainly be categorized as: methods based on expert opinion, meth-ods based on observed damage distributions after earthquakes and methods based on structuralresponse analysis. A review of the major approaches related to seismic vulnerability assessmentof structures is available in Porter (2003). Methods based on structural response analysis, suchas Mosalam et al. (1997), Singhal and Kiremidjian (1997), provide the advantage of investigat-ing the effect of individual parameters on the structural response in an analytical manner. Due tothe term dP(D|EDP), the dependency between D and EDP also plays an important role in theevaluation of the usefulness of EDPs. In many of the available vulnerability assessment studiesthis link is established based on empirical methods, such as experiments and post-earthquakedamage observations. The updatability of BPNs makes them an efficient tool, provided that theselected EDPs can be updated.

Robustness, i.e. the complement of dP(CID|D), is an indicator of the indirect consequencesCID due to the damages D of the system under consideration. Considering seismic events, ro-bustness is associated with the conditional probability of losses of various degrees conditionalon the damage state. For a city, societal losses including loss of lives as well as economicallosses depend strongly on the specific time of the year, week and day when an earthquake oc-curs. To ensure consistent decision-making, the uncertainty of any type of economic impact dueto earthquakes has to be included. These economic impacts are direct economic losses due todamaged structures, their contents and lifelines (Geipel, 1990), indirect economic losses due tobusiness interruption (Benson and Clay, 2004), loss of revenues and increases of costs in thepublic sector, expenses and losses of individuals and loss of household incomes due to death,injury, or job interruption (NRC, 2004). Loss of individuals may be quantified by the SocietalLife Saving Cost (SLSC) (Rackwitz, 2006) or by the Implied Cost of Averting Fatalities (ICAF)(Skjong and Ronold, 1998), which are derived on the basis of the Life Quality Index (Nathwaniet al., 1997).

Bayesian probabilistic networks

The proposed framework allows for the utilization of any type of quantifiable characteristic,denoted as indicator, in regard to the exposure, vulnerability and robustness of the consideredsystem. Within the proposed framework risk reduction measures can also be implemented atdifferent levels in the system representation, in regard to exposure, vulnerability and robustness(Figure 3.2). The hierarchical structure of risk assessment can effectively be mapped by modernrisk assessment tools such as Bayesian probabilistic networks (BPN) and influence diagrams.In this subsection the main concepts for the construction of BPNs are introduced. The mostimportant algorithms for the evaluation of BPN’s are given in Annex A.

BPNs constitute a flexible, intuitive and strong model framework for Bayesian probabilisticanalysis. They were originally developed as an extension to predicate logic based on determinis-tic production rules in the field of artificial intelligence. The advantage is that the variables mayhave values other than binary cases and that relations among variables need not be deterministic.Despite the modeling power, the popularity of BPNs as a modeling tool did not increase untilefficient inference algorithms were developed (Lauritzen and Spiegelhalter, 1988), (Jensen et al.,1990).

26


Bayesian probabilistic networks, alternatively referred to as belief networks or probabilisticcausal networks, have become popular during the last two decades in the research areas of artifi-cial intelligence, probability assessment and uncertainty modeling (Pearl, 1988). The ideas andtechniques have also gained recognition in other engineering disciplines and natural sciences,especially in problems involving high complexities and large uncertainties, see also Faber et al.(2002). An application of causal networks for the purpose of aiding technicians to assess histor-ical buildings subject to earthquake hazards is given in Salvaneschi et al. (1997). As an examplethe seismic vulnerability is evaluated by modeling the available knowledge in the form of logicaltrees in Miyasato et al. (1986), Ishizuka et al. (1981) and Pagnoni et al. (1989). Furthermore,in Zhang and Yao (1988) conceptual networks and frames are applied to map observable infor-mation into damage states. In Friis-Hansen (2000) BPNs were developed as a decision supporttool in marine applications. The description and assessment of natural hazards and the quantifi-cation of their related risks appears to be a problem for which BPNs can be a helpful tool. InBayraktarli et al. (2005) BPNs are applied to assess the earthquake risk for single structures, inBayraktarli et al. (2006) to assess the earthquake risks for cities, in Bayraktarli and Faber (2007)for value of information analysis, in Straub (2005) to natural hazards risk assessment in generaland in Gret-Regamey and Straub (2006) BPNs are linked to a Geographic Information System(GIS) for natural hazards risk assessment.

BPNs are designed as a knowledge representation of the problem domain, explicitly encod-ing the probability dependence between the variables in the model. Model building intuitivelyfocuses on causal relationships between variables. Hence a BPN automatically reveals the ana-lyst’s analytical understanding of the problem. This enables validation of the models and com-munication between different parties especially useful when dealing with interdisciplinary prob-lems.

In probabilistic terms a BPN represents the joint probability density function of all variablesexplicitly considered in the problem. Considering causal relations among the variables lead tothe most compact representation of the joint probability density function (Jensen, 2001). Theoutcome of the compilation of the BPN is the marginal probability distribution of all variables.An important feature of BPNs is that easy inference based on observed evidence is allowed.Observing one variable in the domain, the probability distributions of the remaining variables inthe model are easily updated. Furthermore, any of the variables in the BPN can be conditionedto a certain state and the probability distributions of the remaining variables in the model canbe evaluated. The difference from updating lies in the fact, that through conditioning, a BPN isadjusted to a new configuration.

Several program packages exist for constructing and evaluating BPNs, such as Hugin (2008)or Netica (2008). Many other non-commercial packages are also available, GeNIe-Smile (2008)or WinBUGS (2008). Throughout the dissertation the inference engine of Hugin (2008) hasbeen used.

The basic feature of BPNs may be described by the following steps:

• Formulation of causal interrelationships of events leading to the events of interest (conse-quences). This is graphically shown in terms of nodes (variables) connected by arrows.

27

3 Methodologies

Variables with ingoing arrows are referred to as children. Variables with outgoing arrowsare referred to as parents.

• Assigning to each variable a number of discrete mutually exclusive states.

• Assigning probability structures (tables) for the states of each of the variables (conditionalprobabilities in cases where the variables are children).

• Assigning consequences corresponding to the states represented by the BPN.

Having developed the BPNs, the required probability tables and consequences, risk assess-ment and decision analysis is straightforward.

Elements in Bayesian probabilistic networks

After analyzing the problem at hand, the governing parameters need to be identified. This isalways carried out considering the needs of the analysis, which determines the detailing level.The risk assessment framework introduced makes it possible to perform this step in a structuredmanner. The next step is to identify the dependencies among the parameters. The easiest way toconstruct these dependencies is to rely on causality. A correct model based on causality reducesthe number of connections among the parameters.

It is however not easy to base modeling on causality, as causal relations are not always obvi-ous. Causality is not a well understood concept. The debate in philosophy after David Hume is,whether a causal relation is a property of the real world or a concept in our minds helping us toorganize our perception of the world. Putting the philosophic debate aside a more practical rea-son may hamper our reliance on causality. The acquisition of the conditional probability tablesfor a correct causal model may not be possible; the information may be available for other typesof conditional probabilities.

Although it is an advantage, the structure of a Bayesian probabilistic network need not nec-essarily reflect cause-effect relations. The only requirement is that the so-called directional-separation (d-separation) property of the network for the domain modeled holds true. It isa property of directed graphs which may be used to identify irrelevant information for spe-cific queries in a Bayesian probabilistic network or influence diagram. Two nodes of a net-work are d-separated if they are conditionally independent given a specified set of nodes. IfP(A|B,C) = P(A|C) then A and B are conditionally independent, or d-separated, given C. It isimportant to note that A, B and C may be variables or sets of variables.

Having defined the main construction rules for Bayesian probabilistic networks it is now ap-propriate to consider the three main types of connections found in a Bayesian probabilistic net-work. The d-separation principle will also be discussed for these main types. The BPNs givenin this section and in Appendix A, where the most important algorithms are given for evaluat-ing BPNs, are constructed for a very crude earthquake-related problem. The characteristic ofan earthquake is given only by its magnitude (M). The ground motion intensity parameters arecharacterized by peak ground acceleration (G) and spectral displacement (S). Liquefaction (L)is considered as one type of soil related failure. Structural damage (D) is assumed to be caused

28


by either liquefaction or spectral displacement. In formulating a decision problem the risk re-duction measures are considered in alternative actions (A). The structural damage and the riskreduction measures may result in costs (C).

In many engineering problems, the objective is to model physical phenomena which are inher-ently continuous. The consideration of continuous variables is not directly possible with BPNsas the associated algorithms are tailored to handle discrete variables. Attempts to include contin-uous variables into BPNs were made by Pearl (1988), Lauritzen (1992) and Alag and Agogino(1996).

The consideration of discrete variables is not a limitation, because the state-of-the-practiceis to consider some main variables as earthquake magnitude and structural damage anyway indiscrete states. Earthquake magnitude is evaluated mostly in 0.5 steps down to mostly onesignificant digit in seismic hazard calculations. The damage to structures is mostly consideredwith classes as "immediate occupancy", "life safety" or "collapse prevention" (FEMA, 356).

The approximation of a continuous space of a variable of a BPN is known as discretisation.The continuous space is hereby subdivided into a set of bins or intervals. Discretisation mayalso be understood as a categorization or classification of a data set. The most important pointin a discretisation is that the continuous function or the given data set is compactly representedwithout disregarding the most important properties of the function or data set. The discretestates should represent mutually exclusive ranges. Discretisation of a data set can be madeaccording to equidistant split-points, equal frequency or supervision. In the equidistant split-points approach the range is split into intervals of equal length. This approach gives reasonableaccuracy when the uncertainty of the variable is very large. When the variable at hand hasa very low uncertainty, the equal frequency approach may be applied. An equal number ofdata points is targeted in each interval, so that a fine discretisation of the dense part of thedistribution and fewer and longer intervals in the sparse regions can be achieved. In a superviseddiscretisation the interval number and interval lengths are chosen iteratively until the histogramof the data set is reasonably represented. This step is in principle comparable to the choice of thenumber of bins when drawing the histogram of a data set in a statistical analysis. A BPN mayinclude variables, which follow a known distribution. The task is to determine the length of theindividual intervals so that the discretized distribution represents the probability distribution ina reasonable way. The discretisation approaches are the same as given above. The choice of thediscretisation principle is determined by the purpose of the study and the characteristic of theprobability distribution considered. For a thorough discussion of this subject, see Friis-Hansen(2000).

In Figure 3.3 a very simple BPN with two nodes Magnitude (M) and structural damage (D)is illustrated, where magnitude has an influence on structural damage. In BPN terminology themagnitude node is denoted as a parent node and the structural damage node as a child node.

M D

Figure 3.3: Simplest BPN.

29

3 Methodologies

For each node a probability table is specified. Parent nodes have by definition unconditionalprobability tables. The magnitude node M may be discretized, for example into categoricalstates: small magnitudes, moderate magnitudes and large magnitudes, or into any other numberof discrete states: M<5, 5<M<6, 6<M<7 and M>7. In any case the probabilities of the mutuallyexclusive states must total 1. A child variable requires a conditional probability table with theassignment of a probability to each of the mutually exclusive discrete states of the child nodegiven the state of the parent node. In the example given, the structural damage node may bediscretized into three states: "immediate occupancy", "life safety" and "collapse prevention".The probabilities of being in each of these damage states need to be assigned for each of thegiven magnitude ranges. The same principles apply analogously when there are more than oneparent nodes and/or child nodes in a BPN.

Serial, diverging and converging connections

A typical serial connection is given in Figure 3.4. Magnitude (M) has an influence on Spectraldisplacement (S), which in turn has an influence on Damage (D). Evidence (or observation) onM will influence the certainty of S, which then influences the certainty of D. Similarly, evidenceon D will influence the certainty on M through S. But, if the state of S is known, then M andD become independent. In this case it can be said that M and D are d-separated given S. Wheninformation on spectral displacement can be retrieved, the estimation of the structural damageis no longer dependent on the earthquake magnitude, given that the BPN in Figure 3.4 correctlyreflects the situation.

In a typical diverging connection (Figure 3.4, middle) evidence may be transmitted if the stateof the parents are not known. In this case it can be said that peak ground acceleration (G) andS are d-separated given M. Although not relevant in practical situations, where the magnitudeof an event is mostly known before any other information, the following is assumed: if themagnitude of an earthquake is not known, having information on spectral displacement indicatessomething on the peak ground acceleration; but when the magnitude is known, information onspectral displacement gives no additional information given that the BPN in Figure 3.4 correctlyreflects the situation.

In converging connections (Figure 3.4, right) evidence is transmitted through the child node,when the child node or any one of the parents receive evidence. In this case it can be said thatliquefaction (L) and S are d-connected (opposite of d-separated) given D. If the state of damageis known, information on liquefaction indicates something on the spectral displacement.

Sample BPN and sample influence diagram

BPNs of any size can be constructed using the three main types. A sample BPN is given inFigure 3.5. This is the BPN used in explaining existing algorithms for evaluating BPNs given inAppendix A.

BPNs extended to solve decision problems are also known as influence diagrams. In influencediagrams two additional node types are used: Decision nodes (usually symbolized by rectangles)and utility nodes (usually symbolized by diamonds), see Figure 3.5. A decision node comprises

30


M

S

D

M

G S

L S

D

Figure 3.4: Serial, diverging and converging connections.

the alternative actions considered by the decision-maker. The parents of a decision node definethe available information at the point of the decision. From this it follows that when morethan one decision node exists in an influence diagram (typical sequential decision problem), thedecision nodes need to be connected by arrows and decision analysis needs to be performedconsecutively following the arrows. Utility nodes may be conditional on probabilistic and/ordecision nodes, but do not have child nodes. In the corresponding table utilities which quantifythe decision-maker’s preference for each configuration are given, and not probabilities.

The rational basis for decision-making is established by comparing the expected utilities ofeach considered action alternative in the decision nodes. The construction and evaluation of theinfluence diagram follows very close to the BPNs. Most of the program packages available forBPNs can also be used for influence diagrams. Besides marginal probability distributions of allvariables in the domain, the expected utilities of all decision alternatives are also evaluated.

M

G S

L

D

CA

M

G S

L

D

Figure 3.5: Sample BPN (left) and sample influence diagram (right).

The proposed framework in regard to the modeling basis

Using the sample BPN introduced above, the capability of the proposed framework with regardto the six identified aspects will be discussed.

31

3 Methodologies

Integrality/generality

The proposed framework is applied using indicators representing exposure, vulnerability androbustness. A consistent modeling at all levels is ensured by the modeling with BPNs. Theindicators may be identified and quantified for specific or general situations.

Inference

BPNs have the advantage of allowing inference based on observed evidence. It is not necessaryfor all variables explicitly considered in the BPN to be observable. The model can be updated inline with the observations using Bayes theorem. Considering Figure 3.6, any evidence receivedon any of the variables in the BPN can be used to update the probability distribution of anyvariable. For example, a site may be observed to indicate liquefaction. In this case the nodeLiquefaction receives certainty for the corresponding state "liquefaction is observed" and theother variables are updated following Bayes theorem. This feature allows the exploitation of theBPN model to answer queries and to investigate different scenarios.

M

G S

L

D

CA

Exposure

Vulnerability

Robustness

M

G S

L

D

e

Figure 3.6: Proposed framework with regard to integrality/generality.

Modularity

Modularity enables the easy adaptation of alternative methods and models to the integral model.The application of the framework using BPNs has the advantage that dependencies are explicitlymodeled. Parts of the integral BPN may be considered as a module as soon as it is separatedfrom other parts of the BPN. Considering Figure 3.7, the peak ground acceleration (G) and lique-faction (L) may be considered as a stand-alone module. Any model in geotechnical earthquakeengineering providing an estimate for liquefaction given peak ground acceleration can be appliedto quantifying this module. The only point where caution must be exercised is the interface withthe other modules, which is assured by using the same discrete states of the nodes, here G andL.

32


M

G SG

L SL

D

CA

M

Seismic hazardModule

Structural damage Module

Soil FailureModule

Consequences Module

Figure 3.7: Proposed framework with regard to modularity.

Updateability

One of the strengths of the proposed framework is its ability of systematically update of anyparameter in the model. It is possible that data in the form of reports on the damages on buildingsin an affected area may be available after an earthquake. The state of damage of each buildingin the city is certain. Any model parameter can be updated with this information (Figure 3.8).

G S

L

D

M

G S

L

D

G S

L

D

...eModel

Information on structural damage after an earthquake with M

Figure 3.8: Proposed framework with regard to updating.

Dependencies and hierarchical modeling

Explicit modeling of dependencies is one of the strengths of BPNs. For instance, research onthe correlation of the two ground motion intensity parameters in Figure 3.9 may indicate theirdependency. This is considered by an arrow from G to S. The probability table of the nodewith an incoming arrow needs to be changed accordingly. Another important feature of BPNs

33

3 Methodologies

is their advantage in hierarchical modeling. Not all variables within a model may display onlylocal dependencies. There may be parameters which influence variables globally. For example,when a BPN as given in Figure 3.5 is applied not just for one single building, but for a portfolioof structures, the node magnitude comprising the size and probability of an earthquake affectsall buildings. Hence the node magnitude is treated as a hyper parameter. The state of the nodemagnitude has to be the same for all buildings considered. This is especially important when theindividual damage nodes are combined with joint node costs (C) as illustrated in Figure 3.9.

G S

L

D

M

G S

L

D

G S

L

D

...

C

Figure 3.9: Proposed framework with regard to dependencies and hierarchical modeling.

Multi-detailing level

In principle the same BPN can be applied for different decision-making levels as illustrated inFigure 3.10. An important point is to identify the hyper parameters and to consider hierarchicalmodeling.

G S

L

D

M

G S

L

D

G S

L

D

...

C

M

G S

L

D

C

Building 1 Building 2 Building nBuilding 1

Figure 3.10: Proposed framework with regard to application for multi-detailing levels, e.g. pri-vate owner (left), insurer (right).

34

4 Models

The earthquake risk is dissected into its constituent steps (Figure 4.1). Earthquake risk beginswith rupture along a surface. Phenomena related to the rupture process are summarized into"source". The rupture process results in a number of earthquake hazards. Most fundamentally isfaulting, i.e. the surface expression of the differential movement of blocks of the Earth’s crust.Faulting is typically a long narrow feature affecting a geographically small area. A much greaterarea is affected by ground shaking, typically the primary hazard due to earthquakes. The seismicwaves resulting from the rupture process propagate along the "path" reflecting and refracting atboundaries between different rock types. The transfer of seismic waves from hard deep soilsto softer superficial soils may result in seismic energy focalization and hence higher levels ofground shaking. These are known as "site" effects, which may also result in soil failure.

Depending on the earthquake and site characteristics, liquefaction, other forms of soil failure,tsunamis, or other types of hazards may be significant source of damage. Buildings, infrastruc-ture or other structures may not fully resist to these hazards, and sustain some degree of damage.Damage to the structural components of the buildings can vary from minor cracking to collapseand/or contents of the building may be severely damaged. Structural damage has consequences,i.e.loss. Primary losses are life loss or injury, financial loss (e.g. repair or replacement cost of thebuildings) or loss of function (e.g. outfall of hospitals or emergency services). These primarylosses lead to secondary losses, such as loss of revenues resulting from business interruption.

The application of the proposed framework is prepared in this chapter by adapting state-of-

Seismic hazard

Structural Damage

Soil failure

ConsequencesFestgestein

Sediment-schichten

Source

Soil failure

Path

Damage

Site

Figure 4.1: Earthquake loss process.

35

4 Models

the-art earthquake-related models to the proposed methodology. BPN models for seismic hazard,for soil failure, for structural damage and for consequence assessment are developed. TheseBPN models form the basis of the BPN models in Chapter 5, where four example applicationsillustrate the application.

4.1 Seismic hazard

Phenomena related to source, path and site (see Figure 4.1) are considered within seismic haz-ard studies. State-of-the-art seismic hazard studies calculate the probability of occurrence of aground motion intensity parameter within a given time period due to an earthquake using earthscience models for the characteristics of earthquakes in the region of interest. Uncertainty aboutthe causes and effects of earthquakes and about the seismic characteristics of potential activefaults lead to uncertainties in the input parameters for the seismic hazard analysis. Cornell(1968) proposed a mathematical approach for systematically incorporating the uncertainties forcalculating probability of exceeding some level of ground shaking at a site. The methodology isknown as Probabilistic Seismic Hazard Analysis (PSHA) and comprises in summary four steps:

• Identification of all earthquake sources capable of producing ground motions, specifyingthe uncertainty in location. Source-to-site distance is used to characterize the decrease inground motion as it propagates away from the earthquake source. As a distance measurethe Joyner-Boore distance is used throughout the dissertation. The Joyner-Boore distanceis the closest horizontal distance to the surface projection of the rupture area.

• Characterization of the temporal distribution of earthquake recurrence, specifying the un-certainty in size and time of occurrence. The moment magnitude scale, Mw, is used tomeasure the size of earthquakes in terms of the energy released. Unlike the local mag-nitude scale, ML, which is also known as Richter scale, there is no upper limit on thehighest measurable magnitude, nor are there problems in measuring earthquake sizes atlarge distances. Throughout the dissertation the size of an earthquake is characterized bythe moment magnitude, Mw. For brevity it is however termed as "magnitude".

• Prediction of the resulting ground motion intensity as a function of location and magni-tude.

• Combination of the uncertainties in location, magnitude, time and ground motion inten-sity, using the total probability theorem.

In more advanced seismic hazard studies two types of uncertainty, namely aleatory variabilityand epistemic uncertainty, are distinguished. The uncertainty in size, location and time to thenext earthquake and the resulting ground motion are considered to be inherent in the naturalphysical process and indifferent to change in our knowledge. Hence they are called aleatoryvariability or sometimes also randomness. Epistemic uncertainty results from our imperfectknowledge of earthquakes and can be reduced with a better knowledge basis and additional data(NRC, 1997).

36

4.1 Seismic hazard

The combination of the aleatory variability in location, size and time and ground motion in-tensity yields a single hazard curve (i.e., a curve reporting annual rates of exceedance for varyinglevels of ground motion intensity). Considering different assumptions, hypotheses, models andparameter values for the location, size and time to the next earthquake and for the predictedground motion yields a suite of hazard curves representing the epistemic uncertainty. Theseepistemic uncertainties are typically organized and displayed by means of logic trees (Kulkarniet al., 1984; Coppersmith and Youngs, 1986). Quantification and treatment of these epistemicuncertainties is carefully considered in the SSHAC Level 4 methodology, where a systematicand well-balanced integration of experts was a central issue (NRC, 1997; Stepp et al., 2001;Abrahamson et al., 2002).

In this section the application of the state-of-the-art PSHA methodology using Bayesian Prob-abilistic Networks (BPN) is considered (Bayraktarli et al., 2009). After illustrating a genericapplication, aspects regarding the correlation of several ground motion intensity measures, in-corporation of epistemic uncertainties and non-Poisson earthquake recurrence are discussed. Intraditional PSHA studies, the primary analysis output is the annual frequency of the exceedanceof some level of ground shaking at a particular site. In contrast, the main output of the BPN-based seismic hazard calculation is the probability distribution of the maximum ground motionintensity measurements observed during a certain time frame. The two output formats have aone-to-one relationship. This transformation from one to the other is introduced because of theneed for these distributions in the examples given in Chapter 5.

4.1.1 A Bayesian probabilistic network for a generic seismic source

The application of the BPN approach for seismic hazard analysis is described using a genericline source as specified in Kramer (1996) (Figure 4.2a). Line sources are tectonic faults capableof producing earthquakes of different sizes. In cases where individual faults cannot be identified,the earthquake sources may be described by area zones. The application of the approach to areasources is analogous to the line source described here.

To predict the ground shaking at a site, the distribution of distances from the earthquakeepicenter to the site of interest is necessary. The seismic sources are defined by epicentersassumed to have equal probability. In a line sources these equal probability locations fall along aline; in a point source they fall on a single point; in other cases area sources are postulated. Usingthe geometric characteristics of the source, the distribution of distances can be easily calculatedfor a chosen number of discrete states (Figure 4.2b).

Gutenberg and Richter (1944) observed that the distribution of earthquake sizes in a regiongenerally follows a distribution given by:

logλm = a−bm (4.1)

where λm is the rate of earthquakes with magnitude greater than m, and a and b are constants(Figure 4.2c). a and b are generally estimated using statistical analysis of historical observations.a indicates the overall rate of earthquakes in a region, and b the relative ratio of small to largemagnitudes.

37

4 Models

The Gutenberg-Richter recurrence law mentioned above is sometimes applied with a lowerand upper bound. The lower bound is represented by a minimum magnitude mmin below whichearthquakes are ignored due to their lack of engineering importance. The upper bound is givenby the maximum magnitude mmax that a given seismic source can produce. Setting a lowerand upper magnitude for a bin, mL and mU respectively, Equation 4.2 can be used to computethe probability for the earthquake magnitudes that are between a lower and upper earthquakemagnitude (Figure 4.2d).

Rat

e of

exc

eeda

nce l

m

lmL

lmmin

lmU

lmmax

mLmmin mU mmaxMagnitude Mw

Source 1(Mw=7.3)

Site(0,0)

(-15,-30)

(-50,75)

ri

P(D

ista

nce=

r)

34 47 67

Distance [km]

0.2

0.4

0.6

5440 6027 73 80 87

P(e

)

-2 0 3e

1-1 2-3

P(M

agni

tude

=m)

4.5 5.2 6.1

Magnitude Mw

5.54.8 5.84.2 6.5 6.8 7.1

0.3

0.1

0.0

0.5

0.2

0.4

0.6

0.3

0.1

0.0

0.5

0.02

0.04

0.06

0.03

0.01

0.00

0.05

a)

b)

c)

d)

e)

f)

0.100.20

1.00

0.020.01

1 2 10 20 100Distance [km]

Peak

Acc

eler

atio

n [g

]

Faulting mechanism unspecifiedSoil, shear wave velocity=310 m/s

M=6.5

Figure 4.2: Specification of the generic line source and discrete probabilities of distance (a, b),Gutenberg-Richter recurrence law and discrete probabilities of magnitude (c, d), anda sample ground motion prediction equation and discrete probabilities for 50 statesof epsilon (e, f).

P(mL ≤ M ≤ mU | mmin ≤ M ≤ mmax) =λmL −λmU

λmmin −λmmax

(4.2)

The ground shaking resulting from an earthquake at any site can be characterized by groundmotion intensity measures, e.g. the maximum value of the acceleration time history called peakground acceleration (PGA). Alternatively, the maximum value of the acceleration, velocity ordisplacement response of a single-degree-of-freedom system (SDOF) can be used as a groundmotion intensity measure. The SDOF is represented by a mass and a stiffness, characterizing itsfundamental period or frequency. The maximum value of the response of the SDOF is termedas the spectral value at the fundamental period of that SDOF.

Earthquakes in a source result in different ground motion intensity measures at the surfacedepending on the magnitude, source-to-site distance, faulting mechanism, near-surface site con-dition, etc. Ground motion prediction equations, also referred to as attenuation functions, are

38

4.1 Seismic hazard

generally developed using statistical analysis of measurements from past earthquakes (Figure4.2e). As an example the ground motion prediction equation proposed by Boore et al. (1997) isgiven:

lnY = b1 +b2(M−6)+b3(M−6)2 +b5 ln(√

r2jb +h2)+bv ln(

Vs

VA) (4.3)

where Y is the predicted mean of ground motion intensity parameter, M is the moment mag-nitude, r jb is the Joyner-Boore distance and Vs is the average shear wave velocity to 30m. Thecoefficients b1, b2, b3, b5, h, bv and VA were determined by regression and provided in tabulatedform for certain fundamental periods. As there is significant scatter in the measured groundmotion intensities, the standard deviation of the overall regression, σlny is also provided for eachfundamental period (Figure 4.2e). The distribution of the ground motion intensity measure canbe calculated then by adding a normalized residual (i.e., a factor times σlny) to the predictedmean. This standard normal distributed factor is often denoted as epsilon, ε .

A BPN for the generic line source is constructed by conditioning the ground motion inten-sity parameter (here, spectral displacement (SD)) on magnitude M, distance R and epsilon ε .Equation 4.2 is used to calculate the probability distribution of magnitudes for 10 bins of equalintervals. Using simple geometric considerations, the distribution of distance of the site to thegeneric line source is also calculated for 10 bins. The distribution of the standard normal dis-tributed epsilon is discretized into 50 bins. The spectral displacement node is also discretizedinto 10 bins. For any of the combination of the 10 magnitudes, 10 distances and 50 epsilonsthe corresponding spectral displacement value is calculated using the ground motion predictionequation (Boore et al., 1997). The vector with the size of the number of bins for spectral dis-placement has an entry of unity in the bin, where the ground motion intensity value falls. The5000 vectors (multiplying 10x10x50) form the conditional probability table of spectral displace-ment. Having constructed the structure of the BPN and the corresponding probability tables, theBPN can be evaluated to yield the marginal distributions of any parameter in the network. Theseprobability distributions may now be used to calculate the joint distribution of all or any set ofthe parameters in the BPN using simple statistical calculation schemes. In Figure 4.3 the BPNfor seismic hazard analysis of the generic line source is illustrated with the marginal distributionof the spectral displacement. Once constructed, the BPN can be used to find the conditional dis-tribution of any parameter, given knowledge of the state of any other parameter in the BPN. InFigure 4.3 the distribution of spectral displacement is given for the situation that the magnitudeis known to be 5.5 and the distance to be 60 km. The software code for the construction andevaluation of the BPN is given in Appendix C.

The information about which earthquake scenarios are most likely to produce a specific levelof ground motion intensity can be retrieved from a PSHA computation through a process knownas deaggregation (McGuire, 1995). Using the constructed BPN and by instantiating, i.e. byassigning certainty to any state of any node, the conditional probabilities of the other nodes or thejoint probability of any node combination can easily be retrieved. A sample deaggregation resultfor the magnitude - distance deaggregation given a SD (T=0.5 s) between 4 mm and 5 mm isgiven in Figure 4.4. To verify the constructed BPN, the same magnitude-distance deaggregationis computed using a traditional PSHA analysis procedure (Figure 4.4). The small discrepancies

39

4 Models

SD

Magnitude Distance e

P(SD

=sd)

Spectral Displacement SD [mm]

0<=S

D<1

P(SD

=sd|

m=5

.5,r=

60km

)


SD

0.2

0.4

0.6

0.3

0.1

0.0

0.5

0.2

0.4

0.6

0.3

0.1

0.0

0.5

a)

b) c)1<

=SD

<22<

=SD

<33<

=SD

<44<

=SD

<55<

=SD

<66<

=SD

<10

10<=

SD<4

0

40<=

SD<7

070

<=SD

<100

0<=S

D<1

1<=S

D<2

2<=S

D<3

3<=S

D<4

4<=S

D<5

5<=S

D<6

6<=S

D<1

010

<=SD

<40

40<=

SD<7

070

<=SD

<100

Figure 4.3: BPN for a generic seismic line source (a), discrete probabilities of SD (T=0.5 s)evaluated using the BPN for the probability distributions given in Figure 4.2 (b, c).

in the probabilities arise from differences in the discretisation schemes between the traditionalPSHA analysis and the BPN.

4.1.2 Incorporating correlation of ground motion intensity parameters

Reliability assessments that attempt to simultaneously consider structural and geotechnical fail-ures are currently not applicable, because structural and geotechnical responses are generallypredicted using different ground motion intensity measures, and the tools are not available fordetermining a probabilistic characterization of the joint occurrence of these parameters (Baker,2007). Structural response (and structural failure) is often predicted using elastic spectral dis-placement (SD) (Pinto et al., 2004). Liquefaction failure, on the other hand, is typically predictedusing peak ground acceleration (PGA) (Cetin et al., 2004; Youd et al., 2001).

The correlation of the ground motion intensity parameters PGA and SD(T ), which is usedin developing BPN models, is developed based on Baker (2007). First a large set of recordedground motions is selected. For each recorded ground motion (a vector of PGA and spectral pa-rameters for given fundamental periods, T ), the corresponding ground motion intensity measuresare calculated using ground motion prediction equations. For SD the ground motion predictionequation of Abrahamson and Silva (1997) and for PGA the ground motion prediction equationof Boore et al. (1997) is used.

40

4.1 Seismic hazard

Figure 4.4: Deaggregation by magnitude and distance for SD= 4.4mm using traditional PSHA(left), and for 4 mm<=SD<5 mm using BPN (right).

Once the means and standard deviations of the intensity for a given ground motion is com-puted, a normalized residual, ε , that indicates the number of standard deviations, the givenobservation is away from the mean prediction can be evaluated:

ε =lnx−µlnX

σlnX(4.4)

where x is the observed ground motion intensity (defined using one of the above parameters,and here denoted as X), µlnX is the predicted mean value of the logarithm of that intensity (givenmagnitude, distance, etc.) and σlnX is the predicted standard deviation of the log intensity.These "epsilons" represent the record-to-record aleatory variability that is not considered bythe ground motion prediction equation. This variability is explicitly considered in probabilisticassessments such as probabilistic seismic hazard analysis (PSHA). A given ground motion willhave a different ε value for each ground motion intensity measure considered, and it is thecorrelation among these different ε values that must be considered if seismic reliability analysisis to be performed (Baker and Cornell, 2006).

Using this approach, empirical correlation coefficients are computed for the large database ofε values. The following piecewise linear equation provides a good fit to the observed values, asa function of the fundamental periodof a single-degree-of-freedom system, T :

ρPGA,SD(T ) =

0.500−0.127lnT if 0.05 ≤ T < 0.110.968+0.085lnT if 0.11 ≤ T < 0.250.568−0.204lnT if 0.25 ≤ T < 5.00

(4.5)

41

4 Models

The correlation of the ground motion intensity parameters PGA and SD(T ) is considered inthe BPN by conditioning the epsilon values on each other. εPGA is discretized from the standardnormal distribution into 10 states. εSD(T ) on the other hand is a conditional normal distribution

with a mean of ρPGA,SD(T )εPGA and a standard deviation of√

1−ρPGA,SD(T )2 and discretized also

into 10 states. Since the correlation coefficient depends on the fundamental period, the node εSD

is also dependent on the fundamental period. The extended BPN is given in Figure 4.5 and asample output in the form of a magnitude-distance deaggregation and a PGA-SD deaggregationis given in Figure 4.6.

SDPGA

Magnitude Distance

εSDεPGA

Period

Figure 4.5: BPN considering correlation of ground motion intensity parameters PGA and SD.

0.03 0.08 0.13 0.18 0.23 0.28 0.33 0.38 0.43 0.48 1.03.0

2.04.0 5.0 6.0 10.0 40.0 70.0 80.0

Spectral Displacement [mm]PGA [g]

Like

lihoo

d0.1

00.2

00.3

00.4

0

Figure 4.6: Deaggregation by magnitude and distance using BPN for PGA=0.05g,SD(T=0.5s)=3mm (left) and contribution by PGA and SD for M=6.8, R=27km(right).

4.1.3 Incorporating model uncertainties

For one particular seismic hazard model (defined by specifying a source model, a recurrencemodel and a ground motion prediction model) the aleatory variability described by that model issystematically considered. But there is still uncertainty about the best choices for elements of the

42

4.1 Seismic hazard

seismic hazard model itself. This is now commonly addressed by combining the uncertaintiesabout the various inputs in logic trees (Kulkarni et al., 1984; Coppersmith and Youngs, 1986;NRC, 1997). Each branch of a logic tree represents a set of chosen elements for a seismic hazardmodel. For each of the seismic hazard models the hazard calculations are performed and a singlehazard curve representing ground motion versus annual frequency of exceedance is produced.The relative weighting of each hazard curve is then determined by multiplying the assignedweights in each of the branches. From this set of hazard curves a mean, a median and curves fordifferent fractiles can be defined.

The BPN for the seismic hazard model introduced before will now be extended to incorporatemodel uncertainties. For each of the elements producing branches in a logic tree, a node isintroduced into the network and the required dependencies with the existing nodes are set usingadditional arrows. The simple logic tree shown in Figure 4.7 allows uncertainty in the selectionof models for ground motion prediction equations and maximum magnitude to be considered.The ground motion prediction equations of Boore et al. (1997) and Abrahamson and Silva (1997)are considered. For illustrative purposes weights of 0.7 and 0.3 assigned to these, respectively.At the other level of nodes, again for illustrative purposes weights of 0.4 and 0.6 are assigned tothe maximum magnitudes which the single line source is capable of producing. A sample outputof the BPN in Figure 4.7 is given in Figure 4.8.

SDPGA

Magnitude Distance

eSDePGA

Period

PredictionModel

Max MModel

Maximum Magnitude Model

Ground Motion PredictionModel

Abrahamson & Silva

Boore et al.

MW =7.3

MW =7.7

MW =7.3

(0.3)

(0.4)

(0.6)

(0.4)

(0.6)

(0.7)MW =7.7

Figure 4.7: Logic tree and corresponding BPN considering modeling uncertainties.

4.1.4 Incorporating time-dependent seismic hazard

Earthquake occurrences are stochastic in nature, both in time and space. Small and mediummagnitude earthquakes may occur independently implying a Poisson model. Large magnitudeearthquakes on a particular fault segment, however, should not be independent from each otheraccording to the elastic rebound theory (Reid, 1911). As earthquakes occur in order to releasethe stress accumulation in a fault, the occurrence of a large earthquake should reduce the chancesof the occurrence of a following independent large earthquake in the same source. Paleoseismicstudies on fault slip data led to the ’characteristic earthquake’ recurrence model (Kramer, 1996).The seismic sources tend to regularly generate earthquakes of similar sizes near to the maxi-mum magnitude known as characteristic earthquakes. This tendency is not observed for smallerearthquakes, which occur more or less randomly. Hence, earthquakes are classified into two

43

4 Models

Boore et al.

7.7

Like

lihoo

d0.2

50.5

00.7

51.0

0

Ground Motion Prediction EquationMaximum Magnitude Mw

Abrahamson et al.7.3

Boore et al.

7.7

Like

lihoo

d0.2

50.5

00.7

51.0

0

Ground Motion Prediction EquationMaximum Magnitude Mw

Abrahamson et al.7.3

Figure 4.8: Contribution by ground motion prediction equation and maximum magnitude choicefor PGA=0.15g, SD=6mm (left) and for PGA=0.25g, SD=5mm (right).

groups; small/ medium size earthquakes and characteristic earthquakes. For small and mediumsize earthquakes a time-independent recurrence model and for characteristic earthquakes a time-dependent recurrence model is assumed. A review of non-Poisson models is presented in Anag-nos and Kiremidjian (1988). The application of BPNs for non-Poisson recurrence models isillustrated using the Brownian Passage Time (BPT) model developed by Matthews et al. (2002)and used in Takahashi et al. (2004).

The occurrence of earthquakes of magnitude m constitutes a renewal process with fT (t,m) de-noting the probability density function (PDF) of the interarrival times, t. Such a process becomesa Poisson process, if fT (t,m) is assumed as an exponential distribution. For all non-characteristicearthquakes in the source a Poisson model and for characteristic earthquakes a more generalizedrenewal model is assumed. Setting the origin of time to the most recent occurrence of the char-acteristic earthquake and denoting the waiting time to the n-th occurrence of the characteristicevent by Wn, the conditional PDF of the waiting time to the n-th characteristic event, given thatno characteristic earthquake has happened before the start of the operating time of the struc-tures of interest (t = t0) is denoted by fWn(t,m|W1 > t0). In cases where analysis is performedfor structures with a life span much shorter than the mean interarrival time of the characteristicearthquake, it is reasonable to consider the contribution to seismic activity only from the firstcharacteristic earthquake. As the results of this section will be used in Chapter 5 on earthquakerisk for cities with residential buildings, the contributions of the 2nd and subsequent events tothe rate of activity are neglected (Takahashi et al., 2004).

A special case of the renewal process for which the interarrival times are exponentially dis-tributed is the Poisson process. The interarrival times for the non-characteristic earthquakes aremodeled by the exponential distribution:

fT (t,m) = λ (m)exp[−λ (m)t] (4.6)

44

4.1 Seismic hazard

where λ (m) is the constant mean occurrence rate of earthquakes greater than m. For char-acteristic earthquakes, the interarrival times are modeled by the Brownian Passage Time (BPT)distribution. This model requires two parameters: mean recurrence interval (µ) and aperiodicity(α) of the mean period between events (coefficient of variation):

fT (t) =√

µ

2πα2t3 exp[−(t −µ)2

2µα2t3 ] (4.7)

Since the Poisson process is memoryless, the conditional distribution of the waiting time tothe first event, given that no event has occurred prior to the operating time of the structure,remains exponential with the time origin shifted to t0.

fW1(t,m |W1 > t0) = ν(m)exp[−ν(m)(t − t0)] (4.8)

The conditional PDF for the first characteristic earthquake, given that no characteristic earth-quake has occurred prior to the operating time of the structure, t0 is given by

fW1(t,mchar |W1 > t0) =fW1(t,mchar)

1−∫ t0

0 fW1(τ,mchar)dτ(4.9)

For the generic line source a mean recurrence time of 100 years and an aperiodicity of 0.5 isassumed. It is further assumed that the last characteristic earthquake occurred 10 years ago. Aperiod of 50 years from now is considered as this is the residential building lifespan. Figure 4.9illustrates the rate of occurrence of the next characteristic earthquake. This time-varying rate isthen combined with the Gutenberg-Richter recurrence relation (Figure 4.9, right). For each yearof the building lifespan, the rate of occurrence of the characteristic earthquake (λmchar ) is equalto the time-varying rate, whereas the rate of smaller and medium size earthquakes is constant(following the Poisson model). The distribution of magnitudes is then computed using equations4.10 and 4.11 for each of the 50 years.

The mean occurrence rate of the characteristic earthquake for each year of the lifespan of thebuilding is read from the conditional BPT distribution and assigned to the magnitude-Rate ofexceedance curve. Equation 4.10 can be used to compute the probability for the magnitudesother than the characteristic earthquake for each year.

P(mL ≤ M ≤ mU | mmin ≤ M ≤ mmax′ ∩M 6= mchar) =λmL −λmU

λmmin −λmmax′ +λmchar

(4.10)

The probability of a characteristic earthquake occurring in that year, given that there is anearthquake larger than a minimum magnitude, can be computed using Equation 4.11.

P(M = mchar) =λmchar

λmmin −λmmax′ +λmchar

(4.11)

For each year of the lifespan of the building a BPN is constructed (Figure 4.10). The prob-ability distribution of the magnitude for that year is assigned to the magnitude node and thedistributions of PGA and SD are calculated using the BPNs. In Figure 4.11 sample results of the

45

4 Models

Rat

e of

exc

eeda

nce l

m

lmL

lmmin

lmU

lmchar

mLmmin mU mchar

Time [year]50 150 300

0.00

0.01

0.02

200100 2500

Magnitude Mw

Occ

urre

nce

rate

lm

char t0=5 years

t0=50 years

lmmax’

mmax’

Figure 4.9: Mean occurrence rates using BPT model for mchar = 7.3 and extended Gutenberg-Richter equation considering characteristic earthquakes.

BPNs for t0 = 10 years, t0 = 50 years and for comparison for a Poisson process for all magni-tudes are given. Minor changes in the probabilities can be observed for the lower PGA and SDvalues in the renewal model as they are caused by the (Poissonian) small magnitude events. Incontrast, the probabilities for higher PGA and SD values change more over time as the proba-bility of occurrence of a characteristic earthquake changes. Here, only the methodical issues arediscussed, and the application is shown. In the following section an application to a real casewill be illustrated, and in Chapter 5 the influence of considering the renewal process on a riskmanagement problem will be discussed.

SDPGA

Magnitude Distance

εSDεPGA

Period

T=1 year

SDPGA

Magnitude Distance

εSDεPGA

Period

T=2 year

...

Figure 4.10: BPN considering epistemic uncertainty.

4.1.5 PSHA using BPN for Adapazari, Turkey

The application of PSHA using BPN on a real case is considered for the city of Adapazari. Thisregion in Northwestern Turkey has been a site of many severe earthquakes. In Chapter 5 theseismic risk for the city is considered. The city includes the most affected region during the Ko-caeli Mw7.4 earthquake as well as areas with liquefaction during the same event (DRM, 2004).

46

4.1 Seismic hazard


Peak Ground Acceleration PGA [g]

100

10-1

10-2

10-3

10-4

10-5

10-6

100

10-1

10-2

10-3

10-4

10-5

10-6

0.50

0

0.42

50.

375

0.32

50.

275

0.22

5

0.17

50.

125

0.07

50.

025

100.

0

55.0

25.0

8.00

5.50

4.50

3.50

2.50

1.50

0.50


0.50

0

0.42

50.

375

0.32

50.

275

0.22

5

0.17

50.

125

0.07

50.

025


0.50

0

0.42

50.

375

0.32

50.

275

0.22

5

0.17

50.

125

0.07

50.

025


100.

0

55.0

25.0

8.00

5.50

4.50

3.50

2.50

1.50

0.50


100.

0

55.0

25.0

8.00

5.50

4.50

3.50

2.50

1.50

0.50

a) c) e)

b) d) f)

Figure 4.11: Discrete probabilities of PGA and SD evaluated using the BPN in Figure 4.10 as-suming a Poisson process for all magnitudes (a, b), assuming a more generalizedrenewal model for the characteristic earthquake, results given for t0 = 10 years (c,d) and for t0 = 50 years (e, f).

Hence, the output in this section in the form of probability distributions of peak ground acceler-ation (for liquefaction analysis) and spectral displacement (for structural response analysis) willbe used as input for the earthquake risk studies.

In Figure 4.12 the northwestern part of Turkey with the city of Adapazari (in red) is illustrated.The spatial distribution of the seismicity using the earthquake catalogue from the InternationalSeismological Centre (ISC) is given in the top-left figure. The fault segmentation model for theregion as shown in Figure 4.12 top-right by Erdik et al. (2004) is used for the characteristic earth-quake recurrence model. The characteristic earthquake parameters associated with the segmentsare given in Table 4.1. For the non-characteristic magnitudes the zonation model proposed byAtakan et al. (2002) is used. There, the earthquake sources are based on a gross zonation takinginto account the entire North Anatolian fault zone as a single zone. In the west of Adapazari thezone is divided into a northern and a southern strand following the general trend of the fault sys-tem (Figure 4.12 bottom-left). The relevant source parameters with the areas are given in Table4.2. Thereby, the regional rate of earthquake activity, a,, is calculated by relating the area of thesource within the considered 100km radius to the area of that source. In Figure 4.12 bottom-right the two zonation models are combined in order to use the fault segmentation model for thecharacteristic and the area sources for the non-characteristic earthquakes. Earthquakes within100 km of the city center are included in the analysis.

The BPN in Figure 4.5 is applied to the example application. The earthquakes are classifiedinto six states according to their magnitudes, 4.75 ≤ Mw <5.25, 5.25 ≤ Mw <5.75, 5.75 ≤ Mw<6.25, 6.25 ≤ Mw <6.75, 6.75 ≤ Mw <7.25, 7.25 ≤ Mw <7.75, and their representative valuesas Mw=5, Mw=5.5, Mw=6, Mw=6.5, Mw=7, Mw=7.5. The magnitude range 7.25 ≤ Mw <7.75

47

4 Models

41O

30O 32O40O

42O

31O29O

30O 32O31O29O

41O

40O

42O

41O

30O 32O40O

42O

31O29O

30O 32O31O29O

41O

40O

42O

41O

30O 32O40O

42O

31O29O

30O 32O31O29O

41O

40O

42O

S1S4 S3 S2S5

S25

S15

S14

S13

S21S19

S22S12

41O

30O 32O40O

42O

31O29O

30O 32O31O29O

41O

40O

42O

S1S4 S3 S2S5

S25

S15

S14

S13

S21S19

S22S12

A5

A1

A2 A4

Figure 4.12: Spatial distribution of the seismicity in the region (Atakan et al., 2002) (top-left),seismic zonation models proposed by Erdik et al. (2004) (top-right) and Atakanet al. (2002) (bottom-left), hybrid zonation with the considered area around Ada-pazari (bottom-right).

Table 4.1: Characteristic earthquake parameters associated with the segments Erdik et al. (2004).Segment Last charac− COV Mean recurrence Characteristic Time since last

teristic EQ time magnitude characteristic EQS1 1999 0.5 140 7.2 9S2 1999 0.5 140 7.2 9S3 1999 0.5 140 7.2 9S4 1999 0.5 140 7.2 9S12 1967 0.5 250 7.2 41S13 − 0.5 600 7.2 1000S14 − 0.5 600 7.2 1000S21 1999 0.5 250 7.2 9S22 1957 0.5 250 7.2 51

is assumed to represent the characteristic earthquakes. The occurrence of events belonging to thefirst five states is modeled as Poisson events, while the occurrence of characteristic earthquakesclassified into the last state are modeled by a non-Poisson renewal model. The probability distri-butions of the magnitudes are calculated for each year as described in the preceding section. Theearthquake distance node is discretized into five states; R=10km, R=30km, R=50km, R=70km,R=90km. Simple geometrical considerations as illustrated above are used to calculate the prob-ability distributions of the earthquake distance, R. The node of the standard normal distributed

48

4.1 Seismic hazard

Table 4.2: Parameters of the area sources.Segment Area source a a

′b Characteristic/Max.

magnitudeS1 A1 2.14 0.98 1.12 7.2S2 A1 2.14 1.28 1.12 7.2S3 A1 2.14 1.55 1.12 7.2S4 A1 2.14 1.49 1.12 7.2S12 A1 2.14 1.31 1.12 7.2S13 A2 2.85 2.48 1.00 7.2S14 A2 2.85 2.61 1.00 7.2S21 A1 2.14 0.92 1.12 7.2S22 A1 2.14 1.15 1.12 7.2

Background 0.47 0.47 1.00 5.5

parameter εPGA is discretized into 10 equally spaced states between -3.5 and 3.5. The correlationof the ground motion intensity parameters PGA and SD is considered in the BPN by condition-ing the εSD on εPGA. The correlation coefficient ρ is calculated using Equation 4.5 for T = 0.64s,which is the fundamental period of the 5-story buildings considered in Chapter 5 where the re-sult of these analyses are used. εSD is a conditional normal distribution with a mean of ρεPGA

and a standard deviation of√

1−ρ2 and discretized also into 10 states. Evaluating each statein εPGA from -3.5 to 3.5 with the corresponding correlation coefficient results in a probabilitydistribution of εSD, which is flatter than for εPGA.The range of values for the equally spaced 10discrete states for the node εSD is hence taken from -5 to 5.

For each of the nine segments in the zonation model and each of the 50 years a BPN as givenin Figure 4.5 is constructed. The probability tables of the five nodes other than the magnitudenode are constructed with the specification of the segments. For the magnitude node the prob-ability tables are calculated for each year according to the equations 4.10 and 4.11 (see Figure4.9). Thus 450 BPNs are constructed, which yield a marginal distribution for PGA and SD foreach segment and each year through evaluation of the corresponding BPN. In Figure 4.15 andFigure 4.14 sample results for each segment for the years 2018, 2038 and 2058 are given. Thedistributions of PGA and SD for each segment are also given for the case, where the occur-rence of all the magnitudes is modeled as Poisson events. As the distribution of PGA and SDare calculated with the condition that at least one earthquake larger than Mw=5 will occur, thefinal results when using the output for further analyses will have to be multiplied by the rate ofexceeding Mw=5.

Calculation scheme for seismic hazard model

The discrete probabilities of PGA and SD (Figure 4.14 and 4.15) are calculated using the schemegiven in Figure 4.13. The hazards from the seismic line sources at the site are assumed to beindependent, i.e. rupturing in cascades (immediately one line source after the other) and henceproducing larger rupture areas is disregarded.

49

4 Models

The BPN in Figure 4.5 is constructed within the software HUGIN. Only the nodes and arrowsneed to be specified at this step. The number of discrete states in the nodes and the probabilitytables are controlled in MATLAB. The main file BPN_PSHA_Adapazari.m calculates the proba-bility distributions for PGA and SD for each seismic source as illustrated in Figure 4.14 and 4.15.The file EPS_PGA_5.m discretizes the standard normal distributed parameter εPGA. The corre-lation coefficient between PGA and SD is calculated with the file SD_PGA_cor.m. Given thecorrelation coefficient which depends on the fundamental period of the structures considered inthe city, the file EPS_SD_5.m discretizes the node εSD. For both magnitude-recurrence relation-ships considered, the time-independent (Poisson) and time-dependent (Non-Poisson) cases, thefiles EQ_M_5.m and EQ_M_NonPoisson_5.m calculate the discrete probabilities for the statesin the node ’Magnitude’. The probability distribution of the distances from the seismic sourceto the site of interest is calculated in the file Line_EQ_R_5.m. Files calculating the probabilitydistribution of the distances from area and point sources may also be incorporated. For anycombination of magnitude, distance and epsilon given the seismic source, the ground motions atthe site are calculated in the files PGA_5.m and SD_5.m. The conditional probability tables ofPGA and SD are then specified in the main file by classification of each of these ground motionsinto the discrete states of the nodes PGA and SD. All files are given in Annex C.

The main file evaluates the given BPN with the assigned conditional and unconditional prob-ability tables. The inference engine of HUGIN which is called in the MATLAB environmentevaluates the BPN. The probability distribution for the time-independent and the time-dependentcases are illustrated in Figure 4.14 and 4.15.

50

4.1 Seismic hazard

Construction of BPN in Figure 4.4

Discretisation of node 'SD' Discretisation of the node'PGA'

Correlation of SD withPGA (or 'eSD ' with

‘ePGA‘) given 'Period'

Discretisation of the node'eSD '

Discretisation of thestandard normal distributed

node 'ePGA '

Discretisation of node'Magnitude' for each year

in the considered time

Discretisation of the node'Magnitude'

Magnitude-RecurrenceRelation ?

Ground motion predictionequation

Discretisation of node' Distance'

BPN_PSHA_Adapazari.m

SD_PGA_cor.m

EPS_SD_5.m

EPS_PGA_5.m

EQ_M_NonPoisson_5.mEQ_M_5.m

Time-dependantPoisson

bjf_atten_5.m

Line_EQ_R_5.m

SD_5.mPGA_5.m

BPN_PSHA_Adapazari.net

Figure 4.13: Calculation scheme for the BPN in Figure 4.5.

51

4 Models

0

.002

.004

.006

.008.01

Occ

urre

nce

rate

2009 2509 0.1

Year PGA [g] PGA [g] PGA [g] PGA [g]

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

1.8

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.8

1

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

0

.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

0.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

T=2018 T=2038 T=2058

1509

2209 2409Year

2009

Sour

ce S

2

Poisson Model Non-Poisson Renewal Model

Sour

ce S

1So

urce

S4

Sour

ce S

3So

urce

S21

Sour

ce S

12So

urce

S22

Sour

ce S

13So

urce

S14

0.3

0.5

0.7

0.9

1.1

1.3

0.1

0.3

0.5

0.7

0.9

1.1

1.3

0.1

0.3

0.5

0.7

0.9

1.1

1.3

0.1

0.3

0.5

0.7

0.9

1.1

1.3

Figure 4.14: For each seismic source the mean occurrence rates using BPT model for the charac-teristic earthquakes (first block), discrete probabilities of PGA evaluated using theBPN in Figure 4.5 assuming a Poisson process for all magnitudes (second block),assuming a more generalized renewal model for the characteristic earthquake, re-sults given for T=10 years (third block), for T=30 years (fourth block) and for T=50years (fifth block).

52

4.1 Seismic hazard

0

.002

.004

.006

.008.01

Occ

urre

nce

rate

2009 2509 1.25

Year SD [cm] SD [cm] SD [cm] SD [cm]

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Prob

abili

ty

0

.2

.4

.6

.81

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

Occ

urre

nce

rate

Sour

ce S

2

0

.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

0.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

0

.002

.004

.006

.008.01

Poisson ModelT=2018 T=2038 T=2058

Non-Poisson Renewal Model

1509

2209 2409Year

2009

Occ

urre

nce

rate

Sour

ce S

1So

urce

S4

Sour

ce S

3So

urce

S21

Sour

ce S

12So

urce

S22

Sour

ce S

13So

urce

S14

0.25

3.50

7.50

20.0

40.0

75.0

1.25

0.25

3.50

7.50

20.0

40.0

75.0

1.25

0.25

3.50

7.50

20.0

40.0

75.0

1.25

0.25

3.50

7.50

20.0

40.0

75.0

Figure 4.15: For each seismic source the mean occurrence rates using BPT model for the char-acteristic earthquakes (first block), discrete probabilities of SD evaluated using theBPN in Figure 4.5 assuming a Poisson process for all magnitudes (second block),assuming a more generalized renewal model for the characteristic earthquake, re-sults given for T=10 years (third block), for T=30 years (fourth block) and for T=50years (fifth block).

53

4 Models

4.2 Soil failure

Earthquakes generally cause damage to buildings through ground shaking. They may also causedamage locally through ground deformations to pipelines, sewers and buildings. Ground failuresrange from slope failures in mountains to ground cracking in alluvium-filled valleys (Holzeret al., 1999).

In Section 4.1 the probabilities of the occurrence of ground shaking intensity parameterswithin a given time period were estimated. In doing so, the influence of the soil conditionswere taken into account within the ground motion prediction equations. However, as extensiveliquefaction problems occurred in the city considered for the examples in this dissertation dur-ing past earthquakes, the soil failure mechanism liquefaction is explicitly considered. Duringan earthquake, loosely packed water-saturated sediments near the ground surface may lose theirstrength and stiffness as a result of pore water pressure increase. This phenomenon, knownas liquefaction, may cause serious damage to the built environment as experienced during theearthquakes in Niigata (1964), Loma Prieta (1989) and Kocaeli (1999) (Geoengineer, 2006).

Soil liquefaction may be quantified using deterministic or on probabilistic techniques basedeither on laboratory tests or empirical correlations of in-situ index tests with field case perfor-mance data. The deterministic empirical correlation in using Standard Penetration Test (SPT)blow number count proposed by (Seed and Idriss, 1971) is widely used in practice to evaluatethe potential for soil liquefaction. Here an empirical liquefaction criterion proposed by Cetinet al. (2004) is used. The limit state function is expressed as:

g(N1,60,CSReq,Mw,FC,σ′v,εL) = N1,60(1+0.004FC)−13.32lnCSReq (4.12)

−29.53lnMw −3.70lnσ

′v

Pa+0.05FC +16.85+ εL

where N1,60 is the SPT blow count, CSReq is the equivalent cyclic stress ratio, Mw is themoment magnitude of the earthquake, FC is fines content, σ

′v is effective vertical stress, Pa is

the standard atmospheric pressure in the same units used for σ′v and εL is a random variable

representing model uncertainty. The CSReq is calculated with:

CSReq = 0.65PGA

gσv

σ′vrd (4.13)

where g is the acceleration of gravity, σv is total vertical stress, and rd is the nonlinear shearmass participation factor. For rd , the model of Cetin et al. (2004) is adopted:

rd =1+

−23.013+2.949PGA+0.999Mw+0.0525V ∗s,12m

16.258+0.201exp0.341(−d+0.0785V∗

s,12m+7.586)

1+−23.013+2.949PGA+0.999Mw+0.0525V ∗

s,12m

16.258+0.201exp0.341(0.0785V∗

s,12m+7.586)

+ εrd (4.14)

σrd ={

d0.850.0198 if d ≤ 12[m];120.850.0198 if d > 12[m].

(4.15)

54

4.2 Soil failure

where V ∗s,12m is the representative shear wave velocity over the top 12m soil at the site and is

assumed to be deterministically defined as 150m/s, εrd is a Gaussian random variable represent-ing model error and d is the depth of the critical liquefaction susceptible layer. The soil layerwith the lowest SPT blow number is assumed to be the critical liquefaction susceptible layer forthe entire site. The soil profile may be considered as a serial system and liquefaction occurrencein one layer may cause liquefaction on the site. Cetin et al. (2004) specify the following distri-butions for the model errors: εL has a Gaussian distribution with a mean of zero and a standarddeviation of 2.7, and εrd is Gaussian with zero mean and standard deviation equal to σrd . Theparameters required for the evaluation of liquefaction triggering are thus the SPT blow numberNm, the fines content FC, the soil classification according to the Unified Soil Classification Sys-tem USCS, the ground water table GW and the depth of the critical layer d. The last three ofthese five parameters are required for calculating the total and effective stress within the layer.The joint probability distribution of magnitude (M) and peak ground acceleration (PGA) are ob-tained from the results of the seismic hazard model and are assumed to be perfectly correlatedin the city (i.e. one value is assumed throughout the city for M and PGA).

The evaluation of probability of liquefaction triggering for a single point thus requires theevaluation of equations 4.12 through 4.15. The soil database for the region Adapazari consistsof 312 borings (DRM, 2004). Not all borings provide data on all of the five parameters (seeAppendix B). For 312 of the borings the SPT blow number of the critical layer is available.The depth of the critical layer is available for 312, the ground water table for 254, the USCSclassification for 120 and fines content for 51 of these borings. For those locations where allof the required parameters are available, the probability of liquefaction may be calculated. Inrisk assessment and management studies estimates for locations in the city where no data isavailable, are also required. There are several ways of estimating unknown values at specificlocations, which will be briefly discussed below (Isaaks and Srivastava, 1989).

Polygonal method

In the polygonal method the sample value that is closest to the point to be estimated is assigned asthe unknown value. The polygonal estimator can be regarded as a weighted linear combinationthat gives all the weight to the closest sample value (Isaaks and Srivastava, 1989).

Triangulation

The polygonal method may result in sharp discontinuities, as with changing distance a differ-ent sample value with a different value may become the closest value. Discontinuities in theestimated values are generally not desirable. Real values may show discontinuities, but the dis-continuities mentioned arise from the procedure itself and have nothing to do with reality. Thetriangulation method overcomes this problem by fitting a plane through the three closest samplesand assigning the unknown value of the location by substituting the appropriate coordinates.

55

4 Models

Local sample mean

The polygonal method uses only the nearest sample and the triangulation method the closestthree samples to the location to be estimated. The local sample mean method considers all nearbysamples by assigning equal weight to them. This estimate is heavily influenced by extremevalues and hence may result in much higher estimates than the aforementioned two methods.

Inverse distance methods

Local sample mean method gives naively equal weight to all nearby samples disregarding dis-tance to the point to be estimated. An intuitive way which also follows the first law of geology’near things are more closely related than distant things’, lends weight to each sample inverselyproportional to its distance from the point to be estimated. A general formulation of the estimatorfor the unknown point can be written as:

x̂ =∑

ni=1

1dp

i xi

∑ni=1

1dp

i

(4.16)

where di are the distances from each of the n sample locations to the point being estimated, xi

are the sample values and p is a power factor. With a decreasing p the weights assigned to thesamples become more uniform. The local sample mean is the extreme case when p approaches 0,i.e. equal weight is assigned to all nearby samples. The polygonal method is the other extreme,where p approaches infinity, i.e. all the weight is assigned to the closest sample. Generally p ischosen as 2 due to efficiency in the calculations (Isaaks and Srivastava, 1989).

The appropriateness of these deterministic methods in estimated values for several locationsat the same time can be judged based on a number of different criteria; the similarity of the dis-tribution of the estimates to the estimates of true values, the proximity of the means or mediansthereof, and the proximity of the variability of the estimates to the true variability. The choiceof criteria depends on the application. Different methods may be more suitable for differentestimation criteria.

Kriging

Kriging is a geostatistical approach to modeling. The theory behind kriging was developed byMatheron (1963) based on the work of Krige (1951). Instead of weighing nearby samples bysome factors depending on distance, kriging relies on the spatial correlation structure of the datato determine the weighing factors. Kriging aims to reduce the standard deviation of the differ-ence between the estimated and the true value. Different types of kriging exist. Simple kriginguses the assumption of a known and constant mean value for the parameter to be estimated.In ordinary kriging the mean value is assumed to be constant and unknown. Universal krigingassumes a linear mean value. Indicator kriging uses indicator functions instead of the processitself.

The spatial correlation can be described by variograms, covariograms or correlograms (Akinand Siemens, 1988). In principle the same information is given in all three representations. The

56

4.2 Soil failure

means most commonly used to represent spatial dependency in geostatistics is the variogram.The variogram, denoted by γ(h) is equal to the variance of the increment in data points separatedby a distance h. The notation "semivarigram" is also adopted, where one half of the variance ofthe increment is used:

γ(h) =12

var[Z(u)−Z(u+h)] (4.17)

where Z(u) is the distribution of the random variable at location u. The vector distance haccounts for both length and direction. For each distance and direction pairs from the samplesare derived and the half of the variance of the differences in the data pairs is calculated. Re-peating this for multiple distances and directions provides data pairs, which when plotted in asemivariance to distance plot, yields the semivariogram. This empirical semivariogram can beused to assign an analytical formulation for a semivariogram for computational convenience.

The covariogram illustrates the expected value of the covariances given the vector distance h:

C(h) = E[[Z(u)−E[Z(u)]][Z(u+h)−E[Z(u+h)]]] (4.18)

With second-order stationarity the relation between covariogram and semivariogram is:

γ(h) = γ(∞)−C(h) = C(0)−C(h) (4.19)

A covariogram normalized by the variances in Z(u) and Z(u+h) is referred to as a correlo-gram.

The empirical semivariograms for the parameters Nm, FC, GW and d are estimated based onthe data given in Appendix B. Figure 4.16 illustrates the empirical semivariogram for Nm. Here,a semivariogram function referred to an exponential model was chosen for Nm:

γ(h) = C · (1− exp(−3h

a)) (4.20)

where C is the sill and a is the range of the semivariogram. Sill is the semivariance value atwhich the variogram levels off. Range is the distance at which the semivariogram reaches thesill value. Beyond the range, correlation is assumed to vanish. The empirical semivariogram forNm is modeled by an exponential model with a sill value of 0.85 and a range of 5000. This expo-nential model is also plotted in Figure 4.16. The spatial characteristics of the other parametersFC, GW and d are analogously modeled by an exponential semivariogram with a sill of 0.9 anda range of 5000.

Stochastic simulation

In the interpolation algorithms described briefly above, the goal is to provide a best estimate ofthe variable without specific regard to the resulting spatial characteristics of the estimates takentogether. For instance, in kriging a set of local values is provided with emphasis on local accu-racy. Furthermore, only an incomplete measurement of this local accuracy is provided as joint

57

4 Models

***

***

***** ** * ***

*

***

**

**

**

** *****

***

****

*

*** **** * *** *

**

****

*

*

100000

1

2000 3000 4000 5000 6000

Separation distance [mm]N

orm

aliz

ed c

ovar

ianc

e

0.85

*** *****

********

** *

*

***

**

***

****

****

*

Figure 4.16: Empirical semivariogram (crosses) with an exponential model (line) for Nm.

accuracy is not considered when several locations are under consideration. In stochastic simula-tion the missing soil variables are simulated, accounting for the uncertainty of the variables, fortheir spatial dependency and for the observed values obtained in the borings.

In the region of Adapazari, Turkey, for each of the grid point of 100 x 150 elements witha dimension of 100 m x 100 m the soil properties are required. Using a sequential Gaussiansimulation approach (Deutsch and Journel, 1997) a set of random fields is constructed. Usingthe semivariogramms of Nm, FC, GW and d random fields are modeled. All of these randomfields are used in a crude Monte Carlo scheme to calculate the probability of liquefaction foreach grid point in the region using the equations 4.12 to 4.15. Details on this approach aregiven in Baker and Faber (2008). An open source software code for the sequential Gaussiansimulation procedure is available as part of the Geostatistical software library GSLIB (Deutschand Journel, 1997). For each of the (M,PGA) combinations the probability of liquefaction foreach grid point is calculated. The probability of liquefaction for each grid point is implementedin the GIS platform from where the corresponding probabilities are incorporated into the BPNfor the soil response model (Figure 4.18).

Calculation scheme for soil failure model

The calculation scheme for the soil liquefaction modeling is given in Figure 4.19. For each ofthe grid point (100 x 150 elements with a dimension of 100 m x 100 m) the depth, soil class ac-cording to USCS, the Fines Content and the SPT blow count of the critical layer, and the groundwater level is simulated assuming independence between the parameters. The simulations areperformed using the code SGSIM from the program package GSLIB. The critical layer is de-fined as the layer with the lowest number of SPT blow count. The input files for simulating thefive parameters Depth.par, USCS.par, GW.par, FC.par and Nm.par are given in Annex C.

The output files of the simulations are read in MATLAB with the files Depth_field.m, USCS_field.m, GW_field.m, FC_field.m and Nm_field.m. Using these parameters, the stress, effectivestress and the shear wave velocity are calculated for each simulation and each grid point using the

58

4.2 Soil failure

NmFC(%)

GW(m)

d(m)d GW FCNm

Figure 4.17: Conditional simulations of a set of soil parameters (top) and probability of lique-faction for a given magnitude of 7.5 with a PGA of 0.1g (bottom).

Liquefaction

Magnitude

PGA

Figure 4.18: Bayesian probabilistic network for the soil response model.

Matlab files Vs_field.m, Sigma_field.m and Sigmaeff_field.m, respectively. The corrected numberof SPT blow counts and cyclic stress ratio for each simulation and each grid point are calculatedby the Matlab files CSR_field.m and N1_60_field.m. Using the simulated and calculated fieldsthe probability of liquefaction for each grid point is calculated for each pair of magnitude andPGA by the file Liq_field.m. All files are given in Annex C.

59

4 Models

USCS ofthe

criticallayer

SPT blownumber

Nm

Depth tothe

criticallayer

Groundwaterlevel(GW)

Fines Content

of thecriticallayer

USCS.par GW.par FC.par Nm.parDepth.par

Effectivestress in

thecriticallayer

Stress in the

criticallayer

Shearwave

velocityin thecriticallayer

Sigmaeff_field.m Vs_field.mSigma_field.m

Corr. SPT blow

numberN1_60

Cyclicstress

ratio in the

criticallayer

N1_60_field.mCSR_field.m

Cyclicstress

ratio in the

criticallayer

Liq_field.m

For each state ofthe nodes 'Magnitude' and ' Distance'

Cal

cula

ted

rand

om fi

elds

Cal

cula

ted

rand

om fi

elds

Sim

ulat

ed ra

ndom

fiel

ds

Depth_field.m USCS_field.m GW_field.m FC_field.m Nm_field.m


60

4.3 Structural damage


The risk due to earthquake results mainly from its effect on the built environment. The builtenvironment may suffer damage and losses due to earthquake-induced ground failure. One of themost important type of such failures, liquefaction is considered in Section 4.2. In this section,the response of the structures to earthquakes and the damage which may occur is considered.Based on state-of-the-art methods in structural response calculation and damage assessment, theconstruction of BPNs is discussed. Finally, a BPN is set up for further use in the examples ofChapter 5.

Two terms related to seismic induced damage to structures are distinguished: vulnerabilityand fragility. Seismic vulnerability defines loss as a function of a ground motion intensity pa-rameter, whereas seismic fragility curves define the probability of being in a damage state as afunction of ground motion intensity parameter. A fragility function can provide the probabilitythat a building will collapse given a ground motion intensity measure. Analogously, vulnerabil-ity functions would provide the direct damage factor (repair cost divided by replacement costas discussed in Chapter 3), given the intensity of ground shaking. A better modularization isattained when the consequences are excluded from the structural response assessment. This ismore feasible when the structural damage is covered using seismic fragility curves.

There are three fundamental approaches to developing seismic fragility curves:

• statistical

• expert opinion

• analytical

The statistical method uses loss data from past earthquakes. The data includes loss and theintensity of ground motion experienced by individual buildings or class of buildings. The seis-mic vulnerability function is estimated by regression analysis of the loss and the ground shakingintensity. With the expert opinion approach no data about historical loss or detailed structuralcharacteristics is required. The main problem is to find experts who are willing to judge theloss value concerned. Eliciting expert opinion may result in disagreements leading to unaccept-ably high uncertainties, heuristic biases or vulnerability functions that are too low or too high.With the analytical method vulnerability functions are generally estimated in three steps: struc-tural analysis, damage analysis, and loss analysis. Structural analysis estimates the responseof the structure to a ground motion intensity, in terms of internal forces and deformations. Thestructural response parameters are then classified into performance levels calibrated through lab-oratory testing or past earthquake observations. The uncertainty of the ground motion intensityparameters leading to the same performance levels are finally modeled resulting in analyticalfragility curves.

The present dissertation discusses within a Bayesian perspective the construction and applica-tion of BPNs for seismic risk in a city. Since multiple parameters may be handled in a large scalerisk analysis, the analytical method is chosen for estimating the fragility curves. The fragilitycurves estimated for the dissertation provide a prior estimate in the risk analysis in Chapter 5.

61

4 Models

There, it will also be outlined how these prior estimations may be updated within a Bayesianframework using damage data from earthquakes. The focus of the dissertation is not to provideseismic vulnerability curves for different structure classes in a city. Therefore a rather rigorousway of modeling is chosen for modeling earthquake demand, for structural modeling and fordamage assessment. The proposed framework allows the incorporation of alternative seismicfragility curves.

Earthquake demand modeling

Demand is simulated by synthetically-generated ground motions representing probable earth-quakes in the western part of the North Anatolian fault system. For earthquake demand mod-eling a set of acceleration time histories is generated. The ground motion prediction equationproposed by (Boore et al., 1997) is used for estimating the pseudo-acceleration response spectrafor the random horizontal component at 5% damping. A number of 16 pairs of magnitudes Mw(5.5, 6.5, 7.0, 7.5), epicenter distances (10, 20, 40, 80 km) and site class (sand) are selected anddepending on the fundamental periods of the structures, the spectral displacement (SD) and peakground acceleration (PGA) values are estimated. Using a version of SIMQKE (Gasparini andVanmarcke, 1976) modified by Lestuzzi (2000) 20 samples of accelerogram time histories foreach pair of these (Mw,R) are generated, resulting in 320 simulations (Figure 4.20). These 320simulations are then used in the vulnerability model (Bayraktarli et al., 2005).

0 5 10 15 20 25−1.5

−1

−0.5

0

0.5

1

Time [s]

Acc

eler

atio

n [m

/s2 ]

0 2 4 6 8 10 120

1

2

3

4

Frequency [Hz]

Sa [m

/s2

]

Figure 4.20: Target acceleration response spectrum calculated using ground motion predictionequation by Boore et al. for Mw=5 and R=30km (bottom), simulated earthquaketime history (top).

62


Structure types

Assessment of the performance of an individual structure is an intricate task in itself. Therefore,in investigating large building stocks it is inevitable that the process be simplified by groupingthe structures which are expected to have a similar seismic performance. Each building is on theother hand unique. The properties which dominate the seismic performance need to be specifiedand generic building types containing those properties should be established. The uncertaintyinduced in reducing the unique structure to the generic structure type needs to be considered.The identified properties are, in the notation of the dissertation, referred to as indicators. Bydoing so, the seismic performance of each structure can be estimated.

Common classification schemes define structure classes by various combinations of use, yearof construction, construction material, lateral load-resisting system, number of stories and ap-plied building code. One of the most comprehensive classifications of structures with regardto seismic behavior was developed by the Applied Technology Council (ATC-13, 1985). Thestructures in California were classified into 78 classes of industrial, commercial, residential, util-ity and transportation structures. The classification was mainly based on construction material,lateral load-resisting system and number of stories. Similarly HAZUS (2001) provides 36 struc-tural classes covering about 99% of all US structures. The main classification criteria in HAZUS(2001) is the number of stories, the seismic design level and the construction type.

The classification of the structures in Adapazari, Turkey, the city under consideration in thisdissertation, is based on a similar approach. The city center is assumed to be bounded in thewest by the river Cay, in the east by the river Sakarya, in the north by Ankara Avenue and inthe south by the Istanbul-Ankara highway. In total there are 22492 buildings in the city (oneto six-story reinforced concrete moment resisting frames and one to two-story masonry struc-tures). The structural performance of the buildings is assumed to be adequately represented bygeneric reinforced concrete frames (Figure 4.21). The construction year of the individual build-ings, which indicates the seismic design code the structures are originally designed for and theoccupancy class of the building (e.g. hospitals), are also considered in designing the genericframes. For illustration purposes it is assumed that all buildings constructed before 1980 are de-signed according to the Turkish Standards for reinforced concrete structures TS500-1975 (1974)without taking into account lateral loads. The buildings constructed after 1980 are assumed tobe designed according to the Turkish Standards for reinforced concrete structures TS500-1984(1983) and the Turkish Seismic Code (TDY, 1975). These generic structures constitute the struc-tures whose structural performances are further investigated. The proposed framework in thisdissertation is illustrated using one structure class: 5-story reinforced concrete moment resist-ing frames. The naming convention for the structure class is as follows: "Typ5" for 5-story,"NR"/"R" for not retrofitted/retrofitted, "O"/"N" for constructed before 1980 (old)/constructedafter 1980 (new), "Res"/"Hos" for residential/hospital use. E.g. Typ5-R-O-Res is the structureclass "5-story retrofitted residential building constructed before 1980".

Structural damage assessment

For the calculation of the structural response the open source finite element program OpenSees isapplied (OpenSees, 2008). OpenSees is used to perform nonlinear dynamic analysis of the struc-

63

4 Models

Beams300x500 13φ20

Columns retrofitted700x700 48φ20

stirrups φ10@75 stirrups φ10@75

Designed according to the TS500-1984 and TDY 1975

stirrups φ10@75

Columns 500x500 32φ20

Columns retrofitted700x700 40φ16

stirrups φ8@120stirrups φ8@120

Columns 500x500 24φ16

Beams300x500 11φ16

stirrups φ8@150

Generic frame

Designed according to the TS500-1975

3000

50005000 50005000z

3000

3000

3000

3000

y

Figure 4.21: Generic frame representative for the 5-story reinforced concrete moment resistingframes (residential use); construction year before 1980 (top right), after 1980 (bot-tom right).

tures. The structures are subjected to the 320 ground motion time histories and the maximuminter-story drift ratio (MIDR) is calculated using OpenSees.

The beams and columns are modeled with non-linear beam-column elements, which are basedon the non-iterative force formulation and consider the distribution of plasticity along the ele-ment. The reinforced concrete cross sections are modeled using a fibre cross section model.Rigid diaphragms are used to model the slabs. The concrete material is modeled after Parket al. (1972) with degraded linear unloading/reloading stiffness, based on the work of Karsanand Jirsa (1969), and assuming no tensile strength. The confined and unconfined concrete ismodeled by applying two different sets of material parameters. The damping is modeled asRayleigh damping, which implies a combination of the mass and the stiffness matrices at thecurrent state. Masses are lumped in the rigid diaphragms. All material parameters are assumedto be deterministic. The columns on ground level are fixed for all degrees of freedom. Groundacceleration is applied in the y-direction. The solution algorithm is of the Newton type with aconvergence criteria expressed in terms of the norm of the displacement increment vector. Theintegrator is of the Newmark type. The time step is set to 0.01 s and the length of the time seriesis 25 s, resulting in 2500 time steps for each series. It takes about 500 s to calculate one time

64


series for the ex-ample structure on an Intel Pentium PC.

Adapting the performance limit states given in Huo and Hwang (1996), the MIDR’s are clas-sified into three damage states (Green-Safe, Yellow-Limited use and Red-Unsafe). MIDR’slarger than 1.0 are classified as Red-Unsafe, smaller than 1.0 and larger than 0.5 are classifiedas Yellow-Limited use, and smaller than 0.5 are classified as Green-Safe. This damage clas-sification is in line with ATC-20 (1989) methodology which specifies the procedure for post-earthquake inspections of buildings.

The fragility of a structural system is modeled, following the state-of-practice, using a lognor-mal distribution. The justification for this lies in the fact that MIDR and hence SD is a result of amultiplicative influence of several parameters. Furthermore the distribution of the input parame-ters for a certain limit state is skewed to the right and no negative parameters are allowed. Eachfragility curve is characterized by lognormal median (λ ) and lognormal standard deviation (ζ )of the ground motion intensity parameter (e.g. spectral displacement). The probability of beingin or exceeding a given damage state, D, is modeled as a cumulative lognormal distribution:

P(D|Sd) = Φ

(lnSd − lnλ

lnζ

)(4.21)

where Φ(·) is the standard normal cumulative distribution function, λ is the logarithmic me-dian capacity, and ζ is the standard deviation of capacity. The estimation of the parameters isperformed using the maximum likelihood method (Table 4.3). The fragility curves are given inFigure 4.22.

Table 4.3: Parameters of the fragility curves.λYellow ζYellow λRed ζRed

Typ5-NR-O-Res 3.758 0.390 4.215 0.346Typ5-NR-N-Res 3.742 0.226 4.337 0.235Typ5-R-O-Res 3.806 0.347 4.277 0.289Typ5-R-N-Res 3.845 0.299 4.328 0.265

Typ5-NR-N-Hos 3.873 0.372 4.314 0.272Typ5-R-N-Hos 4.029 0.323 4.384 0.222HAZUS C3M 2.906 0.900 4.515 0.900HAZUS C1M 3.640 0.680 5.432 0.680

The uncertainty of the distribution is explicitly considered in the model (node ’Model uncer-tainty’) using a discretisation with three states. Given the ground motion intensity parameter asspectral displacement from the attenuation model, the probabilities of being in a particular dam-age state are read from the fragility curves for the states of node ’SD’. These probabilities formthe conditional probability table of the node ’Damage’. The BPN for the vulnerability model isgiven in Figure 4.23.

65

4 Models

0 20 60 100 140 180 220 260 3000

0.10.20.30.40.50.60.70.80.9

1

Spectral displacement, Sd (T=0.64s, ζ=5%) [mm]

Prob

abili

ty

Typ5 Residential constructed before 1980

Typ5 Residential constructed before 1980 - Retro�tted

HAZUS C3M - PreCode

Typ5 Residential constructed after 1980

Typ5 Residential constructed after1980 - Retro�tted

HAZUS C1M - HighCode

limited use (yellow)Collapse (red)






0 20 60 100 140 180 220 260 3000

0.10.20.30.40.50.60.70.80.9

1


Prob

abili

ty

0 20 60 100 140 180 220 260 3000

0.10.20.30.40.50.60.70.80.9

1

Spectral displacement, Sd (ζ=5%) [mm]

Prob

abili

ty

0 20 60 100 140 180 220 260 300

00.10.20.30.40.50.60.70.80.9

1

Spectral displacement, Sd (ζ=5%) [mm]

Prob

abili

ty

0 20 60 100 140 180 220 260 3000

0.10.20.30.40.50.60.70.80.9

1


Prob

abili

ty

0 20 60 100 140 180 220 260 3000

0.10.20.30.40.50.60.70.80.9

1


Prob

abili

ty

00.10.20.30.40.50.60.70.80.9

1

Prob

abili

ty

00.10.20.30.40.50.60.70.80.9

1Pr

obab

ility

0 20 60 100 140 180 220 260 300Spectral displacement, Sd (T=0.64s, ζ=5%) [mm]

0 20 60 100 140 180 220 260 300Spectral displacement, Sd (T=0.64s, ζ=5%) [mm]



Typ5 Hospital constructed after1980 Typ5 Hospital constructed after1980 - Retro�tted

Figure 4.22: Seismic fragility curves for generic 5-story buildings.

DamageLiquefaction

SD EpsilonDamage

Structure Class

Figure 4.23: Bayesian probabilistic network for the vulnerability model.

66


Alternative BPN model

Single-degree-of-freedom (SDOF) systems with modified Takeda hysteresis can be used as sub-stitute systems for representing the dynamic characteristics of the structural class. The dynamicproperties of the substitute SDOF systems are assigned according to the relationships proposedby Priestley (1998) and the TDY (1998). The yield displacements and base-shear capacities ofthe retrofitted buildings are estimated according to the procedure by Dazio (2000). The responseof substitute SDOF systems to the set of acceleration time histories, is simulated.

It is assumed that the maximum and residual displacements attained by the substitute SDOFsystems can be used as an estimate of those to be attained by the real structures themselves(Figure 4.24, left). In general terms, for the case of maximum displacements of buildings exciteddynamically predominantly in their first mode, this assumption has been verified. However, forthe case of residual displacements it should be noted that the adopted modeling approach has asignificant impact on the computed results (Yazgan and Dazio, 2006). It will be assumed thatthe residual displacements computed for a substitute SDOF system can be used as an estimatefor the residual displacements of the buildings in the corresponding structural class. Maximumdisplacements are known to be well correlated with structural damage. Furthermore, residualdisplacements are critical for the post-earthquake usability/reparability of a structure. These twostructural response parameters together can provide a good picture of the seismic performanceof a structure. The performance level definitions provided in the Vision 2000 (SEAOC, 1995)can be adopted to relate the structural response parameters to damage states.

3000

50005000 50005000z

3000

3000

3000

3000

DamageLiquefaction

SD

Structure Class

ResidualDispl.

MaximumDispl.

Periode

Figure 4.24: Generic frame representative for the 5-story reinforced concrete moment resistingframes and the substitute SDOF system (left). Bayesian probabilistic network forthe alternative vulnerability model (right).

A BPN explicitly considering residual displacements (Figure 4.24, right) is advantageous,as residual displacements could be measured after an earthquake and with these measures themodels could be updated (Bayraktarli et al., 2006). This requires the availability of efficient waysfor measuring the residual displacements with an acceptable precision, e.g. Altan et al. (2001),as well as structural response assessment methods explicitly using residual displacements fordamage assessment, e.g. Yazgan (2009).

For further analysis the BPN in 4.23 for seismic vulnerability will be used. Information fromdamage surveys after earthquakes on the state of damage to structures will be used for updating.

67

4 Models

Calculation scheme for structural damage model

The calculation scheme of the seismic fragility curve for the structure classes are given in Figure4.25. In the calculation scheme, the description in this section and the files in Annex C onlyone structure class is illustrated. The calculations are analogous for the other structure classesconsidered in this dissertation.

The MIDRs for each of the simulated acceleration time histories for the structure class consid-ered are calculated in OpenSees using the files Typ5.Nr.O.Residential.tcl and BuildRCrectSec-tion.tcl. These MIDRs and the corresponding SDs of the simulated acceleration time historiesare classified into damage states given the performance limit MIDRs of the damage states. Theresulting vectors of SDs within each damage state are assumed to be lognormal distributed andmodeled using Maximum Likelihood Method.

Estimation of the seismicfragility curve in

Figure 4.19

Design of the genericstructure class

Calculation ot the 320 MIDRs with the FE-code OpenSees

Generation of total 320 acceleration time histories

for each (Mw,R) pair

Estimation of the parameters of the lognormal distribution for the set of SDs in each

damage state using the Maximum LikelihoodMethod

Classification of the 320 MIDRs and thecorresponding SD‘s of the time histories

given the performance limits of the damagestates

Typ5.NR.O.Residential.tclBuildRCrectSection.tcl


68

4.4 Consequences

4.4 Consequences

A consistent treatment of risk requires a realistic estimation of consequences. The goal should beto consider all types of consequences which may occur if an adverse event takes place. Besidesconsequences resulting from the failure of the objects at risk, in Section 3.2 denoted as directconsequences, indirect consequences which go beyond the costs related to the objects also needto be considered. Depending on the decision-making level at which the risks are consideredand consequently depending on the system definition, the border between direct and indirectconsequences is defined.

The consequences of events may be expressed in different units, in monetary units or in thenumber of casualties, etc. A consistent treatment of different decision alternatives with differ-ent types of consequences is possible when using the same units. In the present dissertationmonetary units will be used for all kinds of consequences. This also enables the aggregation ofconsequences.

Natural hazards generate a variety of socio-economic impacts. These may be classified asstructural losses (repair, replacement and retrofit of structures), nonstructural losses (buildingcontents and inventory), direct economical losses (business interruption), indirect economicallosses (supply shortages and demand reductions) and demographical consequences (loss of lives,injuries) (Faizian and Faber, 2004; King et al., 1997; Brookshire et al., 1997).

Depending on the decision-making level, an inventory of different consequences must bedrawn up to obtain a consistent assessment and management of earthquake risks. For example,a building owner may consider only structural losses as direct consequences and nonstructurallosses as indirect consequences, whereas a city governor classifies the structural, nonstructuraland direct economical losses as direct consequences, and indirect economical losses and demo-graphical losses as indirect consequences.

The proposed methodology in the present dissertation is illustrated on residential and hospitalbuildings in the city of Adapazari, Turkey. In this context direct consequences comprise ofstructural repair, replacement and retrofitting costs. Loss of lives in these structures due topossible earthquakes are considered as indirect consequences.

For Adapazari, the average unit rebuilding cost is estimated as 175 USD/m2, the average unitretrofitting cost for column jacketing as 250 USD/column and the unit repair costs for "limiteduse" damage state as 10% of the unit rebuilding cost (Has-Insaat, 2007).

Statistical value of life

The value of goods and services is estimated in the free market. Hence, the quantification ofmost of the above mentioned consequences in monetary units poses no problems. However,the quantification of the loss of lives, which is decisive in decision problems related to safetyissues, forms the subject of controversial debate. When evaluating the value of life the ques-tion is not the value of a single life, but rather the value people are willing to invest in riskreduction measures to save life. The question has no ethical dimension as long as undefined andimpersonalized life is considered. That is the reason for the wording "statistical life".

69

4 Models

Various methods exist for evaluating the statistical value of life (Schubert, 2009). In methodsbased on compensation, the value of life is estimated through the loss of income of a person whohas died. This human capital compensation approach neglects both immaterial damages likepain and suffering as well as material damages to restore the original situation. The approachusing compensation granted by the courts as a measure for the value of statistical life is moredifferentiated (Miller, 1990). When deriving the value of statistical life, the temporal shortageof benefit resulting from psychological and physical impairment is considered (Hofstetter andHammitt, 2002). Compensation methods are mainly used in the US.

Contingent valuation methods use public opinion polls where people are asked to estimate theamount they are willing to pay or they are willing to accept for a specific immaterial good in aspecific hypothetical scenario. The method was first developed by Ciriacy-Wantrup (1947) andapplied for estimation of the statistical value of life, e.g. in Blaeij et al. (2003). The supportersof this method argue that a direct estimation of the willingness of society to pay is possible(Hanemann, 1994). The opponents point to the fact that different estimates result when thesurvey is done individually for goods in a set or for the set as a whole (Kahnemann and Kentsch,1992). Different estimates also result in a survey in dependence of the order for goods in a set.Contingent valuation methods assume that the interviewee acts in a credible, reliable and precisemanner.

The theory of revealed preferences was developed by Samuelson (1938). Whereas Contin-gent Valuation Methods are based on the declaration of people on the amount they would payor accept, Revealed Preference Methods try to find out how people would behave in such sit-uations. The Revealed Preference Method estimates the best option based on the behavior ofthe consumers. The fundamental assumption of the consumption theory is that consumers makedecisions which maximize the utility function (Hicks and Allen, 1934a,b). Revealed Prefer-ence Methods provide a means for deriving utility functions based on consumer behavior. Anoverview of the applications of this theory to the estimation of the statistical value of life is givenin Blaeij et al. (2003). The estimation may be calibrated on the development of wages (Viscusi,1993), on the development of the property market (Thaler, 1978) or based on consumption goods(Miller, 1990).

Observable economic and social indicators are considered in the Life Quality Index originallyproposed by Nathwani et al. (1997). This method can also be classified among the RevealedPreference Methods as it is based on observations on the preferences of society reflected ininvestments. Originally, the LQI was formulated as a social indicator for estimating rational andefficient decision-making in regard to safety. LQI is consistent with the consumption theory andprovides a consistent method for estimating the statistical value of life.

An efficient life safety activity may be understood as a measure which cost-effectively re-duces the mortality or equivalently increases the statistical life expectancy. The increase in lifeexpectancy, l, through expenditure on risk reduction measures results in loss of economic re-sources measured by Gross National Product (GNP) per capita, g, together with the time spentat work, w. Based on these indicators which are all assessed for a statistical life in a givensociety, the LQI is formulated:

70

4.4 Consequences

L ∝gq

ql (4.22)

where in the optimum q = w/(1−w). Rackwitz et al. (2005) derived based on the LQI,societal life saving cost (SLSC):

SLSC = g[1− (1+lrl)−

1−ww ]lr (4.23)

where g is the Gross Domestic Product per capita, l is the life expectancy, lr is the number oflife years saved and w is the proportion of life spent at work.

In the illustrative examples given in Chapter 5, loss of lives of individuals is quantified by theSocietal Life Saving Cost (SLSC). For Turkey, based on data for 2006 with a life expectancyof 71.5 years and Gross National Product per capita of 5’500 USD, the SLSC is estimated as150’000 USD per fatality.

Estimation of fatalities

For the illustrative examples given in Chapter 5 fatalities are estimated by determining the’lethality ratio’ for each building class damaged by the earthquake (Coburn and Spence, 2002).The lethality ratio is the number of people killed to the number of occupants present in the build-ing. It is estimated from examination of data from previous earthquakes. Coburn and Spence(2002) give five M-factors to estimate the number of fatalities likely to occur in an earthquake(Figure 4.26). In evaluating the examples, data on the number of story and floor area is usedto estimate the number of fatalities of each building. Data on the number of occupants for eachbuilding is then incorporated in the GIS platform for each building in the city and evaluatedusing the BPN for the consequence model (Figure 4.27).

71

4 Models

Figure 4.26: Factors for estimating the number of casualties (Coburn and Spence, 2002).

Damage

Cost

Structure Class

Occupancy Class

Construction Year

Figure 4.27: Bayesian probabilistic network for consequence model.

72

4.5 Verification and validation of the models


Models provide a means for understanding the real world, rather than for trying to reproduceit. Once a model is set, it may be used for predicting outcomes of interest, provided that acertain confidence in the underlying model exists. Fundamental criticism has been made ofthe possibility of accurate quantitative modeling to predict the outcome of natural processes onthe Earth’s surface (Pilkey and Pilkey-Jarvis, 2007). Oreskes et al. (1994) even argue that thevalidation of numerical models of natural systems is impossible.

Verification means that the calculations within the model are demonstrably correct. Given thescale and the detailing level, the model gives the expected output. Models may be verified bycomparisons with known solutions (e.g. experimental or analytical) and by cross comparisonsbetween different models. In contrast to verification, validation aims to check the consistency ofmodel output with relevant observations. Only a validated model may take the credit for being arepresentation of the real system.

Verification and validation of the seismic hazard model

The plausibility of the Probabilistic Seismic Hazard Analysis (PSHA) methodology used heremay be checked by verifying the methodology itself or by validating the application to a certaincontext. The PSHA methodology is a mathematical procedure which has been subjected toverification by standard mathematical theory.

The main output of a PSHA is the annual rate of occurrence of various ground motion inten-sities at a site. Testing a PSHA application thus means determining whether the occurrence rateof ground motion intensities is consistent with the model predictions. This is not possible forhazard estimates for low annual rates, as this would require hundreds or thousands of years toverify. Varying the spatial extent of the region and time period considered may provide a meansof checking: i.e. observations of ground motion occurrences over a relatively large region overa time period of tens of years may be used to verify the predictions of the rate of occurrence ofthese ground motions (McGuire and Barnhard, 1981). This kind of comparison cannot checkthe consistency of local hazard estimates with local sources of seismicity (NRC, 1988).

The BPN model for seismic hazard presented here is verified by cross comparison betweenknown PSHA codes. Due to limitations imposed by lack of data, the only practical means forevaluating or testing PSHA are this test of reasonableness and consistency.

Verification and validation of the soil failure model

The soil liquefaction potential evaluation in the present dissertation is based on empirical correla-tions in Standard Penetration Test blow counts. The required soil parameters for these empiricalcorrelations are modeled by taking their spatial variability into account. For different earthquakeintensity parameters (moment magnitude and peak ground acceleration) and for each point ofthe test area a probability of liquefaction is assigned.

The soil failure model may be validated using observations of liquefaction events from pre-vious earthquakes. Even though liquefaction events were reported from the 1967 earthquake,

73

4 Models

comprehensive data on observations only exists from the recent Kocaeli Mw7.4 earthquake ofAugust 17, 1999. In downtown Adapazari a region was investigated and liquefied areas reported(DRM, 2004). A peak ground acceleration of 0.41g was recorded at the nearby Sakarya ac-celerograph located in southwestern Adapazari at a distance of 10km. Motions in Adapazariwould differ from those at the accelerograph due to ground response effects associated withthe soft and deep alluvium. Recorded ground motions at similar conditions suggest a PGA inAdapazari of the order of 0.35g to 0.45g.

A direct comparison with liquefaction observations from a single event is not meaningful, asthe model outputs are in the form of probabilities. A comparison is nevertheless made by usingthose grid points for which the model estimates certain liquefaction or certain non-liquefaction,given the Mw=7.4 and PGA=0.3 to 0.5g. In Figure 4.28 the investigated area with observedliquefaction after the Kocaeli Mw7.4 earthquake of August 17, 1999 and the predicted liquefac-tion is illustrated. The reason for the differences may be due to the different practices in SPTtechniques in Europe and in the U.S., where the empirical correlation models are founded.

0 750 1’500 m

/

Figure 4.28: Liquefaction observed (left) and predicted (right) in Adapazari during the Kocaeliearthquake in 1999.

Verification and validation of the structural damage model

A verification of the structural damage model may be performed by again referring to the KocaeliMw7.4 earthquake of August 17, 1999. Post-earthquake damage survey teams have identifiedbuildings with different degrees of damage. The data is summarized in the percentage of dam-aged buildings in each district (Figure 4.29). For the area under investigation, about 27% of the5-story buildings constructed before 1980 are reported as heavily damaged. The seismic hazardBPN model in Figure 4.5 is applied with the characteristics of the Kocaeli earthquake of August17, 1999. The magnitude was Mw7.4; the distance to the activated seismic source S3 was about50 km. With this information the nodes magnitude and distance are conditioned in Figure 4.29.

74


The resulting probability distributions for PGA and SD are used as input for the soil failure BPNand structural damage BPN respectively. The main output of the structural damage BPN is inthe form of discrete probabilities for three damage states (no-damage, repairable damage, col-lapse). For the 5-story buildings constructed before 1980 the probability of damage is estimatedas 21%. This 21% includes the two states indicating damage to the structures. There may beseveral reasons for the difference from the 27% resulting from the post-earthquake survey. Thestructural models may not be suitable for older structures with low ductility capacity resultingin underestimation of the estimated damage. Important structural details, e.g. joints, which arenot modeled may be the reason for the difference.

3

13

21

6

1217

2423

2122

2015

1610

118

9

4

1418

19

75

Structure TypesTyp5_O_Res

Typ5_N_Res

Figure 4.29: Collapsed building statistics in Adapazari during the Kocaeli earthquake in 1999.

Verification and validation of the consequence model

The consequence model may also be validated with the data from the Kocaeli Mw7.4 earth-quake of August 17, 1999. Several studies were conducted to estimate the consequences of theearthquake with different boundaries for the cities considered and different sets of incorporatedconsequences. For example, the total loss for housing in all cities affected by the earthquake isestimated by TÜSIAD (Turkish Industrialization and Businessmen’s Association) to be USD 4billions and by the World Bank (Bibbee et al., 2000) to be USD 1.1 to 3 billions. In the presentdissertation the application of the proposed methodology is illustrated by referring to two occu-pancy and structure classes. A complete practical estimation of consequences of earthquakes inthe region is not pursued. Such an application is however required to validate the consequencemodel. The estimated costs given in the two aforementioned studies could be broken down tothe herein considered structure and occupancy classes. A validation of this type has not beenpursued as this would result in significant bias.

75

5 Examples

The proposed framework is illustrated on earthquake risk problems for a class of structures lo-cated in a test area. First the chosen test area is introduced. Briefly the geology, seismicity, builtenvironment, demography and previous damaging earthquakes within the test area are discussed.Finally the building portfolio considered in the examples is described.

Choice of the test area

The Kocaeli Mw7.4 earthquake of August 17, 1999 caused serious damage to the city centerof Adapazari. Besides damage to the built environment due to severe ground shaking (Figure5.1), foundation-related failures of buildings such as tilting, overturning and sinking were alsoexperienced (USGS, 1999). The foundation-related failures were due to the soft flood-plaindeposits of the Sakarya river on which Adapazari has been built. Furthermore it is a city ofregional influence following the classification in Chapter 2. Finally, data on both structural andsoil-related damage after the recent earthquake in 1999 is available to conduct Bayesian updatingof the probabilistic models.

Geology and seismicity of Adapazari

Adapazari is located at the edge of a former lake basin of about 25x40km2. The lake sedimentsare overlain by Pleistocene and early Holocene alluvium transported from the mountains to thenorth and south of the basin. In some areas Quaternary alluvium deposited by the Sakarya Riverand its tributaries overlay the older lake alluvium (Rathje et al., 2000). The Quaternary alluviumlayer reaches a depth of about 15m and is comprised primarily of silt and fine sand.

The city is mainly constructed over the very flat Quaternary alluvial sediments of the basinwith soft near-surface sediments deposited by the river, making it susceptible to liquefactioninduced damage during earthquakes, as experienced in the 1967 and 1999 events (Gülkan et al.,2003). Many soil profiles are characterized as loose silts and silty sand in the upper 5m overlay-ing clay deposits (Bray and Stewart, 2000; Sancio et al., 2002). The bedrock formation descendssharply through the north and reaches a depth of about 200m within the city center. The citycenter of Adapazari is located on the northeastern foothills, to the west of the Sakarya Riverwhich runs from south to north in the basin and enters the Black Sea and to the east into thesmaller Cark River.

The main fault, the North Anatolian Fault, forms the southern boundary; the Düzce Faultforms the southeastern boundary. During the Kocaeli Mw7.4 earthquake in 1999 surface ruptureswith displacements of up to 5m were observed along the North Anatolian Fault (Komazawaet al., 2002).

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5 Examples

Built environment and demography of Adapazari

Adapazari is an important industrial and agricultural area in the western part of Turkey. It is thecapital of the Sakarya Province with a population of 183’000, according to the 1997 census. Thecenter of the city lies within the fertile plain described above, formed by recent fluvial activityof the Sakarya and Cark Rivers.

The city center comprises both new and old construction. Adapazari has experienced signif-icant growth during recent decades. The population doubled from 1967 to 1990. During theindustrial boom of the early 1980s, the city received immigrants from poorer regions of thecountry. The pressure to develop new housing to accommodate the new arrivals led to a loosen-ing of the rule of not exceeding the two-story construction limit (Green, 2005). From 1990 to1997 the population increased by another 10%.

The main construction types in the city are 3 to 6 story reinforced concrete frame buildings andolder 1 to 2 story timber/brick buildings. The reinforced concrete frame buildings are mostlynon-ductile with large openings in the ground stories for commercial use and masonry infillwalls in the upper stories. The foundations are very stiff compared to foundations of buildingsof similar height. The foundations generally consist of about 30cm thick reinforced concretemat which is stiffened with 30cm wide and 100cm deep grade beams spaced typically about5m in both directions (Sancio et al., 2004). The space between the grade beams is filled withcompacted soil and than covered with a thin concrete floor slab. The previous foundation-relatedfailures which occurred during the earthquake in 1967 are probably the reason for these atypi-cally stiff foundations. The stiffness of the foundations may be the reason for the tilting of manystructures without experiencing significant structural damage during the 1999 earthquake. Manyof the buildings with the stiff foundations responded like a rigid body undergoing significant dif-ferential movement, tilt or lateral translation (Figure 5.1).

Damaging earthquakes in Adapazari

Of all the cities affected by the August 17, 1999 Kocaeli earthquake, Adapazari suffered thelargest level of gross building damage. The Ministry of Public quantified the number of severelydamaged or collapsed building to 5078, which is 27% of the total building stock in Adapazari.The official loss of life in Adapazari was 2627.

The fact that the city center of Adapazari lies only 7km from the ruptured fault, and that thecity center is underlain by 200m deep alluvial sediments contributed to the the severity of thebuilding damage. Building damage concentrated more in the central area of the city, whereasrelatively less damage was observed in the south, where the bedrock is closer to the surface.Another factor for this heterogenous distribution of building damage is the greater density ofthe mid-rise buildings (4 to 6 stories) in comparison of the low-rise residential buildings in theoutskirts of the city (Bird et al., 2004; DRM, 2004).

The building damage can be classified into two failure modes: structural system failures dueto excessive ground shaking, and foundation displacements of various forms and levels due tobearing capacity failure. The two failure modes were observed to be mutually exclusive. Theobserved foundation displacements imply nonlinear response of the soil, which in turn may have

78

provided a means of natural base isolation for some buildings during the 1999 earthquake (Bakiret al., 2005; Mollamahmutoglu et al., 2003).

In the last century, Adapazari suffered damage from two other major earthquakes associatedwith the North Anatolian Fault. The first of the two was in 1943 with a magnitude of 6.6 on theRichter scale and an epicenter 10km east of the city. About 70% of the buildings were damagedduring the event. The second major earthquake was in 1967 with a magnitude of 7.1 on theRichter scale and an epicenter of 27km in the southeast of the city. The structural damage wasnot severe during the 1967 earthquake (Bakir et al., 2005; Ambraseys and Zatopek, 1969).

GIS for Adapazari

Adapazari is bound by the Sakarya River in the east, the Cark River in the west, the main road"Cevre Yolu" in the north and the Istanbul-Ankara Highway in the south. The municipality ofAdapazari provided a CAD drawing with the buildings indicated as polygons. The number ofstories of each polygon along with the occupancy were also given for some of the polygons. Thepolygons with a size less than 50m2 were screened out assuming them to be objects other thanbuildings. The polygons without an indication of occupancy were assumed to be residentialbuildings. In this way a GIS database was established comprising a total of 22489 buildings.These 22489 buildings were classified into six occupancy classes (residential, industry, schools,official, sacral and hospital) and six structural classes (1 and 2-story masonry structures, 3 to6-story reinforced concrete frames). Table 5.1 summarizes the building types in the test area andFigure 5.2 illustrates the GIS map of the buildings in Adapazari.

For illustration purposes the 1246 5-story residential buildings were chosen. Since seismicdesign codes applicable to Turkey evolved particularly after 1975, the construction year of thebuildings is classified as pre-1980 and post-1980. The generic reinforced concrete bare framesare designed according to the TS500-1975 (1974) for the pre-1980 structures and according tothe TS500-1984 (1983) and the Turkish Seismic Code TDY (1975) for the post-1980 structures(see also Section 4.3. It is assumed that all the buildings in the old districts of Adapazari belongto the pre-1980 class (534 buildings) and the remaining buildings belong to the post-1980 class(712 buildings). All hospital buildings are from the post-1980 construction period. Figure 5.3illustrates the buildings considered.

Table 5.1: Building stock in the test area in Adapazari.Structure class Residential Industry Schools Official Sacral Hospital Total

Masonry 1-Story 6932 342 6 13 1 0 7294Masonry 2-Story 7690 93 27 24 57 0 7891RC frame 3-Story 3734 14 32 17 3 0 3800RC frame 4-Story 2100 8 16 20 1 0 2145RC frame 5-Story 1246 7 4 6 0 5 1268RC frame 6-Story 90 0 0 1 0 0 91

Total 21792 464 85 81 62 5 22489

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5 Examples

Figure 5.1: Building failures in Adapazari (from http://www.sdr.co.jp/damege_tr).

80

/Residential

Hospital

Religious

Official

Schools

Industry

River

0 1.5 3.00.75 km

Figure 5.2: GIS map of the city of Adapazari (Occupancy classes).

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5 Examples

/

Typ5_O_Res

Typ5_N_Res

Hospital

River0 1'500 3'000750 m

Figure 5.3: GIS map of the considered buildings.

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5.1 Example 1: Decision for retrofitting structures


Two decision alternatives are considered: strengthening the reinforced concrete moment resist-ing frames by column jacketing, or no action. First the BPNs for the seismic hazard (Section4.1), the BPN for the soil response (Section 4.2), the BPN for structural response (Section 4.3)and the BPN for the consequence assessment (Section 4.4) are integrated. For the identifieddecision situation, possible activities are given in the ’Retrofit’ node. In the ’retrofit’ node ’noaction’ and ’retrofitting by column jacketing’ are considered as decision alternatives. The mainBPN for the risk management problem is constructed in this way (Figure 5.4).

DamageLiquefaction

SD

DistanceMagnitude

PGA

Retrofit ? Cost

Epsilon PGA

EpsilonDamage

Structure Class

T

Epsilon SD

Occupancy Class

Construction Year

Figure 5.4: Bayesian probabilistic network for earthquake risk management.

The BPN given in Figure 5.5 is applied for each of the 534 5-story buildings constructed be-fore 1980. First the specific data for each building concerning the story area and the probabilityof liquefaction for the given combination of magnitude and peak ground acceleration for thelocation of the building are incorporated into the BPN from the GIS database. For the presentexample the structure class is either a "Typ 5 residential building constructed before 1980" ora "Typ5 retrofitted residential building constructed before 1980" depending on the action in the"Retrofit" node. In the "Damage" node the fragility curves, which were developed in Section4.3 are implemented including the uncertainty given the state of liquefaction and the spectraldisplacement level.

Case 1: Time-independent seismic hazard (Poisson)

Nine seismic sources are considered within a radius of 100km from the city center of Adapazari(see Section 4.1). Assuming that only one of the seismic sources activates during an event, i.e.the rupture of two and more sources as well as cascading effects are disregarded; the earthquakerisk for each building due to each seismic source is calculated.

For each of the seismic sources the distribution of magnitudes and distances are incorporated

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5 Examples

in Figure 5.5. As explained in Section 4.1 a minimum magnitude for damaging earthquakes ofMw=5.0 is considered when calculating the seismic hazard. This means that the incorporatedmagnitude distributions are calculated for the occurrence of one earthquake with a magnitudegreater than Mw=5.0. Hence the calculated earthquake risks must be multiplied by the rate ofoccurrence of earthquakes greater than Mw=5.0 for each of the seismic sources.

The consequences are implemented in terms of costs, conditional on the damage, story areaand number of fatalities. Direct consequences as well as indirect consequences, i.e. the numberof fatalities, are considered when implementing the table for the ’Cost’ node. Damage caninfluence the cost directly, i.e. for the damage state "Yellow-Limited use" repair of the buildingand for the damage state "Red-Unsafe or collapse" rebuilding costs are considered or indirectlyvia the number of fatalities. Details for the calculation of the consequences are given in Section4.4. For the present example the future costs for each of the 50 considered years have to bediscounted to the present value, when the decision has to be made. The discount factor given inEquation 2.2 must be applied when calculating the costs.

In the literature, mainly two reconstruction strategies are considered (Rosenblueth and Men-doza, 1971); i) surrendering the structure after first failure and ii) reconstructing the structureafter failure. In this example a combination of these is considered: collapsed structures are sur-rendered and damaged structures are renewed. Using the BPN given in Figure 5.4 according to

Structure dataSoil information

Prob

abili

ty o

f Liq

uefa

ction

Seismicity data

S1S4 S3 S2S5

S25

S15

S14

S13

S21S19

S22S12

DamageLiquefaction

SD

DistanceMagnitude

PGA

Retrofit ? Cost

Epsilon PGA

EpsilonDamage

Structure Class

T

Epsilon SD

Occupancy Class

Construction Year

Figure 5.5: Bayesian probabilistic network for earthquake risk management.

84


the scheme given in Figure 5.5 the expected value of the earthquake risk is calculated for bothdecision alternatives in the "Retrofit" node for each of the considered buildings. These expectedvalues are summed up for all nine seismic sources. The optimal decision is based on a compari-son of the expected total costs of the decision alternatives given the reference period of 50 years.

Formally, the expected value of the costs for the two decision alternatives is calculated by:

E[u|a = a1] =9

∑i=1

3

∑j=1

P(DSi j,a1)C j,a1e−0.02tνi (5.1)

E[u|u = a2] =9

∑i=1

3

∑j=1

P(DSi j,a2)C j,a2e−0.02tνi (5.2)

a∗ = argmina∈a1,a2E[u|a] (5.3)

Here, a1 is the decision alternative "No Action", a2 is the decision alternative "Retrofitting",a∗ is the optimal decision, E[u|a = a1] is the associated expected cost for a1, E[u|a = a2] is theassociated expected cost for a2, DSi j is the j-th damage state of the building due to the seismicsource i, C is the associated cost for the damage state, the term e−0.02t is the discounting factor,and νi is the rate of occurrence of earthquakes with magnitude greater than Mw=5.0 for eachseismic source i. The summation over j is for the three damage states "Green-No damage","Yellow-Repair needed" and "Red-Collapsed". The summation over i is for the nine seismicsources considered.

The calculation of the marginal probabilities of the damage states in equations 5.4 and 5.2is carried out using the BPNs for each building and seismic source with the structure and site-specific information. The evaluation of the BPNs was carried out in the GIS environment usingthe commercial software Hugin (2008). The maximum expected costs are thus calculated foreach building and an optimal action is proposed. For all 534 5-story buildings constructed before1980 the optimal decision with a time-independent seismic hazard is calculated. In Figure 5.6the optimal action for each building is illustrated.

Case 2: Time-dependent seismic hazard (characteristic earthquakes)

For 14 of the 534 5-story buildings constructed before 1980 the optimal decision is also cal-culated with a time-dependent seismic hazard. The calculation scheme when using a time-dependent seismic hazard is in principle the same. The distribution for magnitude is differentfrom the time-independent seismic hazard case which is no longer assumed to be constant foreach year. In Section 4.1 a magnitude frequency distribution for time-dependent characteristicearthquakes is considered. Given the time-dependent seismic hazard model, the distribution ofmagnitudes is calculated for each of the 50 years of the reference period as described in Sec-tion 4.1. The aforementioned calculation scheme needs to be extended by one additional loopconsidering the 50 years, each with the corresponding distribution of magnitude.

Formally, the expected value of the costs for the two decision alternatives is calculated by:

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5 Examples

E[u|a = a1] =9

∑i=1

3

∑j=1

50

∑t=0

P(DSi jt,a1)C j,a1e−0.02tνi (5.4)

E[u|u = a2] =9

∑i=1

3

∑j=1

50

∑t=0

P(DSi jt,a2)C j,a2e−0.02tνi (5.5)

a∗ = argmina∈a1,a2E[u|a] (5.6)

Here, a1 is the decision alternative "No Action", a2 is the decision alternative "Retrofitting",a∗ is the optimal decision, E[u|a = a1] is the associated expected cost for a1, E[u|a = a2] is theassociated expected cost for a2, DSi j is the j-th damage state of the building due to the seismicsource i, C is the associated cost for the damage state, the term e−0.02t is the discounting factor,and νi is the rate of occurrence of earthquakes with magnitude greater than Mw=5.0 for eachseismic source i. The summation over j is for the three damage states "Green-No damage","Yellow-Repair needed" and "Red-Collapsed". The summation over i is for the nine seismicsources considered. In Figure 5.7 the optimal action for each building using a time-dependentseismic hazard is illustrated.

In Table 5.2 the number of buildings for which the optimal action is "No Action" and "Retro-fitting" are given. For comparison, the corresponding values when using a time-dependent seis-mic hazard model are also given for the subset of 14 buildings. The time-dependent seismichazard model leads to a substantial increase in seismic activity, which is the reason that for all14 buildings of the subset being considered the optimal action is "Retrofitting".

Table 5.2: Buildings with optimal decision "No action" and "Retrofitting".Time-independent seismic hazard Time-dependent seismic hazardNo Action Retrofitting No Action Retrofitting

All 201 333 - -Subset 3 11 0 14

Calculation scheme for Example 1

The optimal decision regarding retrofitting for each of the buildings in the city is calculatedusing the scheme given in Figure 5.8.

The BPN in Figure 5.5 is constructed in HUGIN for each seismic source. Only the nodesand arrows need to be specified at this step. The number of discrete states in the nodes andthe probability tables are introduced in MATLAB. The main file BPN_PSHA_Adapazari_RM.mcalculates the probability distributions related to seismic hazard as described in Section 4.1.5.The file Fragility_Typ5_O_Res_RM_1.m calculates the discrete probabilities for the states ofthe node ’Damage’ given the states of the node ’Epsilon Damage’ and the states of the node’SD’ specified in BPN_PSHA_Adapazari_RM.m. Upon completion of this step one single BPNis constructed and quantified for the structure class considered. The nodes ’Liquefaction’ and’Cost’ are site and building specific, respectively. They are quantified in GIS.

86


For each of the 42 combinations of magnitude (6 states) and PGA (7 states) the probability ofliquefaction in each grid point was calculated using the scheme provided in Section 4.2. Theseprobability of liquefaction values were assigned to the buildings given the the location and themagnitude-PGA pair. These probabilities were imported into GIS as 42 additional columns inthe attribute table of the corresponding shape file of the building class. In GIS, for each of thebuildings the total story area is calculated and added as an additional column to the attributetable of the shape file. Based on the total story area, an average number of columns (required forestimating retrofitting costs) and the number of people at risk (required for estimating fatalitycosts) are calculated and appended as columns to the attribute table of the shape file.

Using the visual Basic macro files OpenShape.bas, Module1.bas and Module1-NonPoisson.bas,the BPN for the structure class is called and for each building the site and building specific nodes,i.e. node ’Liquefaction’ and ’Cost’, are quantified. By doing this, building and site specific BPNsare generated. These BPNs are evaluated in GIS using Visual Basic as a client and the inferenceengine of HUGIN as a server.

The building and site specific BPNs are evaluated for each decision alternative and each seis-mic source. Totaling the costs for each decision alternative over all seismic sources yields thetotal expected cost due to seismic hazard. The minimum of the total expected costs indicated theoptimal decision for each structure (Figure 5.6).

The above described scheme is applicable when the magnitude-recurrence relationship is as-sumed to be time-independent. Considering a time-dependent magnitude-recurrence relation-ship requires following modifications: Using the file BPN_PSHA_Adapazari_RM_NonPoisson.mfor each seismic source (here, 9 seismic sources) and each year of the time frame considered(here, 50 years) one BPN is quantified. The ’Liquefaction’ and ’Cost’ nodes of each of these450 BPNs are quantified for each of the buildings in GIS. These BPNs are evaluated in GISusing Visual Basic as a client and the inference engine of HUGIN as a server.

The building and site specific BPNs are evaluated for each decision alternative and each seis-mic source. Totaling the costs for each decision alternative over all seismic sources yields thetotal expected cost due to seismic hazard. The minimum of the total expected costs indicated theoptimal decision for each structure (Figure 5.7). For comparison the results are given only for asubset of 14 buildings.

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5 Examples

/

0 1'500 3'000750 km

Retrofit?NoYes

Figure 5.6: Optimal decision for each building (time-independent -Poisson- seismic hazardmodel).

88


/

0 1'500 3'000750 km

Retrofit?NoYes

Figure 5.7: Optimal decision for each building (time-dependent seismic hazard model).

89

5 Examples

Construction of the BPN in Figure 5.4 for the structure

class Typ5-NR-O-Res

Given the spectraldisplacement state, the

discrete probabilities foreach damage state are

calculated

Nodes related to seismichazard are incorporated as

described in the calculationscheme in Section 4.1

For each decision alternative the the structureand site specific BPN is evaluated and the

optimal actions are identified

BPN_PSHA_Adapazari_RM_Poisson.m Fragility_Typ5_O_Res_RM_1.m

For the given story area ofeach building, the number

of people at risk, thenumber of fatalities and thevalue at risk are calculated

For each (Mw,R) pair theprobability of liquefaction

for the location of eachstructure is incorporated in

GIS

Module1.basModule1-NonPoisson.bas

GIS

env

ironm

ent

GIS

env

ironm

ent

Mat

lab

envi

ronm

ent

Module1.bas

BPN_PSHA_Adapazari_RM.m

Figure 5.8: Calculation scheme for Figure 5.6 and 5.7.

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5.2 Example 2: Assessment of seismic risk


Earthquake risk assessment for each building in the city is performed by using the BPN givenin Figure 5.5 with site and structure specific information available in the GIS platform. Havingquantified the probability tables and consequences, the Bayesian probabilistic network is eval-uated for each of the buildings in the city (Bayraktarli and Faber, 2009). For the analysis, thecommercial software package Hugin (2008) is used; however, freeware is now also available forthis type of analysis. With information for each building in the city on story area and number ofstories, the earthquake risk is calculated as the expected total annual cost.

SD

DistanceMagnitude

PGA

Epsilon PGA

Period

Epsilon SD

Liquefaction

Magnitude

PGA

DamageLiquefaction

SD EpsilonDamage

Structure Class

Damage

Cost

Structure Class

Occupancy Class

Construction Year

Cost

Structure dataSoil information

Prob

abili

ty o

f Liq

uefa

ction

Seismicity data

S1S4 S3 S2S5

S25

S15

S14

S13

S21S19

S22S12

Figure 5.9: BPN for calculating the earthquake risk for each building.

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5 Examples

The total expected cost for the portfolio is calculated by totaling the expected costs of thebuildings. This approach has a couple of shortcomings. First of all, simply aggregating the indi-vidual expected cost gives the total expected loss for the portfolio but not the distribution of thelosses. Furthermore, as the same structural model is applied for each building the same modelinguncertainties should be applied for each building. The fact that a common event is affecting theregion is also disregarded here. A structured way of modeling complex systems with Bayesianprobabilistic networks is to define the different underlying models as object classes. An ob-ject oriented Bayesian probabilistic network is generated using the subnetworks introduced inChapter 4 (Figure 5.9).

Careful examination of the main BPNs for the Example 1 reveals that most of the nodesaffect only the building they belong to. There are, however, also nodes which are commonto all buildings and nodes which are common to a subset of buildings. Nodes belonging todifferent levels are modeled within a hierarchical level. For the present case the nodes affectingall buildings comprise the highest modeling level.

Given a seismic source, the distribution of magnitude and distance should be conditioned atthe same state, e.g., when performing a probabilistic analysis over different magnitudes and dis-tance combinations, it cannot be assumed that one building is experiencing a Mw=6 earthquakewhile another building experiences a Mw=7 earthquake. This can be assured by modeling onesingle node for magnitude and distance.

The second modeling level comprises the uncertainty node for the fragility curves. In thepresent example two structure classes are considered: 5-story buildings constructed before 1980and 5-story buildings constructed after 1980. Two sets of fragility curves are modeled in Section4.3 for the two structure classes. As all the buildings in the two structure classes are assumedto respond as the generic building of that structure class, the uncertainty nodes should be in thesame state, e.g., when performing a probabilistic analysis of the structural response, it cannotbe assumed that for one building the structural response is given by the fragility curve with anuncertainty ε = +2σ while for an another building with an uncertainty ε =−2σ . Consistency isassured by modeling a common uncertainty node for each considered structure class. The thirdand last modeling level comprises all other nodes which affect only the building they belong to.In other words, these nodes are independent of the corresponding nodes in the other buildings.This important distinction of the nodes into hierarchical levels does not need to be considered inExample 1 since the optimal decision is based on the expected value of the costs.

When modeling the dependency in a serial system the expected values of the parametersare invariant, while the variances change. For residential buildings, it can be assumed that thecollapse of any residential building has a negligible influence on the cost of the collapse ofanother residential building. In reality this may not be true, especially when a very high numberof residential buildings collapse and the economy is affected, e.g. the unit prices establish adependency. In contrast, when considering lifelines, e.g. hospitals, the collapse of one hospitalhas an influence on the cost of the collapse of another hospital. This may be modeled as aparallel system. When modeling the dependency in a parallel system both the expected valuesand the variances of the parameters are affected (Schubert, 2009).

Figure 5.9 illustrates the application of the BPN with site and structure specific information

92


DistanceMagnitude

Cost

EpsilonDamage

„Typ5 O“ Building 1

DistanceMagnitude

Cost

EpsilonDamage


DistanceMagnitude

Cost

EpsilonDamage

„Typ5 N“ Building 2

DistanceMagnitude

Cost

EpsilonDamage

„Typ5 N“ Building 712

...

...

Cost S2

Magnitude Distance

EpsilonDamage

Epsilon Damage

Figure 5.10: Earthquake risk for the portfolio of 5-story residential buildings from one seismicsource.

for each building. To overcome the aforementioned shortcomings, the Bayesian probabilisticnetwork in Figure 5.9 is modeled as an object class and applied for each building. In addition tothe three input nodes ’Liquefaction’, ’Occupancy class’ and ’Structure class’ which will alwaysbe conditioned, three common nodes are modeled: the common hazard event with the magnitudeand distance of the earthquake, and the modeling uncertainty of the fragility curves for thedamage assessment (node ’Epsilon Damage’). A node ’Costs portfolio’ is modeled conditionedon the node ’Costs’ of the individual buildings. The application of the object oriented Bayesianprobabilistic network is illustrated in Figure 5.10. The results for the two cases, aggregatingthe risk with and without considering common cause effects are given in Figure 5.11 for theseismic source S2. It can be seen that, when aggregating individual risks considering commoncause effects the loss exceedance curve is relatively smooth and centered. It can be seen that,when the individual risks are aggregated without considering common cause effects, the lossdistribution curve underestimates the probability of exceedance of higher total portfolio losses.This is in line with the expectation, that there will always be a certain "leveling out" betweenlosses generated by individual objects within an area given an event. For the three commonnodes considered (i.e. magnitude, distance and fragility curve uncertainty) the composition ofthe loss exceedance curve is illustrated in Figure 5.12. The magnitude node was discretized intosix states, the distance node into five and the fragility curve uncertainty into five states. The totalnumber of combinations is hence 150.

For all of the 150 combinations the loss distribution is calculated. These 150 loss distributionsare multiplied by the joint occurrence probability of the Magnitude-Distance-Fragility curveuncertainty triple and totaled according to the total probability theorem. In Figure 5.12 the

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5 Examples

loss distribution of the individual combinations is illustrated in the table and the aggregation atthe top. The corresponding loss exceedance curve is illustrated in Figure 5.11. It is also veryclear that the ’waves’ in the loss distribution and the loss exceedance curve result mainly fromthe small number of states in the magnitude and distance nodes. Figure 5.13 illustrates lossexceedance curves for the 5-story residential buildings in the city due to the seismic source S2for different parameters. For the sake of completeness the influence of considering commoncause effects in the portfolio loss estimation is illustrated in a). The effect of considering a time-dependent earthquake recurrence model instead of a Poisson recurrence model is illustrated inb). The details of the seismic hazard model considered are given in Section 4.1. There is aminor influence on the loss exceedance curve when considering the time-dependent recurrencemodel, since the risks are calculated per annum. The probability distribution of the magnitudebarely differs for the first years, here for the year 2009. In c) the loss exceedance curve forthe 5-story residential buildings in the city due to the seismic source S2 is given for the totallosses and the property losses. Figure 5.13 illustrates the effect of using different discretisationschemes when calculating and implementing the probability tables, here the ’SD’ node. Firstthe ’SD’ node is discretized by using only 7 states. The bounds of these 7 discrete states areadapted considering the relative frequencies of each quadruple (magnitude, distance, epsilon SDand period T). Alternatively, other discretisation schemes with equal spacing of the state boundsof the ’SD’ node are evaluated. Here, three cases with an ’SD’ node discretized with 10, 20 and50 states are considered. The results clearly show that a very fine equal spaced discretisation (50states) converges to the adapted discretisation scheme with 7 states. That means in constructinga BPN, either a fine disretization or an adaptive discretisation scheme needs to be chosen.

Since the choice for a very fine discretisation easily grows the computation effort involved inevaluating a BPN often adaptive discretisation schemes is unavoidable. This situation may how-ever change when computational efficiency increases. In Figure 5.14 the aggregation schemefor all seismic sources affecting the considered region is illustrated. Figure 5.15 and Figure5.16 illustrate the loss exceedance curves for the 5-story residential building portfolio due toeach seismic source and the aggregation of all seismic sources for the portfolio total losses and

Figure 5.11: Loss exceedance curves for 5-story residential buildings.

94


Cos

t [U

SD]

Loss

D

istr

ibut

ion

e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5e1e2e3e4e5

15’

000

900

62’

000

160

’000

300

’000

Figure 5.12: Illustration of the composition for the loss exceedance curve considering commoncause.

95

5 Examples

0 5 1 5 2x 10

10−3

10−2

10−1

100Seismic source S2

Portfolio Total Loss [USD]

Prob

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[a−1 ]

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[a−1 ]

5

5

5 5

5

5

Common causeNo common cause

Total lossProperty loss

SD with adapted discretisationSD with 20 equal spaced bins



Time-independent hazardTime-dependent hazard

Figure 5.13: Parametric study of the loss exceedance curves for 5-story residential buildings.

portfolio property losses respectively. Figure 5.17 and Figure 5.18 illustrate the loss exceedancecurves for the 5-story hospital building portfolio due to each seismic source and the aggregationof all seismic sources for the portfolio total losses and portfolio property losses respectively. Itcan be seen that the influence is not accentuated compared to the residential buildings as thereare only five hospitals which do not lead to pronounced leveling out. In Figure 5.17 and Figure5.18 also the influence of non-proportional increase of indirect consequence for lifelines such ashospitals is illustrated. The total loss of each hospital is calculated for two damage states, i.e.no damage or collapse. In contrast to the costs of losses for residential buildings, the collapse ofone hospital results in an increase of the costs of the collapse of the remaining hospitals. Here avery simple assumption is applied. The portfolio losses of the five hospitals are multiplied by 1,1.5, 2, 2.5, 3 for the total number of collapsed 1, 2, 3, 4, 5 hospitals respectively.

Cost S1Source 1

Cost S2Source 2

Cost S9Source 9

...

Cost Portfolio

Figure 5.14: Earthquake risk for the portfolio of 5-story buildings (all seismic sources).

96


0 1 2 3 4 5x 106

10−3

10−2

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100All seismic sources


Prob

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[a-1 ]


Figure 5.15: Total loss exceedance curves with and without considering common cause effectsfor the portfolio of 5-story residential buildings.

97

5 Examples

0 1 2 3 4 5x 106

10−3

10−2

10−1


Portfolio Property Loss [USD]

Prob

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[a−1 ]


0 0.5 1 1.5 2x 105

10−3

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[a−1 ]


Figure 5.16: Property loss exceedance curves with and without considering common cause ef-fects for the portfolio of 5-story residential buildings.

98


0 0.5 1 1.5 2 2.5x 106

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Common causeNo common causeCommon cause & NL

0 2 4 6 8 10x 104

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[a−1

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Figure 5.17: Total loss exceedance curves with and without considering common cause effectsfor the portfolio of 5-story hospital buildings.

99

5 Examples

0 0.5 1 1.5 2 2.5x 106

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[a−1

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Figure 5.18: Property loss exceedance curves with and without considering common cause ef-fects for the portfolio of 5-story hospital buildings.

100



The calculation scheme for loss exceedance curves given in Figure 5.11 is given in Figure 5.19.The BPN in Figure 5.9 is constructed in HUGIN for seismic source S2. Only the nodes andarrows need to be specified at this step. The number of discrete states in the nodes and theprobability tables are introduced in MATLAB. The main file BPN_PSHA_Adapazari_RA_Res.mcalculates the probability distributions related to seismic hazard as described in Section 4.1.5.The file Fragility_Typ5_O_Res_RA_1.m calculates the discrete probabilities for the states of thenode ’Damage’ given the states of the node ’Epsilon Damage’ and the states of the node ’SD’specified in BPN_PSHA_Adapazari_RA_Res.m. Upon completion of this step one single BPNis constructed and quantified for the structure class considered. The nodes ’Liquefaction’ and’Cost’ are site and building specific, respectively. They are quantified in GIS.

For each of the 42 combinations of magnitude (6 states) and PGA (7 states) the probabilityof liquefaction in each grid point is calculated using the scheme provided in Section 4.2. Theseprobabilities were assigned to the buildings given the the location and the magnitude-PGA pair.These probabilities were imported into GIS as 42 additional columns in the attribute table of thecorresponding shape file of the building class. In GIS, for each of the buildings the total storyarea is calculated and added as an additional column to the attribute table of the shape file. Basedon the total story area, the number of people at risk (required for estimating fatality costs) arecalculated and appended as columns to the attribute table of the corresponding shape file.

Using the visual Basic macro files Module1-Ex2.bas and OpenShape.bas, the BPN for thestructure class is called and for each building the site and building specific nodes, i.e. node’Liquefaction’ and ’Cost’, are quantified. By doing this, building and site specific BPNs aregenerated. These BPNs are evaluated in GIS using Visual Basic as a client and the inferenceengine of HUGIN as a server.

These BPNs are applied individually when the analyst is interested in risk of individual build-ings. When a portfolio is considered, however, those nodes need to be identified which influencethe individual buildings commonly at different hierarchical levels. As illustrated in Figure 5.10,the nodes ’Magnitude’ and ’Distance’ represent one hierarchical level, and the nodes ’EpsilonDamage’ represents another hierarchical level. This large BPN is decoupled by conditioningeach of the states of the common nodes and evaluating individual BPNs.

For each of the 150 combinations of the states of the common nodes (nodes ’Magnitude’,’Distance’ and ’Epsilon Damage’ with 6, 5 and 5 states, respectively) the distributions of thenodes ’Cost’ for each of the 1246 buildings (534 Typ5-O and 712 Typ5-N) are calculated us-ing the file CostDistS2.m. For each of the 150 combination the distribution of node ’CostS2’ iscalculated by sampling the state of the node ’Cost’ for each of the 1246 buildings using file Ag-gregation_Poisson_S2.m. These distributions are given in Figure 5.12. These 150 distributionsare multiplied by the joint occurrence probability of the Magnitude-Distance-Epsilon Damagetriple and totaled according to the total probability theorem resulting in the distribution of thenode ’CostS2’. The results are given in Figure 5.11.

For comparison, the existence of common nodes are disregarded and the distribution of thenode ’Cost’ for each building is evaluated for the stand-alone BPNs by the file CostDistS2_Integrated.m. The distribution of the node ’CostS2’ is calculated by sampling the distributions ofthe nodes ’Cost’ for each of the 1246 buildings using file Aggregation_Poisson_S2_Integrated.m.

101

5 Examples

The same calculation scheme is used for each seismic source, for different structure classes(residential/hospital), different magnitude-recurrence relationships (Poisson/time-dependent), dif-ferent loss types (total loss/property loss) and different numbers of discrete states (Figures 5.13and Figures 5.15 to 5.18).

Construction of the BPN in Figure 5.8 for the structureclass Typ5-NR-O-Res and

Seismic source S2



calculated



BPN_PSHA_Adapazari_RA_Res.m Fragility_Typ5_O_Res_RA_1.m

The discrete probabilitiesof the node ' Cost S2'

representing the portfolioloss due to seismic sourceS2 are simulated using a

Monte Carlo scheme

The discrete probabilitiesof the node ' Cost S2'

representing the portfolioloss due to seismic sourceS2 are simulated using a

Monte Carlo schemeAggregation_Poisson_S2_Integrated.mAggregation_Poisson_S2.m

For the given story area ofeach building, the number

of people at risk, thenumber of fatalities and thevalue at risk are calculated



GIS

Module1-Ex2.basModule1-Ex2.bas

The distribution of thenode ' Cost' for eachbuilding is evaluated

For each combination ofthe states of the commonnodes the distribution ofthe node ' Cost' for each

building is evaluated

CostDistS2_Integrated.mCostDistS2.m

GIS

env

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Mat

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GIS

env

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OpenShape.basOpenShape.bas

Figure 5.19: Calculation scheme for Figure 5.11.

102

5.3 Example 3: Update of fragility curves


One of the strengths of the proposed framework using BPNs is their ability to systematicallyupdate any parameter in the model with new information. It is possible that data in the formof reports on damage to buildings in an affected area is available after an earthquake. A post-earthquake damage inspection team could have been instructed to utilize ATC-20 (1989) method-ology and assign a color tag (Green-Safe, Yellow-Limited use or Red-Unsafe/Collapsed) to theinspected building. Based on the damage distribution of the buildings after the Kocaeli Mw7.4earthquake of 17th August, 1999, a map indicating the percentage of damaged buildings within adistrict in Adapazari is available (DRM, 2004). The same report also provides a map indicatingthe areas in the city which are liquefied.

The BPN applied for the Bayesian update problem is in principle the same as for the riskmanagement and risk assessment problems illustrated in Example 1 and Example 2. Besidethe fact that all cost-related nodes have been removed, an important modification is performedin handling the uncertainty of the fragility curves which are to be updated using data capturedby post-earthquake damage surveys. In the foregoing examples the uncertainty in the fragilitycurves was modeled by a single node (i.e. ’Epsilon fragility’ node).

To enable updating in the present case, the uncertainty distribution of the fragility curves ismodeled by four nodes: The parameters of the lognormal distribution λ and ζ for each of the twodamage states. The four uncertainty nodes are discretised into three states (mean+sigma, mean,mean-sigma) with the probabilities (0.159,0.682,0.159) respectively. The BPN is illustrated inFigure 5.20. The map available indicating the percentage of damaged buildings is used to assignto each of the 201 "5-story residential buildings constructed before 1980" at random the twodamage states (Green-No damage and Red-Collapsed). It should be noted that for only 201 of the534 "5-story residential buildings constructed before 1980" information on damage is available.It is also important to note that building-specific information on damage is not necessary as thebuildings are grouped into structural classes and the structural behavior of generic buildings isassumed to represent their behavior. Furthermore, from the map indicating liquefaction for eachof the 201 buildings, a tag on the liquefaction state is assigned.

DamageLiquefaction

SD

DistanceMagnitude

PGA

Epsilon PGA

Lambda Yellow

Structure Class

T

Epsilon SD

Zeta Yellow

Lambda Red

Zeta Red

Figure 5.20: Main Bayesian probabilistic network constructed for Bayesian updating.

103

5 Examples

Liquefaction

Damage


LambdaYellow

ZetaYellow

Lambda Red

ZetaRed

Liquefaction

Damage


LambdaYellow

ZetaYellow

Lambda Red

ZetaRed...

LambdaYellow

ZetaYellow

Lambda Red

ZetaRed

10 %

30 %20 %

40 %50 %60 %

Figure 5.21: Collapsed building statistics for buildings with four and more stories in Adapazariafter the Mw7.4 Kocaeli earthquake of 17th August 1999 (Bakir et al., 2005) andBayesian probabilistic network for the update of fragility curves for the hospitalbuildings.

The BPN for the prior case along with information on the liquefaction state and structuraldamage state for 201 buildings is thus available. The characteristics of the earthquake are no

104


longer uncertain. The magnitude was Mw7.4, the distance to the activated seismic source S3was about 50 km. This information can be used to condition the nodes magnitude and distance.Starting with the first of the 201 buildings, the BPN is further conditioned using the informationon the liquefaction state and structural damage state of the first building. The two nodes ’Liq-uefaction’ and ’Damage’ are instantiated correspondingly and the parameters of the fragilitycurves are updated using the BPN. The calculation scheme is illustrated in Figure 5.21.

The probability distributions of the uncertainty nodes are updated using the software Huginwithin a GIS environment. The BPN is modified implementing these updated probability distri-butions for the four uncertainty nodes. The same procedure is applied successively for all of the201 buildings. The probabilities of the uncertainty nodes are given in Table 5.3 and 5.4.

Table 5.3: Probability distribution of the uncertainty nodes.Prior Posterior

µ +σ µ µ −σ µ +σ µ µ −σ

YellowLambda 0.159 0.682 0.159 0.662 0.330 0.008

Zeta 0.159 0.682 0.159 0.088 0.640 0.272

RedLambda 0.159 0.682 0.159 0.000 0.002 0.999

Zeta 0.159 0.682 0.159 0.904 0.095 0.001

The posterior distribution parameters are calculated by weighted averaging:

λYellow = 0.662(3.758+0.034)+0.330(3.758)+0.008(3.758−0.034) = 3.780

ζYellow = 0.088(0.390+0.024)+0.640(0.390)+0.272(0.390−0.024) = 0.386

λRed = 0.000(4.215+0.055)+0.002(4.215)+0.999(4.215−0.055) = 4.160

ζRed = 0.904(0.346+0.040)+0.095(0.346)+0.001(0.346−0.040) = 0.382

The updated fragility curve for the building class is given in Figure 5.22.

Table 5.4: Parameters of the prior and posterior lognormal distribution for the fragility curves.Prior Posterior

Mean Sigma Mean

YellowLambda 3.758 0.034 3.780

Zeta 0.390 0.024 0.386

RedLambda 4.215 0.055 4.160

Zeta 0.346 0.040 0.382


The calculation scheme for fragility curves update given in Figure 5.11 is given in Figure 5.23.The BPN in Figure 5.20 is constructed in HUGIN for seismic source S3 which is assumed

to be ruptured in the Mw7.4 Kocaeli earthquake 1999. Only the nodes and arrows need to be

105

5 Examples

specified at this step. The number of discrete states in the nodes and the probability tables areintroduced in MATLAB. The main file BPN_PSHA-_Adapazari_BU.m calculates the probabilitydistributions related to seismic hazard as described in Section 4.1.5. The file Fragility_Typ5_O_Res_RA_2.m calculates the discrete probabilities for the states of the node ’Damage’ giventhe states of the nodes ’Lambda Yellow’, ’Zeta Yellow’, ’Lambda Red’ and ’Zeta Yellow’, andthe states of the node ’SD’ specified in BPN_PSHA_Adapazari_BU.m. Upon completion of thisstep one single BPN is constructed and quantified for the structure class considered. The nodes’Liquefaction’ and ’Cost’ are site and building specific, respectively. They are quantified in GIS.

For each of the 42 combinations of magnitude (6 states) and PGA (7 states) the probabil-ity of liquefaction in each grid point are calculated using the scheme provided in Section 4.2.These probability of liquefaction values were assigned to the buildings given the location andthe magnitude-PGA pair. These probabilities were imported into GIS as 42 additional columnsin the attribute table of the corresponding shape file of the building class.

Using the visual Basic macro files Module1-Ex3.bas the BPN for the structure class is calledand for each building the site specific node ’Liquefaction’ is quantified. By doing this sitespecific BPNs are generated. These BPNs are evaluated in GIS using Visual Basic as a clientand the inference engine of HUGIN as a server.

The available maps indicating damage to buildings and liquefied areas within the region dur-ing the Mw7.4 Kocaeli earthquake 1999, are used to assign tags (1 or 0) indicating damage/nondamaged and liquefied/non liquefied to each of the 201 buildings. This 201x2 matrix is readfrom the attribute table of the corresponding shape file in GIS.

The BPN given in Figure 5.20 is evaluated using MATLAB as client and the inference engineof HUGIN as a server in file UpdateBPN.m. Using the tags of the first building the BPN isconditioned and the nodes ’Lambda Yellow’, ’Zeta Yellow’, ’Lambda Red’ and ’Zeta Yellow’ ofthe fragility curve are updated. Applying this successively for each of the 201 tags the parameters

0 20 40 60 80 100 120 140 1600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spectral displacement, Sd

(T=0.64s, ζ =5%) [mm]

Prob

abili

ty

limited use (yellow)Collapse (red)limited use updated (yellow)Collapse updated (red)

Figure 5.22: Fragility curves updated with data from the Mw7.4 Kocaeli earthquake 1999.

106


of the fragility curve is updated.

Estimation of the seismicfragility curves in

Figure 5.20



calculated



BPN_PSHA_Adapazari_BU.m Fragility_Typ5_O_Res_RA_2.m

Successive update of thenodes 'Lambda Yellow', 'Zeta Yellow', 'Lambda

Red' and 'Zeta Red'

UpdateBPN.m



GIS

Information given in Figure 5.19 regarding theobserved damage for each

of the 201 buildings isincorporated

Information given in Figure 5.19 regarding theoccurence of liquefaction

is incorporated

Module1-Ex3.bas

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Figure 5.23: Calculation scheme for Figure 5.21 and 5.22.

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5 Examples

5.4 Example 4: Index of robustness

This example illustrates the use of the index of robustness as an indicator for the comparison ofrisk reduction measures. The output given in the Example 2 is used for illustration. As a regionalcity, the damage to the buildings in Adapazari affects the city as well as the region.

Robustness of a system is defined as the relationship of the direct risks to the total risks(Faber, 2008). It is thus a measure indicating the relative importance of a system with regardto a hierarchically upper level system. The derivation of the index of robustness for differenthierarchical levels is illustrated in this example. More formally, robustness is quantified bymeans of an index of robustness IR expressed through the ratio between direct risks and totalrisks.

IR =RD

RD +RID(5.7)

where RD and RID represent the direct and indirect risks respectively.

The index of robustness is illustrated for different system levels for the 5-story residentialbuildings in Adapazari. The direct and indirect risk due to earthquakes for the 5-story residentialbuildings are calculated in Example 2 and given in Figure 5.15 and Figure 5.16. The estimationof the index of robustness is given for three different system characterizations, see Figure 5.24.

Typ5River

Adapazari

/

0 75 150 km

Istanbul

0 1.5 3.0 km

0 50 100 m

/

/System 1: Individual building

System 2: City of Adapazari

System 3: Sakarya Region

Sakarya

Figure 5.24: System characterizations for seismic risk calculations for buildings of differentscales.

108

5.4 Example 4: Index of robustness

For the individual building in Figure 5.24 the indirect and direct losses are USD 46’000 andUSD 15’800. It should be noted that when calculating the direct losses for the system "individualbuilding" repair costs are considered as direct costs, rebuilding and loss of lives as indirect losses.Using Equation 5.7, the index of robustness is calculated as:

IR1 =RD

RD +RID=

1580015800+46000

= 0.255

The situation is different when considering the portfolio of 5-story residential buildings as asystem (Figure 5.24). Hence, repair and rebuilding costs are considered as direct costs and lossof lives as indirect costs. The indirect and direct losses are estimated as USD 1’239’500 andUSD 1’148’600 for the portfolio of 5-story residential buildings (Figure 5.15 and Figure 5.16).Using Equation 5.7, the index of robustness is calculated as:

IR2 =RD

RD +RID=

1′148′6001′148′600+1′239′500

= 0.481

Defining the system as the whole region where the city of Adapazari is embedded results ina different composition of direct and indirect loss, and hence a different index of robustness.When considering System 3 in Figure 5.24, i.e. the Sakarya region, the estimation of the macro-economic costs of the the Kocaeli Mw7.4 earthquake of August 17, 1999 is assumed to apply.In Bibbee et al. (2000) estimates for the whole affected region of the Kocaeli Mw7.4 earthquakeare given. Using the estimation of the Türkish Industrialisation and Businessmen’s Association(TÜSIAD), i.e. an indirect loss estimate of USD 6.8 billion and a direct loss estimate of USD10.0 billion, the index of robustness is calculated as:

IR3 =RD

RD +RID=

10′000′000′00010′000′000′000+6′800′000′000

= 0.595

This example illustrates that the relative importance of the propagation of losses of an adverseevent is dependent on the system representation level, i.e. on the decision-making level. Theindex of robustness provides an alternative way of presenting seismic risk by explicitly indicatingthe fraction of the direct effects to the total effects. For different risk reduction measures, theindex of robustness may lead to different optimal decisions, given the system characterizationlevel. For each decision alternative, the index of robustness could be calculated given the systemcharacterization level. The increase in the index of robustness through retrofit may be feasiblefor one decision level, but not feasible for another. A thorough consideration of this new conceptof index of robustness is anyway necessary in the research community.

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Summary

In this dissertation, a framework for risk assessment has been proposed for use in earthquake-related problems in cities. The framework aims to represent the problem complex of earthquakerisk by modeling the prevailing parameters, to explicitly model the dependencies among theparameters for considering system effects and to provide a framework for systematic update. Ituses a modular structure in modeling seismic hazard, soil response, structural response and dam-age and loss assessment. The framework is applied by Bayesian probabilistic networks (BPN).The applicability and pros and cons of BPNs are discussed for earthquake risk management, forportfolio loss distribution and for the systematic update of model constituents within a Bayesianperspective.

In the proposed risk assessment framework, three levels are distinguished: exposure, vul-nerability and robustness. The proposed framework allows for utilization of any type of riskindicators with regard to exposure, vulnerability and robustness of the considered system. Riskindicators are any observable or measurable characteristics of the system or its constituents con-taining information about risk.

Existing earthquake loss estimation methodologies were reviewed and six main characteris-tics were identified, thus ensuring a consistent treatment of complex problems subject to un-certainties; integrality/generality, modularity, inference, dependencies, updateability, and multi-detailing. Although the first two, and partly the third characteristics are covered by some of theexisting methodologies, none of the existing methodologies comprehends the latter three.

An integral approach to risk assessment ensures that significant risk contributors originatingfrom interactions between the different agents are accounted for. To some degree all of the ex-isting loss estimation methodologies can be regarded as integral approaches. On the other hand,generality, i.e. context independence, is especially important when the loss estimation method-ology is to be applied to other regions as well as for other hazard types. The proposed frameworkis applied using indicators representing exposure, vulnerability and robustness. Consistent mod-eling at all levels is ensured by modeling using BPNs. The indicators may be identified andquantified for specific or general situations.

Earthquake loss estimation requires an interdisciplinary approach. Research on individualdisciplines such as seismology, soil dynamics or earthquake engineering provide scientific andtechnological improvements which may result in the existing loss estimation methodologies be-coming obsolete, if these improvements are not considered. A modular structure enables theimplementation of new models in any of the disciplines without resetting the overall methodol-ogy. Just like the other existing earthquake loss estimation methodologies, the proposed frame-work has as a modular structure. It comprises modules for seismic hazard, for soil response,

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6 Conclusions

for structural response, for damage and losses. The interfaces of these modules is defined andcalculations are executed from end-to-end. Modularity enables the easy adaptation of alternativemethods and models to the integral model.

Earthquake risk is evaluated in the existing methodologies in forward direction only. Theanalysis on seismic hazard, soil response, structural response, damage and loss assessment isperformed end-to-end leading to a quantification of the risk or to a damage scenario. Using theproposed methodology it is also possible to perform diagnostic analysis. Typical queries are ofthe kind "Which magnitude of earthquakes would lead to complete unavailability of importantstructures such as hospitals?" None of the existing loss estimation methodologies considers theprevailing parameters explicitly by considering their uncertainties. Explicit consideration of theuncertainty of the parameters enables inference in both directions. It is shown that BPNs allowinference based on observed evidence.

The lack of explicit consideration of the dependencies of the prevailing parameters is an-other pitfall in existing methodologies. Disregarding dependencies leads to a suppression ofimportant system effects when considering loss estimation to portfolios of buildings. Statisti-cal dependency may be appropriately represented through correlation. Functional dependencyor common cause dependency is appropriately represented through hierarchical probabilisticmodels. Explicit modeling of dependencies is one of the strengths of BPNs, especially when de-pendencies at different modeling levels are considered, as in hierarchical models. Modeling thedependencies at different hierarchical levels makes it possible to calculate risks for portfolios.

The continuous adaptation of engineering models is a major challenge. In Bayesian un-derstanding the constructed models represent the present state of knowledge and with everyincoming information, the underlain models could be updated. The existing loss estimationmethodologies are in some ways considering the knowledge gained through new earthquakes byremodeling, but none of them provides a framework for systematic update. The knowledge andinformation basis is not very broad when assessing earthquake risks. Damaging earthquakes arenot very frequent, data on the built environment is usually very scarce and research on the occur-rence of earthquakes and their effects on soil and structures is ongoing. Hence, the processingof scientifically verified knowledge over statistically representative data to experience reflectingexpert opinions is necessary. An update is facilitated especially when the problem complex ismodeled explicitly by observable characteristic descriptors referred to as indicators. It is shownthat the Bayesian perspective as proposed in this dissertation provides a sound and thoroughmeans for this.

Considering risks due to earthquakes, the decision-makers are to some degree all individualswithin the earthquake-prone region. The formulation of a specific decision problem depends,however, on the decision-making level. It is illustrated that in principle the framework can beapplied for different decision-making levels. This is possible as the dependencies at differenthierarchical levels are modeled explicitly. None of the existing loss estimation methodologies iscapable of application for different decision makers with different levels of detailing.

In this dissertation the construction and application of BPNs for "seismic hazard", "soil be-havior", "structural response" and "consequence assessment" are also discussed. The applicationof the proposed framework is illustrated using four examples on earthquake risk problems for

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a class of structures located in Adapazari, Turkey. The city center of Adapazari was affectedby the Kocaeli Mw7.4 earthquake of August 17, 1999. Adapazari is chosen as a test area, asit has suffered damage due both to ground shaking and soil failures such as tilting, settlementand lateral displacements. In the first example, a risk management problem is considered. Twodecision alternatives, namely strengthening the reinforced concrete moment resisting frames bycolumn jacketing or no action, are considered. For a chosen structure class, namely five-story re-inforced concrete moment resisting frames, the optimal decision of each of the buildings withinthe structure class is identified. Here, both time-independent and time-dependent seismic hazardmodels are considered. The time-dependent seismic hazard model leads to a substantial increasein seismic activity which results in the optimal action being "retrofitting" for the buildings. In thesecond example, earthquake risk assessment for each building in the city is performed using theproposed methodology. The example illustrates that the framework facilitates the assessment ofgroups of structures and of the portfolio loss exceedance probability function. The third exampleillustrates one of the strengths of the proposed methodology in their ability for straightforwardupdating of any parameter in the model. It is possible that data in the form of reports on thedamages on buildings in an affected area is available after an earthquake. A post-earthquakedamage inspection team could have assigned a color tag (Green-Safe, Yellow-Limited use orRed-Unsafe/Collapsed) to the inspected building on the basis of the building’s safety. Based onthe damage distribution of the buildings after the 17th August, 1999 Kocaeli Mw7.4 earthquake,a map indicating the percentage of damaged buildings within a district in Adapazari and a mapshowing the areas in the city which are liquefied are used and the parameters of the fragilitycurves are updated using the BPN. The fourth example illustrates the use of the index of robust-ness as an indicator. The example illustrates that the relative importance of the propagation oflosses of an adverse event is dependent on the system representation level, i.e. on the decision-making level. The index of robustness provides an alternative way of presenting seismic risk byexplicitly indicating the fraction of the direct consequences related to the total consequences.

Originality

The dissertation has two main objectives: The construction and application of BPNs in individ-ual discipline and the systematic application of BPNs to cities.

The first objective is met by "translating" the state-of-the-art models for probabilistic seismichazard assessment, for seismic-induced soil liquefaction potential evaluation and for structuralassessment into BPN models. In detail:

• For the seismic hazard model an alternative calculation and representation scheme for thestandard Probabilistic Seismic Hazard Analysis (PSHA) using Bayesian Probabilistic Net-works (BPN) is presented. The BPN can easily be extended to compute joint probabilitydistributions for multiple ground motion parameters, which is a feature not easily imple-mented in standard PSHA. Backward calculation, as implemented using deaggregationin standard PSHA, is also easily performed using BPNs. Incorporation of model choiceuncertainties and time-dependent seismic hazard into the BPN model for seismic hazardwas also discussed.

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6 Conclusions

• The soil liquefaction potential evaluation is generally based on deterministic empiricalcorrelations using Standard Penetration Test blow counts. The required soil parameters forthese empirical correlations are modeled by taking their spatial variability into account.For different earthquake intensity parameters (moment magnitude and peak ground accel-eration) and for each point of the test area a probability of liquefaction is calculated. Thisinformation is represented in a BPN, which is linked to the seismic hazard BPN and thestructural damage BPN.

• BPN models for structural damage are developed based on state-of-the-art seismic fragilityassessment procedures. To illustrate the application of the methodology to risk manage-ment problems, seismic fragility curves are also developed for a retrofitting scheme usingjacketing of the columns. The fragility curves developed are compared with the corre-sponding HAZUS fragility curves as well as with post-earthquake survey data from therecent earthquake in the region.

The second objective of the dissertation has been addressed through a systematic definitionof a city and development of a risk assessment framework. The discussions are based on theapplication of the proposed methodology to a specific city. The new developments are:

• A risk assessment framework for a generic application to earthquake risks for cities isestablished. This includes a system theoretic definition of cities and a hierarchical model-ing of risks. The framework can be applied for different decision-making levels. This ispossible as the dependencies at different hierarchical levels are modeled explicitly.

• Methods and rules for the consistent representation of the effect of dependencies in theestimation of losses are developed. It is shown that inclusion of such effects may have avery significant impact on portfolio loss estimates.

• A framework for systematic update of the models considering the knowledge and datagained through new earthquakes is proposed.

Limitations

Mainly two issues were identified as potential shortcomings of using the proposed frameworkfor earthquake risk problems. The first one is the discretisation of the parameters within themodel. The effect of different discretisation schemes on the portfolio loss exceedance curve isevaluated. The results clearly show that an automated equal spaced discretisation converges toa supervised adapted discretisation scheme with an increasing number of bins. This means, inconstructing a BPN, either a fine discretization or a supervised adaptive discretisation schemehas to be chosen. Since the choice of very fine discretisation increases the computation effortinvolved in evaluating a BPN, adaptive discretisation schemes may often be unavoidable.

The second shortcoming is related to the computational efficiency when considering a city orregion with thousands of buildings. It is not mainly the number of buildings but the complexdependency structure among the variables in the BPN models and especially the different hier-archical levels in the model which results in computational difficulties. The number of buildings

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has no affect when considering management problems as illustrated in the first example. Theoptimal decision is based on the expected value of risk, and the expected value of risk is cal-culated "decoupling" all the hierarchical dependencies within the model. However, when thedistribution of risk is an issue, as illustrated in the second example, these dependencies need tobe considered. This results in very large BPNs for risk assessment problems. It is illustrated thatthis kind of problem can be solved by decoupling the BPNs at the hierarchical levels. By doingso, the distribution of risk can be calculated; however, other important features of BPNs such asbackward inference, update or diagnostics cannot be applied.

Recommendations

When applying BPNs for large scale problems with thousands of elements at risk which havedependencies at different hierarchical levels, the existing software tools reach their limit. In thepresent dissertation, the very large integral BPN models were decoupled, evaluated and finallycoupled. By doing so, the useful features of BPNs such as backward inference, update anddiagnostics are not fully applicable. More efficient programming may be targeted to enable themodeling of larger BPNs.

The soil BPN model in the present dissertation was constructed so that the input parametersfor the hazard (PGA,Mw) and the main output in the form of liquefaction triggering are explicitlymodeled as nodes. The underlying empirical models for liquefaction triggering prediction maybe considered using the soil parameters explicitly within the BPN model. The spatially explicitapplication of these BPN models may be used to update the underlying empirical models withdata from earthquake events.

The update of seismic fragility curves with damage data from past earthquakes using BPNswas illustrated. Besides the damage observed, measurements from seismic stations may alsobe used when updating the structural damage models as well as the ground motion predictionequations in the seismic hazard models.

The BPNs in the present dissertation are based mainly on existing engineering models. Thegoverning parameters in these models are explicitly modeled in the BPNs taking the causal re-lations into account. Another approach would be to set up the parameters and construct BPNmodels based solely on data, without reflecting existing knowledge using causal relations. Thearrows in the BPNs may be defined based on the data using so-called structural learning meth-ods. It is thus possible to reveal former not considered dependencies among parameters to theanalyst. An application to ground motion prediction equations or soil liquefaction models wouldprobably be feasible.

Sustainable and consistent societal decision making requires that a framework for risk man-agement is developed, which, at a fundamental level allows for the comparison of risks fromdifferent natural hazards such as the comparison between risks due to earthquakes with the risksdue to flooding or due to draughts. The main characteristics of the proposed methodology likeits generality and modularity makes it a strong candidate for the modeling of risk due to multi-hazards.

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128

A BPN Algorithms

Algorithms for evaluating BPNs are presented in this Annex. Efficient algorithms are importantfor the applicability of BPNs, especially for comprehensive and complex problems as dealt within the present dissertation. The joint probability table increases exponentially with the number ofvariables and the number of states in these variables in each variable. The presented algorithmshere are among the most efficient methods known. Adhering closely to Jensen (2001), threemethods are presented.

The methods are illustrated via a numerical example with the BPN introduced in Chapter 3.The unconditional and conditional probability tables (hereafter referred to as potentials) of thevariables in the BPN under consideration are given in Figure A.1.

M

G S

L

D

M=0 0.9

M=7 0.1M=0 M=7

G=0 0.9 0.2

G=0.5g 0.1 0.8

G=0 G=0.5g

L='yes' 0.1 0.7

L='no' 0.9 0.3

M=0 M=7

S=0 0.9 0.1

S=10cm 0.1 0.9

L=yes L=

S=0 S=10 S=0 S=10

D='no' 0.2 0.1 0.9 0.3

D='collapse' 0.8 0.9 0.1 0.7

Figure A.1: Considered BPN with the probability tables.

Bucket elimination

For a BPN over U = M,G,S,L,D, the joint probability distribution P(U) is the product of allpotentials specified in the BPN. According to the chain rule for BPNs the joint probability dis-tribution is:

129

A BPN Algorithms

P(U) = ∏i

P(M)P(G|M)P(S|M)P(L|G)P(D|S,L) (A.1)

Using the notation with potentials:

P(U) = ∏i

φ1(M)φ2(G,M)φ3(S,M)φ4(L,G)φ5(D,S,L) (A.2)

When any of the variables receive specific information, i.e. evidence, the BPN is used tocalculate the updated probabilities. For example, when the state of the node "Liquefaction" inthe BPN in Figure A.1 is no longer uncertain, i.e. it is known that liquefaction of the soil isobserved, the joint probability distribution P(U,e) is calculated with the evidence of observingliquefaction:

P(U,e) = ∏i

φ1(M)φ2(G,M)φ3(S,M)φ4(L,G)φ5(D,S,L)e (A.3)

The marginal distribution of any variable in a BPN can be calculated by marginalizing allother variables out of the joint probability distribution function. Starting with the set of tables asgiven in equations A.2 or A.3, whenever a variable has to be marginalized, all tables with thatvariable are taken, in multiplied form. The variable to be marginalized is then integrated out.This is called eliminating the variable, and the process of repeatedly eliminating a variable froman initial set of tables is called bucket elimination.

For example, the marginal distribution of the variable ’Damage (D)’ is calculated by:

P(D) = ∑M,G,S,L


The order of marginalization is chosen as M-G-S-L. First the variable M is marginalized. Thenumerical evaluation is given in Figure A.2.

P(G,S,L,D) = ∑M


= φ4(L,G)φ5(D,S,L)∑M

φ1(M)φ2(G,M)φ3(S,M)

= φ4(L,G)φ5(D,S,L)φ′1(G,S)

Next, variable G is marginalized out. The numerical evaluation is given in Figure A.3.

P(S,L,D) = ∑G

φ4(L,G)φ5(D,S,L)φ′1(G,S) (A.6)

= φ5(D,S,L)∑G

φ4(L,G)φ′1(G,S)

= φ5(D,S,L)φ′4(L,S)

130

5 4 1 2 3( ) ( , , ) ( , ) ( ) ( , ) ( , )L S G M

P D D S L L G M M G M S=∑ ∑ ∑ ∑

S=0 S=10

M=0 M=7 M=0 M=7

G=0 0.9*0.9*0.9=0.729

0.2*0.1*0.1=0.002

0.9*0.9*0.1=0.081

0.2*0.1*0.9=0.018

G=0.5g 0.1*0.9*0.9=0.081

0.8*0.1*0.1=0.008

0.1*0.9*0.1=0.009

0.8*0.1*0.9=0.072

Multiply:

Marginalize M:

M=0 0.9

M=7 0.1

M=0 M=7

G=0 0.9 0.2

G=0.5g 0.1 0.8

M=0 M=7

S=0 0.9 0.1

S=10cm 0.1 0.9

S=0 S=10

G=0 0.729+0.002=0.731

0.081+0.018=0.099

G=0.5g 0.081+0.008=0.089

0.009+0.072=0.081

'1 ( , )G Sφ

φ φ φ φ φ

Figure A.2: Marginalizing out variable M.

'5 4 1( ) ( , , ) ( , ) ( , )

L S GP D D S L L G G Sφ=∑ ∑ ∑

L=yes L=no

G=0 G=0.5g G=0 G=0.5g

S=0 0.731*0.1=0.0731

0.089*0.7=0.0623

0.731*0.9=0.6579

0.089*0.3=0.0267

S=10 0.099*0.1=0.0099

0.081*0.7=0.0567

0.099*0.9=0.0891

0.081*0.3=0.0243

Multiply:

Marginalize P: L=yes L=no

S=0 0.0731+0.0623=0.1354

0.6579+0.0267=0.6846

S=10 0.0099+0.0567=0.0666

0.0891+0.0243=0.1134

'4 ( , )L Sφ

G=0 G=0.5g

L=yes 0.1 0.7

L=no 0.9 0.3

S=0 S=10

G=0 0.731 0.099

G=0.5g 0.089 0.081

φ φ

Figure A.3: Marginalizing out variable G.

Finally, variables S and L are marginalized out. The numerical evaluation is given in FigureA.4.

131

A BPN Algorithms

P(L,D) = ∑S

φ5(D,S,L)φ′4(S,L) (A.7)

= φ′5(D,L)

P(D) = ∑L

φ′5(D,L) (A.8)

'5 4( ) ( , , ) ( , )

L SP D D S L L Sφ=∑ ∑

No 0.03374+0.65016=0.6839

Collapse 0.16826+0.07938=0.3161

Multiply:

Marginalize S:

Marginalize L:

L=yes L=no

S=0 0.1354 0.6846

S=10 0.0666 0.1134

'5 ( , )D Lφ

L=yes L=no

S=0 S=10 S=0 S=10

No 0.2 0.1 0.9 0.3

Collapse 0.8 0.9 0.1 0.7

L=yes L=no

S=0 S=10 S=0 S=10

No 0.2*0.1354=0.02708

0.1*0.0666=0.00666

0.9*0.6846=0.61614

0.3*0.1134=0.03402

Collapse 0.8*0.1354=0.10832

0.9*0.0666=0.05994

0.1*0.6846=0.06846

0.7*0.1134=0.07938

L=yes L=no

No 0.02708+0.00666=0.03374 0.61614+0.03402=0.65016

Collapse 0.10832+0.05994=0.16826 0.06846+0.07938=0.14784

( )P D

φ

Figure A.4: Marginalizing out variable S and L.

The calculation order yielding the marginal distribution of D is hence:

P(D) = ∑L

∑S

φ5(D,S,L)∑G

φ4(L,G)∑M

φ1(M)φ2(G,M)φ3(S,M) (A.9)

The steps in marginalizing down to P(D) can be illustrated as in Figure A.5. The circlenodes are buckets containing potentials. The potentials in the buckets are multiplied by theincoming potentials, a variable is marginalized out, and the result is placed in a rectangular box.The rectangular box serves as a mailbox for a neighboring bucket. In Figure A.5 the bucketelimination scheme is given for two elimination orders. It should be noted that the domains forthe elimination order M-G-S-L are smaller than for the elimination order L-G-M-S. As the sizeof the domains to be handled is a good indicator of complexity, choosing an elimination orderyielding the smallest domains to be handled is important.

Marginalizing down to different variables in a BPN yields different elimination frames asillustrated in Figure A.6 for calculating P(D) and P(L). It can easily be seen that most of theelements of the frames are the same and many calculations from the calculation of P(D) as givenabove can be reused for calculating P(L). The so-called junction trees present a systematic wayof exploiting reuse when calculating all marginals.

132

4 ( , )G Lφ

M∑'

1 ( , )G Sφ

1

2

3

( )( , )( , )

MG MS M

φφφ

5 ( , , )D S Lφ

'4 ( , )L Sφ

G∑

S∑ L∑ ( )P D

4 ( , )G Lφ

M∑'

1 ( , )G Sφ

1

2

3

( )( , )( , )

MG MS M

φφφ

5 ( , , )D S Lφ

'4 ( , )L Sφ

G∑

S∑ D∑ ( )P L

Figure A.5: A frame for computing P(D) with an elimination order M-G-S-L (left) and an elim-ination order L-G-M-S (right).

2 ( , )M Gφ

L∑'4 ( , , )G D Sφ

4

5

( , )( , , )L GD S L

φφ

1

3

( )( , )MM S

φφ

'2 ( , , )M D Sφ

G∑

M∑ S∑ ( )P D

4 ( , )G Lφ

M∑'

1 ( , )G Sφ

1

2

3

( )( , )( , )

MG MS M

φφφ

5 ( , , )D S Lφ

'4 ( , )L Sφ

G∑

S∑ L∑ ( )P D

Figure A.6: A frame for computing P(L) with an elimination order M-G-S-D (left) and P(D)with an elimination order M-G-S-L (right).

Junction tree

Evaluating a BPN, especially when multiple pieces of evidences are inserted, quickly becomesinefficient when using straightforward methods such as bucket elimination. These are practicalonly if the BPN is small and each node represents only a few states. To increase efficiency inevaluating BPNs several algorithms have been developed (Shaktar, 1986). Research has paidmost attention to algorithms based on the transformation of the BPN into join trees or junctiontrees. The junction tree algorithms are introduced using a graph-theoretic representation of theBPNs.

133

A BPN Algorithms

A domain graph is an undirected graph with variables of the domain set as nodes and withlinks between pairs of variables being members of the same domain. For the BPN consideredin this Annex, the domain graph is given in Figure A.7. In addition to the undirected links foreach of the arrows, a new link between the nodes S and L is introduced. This link is called amoral link as it connects the parents of a common child node. In eliminating a variable, thepotentials with that variable in their domain are multiplied. The domain of this product consistsof that variable and its neighbors. When the variable is eliminated, the resulting potential has allneighbors of the eliminated variable in its domain. The graph-theoretical meaning of this is thatall the neighbors of the eliminated variable are linked in pairs. For example, when eliminatingthe variable G in the domain graph in Figure A.7, a new link is introduced. These so-calledfill-ins are not favorable as they require to deal with new potentials. That is why an eliminationsequence generating no fill-ins is called perfect elimination sequence.

M

G S

L

D

M

G S

L

D

Figure A.7: BPN and domain graph.

An undirected graph with a perfect elimination sequence for all nodes is called a triangulatedgraph. The domain graph in Figure A.7 is not a triangulated graph; as mentioned above, theelimination of the variable G requires a new link. Introducing a link between the nodes G and Swould result in a triangulated graph (Figure A.8).

M

G S

L

D

M

G S

L

D

Figure A.8: Domain graph and triangulated graph.

The set of domains produced during an elimination is called a domain set, where potentials

134

that are subsets of other potentials are removed. All perfect elimination sequences produce thesame domain set; this is referred to as the set of cliques of the domain graph.

This algorithm is originally developed by Lauritzen and Spiegelhalter (1988) and adapted byJensen et al. (1990), see also Friis-Hansen (2000). In summary, the set of cliques is establishedusing the following procedure:

Moralization Parent nodes with a common child node are connected.

Deletion All arrows for the links are removed.

Triangulation First, a variable with neighbors which are mutually connected is eliminated. Theeliminated variable and its neighbors form a clique. If there are no other variables withmutually connected neighbors, a fill-in link is added to the graph to obtain full connectiv-ity. Then the second variable is eliminated analogously. If at any point a clique is formedwhich is a subset of an existing clique, it is not considered. Eliminating all variables yieldsthe set of cliques.

The cliques are organized in a tree, satisfying the following condition: the cliques on the pathbetween two cliques must contain the intersection set of variables in the two cliques. The treeshaving this property are called join trees. Triangulated graphs can always be organized into ajoin tree (Jensen, 2001). The process of constructing a join tree is illustrated in the domain graphin Figure A.8 for the perfect elimination sequence M-G-S-L.

Starting with the first node to be eliminated, M, the first clique is found M,G,S and denotedas V1. From this clique all nodes having only neighbors in the clique are eliminated. For thisstep this is node M only. After eliminating M, the remaining nodes in the set G,S is denoted asseparator S1 (Figure A.9). The clique set and separator set are given an index according to thenumber of nodes eliminated. The other clique and separator sets are counted on this index.

G S

L

D

M,G,S G,SV1 S1

M

Figure A.9: Clique set and separator while eliminating node M.

Next, node G is eliminated. Again only one variable can be eliminated, as only node G hasneighbors in the clique set. The clique set G,S,L is denoted as V2 and the remaining node set S,Lis denoted as separator S2 (Figure A.10).

135

A BPN Algorithms

S

L

D

G,S,L S,LV2 S2

G

Figure A.10: Clique set and separator while eliminating node G.

Finally, nodes S and L can be eliminated yielding the clique S,L,D denoted as V5 (FigureA.11).

S

L

D

S,L,DV5

Figure A.11: Clique set while eliminating nodes S and L.

Having determined the clique set, the join tree can be constructed satisfying the condition thaton a path between two clique sets, the intersection set of variables are in the separator set. FigureA.12 illustrates the join tree for the BPN under consideration.

M,G,S

G,S

V1

S1

G,S,L

S,L

V2

S2

S,L,DV5

Figure A.12: Join tree for the BPN in Figure A.7.

A junction tree for a set of potentials of a triangulated graph is constructed with the followingadditional structure:

• Each potential is attached to a clique

136

• Each link has the appropriate separator attached

• Each separator contains mailboxes for each direction

1 : , ,V M G S1 2 3, ,φ

2 : , ,V G S L4φ

3 : , ,V S L D5φ

1 : ,S G S 2 : ,S S L

5ψ5ψ

1ψ1ψ

φ φ

Figure A.13: Junction tree after a full propagation.

To calculate P(D), a clique containing D is made a temporary root and messages in the direc-tion of that clique are sent from other leaf cliques. The message ψ1 = ΣMφ1φ2φ3 is placed inthe appropriate S1 mailbox. Next, V2 assembles the incoming message and the potentials held tothe set Φ2 = ψ1φφ4. The variable G is eliminated from Φ2, and the result, Ψ5 = ΣGΣMφ1φ2φ3is placed in the appropriate mailbox (see Figure A.13). V3 collects the incoming message, mul-tiplies it by φ5 and marginalizes S and L to obtain P(D):

P(D) = P(D) = ∑L

∑S

φ5(D,S,L)∑G

φ4(L,G)∑M

φ1(M)φ2(G,M)φ3(S,M) (A.10)

This process is called collect evidence to V3. As expected, it yields the same equation asderived in the bucket elimination algorithm (see Equation A.9). To calculate the marginal forother variables, the messages are collected into a clique containing that variable. The junctiontree can be prepared to calculate all marginals by a process referred to as distribute evidence.First, V3 sends the message ψ5 = ΣDφ5 to S2. This message is assembled in V2, L is eliminatedand the message ψ1 = ΣLφ4ΣDφ5 is sent to S1.

A full propagation is performed, in which evidences are collected and distributed in a junctiontree. To calculate a marginal of a variable, the incoming messages are collected to a cliquecontaining the variable. The incoming message is multiplied by the potential in the clique of thevariable being considered. Eliminating all other variables from this product yields the marginal.

Stochastic simulation

Even though the junction tree algorithm increases the efficiency of evaluating BPNs, it canhappen that for some problems the space requirements cannot be met by the hardware available.In that case approximate methods such as stochastic simulation can be used. The probabilistic

137

A BPN Algorithms

structure in the form of conditional probability tables of BPNs is exploited to draw a randomconfiguration of the variables in a BPN a sufficient number of times.

Again, the BPN given in Figure A.1 is used for illustration of the algorithm. First, the state ofthe node M is sampled. Another random number is drawn and the state of node G is assigned ac-cording to the conditional probability table of the variable G given M. This procedure is repeatedto obtain the states of S,L and D, and a configuration is determined. Repeating this procedure100000 times and sorting yields the Table A.1.

Table A.1: Characteristic earthquake parameters associated with the segments.SLD

MG 111 112 121 122 211 212 221 22211 1505 5921 58790 6492 76 701 2124 524012 1116 4609 2214 247 53 584 83 19821 5 18 154 13 16 154 472 111722 129 453 217 26 492 4639 642 1500

The probability distributions of the variables are calculated, counting in the sample set. Forexample, the node D is in the state ’No’ in the columns 2, 4, 6 and 8, i.e. in 68088 of the cases.This yields the marginal distribution of node D:

P(D =′ No′) =68088

100000= 0.6809

P(D =′ Collapse′) =31912

100000= 0.3192

The marginal distributions of the other variables can be calculated analogously.

138

B Soil Parameters

Table B.1: Data for the most susceptible soil layer to liquefaction in 312 borings in Adapazari.North UTM East UTM GW [m] Depth [m] FC [%] USCS N30

4510586 533557 1 1.725 N.A. ML 134511110 533760 0.4 1.725 41 ML 174518615 536041 0.25 3.725 N.A. ML 114518347 536011 0.5 5.725 N.A. CH 154517897 536071 0.3 4.225 30 CH 74517700 535828 0.5 4.525 N.A. CH 64517525 536048 0.25 5.225 N.A. N.A. 114517298 536405 0.5 13.225 N.A. N.A. 74516769 536228 1.7 5.375 N.A. ML 174509494 530734 1.42 2.225 N.A. N.A. 44508982 530549 1.1 3.725 N.A. N.A. 74507235 530548 4.5 6.225 N.A. N.A. 264506209 530302 4.4 4.725 N.A. N.A. 204509672 529883 0.4 6.725 N.A. N.A. 34508972 529783 1.9 7.725 N.A. N.A. 74507679 529543 3.8 6.225 N.A. N.A. 444507192 529685 7.4 7.725 N.A. CH 284506205 529882 4.1 15.225 N.A. N.A. 164509880 528896 0.16 10.725 N.A. N.A. 54508998 528847 1.4 9.225 N.A. N.A. 34507602 529129 N.A. 1.725 N.A. N.A. 214507200 529083 N.A. 7.725 N.A. N.A. 304506606 529571 5.2 1.725 N.A. N.A. 194509050 528349 0.4 12.725 N.A. N.A. 44507534 528354 N.A. 1.725 N.A. N.A. 294507188 528172 N.A. 2.225 N.A. N.A. 184510816 530058 0.55 7.725 N.A. N.A. 74510221 529937 0.6 2.225 N.A. N.A. 44509426 528836 0.1 10.725 N.A. N.A. 44508605 528761 3.5 1.725 N.A. CL-ML 274508528 528092 15.4 1.725 N.A. N.A. 234506454 528838 N.A. 2.225 N.A. N.A. 23

139

B Soil Parameters


4508215 530236 N.A. 2.225 N.A. N.A. 184514164 533247 0.3 5.225 N.A. N.A. 104513852 532902 2.2 2.725 N.A. N.A. 164514545 532248 1.2 1.725 N.A. N.A. 84514714 532749 2.1 3.225 N.A. N.A. 44515180 533164 1.4 6.725 N.A. N.A. 74515574 533262 N.A. 6.725 N.A. N.A. 64515180 533802 1 3.225 N.A. N.A. 64515111 534130 0.5 1.725 N.A. N.A. 44515320 534503 N.A. 1.725 N.A. N.A. 164515383 534108 1.25 1.725 N.A. N.A. 64515631 533840 0.7 3.225 N.A. N.A. 54515648 534227 0.6 3.225 N.A. N.A. 44516010 534163 0.22 4.725 N.A. N.A. 64516540 534160 N.A. 3.225 N.A. N.A. 64516980 534260 N.A. 1.725 N.A. N.A. 94516860 534830 0.6 1.725 N.A. N.A. 44515900 535220 0.75 6.225 N.A. N.A. 84516817 535562 0.7 3.225 N.A. N.A. 64517910 534440 0.8 6.725 N.A. N.A. 124517870 533910 0.6 3.225 N.A. N.A. 114517860 535250 0.95 2.725 N.A. N.A. 64518800 534910 1.1 3.225 N.A. N.A. 74517590 533420 0.55 3.225 N.A. N.A. 124517620 532720 0.7 6.225 N.A. N.A. 54516980 532820 0.8 4.725 N.A. N.A. 54516990 533350 0.6 3.725 N.A. N.A. 74516310 533050 0.7 3.225 N.A. N.A. 194516540 533480 0.58 2.225 N.A. N.A. 54516330 533440 0.65 3.725 N.A. N.A. 44517180 532110 0.75 4.725 N.A. N.A. 64515959 533500 0.68 3.225 N.A. N.A. 64515127 534636 0.5 3.225 N.A. N.A. 84516510 533850 1.1 3.225 N.A. N.A. 94516331 534493 0.8 3.225 N.A. N.A. 64514854 535263 0.7 4.725 N.A. N.A. 114517410 533840 0.8 6.225 N.A. N.A. 194519040 534253 0.9 15.225 N.A. N.A. 174517290 531640 1.5 1.725 N.A. N.A. 134516220 532150 1 4.225 N.A. N.A. 13

140


4517990 533200 1.2 2.225 N.A. N.A. 104518510 533650 1 2.225 N.A. N.A. 444516760 534980 1.12 1.975 N.A. N.A. 74515020 534860 2.5 1.725 N.A. N.A. 144518170 534310 0.75 1.725 N.A. N.A. 94516040 533870 1.35 3.225 N.A. N.A. 114515920 533700 0.9 3.225 N.A. N.A. 64513820 532380 1.2 2.225 N.A. N.A. 44514220 532510 1.25 5.125 N.A. N.A. 34513970 531980 1.3 2.125 N.A. N.A. 34513490 531780 1.7 3.725 N.A. N.A. 244514250 532110 2 2.225 N.A. N.A. 64515260 532610 2.4 2.225 N.A. N.A. 44516000 532590 3.7 5.725 N.A. N.A. 84516760 531770 N.A. 1.725 N.A. N.A. 64517300 531500 3.1 4.725 N.A. N.A. 94516190 532640 1.7 1.725 N.A. N.A. 74516460 534320 1.1 10.225 N.A. N.A. 24515600 534280 0.9 3.725 N.A. N.A. 124517740 533570 1.95 9.725 N.A. N.A. 114514690 533360 2.1 8.725 N.A. N.A. 64516230 533480 2.05 1.725 N.A. N.A. 24514790 532710 1.9 5.225 N.A. N.A. 84512230 533090 0.9 5.225 N.A. N.A. 114512430 533160 1 4.725 N.A. N.A. 64512510 533260 0.9 1.725 N.A. MH 94512210 533480 0.95 2.225 N.A. N.A. 74512640 533320 0.85 3.725 N.A. N.A. 74512320 533630 0.8 1.725 N.A. N.A. 94512360 533590 0.85 2.225 N.A. N.A. 64511970 533740 0.85 12.225 N.A. N.A. 44512100 534100 0.3 1.725 N.A. N.A. 94512490 534010 0.9 3.725 N.A. N.A. 74512330 533840 0.8 1.725 N.A. N.A. 94512870 533990 0.9 3.225 N.A. N.A. 64512700 534280 N.A. 3.225 N.A. ML 54512340 534570 N.A. 3.725 N.A. N.A. 64512940 534670 N.A. 1.725 N.A. N.A. 44513310 534670 N.A. 1.725 N.A. N.A. 44512450 532860 N.A. 1.725 N.A. N.A. 28

141

B Soil Parameters


4512970 535520 N.A. 3.225 N.A. N.A. 54512890 535890 N.A. 7.725 N.A. MH 94513150 535790 N.A. 2.225 N.A. N.A. 104513510 534760 N.A. 2.225 N.A. ML 34513390 535760 2.1 3.225 N.A. N.A. 24513030 535030 N.A. 1.725 N.A. N.A. 54513850 535490 N.A. 3.225 N.A. N.A. 74512580 534960 N.A. 1.725 N.A. N.A. 84513490 535110 N.A. 2.175 N.A. N.A. 74513590 534470 N.A. 2.225 N.A. N.A. 74513720 535190 N.A. 1.725 N.A. N.A. 114513830 534510 N.A. 3.725 N.A. N.A. 84514160 534690 N.A. 1.725 N.A. N.A. 34514060 535140 N.A. 1.725 N.A. CL 54513840 534790 N.A. 2.025 N.A. N.A. 44515120 535280 N.A. 1.725 N.A. N.A. 24513860 534940 N.A. 2.225 N.A. N.A. 64515060 535040 N.A. 2.225 N.A. N.A. 64514360 534500 N.A. 4.725 N.A. N.A. 64514460 534700 1.2 3.225 N.A. N.A. 54514060 534870 N.A. 2.225 N.A. N.A. 34514170 534360 N.A. 10.725 N.A. N.A. 84514480 534400 0.75 2.225 N.A. N.A. 34514740 534680 N.A. 2.225 N.A. N.A. 24514730 534350 N.A. 1.725 N.A. N.A. 64514440 534120 N.A. 2.225 N.A. N.A. 44513900 534140 N.A. 1.725 N.A. N.A. 74513880 533910 N.A. 1.725 N.A. N.A. 44514660 533940 N.A. 3.225 N.A. N.A. 44514720 533500 N.A. 3.225 N.A. N.A. 64513400 533370 N.A. 4.725 N.A. N.A. 474513620 533830 N.A. 3.225 N.A. N.A. 54513620 533570 N.A. 2.225 N.A. N.A. 44513840 533440 1.1 1.725 N.A. CH 44513850 533690 0.6 2.225 N.A. N.A. 54513800 533280 N.A. 5.225 N.A. N.A. 74514020 533510 0.6 1.725 N.A. N.A. 44514240 533610 0.9 2.225 N.A. N.A. 44514250 533840 0.5 1.725 N.A. CL-ML 34514470 533710 0.65 2.225 N.A. N.A. 5

142


4514230 534610 0.8 3.225 N.A. N.A. 54512060 534980 2 3.725 N.A. MH 64512420 532680 N.A. 1.725 N.A. N.A. 114512170 532710 N.A. 2.225 N.A. N.A. 114514890 534230 1 2.225 N.A. N.A. 44512680 533460 N.A. 1.725 N.A. N.A. 404510847 530771 1.2 3.225 N.A. N.A. 84510295 530515 1.2 5.225 N.A. N.A. 24508300 530584 0.6 5.225 N.A. N.A. 74507747 530569 1.35 2.225 N.A. N.A. 154506414 530583 6.4 2.225 N.A. CL 234508396 529774 5.3 2.225 N.A. N.A. 24506692 529776 4.8 6.225 N.A. N.A. 264506550 528158 N.A. 9.225 N.A. N.A. 284507617 529963 8.4 1.725 N.A. SC 244509134 530507 1.1 1.725 N.A. N.A. 94509413 530483 0.9 1.725 N.A. N.A. 74508726 530068 1 2.225 N.A. N.A. 54515083 532465 1.2 1.725 N.A. CH 44516105 533764 1.1 3.225 N.A. CL 74516260 533319 0.92 2.225 23 ML 24516303 532993 2.2 1.725 N.A. N.A. 34514730 533496 1.1 1.725 N.A. N.A. 44513596 533261 2.1 8.525 N.A. N.A. 144513513 533737 1.2 1.725 N.A. N.A. 24516179 534731 0.1 3.225 N.A. CL 34511749 534967 1.7 1.725 N.A. N.A. 74511748 534666 1.3 1.725 N.A. N.A. 54511872 534928 0.7 3.225 N.A. N.A. 34512113 534173 1.1 3.725 N.A. N.A. 54513324 534448 N.A. 1.725 N.A. N.A. 44511590 534067 0.5 1.725 N.A. N.A. 94513694 534836 3.3 3.225 N.A. ML 74512960 533784 2.2 9.225 N.A. N.A. 304513985 533349 1.2 1.725 N.A. N.A. 84510460 532816 1 1.725 N.A. SC 134510560 534750 2.2 1.725 48 ML 204511130 534850 1 1.725 63 CL 144510172 533281 1.7 1.725 39 CL 124510810 533510 1.4 1.725 N.A. SM 7

143

B Soil Parameters


4511230 533420 1.4 1.725 N.A. ML 144510076 532647 1 1.725 N.A. ML 74510890 532760 1 1.725 N.A. ML 124510411 533901 N.A. 1.725 N.A. MH 114511210 533980 N.A. 1.725 N.A. ML 164510501 533419 N.A. 1.725 N.A. N.A. 144514676 533800 1.2 1.725 N.A. CH 74516652 533622 1.8 1.725 N.A. N.A. 104516705 533107 1.3 4.725 N.A. CL-ML 34516898 534387 0.9 3.225 N.A. CL 44513742 535458 2.6 3.225 N.A. CL 64515103 532498 1.8 1.725 N.A. N.A. 54514572 532838 1.1 4.225 N.A. N.A. 54514657 532077 1.4 1.725 N.A. N.A. 54514682 532635 1.9 5.225 N.A. CH 84514249 532738 0.41 5.225 N.A. N.A. 44515889 532330 2.6 3.225 N.A. CL 34513587 531740 1.7 3.725 N.A. N.A. 244513904 531738 2 2.225 N.A. N.A. 64516704 533921 2 2.225 N.A. N.A. 124509071 534840 1.5 1.725 N.A. SM 114508640 534449 1.3 1.725 7 ML 114508277 533858 2.2 1.725 8 SM 114508543 533388 1.6 1.725 10 ML 124508892 532955 1.8 1.725 7 ML 164508339 532129 1.2 1.725 16 ML 74508106 531594 1.8 6.225 16 CL 104507660 531489 2 14.725 16 ML 184507170 531430 2.2 9.725 18 ML 104507380 531689 2.1 1.725 12 ML 104516743 533935 2 1.725 N.A. N.A. 54516332 533982 1.7 1.725 N.A. CL 44516612 534342 2.8 3.225 15 ML 104517155 533067 1.5 3.225 N.A. CL 84515661 534286 1.8 1.725 11 ML 24516047 533200 1.11 8.15 40 CL 34516118 533651 2.4 9.6 19 ML 44515957 532953 0.37 3.05 69 CL 34515841 532505 2.6 3.55 N.A. CL 34516300 534230 0.74 1.6 15 ML 2

144


4516367 534373 0.62 1.5 17 ML 24516219 533488 0.92 2.2 23 ML 24516197 533291 N.A. 4.35 22 CL 24515721 534498 0.5 1.5 19 ML 34515664 534209 1.16 1.15 32 CL 24516879 534078 1.65 2.85 13 ML 44515919 534284 0.64 3.45 15 ML 24515037 534780 N.A. 4.95 23 ML 34516312 533341 0.9 6.1 18 CL-ML 34516299 533355 0.7 3.725 N.A. CL-CH 24516312 533331 0.87 0.9 N.A. fill 34516307 533355 0.82 5.2 42 CL-MH 34516934 533804 3.3 1.8 28 ML 24516900 533803 1.68 3.4 4 ML 24516841 533166 1.42 1.8 N.A. ML-CL 24516865 533170 1.45 1.4 N.A. CL-ML 24516862 533153 1.3 4.85 N.A. CL-ML 34516857 533152 0.44 2.2 46 CL 24516854 533168 N.A. 3 38 CL 14516858 533168 0.96 2.05 20 ML 14516850 533167 N.A. 2.05 18 ML 34515112 534418 1.68 1.05 30 CL 14515121 534405 N.A. 2.5 N.A. ML 34515121 534425 2.28 2.95 16 ML 44516132 534262 0.7 3.8 N.A. MH-CH 34516130 534262 0.44 1.8 N.A. SP-SM 64516130 534264 0.35 4.5 51 CH 34515410 534437 1.64 3.6 27 ML-CL 34515760 534548 0.67 1.55 26 CL-ML 34515780 534527 0.45 5.15 N.A. ML 64516444 535187 1.72 4.15 50 CH-MH 34516020 533156 0.71 2.75 22 ML 44515803 534714 0.41 1.35 25 ML-CL 24515775 534700 0.69 1.75 N.A. ML 34515790 534706 0.7 2.8 30 ML 64515797 534718 0.4 1.4 36 ML 34516137 534118 0.8 1.3 40 CL 34516317 534242 0.68 3.2 41 ML-CL 24516986 533823 2.3 4.725 N.A. CL 44517606 533441 2.7 1.725 N.A. N.A. 3

145

B Soil Parameters


4518048 533270 2.8 1.725 N.A. N.A. 224517011 532109 0.65 1.725 N.A. N.A. 54517114 532675 0.9 3.225 N.A. CH 34516743 532443 2.8 1.725 N.A. N.A. 34517258 532841 1.65 4.725 N.A. CH 54516081 532671 1.7 1.725 N.A. N.A. 74517198 531661 3.1 3.225 N.A. N.A. 94515582 533257 2.4 4.725 N.A. CL 84515909 533039 1.9 6.225 N.A. CL 74515587 533840 1.7 1.725 N.A. CH 44513949 532538 2.8 3.225 N.A. N.A. 114517830 533760 N.A. 1.725 N.A. N.A. 44517551 535694 1.4 1.725 N.A. N.A. 64518577 534205 3.5 15.225 N.A. N.A. 144517882 534397 2.3 1.725 N.A. N.A. 74514166 534765 1 1.725 N.A. N.A. 84514577 534692 1.1 1.725 N.A. N.A. 24515784 534254 0.7 3.225 N.A. N.A. 44514916 535331 1 1.725 N.A. N.A. 94517612 534620 1 2.225 N.A. N.A. 444518243 534987 2.1 1.725 N.A. CL 24516776 534861 0.9 1.725 N.A. N.A. 44517015 535543 1 3.225 N.A. CL 54516822 535448 0.5 1.725 N.A. N.A. 64515965 534786 0.7 1.725 N.A. ML 84516008 535203 1 1.725 N.A. ML 44515905 535233 N.A. 3.225 N.A. N.A. 54515365 534148 1.87 1.725 52 CH 64515169 533935 1.6 1.725 N.A. CL 34515191 533503 0.4 3.225 N.A. CH 124515117 533114 2.2 3.225 N.A. CL 134515222 534601 0.9 3.225 N.A. CL 44515800 535014 0.4 3.225 N.A. CL 34515034 535010 1.6 1.725 N.A. ML 64515765 534620 0.89 1.725 N.A. ML 34512450 535212 3 1.725 N.A. ML 104513304 535081 1.15 1.725 N.A. N.A. 84512641 535583 4.1 1.725 N.A. ML 144513151 535445 2.8 3.225 N.A. CL 7

146

C Software Codes

C.1 Seismic hazard analysis tools

Matlab files for evaluating the BPN in subsection 4.1.1 - A Bayesian probabilisticnetwork for a generic seismic source

f u n c t i o n BPN_PSHA

% by Yahya Y. Bayraktarli, 30/08/2009% ETH Zürich

5 % [email protected]%% Bayraktarli, Y.Y, Baker, J.W., Faber, M.H., 2009. Uncertainty treatment% in earthquake modeling using Bayesian networks, Georisk, accepted for% publication.

10 %% This script reads a Bayesian probabilistic network for probabilistic% seismic hazard analysis, calculates the probability distribution of the% nodes in the BPN and compiles the BPN with the inference engine of HUGIN.% Output is specified in form of marginal and conditional probability

15 % distributions.%% For controling HUGIN from MATLAB the ActiveX server is loaded and% then the available functions in this library are used to alter objects% created from this library. The HUGIN ActiveX Server is loaded with the

20 % following command and create a HUGIN API object named bpn:% bpn=actxserver(’HAPI.Globals’);

%An object ’domain’ is created which holds the network:domain= in vo ke ( bpn , ’ LoadDomainFromNet ’ , ’C : \ D i s s e r t a t i o n \ Chapte r4 −1\BPN \ . . .

25 . . . BPN_PSHA . n e t ’ , 0 , 0 ) ;

%The nodes to be manipulated are defined:ndEQ_M= in vo ke ( domain , ’ GetNodeByName ’ , ’ EQ_Magnitude ’ ) ;ndEQ_R= i nv ok e ( domain , ’ GetNodeByName ’ , ’ EQ_Dis tance ’ ) ;

30 ndEps_SD= in v o k e ( domain , ’ GetNodeByName ’ , ’ Epsi lon_SD ’ ) ;ndSD= i nv oke ( domain , ’ GetNodeByName ’ , ’SD ’ ) ;

%The number of discrete states of the nodes ’Magnitude’, ’Distance’,%’Eps_SD’ and ’SD’ is set:

35 nM=10;nR=10;nEps_SD =50;

147

C Software Codes

nSD=10;s e t (ndEQ_M , ’ NumberOfSta tes ’ ,nM ) ;

40 s e t ( ndEQ_R , ’ NumberOfSta tes ’ , nR ) ;s e t ( ndEps_SD , ’ NumberOfSta tes ’ , nEps_SD ) ;s e t ( ndSD , ’ NumberOfSta tes ’ , nSD ) ;

% Marginal distribution of node ’Distance’ (Figure 4.1b) is calculated and45 % plotted:

[R , P_R]= Line_EQ_R_1 ( 5 0 0 , nR,−50 ,75 ,−15 ,−30);M=M_EQ;f i g u r ebar ( P_R )

50

% Marginal distribution of node ’Magnitude’ (Figure 4.1d) is calculated and% plotted:[ Nu_Mmin ,M_EQ, P_M]=EQ_M_1 ( 4 , 7 . 3 , 4 . 4 , 1 ,nM ) ;f i g u r e

55 bar (P_M)

% Marginal distribution of node ’e_SD’ (Figure 4.1f) is calculated and% plotted[ Eps_SD , dSD , e ]= EPS_SD_1 ( nEps_SD ) ;

60 f i g u r ebar ( Eps_SD )

%The discrete probabilities are set for node ’Magnitude’ in the BPN:f o r i = 1 : (nM)

65 s e t (ndEQ_M . Table , ’ Data ’ , ( i −1) ,P_M( i ) ) ;s e t (ndEQ_M , ’ S t a t e L a b e l ’ , ( i −1) , [ ’M= ’ num2str (M( i ) ) ] ) ;

end

%The discrete probabilities are set for node ’Distance’ in the BPN:70 f o r i = 1 : ( nR )

s e t ( ndEQ_R . Table , ’ Data ’ , ( i −1) ,P_R ( i ) ) ;s e t ( ndEQ_R , ’ S t a t e L a b e l ’ , ( i −1) , [ ’R= ’ num2str (R( i ) ) ] ) ;

end

75 %The discrete probabilities are set for node ’Eps_SD’ in the BPN:f o r i = 1 : ( nEps_SD )

s e t ( ndEps_SD . Table , ’ Data ’ , ( i −1) , Eps_SD ( i ) ) ;s e t ( ndEps_SD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’ Eps_SD= ’ num2str ( dSD ( i ) ) ] ) ;

end80

%The conditional probability table of the node ’SD’ is initialized with%zeros:f o r i = 1 : (nM∗nR∗nSD∗nEps_SD )

s e t ( ndSD . Table , ’ Data ’ , ( i −1 ) , 0 ) ;85 end

%for all combinations of the states in the node ’Magnitude’ and ’Distance’%the spectral displacements are calculated with the Boore Joyner and Fumal

90 %attenuation model

148

[SDBOORE]=SD_1 ( nEps_SD ,M, R ) ;

%The limits for the discretisation of the node ’SD’ are set:CoeffSD =[0 0 .001 0 .002 0 .003 0 .004 0 .0 05 0 .006 0 . 0 1 0 . 0 4 0 . 0 7 max (SDBOORE ) ] ;

95

%The labels of the states are set for node ’SD’ in the BPN:f o r i = 1 : ( nSD )

s e t ( ndSD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’SD= ’ num2str ( CoeffSD ( i + 1 ) ) ] ) ;end

100

%The discrete probabilities are set for node ’SD’ in the BPN:K=0;f o r i =1 : l e n g t h (SDBOORE)

f o r j =1 :nSD105 i f SDBOORE( i ) <= CoeffSD ( j +1) & SDBOORE( i ) > CoeffSD ( j )

s e t ( ndSD . Table , ’ Data ’ , ( j +K−1 ) , 1 ) ;K=K+nSD ;end

end110 end

% The loaded BPN is compiled with the HUGIN inference engine:i n vo ke ( domain , ’ Compile ’ ) ;

115 % The BPN with the manipulated states and probabilities can be saved% using the command:i n vo ke ( domain , ’ SaveAsNet ’ , ’C : \ D i s s e r t a t i o n \ Chapte r4 −1\BPN \ . . .

BPN_PSHA_Output . n e t ’ ) ;

120 % Marginal distribution of node ’SD’ (Figure 4.2b) is calculated and% plottedf o r i =1 :nSD

% The marginal probability of any state of any node can be read out% from a compiled BPN. Note that the first state is labeled as 0, the

125 % second state as 1, etc.P_SD ( i )= g e t ( ndSD , ’ B e l i e f ’ , ( i −1 ) ) ;

endf i g u r ebar ( P_SD )

130

% Conditional probability distribution of node ’SD’ given M=5.5 and R=60km% (Figure 4.2c) is calculated and plotted

% The states of the nodes with evidences are selected135 i n vo ke (ndEQ_M , ’ S e l e c t S t a t e ’ , 4 ) ;

i n vo ke ( ndEQ_R , ’ S e l e c t S t a t e ’ , 5 ) ;

% With new evidences set, the BPN has to be propagate the findigsi n vo ke ( domain , ’ P r o p a g a t e ’ , ’ hEqui l ib r iumSum ’ , ’ hModeNormal ’ ) ;

140

f o r i =1 :nSDP_SD ( i )= g e t ( ndSD , ’ B e l i e f ’ , ( i −1 ) ) ;

end

149

C Software Codes

f i g u r e145 bar ( P_SD )

f u n c t i o n BPN_PSHA_Deaggregation



10 %% This script reads a Bayesian probabilistic network for probabilistic% seismic hazard analysis, calculates the probability distribution of the% nodes in the BPN and compiles the BPN with the inference engine of HUGIN.% Output is specified in form of deaggregation.

15 %%%For controling HUGIN from MATLAB the ActiveX server is loaded and%then the available functions in this library are used to alter objects%created from this library. The HUGIN ActiveX Server is loaded with the

20 %following command and create a HUGIN API object named bpn:bpn= a c t x s e r v e r ( ’HAPI . G l o b a l s ’ ) ;

%An object ’domain’ is created which holds the network:domain= in vo ke ( bpn , ’ LoadDomainFromNet ’ , . . .

25 ’C : \ D i s s e r t a t i o n \ Chapte r4 −1\BPN\ BPN_PSHA . n e t ’ , 0 , 0 ) ;

%The nodes to be manipulated are defined:ndEQ_M= in vo ke ( domain , ’ GetNodeByName ’ , ’ EQ_Magnitude ’ ) ;ndEQ_R= i nv ok e ( domain , ’ GetNodeByName ’ , ’ EQ_Dis tance ’ ) ;

30 ndEps_SD= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsi lon_SD ’ ) ;ndSD= i nvo ke ( domain , ’ GetNodeByName ’ , ’SD ’ ) ;

%The number of discrete states of the nodes ’Magnitude’, ’Distance’,%’Eps_SD’ and ’SD’ is set:

35 nM=10;nR=10;nEps_SD =50;nSD=10;s e t (ndEQ_M , ’ NumberOfSta tes ’ ,nM ) ;

40 s e t ( ndEQ_R , ’ NumberOfSta tes ’ , nR ) ;s e t ( ndEps_SD , ’ NumberOfSta tes ’ , nEps_SD ) ;s e t ( ndSD , ’ NumberOfSta tes ’ , nSD ) ;

%The probability distribution of node ’Eps_SD’ is calculated:45 [ Eps_SD , dSD , e ]= EPS_SD_1 ( nEps_SD ) ;

%The probability distribution of node ’Magnitude’ is calculated:[ Nu_Mmin ,M_EQ, P_M]=EQ_M_1 ( 4 , 7 . 3 , 4 . 4 , 1 ,nM ) ;

50 %The probability distribution of node ’Distance’ is calculated:

150

[R , P_R]= Line_EQ_R_1 ( 5 0 0 , nR,−50 ,75 ,−15 ,−30);M=M_EQ;

%The discrete probabilities are set for node ’Magnitude’ in the BPN:55 f o r i = 1 : (nM)

s e t (ndEQ_M . Table , ’ Data ’ , ( i −1) ,P_M( i ) ) ;s e t (ndEQ_M , ’ S t a t e L a b e l ’ , ( i −1) , [ ’M= ’ num2str (M( i ) ) ] ) ;

end

60 %The discrete probabilities are set for node ’Distance’ in the BPN:f o r i = 1 : ( nR )


end65

%The discrete probabilities are set for node ’Eps_SD’ in the BPN:f o r i = 1 : ( nEps_SD )

s e t ( ndEps_SD . Table , ’ Data ’ , ( i −1) , Eps_SD ( i ) ) ;s e t ( ndEps_SD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’ Eps_SD= ’ num2str ( dSD ( i ) ) ] ) ;

70 end

%The conditional probability table of the node ’SD’ is initialized with%zeros:f o r i = 1 : (nM∗nR∗nSD∗nEps_SD )

75 s e t ( ndSD . Table , ’ Data ’ , ( i −1 ) , 0 ) ;end

%for all combinations of the states in the node ’Magnitude’ and ’Distance’80 %the spectral displacements are calculated with the Boore Joyner and Fumal

%attenuation model[SDBOORE]=SD_1 ( nEps_SD ,M, R ) ;

%The limits for the discretisation of the node ’SD’ are set:85 CoeffSD =[0 0 .0 01 0 . 00 2 0 .0 03 0 . 00 4 0 .0 05 0 .006 0 . 0 1 0 . 0 4 0 . 0 7 max (SDBOORE ) ] ;

%The labels of the states are set for node ’SD’ in the BPN:f o r i = 1 : ( nSD )

s e t ( ndSD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’SD= ’ num2str ( CoeffSD ( i + 1 ) ) ] ) ;90 end

%The discrete probabilities are set for node ’SD’ in the BPN:K=0;f o r i =1 : l e n g t h (SDBOORE)

95 f o r j =1 :nSDi f SDBOORE( i ) <= CoeffSD ( j +1) & SDBOORE( i ) > CoeffSD ( j )s e t ( ndSD . Table , ’ Data ’ , ( j +K−1 ) , 1 ) ;K=K+nSD ;end

100 endend

% The loaded BPN is compiled with the HUGIN inference engine:

151

C Software Codes

i n vo ke ( domain , ’ Compile ’ ) ;105

% The BPN with the manipulated states and probabilities can be saved using% the command:i n vo ke ( domain , ’ SaveAsNet ’ , . . .

’C : \ D i s s e r t a t i o n \ Chapte r4 −1\BPN\ BPN_PSHA_Output . n e t ’ ) ;110

% The states of the nodes with evidences are selectedi n vo ke ( ndSD , ’ S e l e c t S t a t e ’ , 4 ) ;

115 % With new evidences set, the BPN has to be propagate the findigsi n vo ke ( domain , ’ P r o p a g a t e ’ , ’ hEqui l ib r iumSum ’ , ’ hModeNormal ’ ) ;

% Deaggregation by magnitude and distance for 4mm<=SD<5mm (Figure 4.3) is% calculated:

120 f o r j =1 :nRP_R2 ( j )= g e t ( ndEQ_R , ’ B e l i e f ’ , ( j −1 ) ) ;

endf o r i =1 :nM

f o r j =1 :nR125 i n vo ke ( ndEQ_R , ’ S e l e c t S t a t e ’ , ( j −1 ) ) ;

i n vo ke ( domain , ’ P r o p a g a t e ’ , ’ hEqui l ib r iumSum ’ , ’ hModeNormal ’ ) ;P_M_R( i , j )= P_R2 ( j ) ∗ ( g e t (ndEQ_M , ’ B e l i e f ’ , ( i −1 ) ) ) ;

endend

130 f i g u r eba r3 (P_M_R)

f u n c t i o n [ Eps_SD , dSD , e ]= EPS_SD_1 ( nEps_SD )%Calculates the probabilites of the standardnormal distribution for the%bins given a number of discretisation ’nEps_SD’

5 emin =−3;emax =3;f o r i = 1 : ( nEps_SD−1)

e ( i )= emin +( i −1)∗( emax−emin ) / ( nEps_SD −2);end

10

f o r i =1 : nEps_SDi f i ==1

Eps_SD ( i )= normcdf ( e ( i ) , 0 , 1 ) ;e l s e i f i ==( l e n g t h ( e ) + 1 )

15 Eps_SD ( i )= normcdf ( e ( 1 ) , 0 , 1 ) ;e l s e

Eps_SD ( i )= normcdf ( e ( i ) ,0 ,1 ) − normcdf ( e ( i −1 ) , 0 , 1 ) ;end

end20 Eps_SD=Eps_SD ’ ;

f o r i =1 : nEps_SDi f i ==1

dSD ( i )= e ( i ) −0 .5 ;

152

25 e l s e i f i ==( l e n g t h ( e ) + 1 )dSD ( i )= e ( i −1)+0 .5 ;

e l s edSD ( i ) = ( e ( i −1)+e ( i ) ) / 2 ;

end30 end

dSD=dSD ’ ;

f u n c t i o n [ Nu_Mmin ,M_EQ, P_M, M l i m i t s ]=EQ_M_1( Mmin , Mmax, a , b ,nM)%Assuming that earthquakes of magnitude less than Mmin do not contribute to%damage the mean rate of exceedance for Mmin is Nu_Mmin. P_M is the%probability of having a magnitude in the magnitude range characterised by

5 %M_EQ. The formula for Gutenberg and Richter law with upper (Mmax) and%lower (Mmin) bounds is applied.%a and b are the parameters of the Gutenberg richter recurrence relation.

Nu_Mmin=10^( a+b ∗ (Mmin ) ) ;10

f o r i = 1 : (nM+1)M l i m i t s ( i )=Mmin+(Mmax−Mmin ) / nM∗ ( i −1);

end

15 f o r i =1 :nMM_EQ( i ) = ( M l i m i t s ( i )+ M l i m i t s ( i + 1 ) ) / 2 ;

endM_EQ=M_EQ’ ;f o r i =1 :nM

20 P_M( i ) = ( exp (−2.303∗b ∗ ( M l i m i t s ( i )−Mmin))− exp (−2.303∗b ∗ ( M l i m i t s ( i +1)−Mmin ) ) ) . . ./(1− exp (−2.303∗b ∗ (Mmax−Mmin ) ) ) ;

endP_M=P_M ’ ;

f u n c t i o n [R , P_R , R l i m i t s ]= Line_EQ_R_1 ( nr , nR , X1 , Y1 , X2 , Y2 )%Coordinates of the site (0,0), Coordinates of the two ends of the line%source are (X1,Y1) and (X2,Y2).%nr is the discretisation of the line source, nR is the discretisation of

5 %the bins for the EQ_Distance node

f o r i = 1 : ( n r +1)x ( i )=X1+(X2−X1 ) ∗ ( i −1)/ n r ;

10 y ( i )=Y1+(Y2−Y1 ) ∗ ( i −1)/ n r ;end

f o r i =1 : n rr ( i )= s q r t ( ( ( x ( i )+ x ( i + 1 ) ) / 2 ) ^ 2 + ( ( y ( i )+ y ( i + 1 ) ) / 2 ) ^ 2 ) ;

15 end

f o r i = 1 : ( nR+1)R l i m i t s ( i )= min ( r ) + ( max ( r )−min ( r ) ) / nR∗ ( i −1);

end20 f o r i =1 :nR

R_EQ( i ) = ( R l i m i t s ( i )+ R l i m i t s ( i + 1 ) ) / 2 ;

153

C Software Codes

endH_R= h i s t ( r , R_EQ ) ;P_R=H_R / n r ;

25 P_R=P_R ’ ;R=R_EQ ’ ;

f u n c t i o n [SD]=SD_1 ( nEps_SD ,M, R)%for all combinations of the states in the node ’Magnitude’ and ’Distance’%the spectral displacements are calculated with the Boore Joyner and Fumal%attenuation model

5 T = 0 . 4 9 ;[ Eps_SD , dSD , e ]= EPS_SD_1 ( nEps_SD ) ;

N1=1;f o r j =1 : nEps_SD

10 f o r m=1: l e n g t h (M)f o r n =1: l e n g t h (R)

[MU_SA SIGMA_SA]= b j f _ a t t e n _ 1 (M(m) ,R( n ) , T , 1 , 3 1 0 , 1 ) ;SD( N1 ) = 9 . 8 0 6∗ ( T∗T / 4 / pi / pi )∗ exp ( l o g (MU_SA)+SIGMA_SA∗dSD ( j ) ) ;i f SD( N1) <=0

15 SD( N1 ) = 0 ;endN1=N1+1;

endend

20 end

f u n c t i o n [ sa , s igma ] = b j f _ a t t e n _ 1 (M, R , T , Fau l t_Type , Vs , a r b )

% by Jack Baker, 2/1/05% Stanford University

5 % [email protected]%% Boore Joyner and Fumal attenuation model (1997 Seismological Research% Letters, Vol 68, No 1, p154).%

10 % This script includes standard deviations for either% arbitrary or average components of ground motion (See Baker and Cornell,% 2005, "What spectral acceleration are you using," Earthquake Spectra,% submitted).%

15 %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INPUT%

20 % M = Moment Magnitude% R = boore joyner distance% T = period: 0.1 to 2s (0.001s is a placeholder for the PGA)% Fault_Type = 1 for strike-slip fault% = 2 for reverse-slip fault

25 % = 0 for non-specified mechanism% Vs = shear wave velocity averaged over top 30 m

154

% (use 310 for soil, 620 for rock)% arb = 1 for arbitrary component sigma% = 0 for average component sigma

30 %% OUTPUT%% sa = median spectral acceleration prediction% sigma = logarithmic standard deviation of spectral acceleration

35 % prediction FOR AN ARBITRARY OR AVERAGE COMPONENT%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% coefficients40 p e r i o d = [ 0 . 0 0 1 0 . 1 0 . 1 1 0 . 1 2 0 . 1 3 0 . 1 4 0 . 1 5 0 . 1 6 0 . 1 7 0 . 1 8 0 . 1 9 0 . 2 0 . 2 2 . . .

0 . 2 4 0 . 2 6 0 . 2 8 0 . 3 0 . 3 2 0 . 3 4 0 . 3 6 0 . 3 8 0 . 4 0 . 4 2 0 . 4 4 0 . 4 6 0 . 4 8 0 . 5 . . .0 . 5 5 0 . 6 0 . 6 5 0 . 7 0 . 7 5 0 . 8 0 . 8 5 0 . 9 0 . 9 5 1 1 . 1 1 . 2 1 . 3 1 . 4 1 . 5 1 . 6 . . .1 . 7 1 . 8 1 . 9 2 ] ;

B1ss = [ −0.313 1 .006 1 .0 72 1 .109 1 .128 1 .135 1 .128 1 .112 1 . 0 9 1 .063 1 .032 . . .45 0 .999 0 . 92 5 0 .8 4 7 0 . 76 4 0 .6 81 0 .598 0 .518 0 .439 0 .361 0 .286 0 .212 . . .

0 . 1 4 0 .073 0 .005 −0.058 −0.122 −0.268 −0.401 −0.523 −0.634 −0.737 . . .−0.829 −0.915 −0.993 −1.066 −1.133 −1.249 −1.345 −1.428 −1.495 . . .−1.552 −1.598 −1.634 −1.663 −1.685 −1.699 ] ;

B1rv = [ −0.117 1 .087 1 .1 64 1 .215 1 .246 1 .261 1 .264 1 .257 1 .242 1 .222 1 .198 . . .50 1 . 1 7 1 . 10 4 1 .0 33 0 . 95 8 0 .8 81 0 .803 0 .725 0 .648 0 . 5 7 0 .495 0 .423 . . .

0 . 352 0 .282 0 .2 17 0 .151 0 .087 −0.063 −0.203 −0.331 −0.452 −0.562 . . .−0.666 −0.761 −0.848 −0.932 −1.009 −1.145 −1.265 −1.37 −1.46 −1.538 . . .−1.608 −1.668 −1.718 −1.763 −1.801 ] ;

B 1 a l l = [ −0.242 1 .059 1 . 1 3 1 .174 1 . 2 1 .208 1 .204 1 .192 1 .1 73 1 .151 1 .122 . . .55 1 .089 1 . 01 9 0 .9 4 1 0 . 86 1 0 . 7 8 0 . 7 0 .619 0 . 5 4 0 .46 2 0 .385 0 .311 0 .239 . . .

0 . 169 0 .102 0 .0 36 −0.025 −0.176 −0.314 −0.44 −0.555 −0.661 −0.76 . . .−0.851 −0.933 −1.01 −1.08 −1.208 −1.315 −1.407 −1.483 −1.55 −1.605 . . .−1.652 −1.689 −1.72 −1.743 ] ;

B2 = [ 0 .527 0 .753 0 .732 0 .721 0 .711 0 .707 0 .702 0 .702 0 .702 0 .705 0 .709 . . .60 0 .711 0 . 72 1 0 .7 3 2 0 . 74 4 0 .7 58 0 .769 0 .783 0 .794 0 .806 0 . 8 2 0 .831 . . .

0 . 8 4 0 .852 0 .863 0 .873 0 .884 0 .907 0 .928 0 .946 0 .962 0 .979 0 .992 . . .1 . 006 1 .018 1 .0 27 1 .036 1 .052 1 .064 1 .073 1 . 0 8 1 .085 1 .087 1 .089 . . .1 . 087 1 .087 1 .085 ] ;

B3 = [ 0 −0.226 −0.23 −0.233 −0.233 −0.23 −0.228 −0.226 −0.221 −0.216 −0.212 . . .65 −0.207 −0.198 −0.189 −0.18 −0.168 −0.161 −0.152 −0.143 −0.136 −0.127 . . .

−0.12 −0.113 −0.108 −0.101 −0.097 −0.09 −0.078 −0.069 −0.06 −0.053 . . .−0.046 −0.041 −0.037 −0.035 −0.032 −0.032 −0.03 −0.032 −0.035 −0.039 . . .−0.044 −0.051 −0.058 −0.067 −0.074 −0.085 ] ;

B5 = [ −0.778 −0.934 −0.937 −0.939 −0.939 −0.938 −0.937 −0.935 −0.933 −0.93 . . .70 −0.927 −0.924 −0.918 −0.912 −0.906 −0.899 −0.893 −0.888 −0.882 . . .

−0.877 −0.872 −0.867 −0.862 −0.858 −0.854 −0.85 −0.846 −0.837 −0.83 . . .−0.823 −0.818 −0.813 −0.809 −0.805 −0.802 −0.8 −0.798 −0.795 −0.794 . . .−0.793 −0.794 −0.796 −0.798 −0.801 −0.804 −0.808 −0.812 ] ;

Bv = [ −0.371 −0.212 −0.211 −0.215 −0.221 −0.228 −0.238 −0.248 −0.258 −0.27 . . .75 −0.281 −0.292 −0.315 −0.338 −0.36 −0.381 −0.401 −0.42 −0.438 −0.456 . . .

−0.472 −0.487 −0.502 −0.516 −0.529 −0.541 −0.553 −0.579 −0.602 . . .−0.622 −0.639 −0.653 −0.666 −0.676 −0.685 −0.692 −0.698 −0.706 −0.71 . . .−0.711 −0.709 −0.704 −0.697 −0.689 −0.679 −0.667 −0.655 ] ;

Va = [ 1396 1112 1291 1452 1596 1718 1820 1910 1977 2037 2080 2118 2158 . . .

155

C Software Codes

80 2178 2173 2158 2133 2104 2070 2032 1995 1954 1919 1884 1849 1816 . . .1782 1710 1644 1592 1545 1507 1476 1452 1432 1416 1406 1396 1400 . . .1416 1442 1479 1524 1581 1644 1714 1795 ] ;

h = [ 5 . 5 7 6 . 2 7 6 . 6 5 6 . 9 1 7 . 0 8 7 . 1 8 7 . 2 3 7 . 2 4 7 . 2 1 7 . 1 6 7 . 1 7 . 0 2 6 . 8 3 6 . 6 2 . . .6 . 3 9 6 . 1 7 5 . 9 4 5 . 7 2 5 . 5 5 . 3 5 . 1 4 . 9 1 4 . 7 4 4 . 5 7 4 . 4 1 4 . 2 6 4 . 1 3 3 . 8 2 . . .

85 3 . 5 7 3 . 3 6 3 . 2 3 . 0 7 2 . 9 8 2 . 9 2 2 . 8 9 2 . 8 8 2 . 9 2 . 9 9 3 . 1 4 3 . 3 6 3 . 6 2 3 . 9 2 . . .4 . 2 6 4 . 6 2 5 . 0 1 5 . 4 2 5 . 8 5 ] ;

s igma1 = [ 0 .431 0 . 4 4 0 .437 0 .437 0 .43 5 0 .435 0 .435 0 .435 0 .435 0 .435 0 .435 . . .0 . 435 0 .437 0 .437 0 .437 0 . 4 4 0 . 4 4 0 .442 0 .444 0 .44 4 0 .447 0 .447 0 .449 . . .0 . 449 0 .451 0 .451 0 .454 0 .456 0 .458 0 .461 0 .463 0 .465 0 .467 0 .467 . . .

90 0 . 4 7 0 .472 0 .474 0 .477 0 .479 0 .481 0 .484 0 .486 0 .4 88 0 . 4 9 0 .493 . . .0 . 493 0 .495 ] ;

s igmac = [ 0 . 1 6 0 0 .134 0 .141 0 .148 0 .153 0 .158 0 .163 0 .166 0 .169 0 .17 3 0 .176 . . .0 . 177 0 .182 0 .185 0 .189 0 .192 0 .195 0 .197 0 .199 0 .200 0 .202 0 .204 . . .0 . 205 0 .206 0 .209 0 .210 0 .211 0 .214 0 .216 0 .218 0 .220 0 .221 0 .223 . . .

95 0 .226 0 .228 0 .230 0 .230 0 .233 0 .236 0 .239 0 .241 0 .244 0 .246 0 .249 . . .0 . 251 0 .254 0 . 2 5 6 ] ;

s i gm ar = [ 0 .460 0 .460 0 .459 0 .461 0 .461 0 .463 0 .465 0 .466 0 .46 7 0 .468 . . .0 . 469 0 .470 0 .473 0 .475 0 .476 0 .480 0 .481 0 .484 0 .487 0 .487 0 .491 . . .0 . 491 0 .494 0 .494 0 .497 0 .497 0 .501 0 .504 0 .506 0 .510 0 .513 0 .515 . . .

100 0 .518 0 .519 0 .522 0 .525 0 . 5 2 7 0 .531 0 . 5 3 4 0 .537 0 . 5 4 1 0 .544 0 . 5 4 6 . . .0 . 550 0 .553 0 .555 0 . 5 5 7 ] ;

s igmae = [ 0 .184 0 0 0 0 0 0 0 0 0 . 002 0 .005 0 .009 0 .016 0 .025 0 .032 0 .039 . . .0 . 048 0 .055 0 .064 0 .071 0 .078 0 .085 0 .092 0 .099 0 .104 0 .111 0 .115 . . .0 . 129 0 .143 0 .154 0 .166 0 .175 0 .184 0 .191 0 . 2 0 .20 7 0 .214 0 .226 . . .

105 0 .235 0 .244 0 .251 0 .256 0 . 2 6 2 0 .267 0 . 2 6 9 0 .274 0 . 2 7 6 ] ;s i g m a ln y = [ 0 . 495 0 .460 0 .459 0 .461 0 .461 0 .463 0 .465 0 .466 0 .467 0 .468 . . .

0 . 469 0 .470 0 .474 0 .475 0 .477 0 .482 0 .484 0 .487 0 .491 0 .492 0 .497 . . .0 . 499 0 .502 0 .504 0 .508 0 .510 0 .514 0 .520 0 .526 0 .533 0 .539 0 .544 . . .0 . 549 0 .553 0 .559 0 .564 0 .569 0 .577 0 .583 0 .590 0 .596 0 .601 0 .606 . . .

110 0 .611 0 .615 0 .619 0 . 6 2 2 ] ;

% interpolate between periods if neccesaryi f ( l e n g t h ( f i n d ( p e r i o d == T ) ) == 0)

index_ low = sum ( p e r i o d <T ) ;115 T_low = p e r i o d ( index_ low ) ;

T_hi = p e r i o d ( index_ low + 1 ) ;

[ sa_low , sigma_low ] = b j f _ a t t e n (M, R , T_low , Fau l t_Type , Vs , a r b ) ;[ s a _ h i , s i g m a _ h i ] = b j f _ a t t e n (M, R , T_hi , Fau l t_Type , Vs , a r b ) ;

120

x = [ l o g ( T_low ) l o g ( T_hi ) ] ;Y_sa = [ l o g ( sa_ low ) l o g ( s a _ h i ) ] ;Y_sigma = [ sigma_low s i g m a _h i ] ;s a = exp ( i n t e r p 1 ( x , Y_sa , l o g ( T ) ) ) ;

125 s igma = i n t e r p 1 ( x , Y_sigma , l o g ( T ) ) ;

e l s ei = f i n d ( p e r i o d == T ) ;

130 % compute median and sigmar = s q r t (R^2 + h ( i ) ^ 2 ) ;

156

i f ( F a u l t _ T y p e == 1)b1 = B1ss ( i ) ;

135 e l s e i f ( F a u l t _ T y p e == 2)b1 = B1rv ( i ) ;

e l s eb1 = B 1 a l l ( i ) ;

end140

l n y = b1+ B2 ( i ) ∗ (M−6)+ B3 ( i ) ∗ (M−6)^2+ B5 ( i )∗ l o g ( r )+Bv ( i )∗ l o g ( Vs / Va ( i ) ) ;s a = exp ( l n y ) ;

i f ( a r b ) % arbitrary component sigma145 s igma = s i g m a l n y ( i ) ;

e l s e % average component sigmas igma = s q r t ( s igma1 ( i ) ^2 + s igmae ( i ) ^ 2 ) ;

endend

Matlab files for evaluating the BPN’s in Subsection 4.1.5 - PSHA using BPN forAdapazari, Turkey

f u n c t i o n [ P_PGA , P_SD]= BPN_PSHA_Adapazari



10 %% This script reads a Bayesian probabilistic network for probabilistic% seismic hazard analysis, calculates the probability distribution of the% nodes in the BPN and compiles the BPN with the inference engine of HUGIN.% Output is specified in form of marginal probability distributions of the

15 % nodes ’PGA’ and ’SD’ (see Figure 4.14 and 4.15).%% For each seismic source, Z and each year, Q a set of BPN’s are% calculated:f o r Z=1:1

20 f o r Q=1:50 %This loop and the relevant parts of the code are skipped%for the "poisson" case

% For controling HUGIN from MATLAB the ActiveX server is loaded and% then the available functions in this library are used to alter% objects created from this library. The HUGIN ActiveX Server is

25 % loaded with the following command and create a HUGIN API object% named bpn:bpn= a c t x s e r v e r ( ’HAPI . G l o b a l s ’ ) ;

% An object ’domain’ is created which holds the network:30 domain= in vo ke ( bpn , ’ LoadDomainFromNet ’ , . . .

157

C Software Codes

’C : \ D i s s e r t a t i o n \ Chapte r4 −1\BPN\ BPN_PSHA_Adapazari . n e t ’ , 0 , 0 ) ;

% The nodes to be manipulated are defined:ndEQ_M= in vo ke ( domain , ’ GetNodeByName ’ , ’ EQ_Magnitude ’ ) ;

35 ndEQ_R= i nv ok e ( domain , ’ GetNodeByName ’ , ’ EQ_Dis tance ’ ) ;ndEps_PGA= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsilon_PGA ’ ) ;ndEps_SD= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsi lon_SD ’ ) ;ndPGA= inv ok e ( domain , ’ GetNodeByName ’ , ’PGA ’ ) ;ndSD= i nv oke ( domain , ’ GetNodeByName ’ , ’SD ’ ) ;

40

%The number of discrete states of the nodes ’Magnitude’, ’Distance’,%’Eps_PGA’, ’Eps_SD’, ’PGA’ and ’SD’ is set:nM=6;nR =5;

45 nEps_PGA =10;nEps_SD =10;nPGA=7;nSD=7;s e t (ndEQ_M , ’ NumberOfSta tes ’ ,nM ) ;

50 s e t ( ndEQ_R , ’ NumberOfSta tes ’ , nR ) ;s e t ( ndEps_SD , ’ NumberOfSta tes ’ , nEps_SD ) ;s e t ( ndEps_PGA , ’ NumberOfSta tes ’ , nEps_PGA ) ;s e t ( ndSD , ’ NumberOfSta tes ’ , nSD ) ;s e t ( ndPGA , ’ NumberOfSta tes ’ ,nPGA ) ;

55

%The probability distribution of node ’Eps_PGA’ is calculated:[ Eps_PGA , dPGA]=EPS_PGA_5 ( nEps_PGA ) ;

%The probability distribution of node ’Eps_SD’ is calculated:60 [ Eps_SD , dSD]= EPS_SD_5 ( nEps_SD , nEps_PGA , 0 . 6 4 ) ;

%The probability distribution of node ’Distance’ is calculated:[R , P_R , R l i m i t s ]= Line_EQ_R_5 ( 5 0 0 , nR , 6 7 , 0 , 4 8 , 9 ) ;

65 %The probability distribution of node ’Magnitude’ is calculated:[ Nu_Mmin ,M_EQ, P_M, M l i m i t s ]= EQ_M_NonPoisson_5 ( 5 , 0 . 9 8 , 1 . 1 2 ,nM, Q, Z ) ;%The probability distribution of node ’Magnitude’ is calculated%for the "poisson" case%[Nu_Mmin,M_EQ,P_M,Mlimits]=EQ_M_5(5,1.15,1.12,nM);

70 M=M_EQ;

%The discrete probabilities are set for node ’Magnitude’ in the%BPN:f o r i = 1 : (nM)

75 s e t (ndEQ_M . Table , ’ Data ’ , ( i −1) ,P_M( i ) ) ;s e t (ndEQ_M , ’ S t a t e L a b e l ’ , ( i −1) , [ ’M= ’ num2str (M( i ) ) ] ) ;

end

%The discrete probabilities are set for node ’Distance’ in the BPN:80 f o r i = 1 : ( nR )


end

158

85 %The discrete probabilities are set for node ’Eps_PGA’ in the BPN:f o r i = 1 : ( nEps_PGA )

s e t ( ndEps_PGA . Table , ’ Data ’ , ( i −1) ,Eps_PGA ( i ) ) ;s e t ( ndEps_PGA , ’ S t a t e L a b e l ’ , ( i −1) , [ ’Eps_PGA= ’ num2str (dPGA( i ) ) ] ) ;

end90

%The discrete probabilities are set for node ’Eps_SD’ in the BPN:f o r i = 1 : ( nEps_SD∗nEps_PGA )

s e t ( ndEps_SD . Table , ’ Data ’ , ( i −1) , Eps_SD ( i ) ) ;end

95

%The conditional probability table of the node ’PGA’ is initialized% with zeros:f o r i = 1 : (nM∗nR∗nEps_PGA∗nPGA)

s e t ( ndPGA . Table , ’ Data ’ , ( i −1 ) , 0 ) ;100 end

%The conditional probability table of the node ’SD’ is initialized%with zeros:f o r i = 1 : (nM∗nR∗nSD∗nEps_SD )


%for all combinations of the states in the node ’Magnitude’ and%’Distance’ the peak ground accelerations and spectral

110 %displacements are calculated with the Boore Joyner and Fumal%attenuation model[PGABOORE]=PGA_5 ( nEps_PGA ,M, R ) ;[SDBOORE]=SD_5 ( nEps_SD , nEps_PGA ,M, R ) ;

115 %The limits for the discretisation of the nodes ’SD’ and ’PGA’ are%set:CoeffPGA =[0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 max (PGABOORE ) ] ;CoeffSD =[0 0 .005 0 . 0 2 0 . 0 5 0 . 1 0 . 3 0 . 5 max (SDBOORE ) ] ;

120 %The labels of the states are set for node ’Eps_SD’, ’PGA’ and ’SD’%in the BPN:f o r i = 1 : ( nEps_SD )

s e t ( ndEps_SD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’ Eps_SD= ’ num2str ( dSD ( i ) ) ] ) ;end

125 f o r i = 1 : (nPGA)s e t ( ndPGA , ’ S t a t e L a b e l ’ , ( i −1) , . . .[ ’PGA= ’ num2str ( ( CoeffPGA ( i )+ CoeffPGA ( i + 1 ) ) / 2 ) ] ) ;

endf o r i = 1 : ( nSD )

130 s e t ( ndSD , ’ S t a t e L a b e l ’ , ( i −1) , . . .[ ’SD= ’ num2str ( ( CoeffSD ( i )+ CoeffSD ( i + 1 ) ) / 2 ) ] ) ;

end

%The discrete probabilities are set for node ’PGA’ in the BPN:135 N=0;

f o r i =1 : l e n g t h (PGABOORE)

159

C Software Codes

f o r j =1 :nPGAi f PGABOORE( i ) <= CoeffPGA ( j +1) & PGABOORE( i ) > CoeffPGA ( j )

s e t ( ndPGA . Table , ’ Data ’ , ( j +N−1 ) , 1 ) ;140 N=N+nPGA ;

endend

end

145 %The discrete probabilities are set for node ’SD’ in the BPN:K=0;f o r i =1 : l e n g t h (SDBOORE)

f o r j =1 :nSDi f SDBOORE( i ) <= CoeffSD ( j +1) & SDBOORE( i ) > CoeffSD ( j )

150 s e t ( ndSD . Table , ’ Data ’ , ( j +K−1 ) , 1 ) ;K=K+nSD ;

endend

end155

% The loaded BPN is compiled with the HUGIN inference engine:i n vo ke ( domain , ’ Compile ’ ) ;

% The BPN with the manipulated states and probabilities can be160 % saved using the commands:

Fi l ename =[ ’C : \ D i s s e r t a t i o n \ Chapte r4 −1\BPN \ . . .BPN_PSHA_Adapazari_Output_T ’ num2str ( Z ) ’ _ ’ num2str (Q) ’ . n e t ’ ] ;i n vo ke ( domain , ’ SaveAsNet ’ , F i l ename ) ;

165 %For each seismic Source the marginal probabilities of each state%and each year are calculatedf o r j =1 :nPGA

P_PGA . SourceS1 (Q ) . P_PGA( j )= g e t ( ndPGA , ’ B e l i e f ’ , ( j −1 ) ) ;end

170 f o r j =1 :nSDP_SD . SourceS1 (Q ) . P_SD ( j )= g e t ( ndSD , ’ B e l i e f ’ , ( j −1 ) ) ;

endend

end

f u n c t i o n [ Eps_PGA , dPGA]=EPS_PGA_5 ( nEps_PGA )%Calculates the probabilites of the standardnormal distribution for the%bins given a number of discretisation

5 emin =−3;emax =3;f o r i = 1 : ( nEps_PGA−1)

e ( i )= emin +( i −1)∗( emax−emin ) / ( nEps_PGA−2);end

10

f o r i = 1 : ( l e n g t h ( e ) + 1 )i f i ==1

Eps_PGA ( i )= normcdf ( e ( i ) , 0 , 1 ) ;e l s e i f i ==( l e n g t h ( e ) + 1 )

160

15 Eps_PGA ( i )= normcdf ( e ( 1 ) , 0 , 1 ) ;e l s e

Eps_PGA ( i )= normcdf ( e ( i ) ,0 ,1 ) − normcdf ( e ( i −1 ) , 0 , 1 ) ;end

end20 Eps_PGA=Eps_PGA ’ ;

f o r i = 1 : ( l e n g t h ( e ) + 1 )i f i ==1

dPGA( i )= e ( i ) −0 .5 ;25 e l s e i f i ==( l e n g t h ( e ) + 1 )

dPGA( i )= e ( i −1)+0 .5 ;e l s e

dPGA( i ) = ( e ( i −1)+e ( i ) ) / 2 ;end

30 enddPGA=dPGA ’ ;

f u n c t i o n [ Eps_SD , dSD]= EPS_SD_5 ( nEps_SD , nEps_PGA , T )%Calculates the probabilites of the standardnormal distribution for the%bins given a number of discretisation nEPS_SD, nEPS_PGA

5 %Setting the array for the discretisation of PGAemin =−3;emax =3;f o r i = 1 : ( nEps_PGA−1)

e ( i )= emin +( i −1)∗( emax−emin ) / ( nEps_PGA−2);10 end

f o r i = 1 : ( l e n g t h ( e ) + 1 )i f i ==1

Eps_PGA ( i )= normcdf ( e ( i ) , 0 , 1 ) ;15 e l s e i f i ==( l e n g t h ( e ) + 1 )

Eps_PGA ( i )= normcdf ( e ( 1 ) , 0 , 1 ) ;e l s e

Eps_PGA ( i )= normcdf ( e ( i ) ,0 ,1 ) − normcdf ( e ( i −1 ) , 0 , 1 ) ;end

20 endEps_PGA=Eps_PGA ’ ;

f o r i = 1 : ( l e n g t h ( e ) + 1 )i f i ==1

25 dPGA( i )= e ( i ) −0 .5 ;e l s e i f i ==( l e n g t h ( e ) + 1 )

dPGA( i )= e ( i −1)+0 .5 ;e l s e

dPGA( i ) = ( e ( i −1)+e ( i ) ) / 2 ;30 end

enddPGA=dPGA ’ ;

%Setting the array for the discretisation of SD35 hmin =−4.5;

161

C Software Codes

hmax = 4 . 5 ;f o r i = 1 : ( nEps_SD−1)

h ( i )= hmin +( i −1)∗(hmax−hmin ) / ( nEps_SD −2);end

40

%Correlation PGA-SD depending on the periodT = [ 0 . 6 4 ] ;[RHO]= SD_PGA_cor ( T ) ;

45 f o r i = 1 : ( nEps_PGA )MU1( i )=dPGA( i )∗RHO( 1 ) ;SIGMA1( i )=(1−RHO( 1 )∗RHO( 1 ) ) ^ 0 . 5 ;

endMU=[MU1] ;

50 MU=MU( : ) ;SIGMA=[SIGMA1 ] ;SIGMA=SIGMA ( : ) ;

f o r j = 1 : ( l e n g t h (MU) )55 f o r i = 1 : ( l e n g t h ( h ) + 1 )

i f i ==1Eps_SD ( i , j )= normcdf ( h ( i ) ,MU( j ) ,SIGMA( j ) ) ;

e l s e i f i ==( l e n g t h ( h ) + 1 )Eps_SD ( i , j )=1−normcdf ( h ( i −1) ,MU( j ) ,SIGMA( j ) ) ;

60 e l s eEps_SD ( i , j )= normcdf ( h ( i ) ,MU( j ) ,SIGMA( j ) ) . . .−normcdf ( h ( i −1) ,MU( j ) ,SIGMA( j ) ) ;

endend

65 endEps_SD=Eps_SD ( : ) ;

%Setting the array for the discretisationf o r i = 1 : ( l e n g t h ( h ) + 1 )

70 i f i ==1dSD ( i )= h ( i ) −0 .5 ;

e l s e i f i ==( l e n g t h ( h ) + 1 )dSD ( i )= h ( i −1)+0 .5 ;

e l s e75 dSD ( i ) = ( h ( i −1)+h ( i ) ) / 2 ;

endenddSD=dSD ’ ;

f u n c t i o n [ Nu_Mmin ,M_EQ, P_M, M l i m i t s ]=EQ_M_5( Mmin , a , b ,nM)%Assuming that earthquakes of magnitude less than Mmin do not contribute to%damage the mean rate of exceedance for Mmin is Nu_Mmin. P_M is the%probability of having a magnitude in the magnitude range characterised by


Nu_Mmin=10^( a−b ∗ (Mmin ) ) ;

162

10

M l i m i t s = [ 4 . 7 5 5 . 2 5 5 . 7 5 6 . 2 5 6 . 7 5 7 . 2 5 7 . 7 5 ] ;M_EQ=[5 5 . 5 6 6 . 5 7 7 . 5 ] ;M_EQ=M_EQ’ ;

15 f o r i = 1 : (nM+1)LambdaM ( i ) = 1 0 ^ ( a−b ∗ ( M l i m i t s ( i ) ) ) ;

end

LambdaTotal=LambdaM(1)−LambdaM (nM+ 1 ) ;20

f o r i = 1 : (nM)P_M( i ) = ( LambdaM ( i )−LambdaM ( i + 1 ) ) / LambdaTotal ;

endP_M=P_M ’ ;

f u n c t i o n [ Nu_Mmin ,M_EQ, P_M, M l i m i t s ]= EQ_M_NonPoisson_5 ( Mmin , a , b , nM, Q, Z )%Assuming that earthquakes of magnitude less than Mmin do not contribute to%damage the mean rate of exceedance for Mmin is Nu_Mmin. P_M is the%probability of having a magnitude in the magnitude range characterised by


Nu_Mmin=10^( a−b ∗ (Mmin ) ) ;10 M l i m i t s = [ 4 . 7 5 5 . 2 5 5 . 7 5 6 . 2 5 6 . 7 5 7 . 2 5 7 . 7 5 ] ;

M_EQ=[5 5 . 5 6 6 . 5 7 7 . 5 ] ;M_EQ=M_EQ’ ;

%Line source with 100 year characteristic EQ return period and COV=0.5.15 %No EQ for 30 years, T=10,30,50 years mean rate of

%exceedanceLambdaNonPoisson =[ . . .9 .77000E−12 9 .77000E−12 1 .22218E−051 .06433E−10 1 .06433E−10 1 .54372E−05

20 7 .68107E−10 7 .68107E−10 1 .92703E−054 .04182E−09 4 .04182E−09 2 .37928E−051 .66051E−08 1 .66051E−08 2 .90773E−055 .59953E−08 5 .59953E−08 3 .51972E−051 .60909E−07 1 .60909E−07 4 .22252E−05

25 4 .05463E−07 4 .05463E−07 5 .02331E−059 .16021E−07 9 .16021E−07 5 .92907E−051 .88818E−06 1 .88818E−06 6 .94652E−053 .60106E−06 3 .60106E−06 8 .08209E−056 .42646E−06 6 .42646E−06 9 .34182E−05

30 1 .08311E−05 1 .08311E−05 0 .0001073131 .73712E−05 1 .73712E−05 0 .0001225572 .66806E−05 2 .66806E−05 0 .0001391973 .94520E−05 3 .94520E−05 0 .0001572725 .64151E−05 5 .64151E−05 0 .000176820

35 7 .83120E−05 7 .83120E−05 0 .0001978690 .000105872 0 .000105872 0.0002204420 .000139790 0 .000139790 0.000244558

163

C Software Codes

0 .000180700 0.000180700 0 .0002702290 .000229165 0.000229165 0 .000297459

40 0.000285658 0 .000285658 0.0003262490.000350550 0 .000350550 0.0003565940.000424109 0 .000424109 0.0003884820.000506497 0 .000506497 0.0004218970.000597765 0 .000597765 0.000456817

45 0.000697864 0 .000697864 0.0004932170.000806644 0 .000806644 0.0005310660.000923867 0 .000923867 0.0005703290.001049210 0 .001049210 0.0006109670.001182280 0 .001182280 0.000652938

50 0.001322619 0 .001322619 0.0006961960.001469719 0 .001469719 0.0007406940.001623027 0 .001623027 0.0007863790.001781957 0 .001781957 0.0008331980.001945898 0 .001945898 0.000881096

55 0.002114225 0 .002114225 0.0009300150.002286303 0 .002286303 0.0009798960.002461495 0 .002461495 0.0010306790.002639171 0 .002639171 0.0010823030.002818710 0 .002818710 0.001134706

60 0.002999504 0 .002999504 0.0011878260.003180968 0 .003180968 0.0012416000.003362536 0 .003362536 0.0012959660.003543667 0 .003543667 0.0013508610.003723848 0 .003723848 0.001406223

65 0.003902595 0 .003902595 0.0014619910.004079453 0 .004079453 0.0015181020.004253997 0 .004253997 0.001574498 ] ;

f o r i = 1 : (nM)70 LambdaM ( i ) = 1 0 ^ ( a−b ∗ ( M l i m i t s ( i ) ) ) ;

end

% Equation 4.9 in the PHD dissertationLambdaTotal=LambdaM(1)−LambdaM (nM)+ LambdaNonPoisson (Q, Z ) ;

75

f o r i = 1 : (nM−1)P_M( i ) = ( LambdaM ( i )−LambdaM ( i + 1 ) ) / LambdaTotal ;

end

80 P_M(nM)= LambdaNonPoisson (Q, Z ) / LambdaTotal ;P_M=P_M ’ ;

f u n c t i o n [R , P_R_Line , R l i m i t s ]= Line_EQ_R_5 ( nr , nR , X1 , Y1 , X2 , Y2 )%Coordinates of the site (0,0), Coordinates of the two ends of the line%source are (X1,Y1) and (X2,Y2).%nr is the discretisation of the line source, nR is the discretisation of

5 %the bins for the EQ_Distance node

f o r i = 1 : ( n r +1)

164

x ( i )=X1+(X2−X1 ) ∗ ( i −1)/ n r ;10 y ( i )=Y1+(Y2−Y1 ) ∗ ( i −1)/ n r ;

end

f o r i =1 : n rr ( i )= s q r t ( ( ( x ( i )+ x ( i + 1 ) ) / 2 ) ^ 2 + ( ( y ( i )+ y ( i + 1 ) ) / 2 ) ^ 2 ) ;

15 end

R l i m i t s =[0 20 40 60 80 1 0 0 ] ;R_EQ=[10 30 50 70 9 0 ] ;H_R= h i s t ( r , R_EQ ) ;

20 P_R_Line=H_R / n r ;P_R_Line=P_R_Line ’ ;R=R_EQ ’ ;

f u n c t i o n [PGABOORE]=PGA_5 ( nEps_PGA ,M, R)%for all combinations of the states in the node ’Magnitude’ and ’Distance’%the peak ground accelerations are calculated with the Boore Joyner and%Fumal attenuation model

5 T1 = 0 . 0 0 1 ;[ Eps_PGA , dPGA]=EPS_PGA_5 ( nEps_PGA ) ;

N1=1;f o r j =1 : nEps_PGA


[MU_SA SIGMA_SA]= b j f _ a t t e n _ 5 (M(m) ,R( n ) , T1 , 1 , 6 2 0 , 1 ) ;PGABOORE( N1)= exp ( l o g (MU_SA)+SIGMA_SA∗dPGA( j ) ) ;i f PGABOORE( N1) <=0

15 PGABOORE( N1 ) = 0 ;endN1=N1+1;

endend

20 end

f u n c t i o n [SDBOORE]= SD_5 ( nEps_SD , nEps_PGA ,M, R)%for all combinations of the states in the node ’Magnitude’ and ’Distance’%the spectral displacements are calculated with the Boore Joyner and Fumal%attenuation model

5 T = 0 . 6 4 ;[ Eps_SD , dSD]= EPS_SD_5 ( nEps_SD , nEps_PGA ) ;

N1=1;f o r j =1 : nEps_SD


[MU_SA SIGMA_SA]= b j f _ a t t e n _ 5 (M(m) ,R( n ) , T , 1 , 3 1 0 , 1 ) ;SDBOORE( N1 ) = 9 . 8 0 6∗ ( T∗T / 4 / pi / pi )∗ exp ( l o g (MU_SA)+SIGMA_SA∗dSD ( j ) ) ;i f SDBOORE( N1) <=0

15 SDBOORE( N1 ) = 0 ;endN1=N1+1;

165

C Software Codes

endend

20 end

f u n c t i o n [ rho ] = SD_PGA_cor ( T )

% computes correlation between PGA and Sa(T) for a given T

5 % if T is a vector, the function will return a vector rho of the same size

rho = z e r o s ( s i z e ( T ) ) ;

f o r i = 1 : l e n g t h ( T )10 i f T ( i ) <0 .05

rho ( i ) = nan ; % outside of fitted rangef p r i n t f ( ’ I n v a l i d p e r i o d ’ )

e l s e i f T ( i ) >5rho ( i ) = nan ; % outside of fitted range

15 f p r i n t f ( ’ I n v a l i d p e r i o d ’ )e l s e i f T ( i ) <0 .11

rho ( i ) = 0 .500 − 0 .127∗ l o g ( T ( i ) ) ;e l s e i f T ( i ) <0 .25

rho ( i ) = 0 .968 + 0 .085∗ l o g ( T ( i ) ) ;20 e l s e

rho ( i ) = 0 .568 − 0 .204∗ l o g ( T ( i ) ) ;end

end

f u n c t i o n [ sa , s igma ] = b j f _ a t t e n _ 5 (M, R , T , Fau l t_Type , Vs , a r b )

% by Jack Baker, 2/1/05% Stanford University

5 % [email protected]%% Boore Joyner and Fumal attenuation model (1997 Seismological Research% Letters, Vol 68, No 1, p154).%

10 % This script includes standard deviations for either% arbitrary or average components of ground motion (See Baker and Cornell,% 2005, "What spectral acceleration are you using," Earthquake Spectra,% submitted).%

15 %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INPUT%

20 % M = Moment Magnitude% R = boore joyner distance% T = period: 0.1 to 2s (0.001s is a placeholder for the PGA)% Fault_Type = 1 for strike-slip fault% = 2 for reverse-slip fault

25 % = 0 for non-specified mechanism

166

% Vs = shear wave velocity averaged over top 30 m% (use 310 for soil, 620 for rock)% arb = 1 for arbitrary component sigma% = 0 for average component sigma

30 %% OUTPUT%% sa = median spectral acceleration prediction% sigma = logarithmic standard deviation of spectral acceleration

35 % prediction FOR AN ARBITRARY OR AVERAGE COMPONENT%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% coefficients40 p e r i o d = [ 0 . 0 0 1 0 . 1 0 . 1 1 0 . 1 2 0 . 1 3 0 . 1 4 0 . 1 5 0 . 1 6 0 . 1 7 0 . 1 8 0 . 1 9 0 . 2 0 . 2 2 . . .

0 . 2 4 0 . 2 6 0 . 2 8 0 . 3 0 . 3 2 0 . 3 4 0 . 3 6 0 . 3 8 0 . 4 0 . 4 2 0 . 4 4 0 . 4 6 0 . 4 8 0 . 5 . . .0 . 5 5 0 . 6 0 . 6 5 0 . 7 0 . 7 5 0 . 8 0 . 8 5 0 . 9 0 . 9 5 1 1 . 1 1 . 2 1 . 3 1 . 4 1 . 5 1 . 6 . . .1 . 7 1 . 8 1 . 9 2 ] ;

B1ss = [ −0.313 1 .006 1 .0 72 1 .109 1 .128 1 .135 1 .128 1 .112 1 . 0 9 1 .063 1 .032 . . .45 0 .999 0 . 92 5 0 .8 4 7 0 . 76 4 0 .6 81 0 .598 0 .518 0 .439 0 .361 0 .286 0 .212 . . .

0 . 1 4 0 .073 0 .005 −0.058 −0.122 −0.268 −0.401 −0.523 −0.634 −0.737 . . .−0.829 −0.915 −0.993 −1.066 −1.133 −1.249 −1.345 −1.428 −1.495 . . .−1.552 −1.598 −1.634 −1.663 −1.685 −1.699 ] ;

B1rv = [ −0.117 1 .087 1 .1 64 1 .215 1 .246 1 .261 1 .264 1 .257 1 .242 1 .222 1 .198 . . .50 1 . 1 7 1 . 10 4 1 .0 33 0 . 95 8 0 .8 81 0 .803 0 .725 0 .648 0 . 5 7 0 .495 0 .423 . . .

0 . 352 0 .282 0 .2 17 0 .151 0 .087 −0.063 −0.203 −0.331 −0.452 −0.562 . . .−0.666 −0.761 −0.848 −0.932 −1.009 −1.145 −1.265 −1.37 −1.46 −1.538 . . .−1.608 −1.668 −1.718 −1.763 −1.801 ] ;

B 1 a l l = [ −0.242 1 .059 1 . 1 3 1 .174 1 . 2 1 .208 1 .204 1 .192 1 .1 73 1 .151 1 .122 . . .55 1 .089 1 . 01 9 0 .9 4 1 0 . 86 1 0 . 7 8 0 . 7 0 .619 0 . 5 4 0 .46 2 0 .385 0 .311 0 .239 . . .

0 . 169 0 .102 0 .0 36 −0.025 −0.176 −0.314 −0.44 −0.555 −0.661 −0.76 . . .−0.851 −0.933 −1.01 −1.08 −1.208 −1.315 −1.407 −1.483 −1.55 −1.605 . . .−1.652 −1.689 −1.72 −1.743 ] ;

B2 = [ 0 .527 0 .753 0 .732 0 .721 0 .711 0 .707 0 .702 0 .702 0 .702 0 .705 0 .709 . . .60 0 .711 0 . 72 1 0 .7 3 2 0 . 74 4 0 .7 58 0 .769 0 .783 0 .794 0 .806 0 . 8 2 0 .831 . . .

0 . 8 4 0 .852 0 .863 0 .873 0 .884 0 .907 0 .928 0 .946 0 .962 0 .979 0 .992 . . .1 . 006 1 .018 1 .0 27 1 .036 1 .052 1 .064 1 .073 1 . 0 8 1 .085 1 .087 1 .089 . . .1 . 087 1 .087 1 .085 ] ;

B3 = [0 −0.226 −0.23 −0.233 −0.233 −0.23 −0.228 −0.226 −0.221 −0.216 −0.212 . . .65 −0.207 −0.198 −0.189 −0.18 −0.168 −0.161 −0.152 −0.143 −0.136 −0.127 . . .

−0.12 −0.113 −0.108 −0.101 −0.097 −0.09 −0.078 −0.069 −0.06 −0.053 . . .−0.046 −0.041 −0.037 −0.035 −0.032 −0.032 −0.03 −0.032 −0.035 −0.039 . . .−0.044 −0.051 −0.058 −0.067 −0.074 −0.085 ] ;

B5 = [ −0.778 −0.934 −0.937 −0.939 −0.939 −0.938 −0.937 −0.935 −0.933 −0.93 . . .70 −0.927 −0.924 −0.918 −0.912 −0.906 −0.899 −0.893 −0.888 −0.882 . . .

−0.877 −0.872 −0.867 −0.862 −0.858 −0.854 −0.85 −0.846 −0.837 −0.83 . . .−0.823 −0.818 −0.813 −0.809 −0.805 −0.802 −0.8 −0.798 −0.795 −0.794 . . .−0.793 −0.794 −0.796 −0.798 −0.801 −0.804 −0.808 −0.812 ] ;

Bv = [ −0.371 −0.212 −0.211 −0.215 −0.221 −0.228 −0.238 −0.248 −0.258 −0.27 . . .75 −0.281 −0.292 −0.315 −0.338 −0.36 −0.381 −0.401 −0.42 −0.438 −0.456 . . .

−0.472 −0.487 −0.502 −0.516 −0.529 −0.541 −0.553 −0.579 −0.602 . . .−0.622 −0.639 −0.653 −0.666 −0.676 −0.685 −0.692 −0.698 −0.706 −0.71 . . .−0.711 −0.709 −0.704 −0.697 −0.689 −0.679 −0.667 −0.655 ] ;

167

C Software Codes

Va = [ 1396 1112 1291 1452 1596 1718 1820 1910 1977 2037 2080 2118 2158 . . .80 2178 2173 2158 2133 2104 2070 2032 1995 1954 1919 1884 1849 1816 . . .

1782 1710 1644 1592 1545 1507 1476 1452 1432 1416 1406 1396 1400 . . .1416 1442 1479 1524 1581 1644 1714 1795 ] ;

h = [ 5 . 5 7 6 . 2 7 6 . 6 5 6 . 9 1 7 . 0 8 7 . 1 8 7 . 2 3 7 . 2 4 7 . 2 1 7 . 1 6 7 . 1 7 . 0 2 6 . 8 3 6 . 6 2 . . .6 . 3 9 6 . 1 7 5 . 9 4 5 . 7 2 5 . 5 5 . 3 5 . 1 4 . 9 1 4 . 7 4 4 . 5 7 4 . 4 1 4 . 2 6 4 . 1 3 3 . 8 2 . . .

85 3 . 5 7 3 . 3 6 3 . 2 3 . 0 7 2 . 9 8 2 . 9 2 2 . 8 9 2 . 8 8 2 . 9 2 . 9 9 3 . 1 4 3 . 3 6 3 . 6 2 3 . 9 2 . . .4 . 2 6 4 . 6 2 5 . 0 1 5 . 4 2 5 . 8 5 ] ;

s igma1 = [ 0 .431 0 . 4 4 0 .437 0 .437 0 .43 5 0 .435 0 .435 0 .435 0 .435 0 .435 0 .435 . . .0 . 435 0 .437 0 .437 0 .437 0 . 4 4 0 . 4 4 0 .442 0 .444 0 .44 4 0 .447 0 .447 0 .449 . . .0 . 449 0 .451 0 .451 0 .454 0 .456 0 .458 0 .461 0 .463 0 .465 0 .467 0 .467 . . .

90 0 . 4 7 0 .472 0 .474 0 .477 0 .479 0 .481 0 .484 0 .486 0 .4 88 0 . 4 9 0 .493 . . .0 . 493 0 .495 ] ;

s igmac = [ 0 . 1 6 0 0 .134 0 .141 0 .148 0 .153 0 .158 0 .163 0 .166 0 .169 0 .17 3 0 .176 . . .0 . 177 0 .182 0 .185 0 .189 0 .192 0 .195 0 .197 0 .199 0 .200 0 .202 0 .204 . . .0 . 205 0 .206 0 .209 0 .210 0 .211 0 .214 0 .216 0 .218 0 .220 0 .221 0 .223 . . .

95 0 .226 0 .228 0 .230 0 .230 0 .233 0 .236 0 .239 0 .241 0 .244 0 .246 0 .249 . . .0 . 251 0 .254 0 . 2 5 6 ] ;

s i gm ar = [ 0 .460 0 .460 0 .459 0 .461 0 .461 0 .463 0 .465 0 .466 0 .46 7 0 .468 . . .0 . 469 0 .470 0 .473 0 .475 0 .476 0 .480 0 .481 0 .484 0 .487 0 .487 0 .491 . . .0 . 491 0 .494 0 .494 0 .497 0 .497 0 .501 0 .504 0 .506 0 .510 0 .513 0 .515 . . .

100 0 .518 0 .519 0 .522 0 .525 0 . 5 2 7 0 .531 0 . 5 3 4 0 .537 0 . 5 4 1 0 .544 0 . 5 4 6 . . .0 . 550 0 .553 0 .555 0 . 5 5 7 ] ;

s igmae = [ 0 .184 0 0 0 0 0 0 0 0 0 . 002 0 .005 0 .009 0 .016 0 .025 0 .032 0 .039 . . .0 . 048 0 .055 0 .064 0 .071 0 .078 0 .085 0 .092 0 .099 0 .104 0 .111 0 .115 . . .0 . 129 0 .143 0 .154 0 .166 0 .175 0 .184 0 .191 0 . 2 0 .20 7 0 .214 0 .226 . . .

105 0 .235 0 .244 0 .251 0 .256 0 . 2 6 2 0 .267 0 . 2 6 9 0 .274 0 . 2 7 6 ] ;s i g m a ln y = [ 0 . 495 0 .460 0 .459 0 .461 0 .461 0 .463 0 .465 0 .466 0 .467 0 .468 . . .

0 . 469 0 .470 0 .474 0 .475 0 .477 0 .482 0 .484 0 .487 0 .491 0 .492 0 .497 . . .0 . 499 0 .502 0 .504 0 .508 0 .510 0 .514 0 .520 0 .526 0 .533 0 .539 0 .544 . . .0 . 549 0 .553 0 .559 0 .564 0 .569 0 .577 0 .583 0 .590 0 .596 0 .601 0 .606 . . .

110 0 .611 0 .615 0 .619 0 . 6 2 2 ] ;

% interpolate between periods if neccesaryi f ( l e n g t h ( f i n d ( p e r i o d == T ) ) == 0)

index_ low = sum ( p e r i o d <T ) ;115 T_low = p e r i o d ( index_ low ) ;

T_hi = p e r i o d ( index_ low + 1 ) ;

[ sa_low , sigma_low ] = b j f _ a t t e n (M, R , T_low , Fau l t_Type , Vs , a r b ) ;[ s a _ h i , s i g m a _ h i ] = b j f _ a t t e n (M, R , T_hi , Fau l t_Type , Vs , a r b ) ;

120

x = [ l o g ( T_low ) l o g ( T_hi ) ] ;Y_sa = [ l o g ( sa_ low ) l o g ( s a _ h i ) ] ;Y_sigma = [ sigma_low s i g m a _h i ] ;s a = exp ( i n t e r p 1 ( x , Y_sa , l o g ( T ) ) ) ;

125 s igma = i n t e r p 1 ( x , Y_sigma , l o g ( T ) ) ;

e l s ei = f i n d ( p e r i o d == T ) ;

130 % compute median and sigmar = s q r t (R^2 + h ( i ) ^ 2 ) ;

168

i f ( F a u l t _ T y p e == 1)b1 = B1ss ( i ) ;

135 e l s e i f ( F a u l t _ T y p e == 2)b1 = B1rv ( i ) ;

e l s eb1 = B 1 a l l ( i ) ;

end140

l n y = b1+ B2 ( i ) ∗ (M−6)+ B3 ( i ) ∗ (M−6)^2+ B5 ( i )∗ l o g ( r )+Bv ( i )∗ l o g ( Vs / Va ( i ) ) ;s a = exp ( l n y ) ;

i f ( a r b ) % arbitrary component sigma145 s igma = s i g m a l n y ( i ) ;

e l s e % average component sigmas igma = s q r t ( s igma1 ( i ) ^2 + s igmae ( i ) ^ 2 ) ;

endend

169

C Software Codes

C.2 Soil analysis tools

GSLIB files for simulating random fields

P a r a m e t e r s f o r SGSIM ( Depth . p a r )∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

START OF PARAMETERS:5 Depth . d a t %file with data

1 2 0 3 0 0 %columns for X,Y,Z,vr,wt,sec.var.−1.0 1 . 0 e21 %trimming limits1 %transform the data (0=no, 1=yes)sgs im . t r n %file for output trans table

10 0 %consider ref. dist (0=no, 1=yes)h i s t s m t h . o u t %file with ref. dist distribution1 2 %columns for vr and wt0 . 0 1 6 . 0 %zmin,zmax(tail extrapolation)1 0 . 0 %lower tail option, parameter

15 1 1 6 . 0 %upper tail option, parameter1 %debugging level: 0,1,2,3sgs imDepth . dbg %file for debugging outputsgs imDepth . o u t %file for simulation output100 %number of realizations to generate

20 100 50 100 %nx,xmn,xsiz150 50 100 %ny,ymn,ysiz1 0 . 5 1 . 0 %nz,zmn,zsiz69069 %random number seed0 8 %min and max original data for sim

25 12 %number of simulated nodes to use1 %assign data to nodes (0=no, 1=yes)1 3 %multiple grid search (0=no, 1=yes),num0 %maximum data per octant (0=not used)10000 .0 10000 .0 5 0 . 0 %maximum search radii (hmax,hmin,vert)

30 0 . 0 0 . 0 0 . 0 %angles for search ellipsoid0 %ktype: 0=SK,1=OK,2=LVM,3=EXDR,4=COLC. . / d a t a / y d a t a . d a t %file with LVM, EXDR, or COLC variable4 %column for secondary variable1 0 . 1 %nst, nugget effect

35 3 0 . 9 0 . 0 0 . 0 0 . 0 %it,cc,ang1,ang2,ang35000 .0 5000 .0 1 0 . 0 %a_hmax, a_hmin, a_vert

P a r a m e t e r s f o r SGSIM (USCS . p a r )∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

START OF PARAMETERS:5 USCS . d a t %file with data


10 0 %consider ref. dist (0=no, 1=yes)h i s t s m t h . o u t %file with ref. dist distribution1 2 %columns for vr and wt

170

0 . 0 2 5 . 0 %zmin,zmax(tail extrapolation)1 0 . 0 %lower tail option, parameter

15 1 2 5 . 0 %upper tail option, parameter1 %debugging level: 0,1,2,3sgsimUSCS . dbg %file for debugging outputsgsimUSCS . o u t %file for simulation output100 %number of realizations to generate





P a r a m e t e r s f o r SGSIM (GW. p a r )∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

START OF PARAMETERS:5 GW. d a t %file with data



15 1 1 6 . 0 %upper tail option, parameter1 %debugging level: 0,1,2,3sgsimGW . dbg %file for debugging outputsgsimGW . o u t %file for simulation output100 %number of realizations to generate


25 12 %number of simulated nodes to use1 %assign data to nodes (0=no, 1=yes)1 3 %multiple grid search (0=no, 1=yes),num0 %maximum data per octant (0=not used)

171

C Software Codes

10000 .0 10000 .0 5 0 . 0 %maximum search radii (hmax,hmin,vert)30 0 . 0 0 . 0 0 . 0 %angles for search ellipsoid

0 %ktype: 0=SK,1=OK,2=LVM,3=EXDR,4=COLC. . / d a t a / y d a t a . d a t %file with LVM, EXDR, or COLC variable4 %column for secondary variable1 0 . 1 %nst, nugget effect


P a r a m e t e r s f o r SGSIM ( FC . p a r )∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

START OF PARAMETERS:5 FC . d a t %file with data



15 1 7 0 . 0 %upper tail option, parameter1 %debugging level: 0,1,2,3sgsimFC . dbg %file for debugging outputsgsimFC . o u t %file for simulation output100 %number of realizations to generate





P a r a m e t e r s f o r SGSIM (Nm. p a r )∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

START OF PARAMETERS:5 N30 . d a t %file with data

1 2 0 3 0 0 %columns for X,Y,Z,vr,wt,sec.var.−1.0 1 . 0 e21 %trimming limits

172

1 %transform the data (0=no, 1=yes)sgs im . t r n %file for output trans table


15 1 5 0 . 0 %upper tail option, parameter1 %debugging level: 0,1,2,3sgsimNeu . dbg %file for debugging outputsgsimNeu . o u t %file for simulation output100 %number of realizations to generate



30 0 . 0 0 . 0 0 . 0 %angles for search ellipsoid0 %ktype: 0=SK,1=OK,2=LVM,3=EXDR,4=COLC. . / d a t a / y d a t a . d a t %file with LVM, EXDR, or COLC variable4 %column for secondary variable1 0 . 1 5 %nst, nugget effect

35 2 0 . 8 5 0 . 0 0 . 0 0 . 0 %it,cc,ang1,ang2,ang35000 .0 5000 .0 1 0 . 0 %a_hmax, a_hmin, a_vert

Matlab files for generating random fields for use in Matlab

f u n c t i o n [ D e p t h _ d a t a ]= D e p t h _ f i e l dfilenameOUT = [ ’ . . / GSLIB / sgs imDepth . o u t ’ ] ;f i lenamePAR = [ ’ . . / GSLIB / sgs imDepth . p a r ’ ] ;%determine number of realisations and size of the fields

5 f i l ePAR = fopen ( fi lenamePAR , ’ r ’ ) ;f o r j =1:18

ans= f g e t l ( f i l ePAR ) ;endN= f s c a n f ( f i lePAR , ’%f ’ , 1 ) ;

10 ans= f g e t l ( f i l ePAR ) ;nx= f s c a n f ( f i lePAR , ’%f ’ , 1 ) ;ans= f g e t l ( f i l ePAR ) ;ny= f s c a n f ( f i lePAR , ’%f ’ , 1 ) ;f c l o s e ( f i l ePAR ) ;

15

%construct field for USCSf i leOUT= fopen ( filenameOUT , ’ r ’ ) ;f o r i =1 :3ans= f g e t l ( f i leOUT ) ;

20 end

173

C Software Codes

f o r i =1 :N[A, np ]= f s c a n f ( fi leOUT , ’%f ’ , [ nx , ny ] ) ;D e p t h _ d a t a ( i ) . Depth=A’ ;

end25 f c l o s e ( f i leOUT ) ;

save Depth D e p t h _ d a t a

f u n c t i o n [ USCS_data ]= USCS_fie ldfilenameOUT = [ ’ . . / GSLIB / sgsimUSCS . o u t ’ ] ;f i lenamePAR = [ ’ . . / GSLIB / sgsimUSCS . p a r ’ ] ;%determine number of realisations and size of the fields




15

%construct field for USCSf i leOUT= fopen ( filenameOUT , ’ r ’ ) ;f o r i =1 :3ans= f g e t l ( f i leOUT ) ;

20 endf o r i =1 :N

[A, np ]= f s c a n f ( fi leOUT , ’%f ’ , [ nx , ny ] ) ;B=round (A ) ;C=B+(B= = 0 ) ;

25 USCS_data ( i ) . USCS=C ’ ;endf c l o s e ( f i leOUT ) ;save USCS USCS_data

f u n c t i o n [ GW_data ]= GW_fieldfilenameOUT = [ ’ . . / GSLIB / sgsimGW . o u t ’ ] ;f i lenamePAR = [ ’ . . / GSLIB / sgsimGW . p a r ’ ] ;%determine number of realisations and size of the fields




15

%construct field for GWf i leOUT= fopen ( filenameOUT , ’ r ’ ) ;

174

f o r i =1 :3ans= f g e t l ( f i leOUT ) ;

20 endf o r i =1 :N

[A, np ]= f s c a n f ( fi leOUT , ’%f ’ , [ nx , ny ] ) ;GW_data ( i ) .GW=A’ ;


save GW GW_data

f u n c t i o n [ FC_data ]= F C _ f i e l dfilenameOUT = [ ’ . . / GSLIB / sgsimFC . o u t ’ ] ;f i lenamePAR = [ ’ . . / GSLIB / sgsimFC . p a r ’ ] ;%determine number of realisations and size of the fields




15

%construct field for FCf i leOUT= fopen ( filenameOUT , ’ r ’ ) ;f o r i =1 :3ans= f g e t l ( f i leOUT ) ;

20 endf o r i =1 :N

[A, np ]= f s c a n f ( fi leOUT , ’%f ’ , [ nx , ny ] ) ;FC_data ( i ) . FC=A’ ;


save FC FC_data

f u n c t i o n [ Nm_data ]= Nm_fie ldfilenameOUT = [ ’ . . / GSLIB / sgsimNm . o u t ’ ] ;f i lenamePAR = [ ’ . . / GSLIB / sgsimNm . p a r ’ ] ;%determine number of realisations and size of the fields




15

%construct field for Nm

175

C Software Codes

f i leOUT= fopen ( filenameOUT , ’ r ’ ) ;f o r i =1 :3ans= f g e t l ( f i leOUT ) ;

20 endf o r i =1 :N

[A, np ]= f s c a n f ( fi leOUT , ’%f ’ , [ nx , ny ] ) ;Nm_data ( i ) .Nm=A’ ;


save Nm Nm_data

Matlab files for calculating random fields for use in liquefaction assessment

f u n c t i o n [ N1_60_data ]= N 1 _ 6 0 _ f i e l dload Nmload S i g m a e f f

5 CR=1;CS=1;CE=1;CB=1;N=100;

10 f o r i =1 :Nf o r j =1:150

f o r k =1:100CN_data ( 1 , i ) . CN( j , k ) = ( 1 0 0 / . . .

( S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k ) ) ) ^ 0 . 5 ;15 i f CN_data ( 1 , i ) . CN( j , k ) >1 .7

CN_data ( 1 , i ) . CN( j , k ) = 1 . 7end

endend

20 end

f o r i =1 :Nf o r j =1:150

f o r k =1:10025 N1_60_data ( 1 , i ) . N1_60 ( j , k )= Nm_data ( 1 , i ) .Nm( j , k )∗ . . .

CN_data ( 1 , i ) . CN( j , k )∗CE∗CR∗CS∗CB;end

endend

30

save N1_60 N1_60_data

f u n c t i o n [ CSR_data ]= CSR_f ie ld% PGA in units of g - scalar input% M Magnitude - scalar input% d depth - matrix with the same size as the area of interest.

5 % Assumed to be less than 12m% Vs matrix with the same size as the area of interest

176

load Depthload S i g m a e f f

10 load Sigmaload Vs

M=[0 5 5 . 5 6 6 . 5 7 7 . 5 8 ] ;PGA= [ 0 . 1 0 . 3 0 . 5 0 . 7 0 . 9 1 . 1 1 . 3 ] ;N=100;

15 n =0;f o r l =1 : l e n g t h (M)

f o r p =1: l e n g t h (PGA)n=n +1;PGA_Area=PGA( p )∗ ones ( 1 5 0 , 1 0 0 ) ;

20 M_Area=M( l )∗ ones ( 1 5 0 , 1 0 0 ) ;f o r i =1 :N

%perfectly correlated variables%simulation of depth reduction factor r_d and CSRd1=min ( 1 2 , D e p t h _ d a t a ( 1 , i ) . Depth ) ;

25 s i g m a _ e p s _ r d =d1 . ^ 0 . 8 5 . ∗ 0 . 0 1 9 8 ;e p s _ r d =normrnd ( 0 , 1 )∗ s i g m a _ e p s _ r d ;n u m e r a t o r = 1 + (−23.013 + 2 . 9 4 9 .∗ PGA_Area + 0 . 9 9 9 .∗M_Area + . . .

0 . 0 5 2 5 . ∗ ( Vs_da ta ( 1 , i ) . Vs ) ) . / . . .( 1 6 . 2 5 8 + 0 .201∗ exp (0 .341∗ ( − ( D e p t h _ d a t a ( 1 , i ) . Depth ) + . . .

30 0 . 0 7 8 5 . ∗ ( Vs_da ta ( 1 , i ) . Vs ) + 7 . 5 8 6 ) ) ) ;d e n o m i n a t o r = 1 + (−23.013 + 2 . 9 4 9 .∗ PGA_Area + . . .

0 . 9 9 9 .∗M_Area + 0 . 0 5 2 5 . ∗ ( Vs_da ta ( 1 , i ) . Vs ) ) . / . . .( 1 6 . 2 5 8 + 0 .201∗ exp ( 0 . 3 4 1 ∗ ( 0 . 0 7 8 5 . ∗ ( Vs_da ta ( 1 , i ) . Vs ) + . . .7 . 5 8 6 ) ) ) ;

35 r_d = n u m e r a t o r . / d e n o m i n a t o r + e p s _ r d ;CSR_data ( n ) . CSR_data ( 1 , i ) . CSR = 0 . 6 5 .∗ PGA_Area .∗ . . .

( S igma_da ta ( 1 , i ) . Sigma ) . / ( S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ) . ∗ r_d ;CSR_data ( n ) .M=M( l ) ;CSR_data ( n ) . PGA=PGA( p ) ;

40 endend

end

save CSR CSR_data

f u n c t i o n [ Vs_da ta ]= V s _ f i e l dload Nm

%construct field for Vs5 N=100;

f o r i =1 :Nf o r j =1:150

f o r k =1:100Vs_da ta ( 1 , i ) . Vs ( j , k )=90∗ ( Nm_data ( 1 , i ) .Nm( j , k ) ) ^ 0 . 3 0 9 ;

10 endend

end

save Vs Vs_da ta

177

C Software Codes

f u n c t i o n [ S igma_da ta ]= S i g m a _ f i e l dload Depthload GWload USCS

5 % CH 1% CH-MH 2% CL 3% CL-CH 4% CL-MH 5

10 % CL-ML 6% MH 7% MH-CH 8% ML 9% ML-CL 10

15 % SC 11% SM 12% SP-SM 13Gamma= [ 1 8 . 6 4 17 .27 2 0 . 9 19 .7 7 18 .39 2 0 . 7 15 .89 . . .

17 .27 19 .52 2 0 . 7 21 .68 20 .31 1 9 . 9 1 ] ;20 N=100;

f o r i =1 :Nf o r j =1:150

f o r k =1:10025 i f USCS_data ( 1 , i ) . USCS( j , k )==1

Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 1 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==2Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 2 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==3

30 Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 3 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==4Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 4 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==5Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 5 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;

35 e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==6Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 6 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==7Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 7 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==8

40 Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 8 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==9Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 9 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==10Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 1 0 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;

45 e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==11Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 1 1 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==12Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 1 2 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;e l s e i f USCS_data ( 1 , i ) . USCS( j , k ) >=13

50 Sigma_da ta ( 1 , i ) . Sigma ( j , k )=Gamma ( 1 3 ) ∗ ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ) ) ;end

endend

178

end55

save Sigma Sigma_da ta

f u n c t i o n [ S i g m a e f f _ d a t a ]= S i g m a e f f _ f i e l dload Depthload GWload USCS

5

% CH 1% CH-MH 2% CL 3% CL-CH 4

10 % CL-MH 5% CL-ML 6% MH 7% MH-CH 8% ML 9

15 % ML-CL 10% SC 11% SM 12% SP-SM 13Gamma= [ 1 8 . 6 4 17 .27 2 0 . 9 19 .7 7 18 .39 2 0 . 7 . . .

20 15 .89 1 7 .2 7 19 . 5 2 2 0 . 7 2 1 .6 8 20 .31 1 9 . 9 1 ] ;N=100;

f o r i =1 :Nf o r j =1:150

25 f o r k =1:100i f USCS_data ( 1 , i ) . USCS( j , k )==1

S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k )=Gamma( 1 )∗ . . .( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ))− . . .( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k)−GW_data ( 1 , i ) .GW( j , k ) ) ∗ 9 . 8 0 6 ;

30 e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==2S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k )=Gamma( 2 )∗ . . .

( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ))− . . .( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k)−GW_data ( 1 , i ) .GW( j , k ) ) ∗ 9 . 8 0 6 ;

e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==335 S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k )=Gamma( 3 )∗ . . .


e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==4S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k )=Gamma( 4 )∗ . . .

40 ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ))− . . .( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k)−GW_data ( 1 , i ) .GW( j , k ) ) ∗ 9 . 8 0 6 ;


( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ))− . . .45 ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k)−GW_data ( 1 , i ) .GW( j , k ) ) ∗ 9 . 8 0 6 ;



179

C Software Codes

50 e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==7S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k )=Gamma( 7 )∗ . . .


e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==855 S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k )=Gamma( 8 )∗ . . .



60 ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ))− . . .( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k)−GW_data ( 1 , i ) .GW( j , k ) ) ∗ 9 . 8 0 6 ;

e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==10S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k )=Gamma( 1 0 )∗ . . .

( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k ))− . . .65 ( D e p t h _ d a t a ( 1 , i ) . Depth ( j , k)−GW_data ( 1 , i ) .GW( j , k ) ) ∗ 9 . 8 0 6 ;

e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==11S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k )=Gamma( 1 1 )∗ . . .


70 e l s e i f USCS_data ( 1 , i ) . USCS( j , k )==12S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k )=Gamma( 1 2 )∗ . . .


e l s e i f USCS_data ( 1 , i ) . USCS( j , k ) >=1375 S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ( j , k )=Gamma( 1 3 )∗ . . .


endend

80 endend

save S i g m a e f f S i g m a e f f _ d a t a

f u n c t i o n [ S o i l _ d a t a , L i q _ d a t a ]= l i q _ f i e l dload Depthload S i g m a e f fload FC

5 load N1_60load CSR

M=[0 5 5 . 5 6 6 . 5 7 7 . 5 8 ] ;PGA= [ 0 . 1 0 . 3 0 . 5 0 . 7 0 . 9 1 . 1 1 . 3 ] ;

10 N=100;n =0;f o r l =1 : l e n g t h (M)

f o r p =1: l e n g t h (PGA)n=n +1;

15 PGA_Area=PGA( p )∗ ones ( 1 5 0 , 1 0 0 ) ;M_Area=M( l )∗ ones ( 1 5 0 , 1 0 0 ) ;i f M( l )==0

L i q _ d a t a ( n ) . P r o b _ l i q = z e r o s ( 1 5 0 , 1 0 0 ) ;

180

L i q _ d a t a ( n ) .M=M( l ) ;20 L i q _ d a t a ( n ) . PGA=PGA( p ) ;

e l s ef o r i =1 :N

%perfectly correlated variableseps_L=normrnd ( 0 , 2 . 7 ) ∗ ones ( 1 5 0 , 1 0 0 ) ;

25 g_x =( N1_60_data ( 1 , i ) . N1_60 ) . ∗ ( 1 + 0 . 0 0 4 . ∗ . . .( FC_data ( 1 , i ) . FC) ) −1 3 . 3 2 .∗ . . .l o g ( CSR_data ( 1 , n ) . CSR_data ( 1 , i ) . CSR) . . .−29.53.∗ l o g ( M_Area ) −3 .7 .∗ . . .l o g ( ( S i g m a e f f _ d a t a ( 1 , i ) . S i g m a e f f ) / 1 0 0 ) + . . .

30 0 . 0 5 . ∗ ( FC_data ( 1 , i ) . FC)+16 .85+ eps_L ;g_x_I =g_x ;g_x_I ( f i n d ( g_x < 0 ) ) = 1 ;g_x_I ( f i n d ( g_x > = 0 ) ) = 0 ;S o i l _ d a t a ( i ) . g_x=g_x ;

35 S o i l _ d a t a ( i ) . g_x_I = g_x_I ;endg_x_I_cum= z e r o s ( 1 5 0 , 1 0 0 ) ;f o r i =1 :N

g_x_I_cum=g_x_I_cum+ S o i l _ d a t a ( i ) . g_x_I ;40 S o i l _ d a t a ( i ) . g_x_I_cum=g_x_I_cum ;

endL i q _ d a t a ( n ) . P r o b _ l i q = S o i l _ d a t a (N ) . g_x_I_cum . / N;L i q _ d a t a ( n ) .M=M( l ) ;L i q _ d a t a ( n ) . PGA=PGA( p ) ;

45 endend

end

save Liq L i q _ d a t a

181

C Software Codes

C.3 Structural damage analysis tools

OpenSees files for calculating maximum interstory drift ratios of the structureclasses

# Typ5 .NR.O. R e s i d e n t i a l . t c l −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# based on t h e examples i n# h t t p : / / o p e n s e e s . b e r k e l e y . edu / OpenSees / manuals / ExamplesManual /HTML/# 5− s t o r y RCMRF, n o t r e t r o f i t t e d , d e s i g n e d b e f o r e 1980 , r e s i d e n t i a l use

5 # nonl inearBeamColumn element , i n e l a s t i c f i b e r s e c t i o n## d e f i n e UNITS −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# p u t s " −− U n i a x i a l I n e l a s t i c M a t e r i a l , F i b e r RC−S e c t i o n , N o n l i n e a r Model −−"# p u t s " −− Uniform E a r t h q u a k e E x c i t a t i o n −−"

10 s e t s e c 1 . ; # d e f i n e b a s i c u n i t s −− o u t p u t u n i t ss e t g 9 . 8 0 6 ; # g r a v i t a t i o n a l a c c e l e r a t i o ns e t Ubig 1 . e10 ; # a r e a l l y l a r g e numbers e t Usmall [ exp r 1 / $Ubig ] ; # a r e a l l y s m a l l number# b a s i c u n i t s a r e : m, s e c and kN

15 # SET UP −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−wipe ; # c l e a r memory of a l l p a s t model %n

d e f i n i t i o n smodel B a s i c B u i l d e r \ # D e f i ne t h e model b u i l d e r , %n

ndm=# dimens ion , ndf =# d o f s20 −ndm 2 −ndf 3 ;

# s e t d a t a D i r OutputICASP ; # s e t up name of d a t a d i r e c t o r y# f i l e mkdir $ d a t a D i r ; # c r e a t e d a t a d i r e c t o r ys e t GMdir " . . / GMfi les / " ; # ground−motion f i l e d i r e c t o r ys o u r c e B u i l d R C r e c t S e c t i o n . t c l ; # p r o c e d u r e f o r d e f i n i n i n g RC f i b e r s e c t i o n

25

# d e f i n e GEOMETRY −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# d e f i n e s t r u c t u r e −geomet ry p a r a m t e r ss e t LCol 3 . ; # column h e i g h ts e t LBeam 5 . ; # beam l e n g t h

30 s e t NStory 5 ; # number o f s t o r i e s above grounds e t NBay 4 ; # number o f bays

# d e f i n e NODAL COORDINATESf o r { s e t l e v e l 1} { $ l e v e l <=[ exp r $NStory +1]} { i n c r l e v e l 1} {

35 s e t Y [ exp r ( $ l e v e l −1)∗$LCol ] ;f o r { s e t p i e r 1} { $ p i e r <= [ exp r $NBay +1]} { i n c r p i e r 1} {

s e t X [ exp r ( $ p i e r −1)∗$LBeam ] ;s e t nodeID [ exp r $ l e v e l ∗10+ $ p i e r ]node $nodeID $X $Y ; # a c t u a l l y d e f i n e node

40 }}# BOUNDARY CONDITIONSf ixY 0 . 0 1 1 1 ; # p i n a l l Y=0.0 nodes

45 # De f i n e SECTIONS −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# d e f i n e s e c t i o n t a g s :s e t ColSecTag 1s e t BeamSecTag 2

182

50 # S e c t i o n P r o p e r t i e s :s e t HCol 0 . 5 0 ; # Column wid ths e t BCol 0 . 5 0 ; # Column h e i g h ts e t HBeam 0 . 5 0 ; # Beam d e p t hs e t BBeam 0 . 3 0 ; # Beam wid th

55

# G e n e r a l M a t e r i a l p a r a m e t e r ss e t G $Ubig ; # make s t i f f s h e a r moduluss e t J 1 . 0 ; # t o r s i o n a l s e c t i o n s t i f f n e s s %n

(G makes GJ l a r g e )60 s e t GJ [ exp r $G∗$J ] ;

# c o n f i n e d and u n c o n f i n e d c o n c r e t e −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# nomina l c o n c r e t e c o m p r e s s i v e s t r e n g t hs e t f c −20000. ; # C o n c r e t e Compress ive S t r e n g t h

65 s e t Ec 1 4 0 0 0 0 0 0 . ; # C o n c r e t e E l a s t i c Moduluss e t nu 0 . 2 ;s e t Gc [ exp r $Ec / 2 . / [ ex p r 1+$nu ] ] ; # T o r s i o n a l s t i f f n e s s Modulus

# c o n f i n e d c o n c r e t e70 s e t Kfc 1 . 3 ; # r a t i o o f c o n f i n e d t o u n c o n f i n e d %n

c o n c r e t e s t r e n g t hs e t Kres 0 . 2 ; # r a t i o o f r e s i d u a l / u l t i m a t e t o %n

maximum s t r e s ss e t fc1C [ ex p r $Kfc∗ $ f c ] ; # CONFINED c o n c r e t e ( mander model ) ,%n

75 maximum s t r e s ss e t eps1C [ e xp r 2 .∗ $fc1C / $Ec ] ; # s t r a i n a t maximum s t r e s ss e t fc2C [ ex p r $Kres∗$fc1C ] ; # u l t i m a t e s t r e s ss e t eps2C [ e xp r 20∗ $eps1C ] ; # s t r a i n a t u l t i m a t e s t r e s ss e t lambda 0 . 1 ; # r a t i o be tween u n l o a d i n g s l o p e a t %n

80 $eps2 and i n i t i a l s l o p e $Ec# u n c o n f i n e d c o n c r e t es e t fc1U $ f c ; # UNCONFINED c o n c r e t e ( t o d e s c h i n i %n

p a r a b o l i c model ) , maximum s t r e s ss e t eps1U −0.003; # s t r a i n a t maximum s t r e n g t h o f %n

85 u n c o n f i n e d c o n c r e t es e t fc2U [ ex p r $Kres∗$fc1U ] ; # u l t i m a t e s t r e s ss e t eps2U −0.01; # s t r a i n a t u l t i m a t e s t r e s s

# t e n s i l e −s t r e n g t h p r o p e r t i e s90 s e t f t C [ exp r −0.14∗ $fc1C ] ; # t e n s i l e s t r e n g t h + t e n s i o n

s e t f tU [ exp r −0.14∗ $fc1U ] ; # t e n s i l e s t r e n g t h + t e n s i o ns e t E t s [ exp r $f tU / 0 . 0 0 2 ] ; # t e n s i o n s o f t e n i n g s t i f f n e s s

# s e t up l i b r a r y o f m a t e r i a l s # s e t v a l u e on ly i f i t has n o t95 i f { [ i n f o e x i s t s ima t ] ! = 1}{ # been d e f i n e d p r e v i o u s l y .

s e t ima t 0} ;s e t IDconcCore 1s e t IDconcCover 2

100 u n i a x i a l M a t e r i a l C o n c r e t e 0 2 $IDconcCore $fc1C $eps1C $fc2C $eps2C \$lambda $ f tC $Et s ; # Core c o n c r e t e ( c o n f i n e d )

183

C Software Codes

u n i a x i a l M a t e r i a l C o n c r e t e 0 2 $IDconcCover $fc1U $eps1U $fc2U $eps2U \$lambda $f tU $Et s ; # Cover c o n c r e t e ( u n c o n f i n e d )

105 # REINFORCING STEEL p a r a m e t e r s−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−#s e t Fy 4 2 0 0 0 0 . ; # STEEL y i e l d s t r e s ss e t Es 2 1 0 0 0 0 0 0 0 . ; # modulus o f s t e e ls e t Bs 0 . 0 1 ; # s t r a i n −h a r d e n i n g r a t i o

110 s e t R0 1 8 ; # c o n t r o l t r a n s i t i o n from e l a s t i c t o p l a s t i cs e t cR1 0 . 9 2 5 ; # c o n t r o l t r a n s i t i o n from e l a s t i c t o p l a s t i cs e t cR2 0 . 1 5 ; # c o n t r o l t r a n s i t i o n from e l a s t i c t o p l a s t i cs e t I D S t e e l 3u n i a x i a l M a t e r i a l S t e e l 0 2 $ I D S t e e l $Fy $Es $Bs $R0 $cR1 $cR2

115

# FIBER SECTION p r o p e r t i e s# Column s e c t i o n geomet ry :s e t c o v e r 0 . 0 4 ; # r e c t a n g u l a r −RC−Column c o v e r

120 s e t numBarsTopCol 7 ; # number o f r e i n f o r c e m e n t b a r s on t o p l a y e rs e t numBarsBotCol 7 ; # number o f r e i n f o r c e m e n t b a r s on bot tom l a y e rs e t numBars In tCol 1 0 ; # number o f r e i n f o r c i n g b a r s on i n t e r m . l a y e r ss e t barAreaTopCol 0 . 0 0 0 2 0 1 ; # l o n g i t u d i n a l −r e i n f o r c e m e n t bar a r e as e t barAreaBo tCo l 0 . 0 0 0 2 0 1 ; # l o n g i t u d i n a l −r e i n f o r c e m e n t bar a r e a

125 s e t b a r A r e a I n t C o l 0 . 0 0 0 2 0 1 ; # l o n g i t u d i n a l −r e i n f o r c e m e n t bar a r e a

s e t numBarsTopBeam 8 ; # number o f r e i n f o r c e m e n t b a r s on t o p l a y e rs e t numBarsBotBeam 3 ; # number o f r e i n f o r c e m e n t b a r s on bot tom l a y e rs e t numBarsIntBeam 0 ; # number o f r e i n f o r c i n g b a r s on i n t e r m . l a y e r s

130 s e t barAreaTopBeam 0 . 0 0 0 2 0 1 ; # l o n g i t u d i n a l −r e i n f o r c e m e n t bar a r e as e t barAreaBotBeam 0 . 0 0 0 2 0 1 ; # l o n g i t u d i n a l −r e i n f o r c e m e n t bar a r e as e t barAreaIn tBeam 0 . 0 0 0 2 0 1 ; # l o n g i t u d i n a l −r e i n f o r c e m e n t bar a r e a

s e t nfCoreY 2 0 ; # number o f f i b e r s i n t h e c o r e patch i n y−d i r .135 s e t nfCoreZ 2 0 ; # number o f f i b e r s i n t h e c o r e patch i n z−d i r .

s e t nfCoverY 2 0 ; # number o f f i b e r s i n t h e c o v e r p a t c h e s %nwi th long s i d e s i n t h e y d i r e c t i o n

s e t nfCoverZ 2 0 ; # number o f f i b e r s i n t h e c o v e r p a t c h e s %nwi th long s i d e s i n t h e z d i r e c t i o n

140

# r e c t a n g u l a r s e c t i o n wi th one l a y e r o f s t e e l e v e n l y d i s t r i b u t e d around t h e# p e r i m e t e r and a c o n f i n e d c o r e .B u i l d R C r e c t S e c t i o n $ColSecTag $HCol $BCol $ c ov e r $c ov e r $IDconcCore \$IDconcCover $ I D S t e e l $numBarsTopCol $barAreaTopCol $numBarsBotCol \

145 $barAreaBotCo l $numBars In tCol $ b a r A r e a I n t C o l $nfCoreY $nfCoreZ $nfCoverY \$nfCoverZB u i l d R C r e c t S e c t i o n $BeamSecTag $HBeam $BBeam $ c ov e r $c ov e r $IDconcCore \$IDconcCover $ I D S t e e l $numBarsTopBeam $barAreaTopBeam $numBarsBotBeam \$barAreaBotBeam $numBarsIntBeam $barAreaIn tBeam $nfCoreY $nfCoreZ \

150 $nfCoverY $nfCoverZ

# d e f i n e ELEMENTS# s e t up g e o m e t r i c t r a n s f o r m a t i o n s o f e l e m e n t# s e p a r a t e columns and beams , i n c a s e o f P−D e l t a a n a l y s i s f o r columns

184

155 s e t IDCo lTrans f 1 ; # a l l columnss e t IDBeamTransf 2 ; # a l l beamss e t ColTrans fType L i n e a r ; # L i n e a r PD e l t a C o r o t a t i o n a lgeomTransf $ColTrans fType $IDColTrans f ; # columns can have P De l t a e f f e c t sgeomTransf L i n e a r $IDBeamTransf

160

# De f i n e Beam−Column Elemen t ss e t np 5 ; # number o f Gauss i n t e g r a t i o n p o i n t s f o r n o n l i n e a r %n

c u r v a t u r e d i s t r i b u t i o n −− np=2 f o r l i n e a r d i s t r i b u t i o n ok# columns

165 s e t N0col 10 0 ; # column e l e m e n t numberss e t l e v e l 0f o r { s e t l e v e l 1} { $ l e v e l <=$NStory } { i n c r l e v e l 1} {

f o r { s e t p i e r 1} { $ p i e r <= [ exp r $NBay +1]} { i n c r p i e r 1} {s e t elemID [ exp r $N0col + $ l e v e l ∗10 + $ p i e r ]

170 s e t nodeI [ exp r $ l e v e l ∗10 + $ p i e r ]s e t nodeJ [ exp r ( $ l e v e l +1)∗10 + $ p i e r ]e l e m e n t nonl inearBeamColumn $elemID $nodeI $nodeJ $np \$ColSecTag $IDColTrans f ; # columns

}175 }

# beamss e t N0beam 200 ; # beam e l e m e n t numberss e t M0 0f o r { s e t l e v e l 2} { $ l e v e l <=[ exp r $NStory +1]} { i n c r l e v e l 1} {

180 f o r { s e t bay 1} { $bay <= $NBay} { i n c r bay 1} {s e t elemID [ exp r $N0beam + $ l e v e l ∗10 +$bay ]s e t nodeI [ exp r $M0 + $ l e v e l ∗10 + $bay ]s e t nodeJ [ exp r $M0 + $ l e v e l ∗10 + $bay +1]e l e m e n t nonl inearBeamColumn $elemID $nodeI $nodeJ $np \

185 $BeamSecTag $IDBeamTransf ; # beams}

}# De f i n e GRAVITY LOADS, w e i gh t and masses# c a l c u l a t e dead load of frame , assume t h i s t o be an i n t e r n a l f rame %n

190 ( do LL i n a s i m i l a r manner )# c a l c u l a t e d i s t r i b u t e d w e i gh t a l o n g t h e beam l e n g t hs e t GammaConcrete 2 5 . ; # R e i n f o r c e d−C o n c r e t e f l o o r s l a b ss e t T s l a b 0 . 1 5 ; # 15 cm s l a bs e t L s l a b [ e xp r 2∗$LBeam / 2 ] ; # assume s l a b e x t e n d s a d i s t a n c e o f %n

195 $LBeam1 / 2 i n / o u t o f p l a n es e t Qslab [ exp r $GammaConcrete∗ $ T s l ab ∗ $L s l ab ] ;s e t QdlCol [ exp r $GammaConcrete∗$HCol∗$BCol ] ; # s e l f w e i gh t o f Column ,%n

we ig h t p e r l e n g t hs e t QBeam [ e xp r $GammaConcrete∗$HBeam∗$BBeam ] ; # s e l f we igh t o f Beam , %n

200 we i g h t p e r l e n g t hs e t QdlBeam [ e xp r $Qslab + $QBeam ] ; # dead load d i s t r i b u t e d %n

a l o n g beam .s e t WeightCol [ ex p r $QdlCol∗$LCol ] ; # t o t a l Column w e i gh ts e t WeightBeam [ e xp r $QdlBeam∗$LBeam ] ; # t o t a l Beam we ig h t

205 # a s s i g n masses t o t h e nodes t h a t t h e columns a r e c o n n e c t e d t o# each c o n n e c t i o n t a k e s t h e mass o f 1 / 2 o f each e l e m e n t f r a m i n g i n t o i ts e t i F l o o r W e i g h t " "

185

C Software Codes

s e t W e i g h t T o t a l 0 . 0f o r { s e t l e v e l 2} { $ l e v e l <=[ exp r $NStory +1]} { i n c r l e v e l 1} { ;

210 s e t Floo rWeigh t 0 . 0i f { $ l e v e l == [ exp r $NStory +1]} {

s e t ColWeigh tFac t 1 ; # one column i n t o p s t o r y} e l s e {

s e t ColWeigh tFac t 2 ; # two columns e l s e w h e r e215 }

f o r { s e t p i e r 1} { $ p i e r <= [ exp r $NBay +1]} { i n c r p i e r 1} { ;i f { $ p i e r == 1 | | $ p i e r == [ e x p r $NBay +1]} {

s e t BeamWeightFact 1 ; # one beam a t e x t e r i o r nodes} e l s e { ;

220 s e t BeamWeightFact 2 ; # two beams e l e w h e r e}s e t WeightNode [ exp r $ColWeigh tFac t ∗$WeightCol / 2 + \

$BeamWeightFact∗$WeightBeam / 2 ]s e t MassNode [ exp r $WeightNode / $g ] ;

225 s e t nodeID [ exp r $ l e v e l ∗10+ $ p i e r ]mass $nodeID $MassNode 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 ; # d e f i n e masss e t Floo rWeigh t [ ex p r $F loo rWeigh t +$WeightNode ] ;

}l a p p e n d i F l o o r W e i g h t $F loo rWeigh t

230 s e t W e i g h t T o t a l [ exp r $ W e i g h t T o t a l + $F loo rWeigh t ]}s e t MassTota l [ exp r $ W e i g h t T o t a l / $g ] ; # t o t a l mass# De f i n e RECORDERS −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# r e c o r d e r Node − f i l e $ d a t a D i r / DFree21 . o u t \

235 −node 21 −dof 1 di sp ; # d i s p l a c e m e n t s o f nodes# r e c o r d e r D r i f t − f i l e $ d a t a D i r / Dr1 . o u t \

−iNode 11 −jNode 21 −dof 1 −p e r p D i r n 2 ; # l a t e r a l d r i f t# r e c o r d e r D r i f t − f i l e $ d a t a D i r / Dr2 . o u t \

−iNode 21 −jNode 31 −dof 1 −p e r p D i r n 2 ; # l a t e r a l d r i f t240 # r e c o r d e r D r i f t − f i l e $ d a t a D i r / Dr3 . o u t \



245 −iNode 51 −jNode 61 −dof 1 −p e r p D i r n 2 ; # l a t e r a l d r i f t#### D ef in e DISPLAY −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# DisplayModel2D NodeNumbers# d e f i n e GRAVITY −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

250 # GRAVITY LOADS # d e f i n e g r a v i t y load a p p l i e d t o beams and columns %n−− e l eLoad a p p l i e s l o a d s i n l o c a l c o o r d i n a t e a x i s

p a t t e r n P l a i n 101 L i n e a r {f o r { s e t l e v e l 1} { $ l e v e l <=$NStory } { i n c r l e v e l 1} {f o r { s e t p i e r 1} { $ p i e r <= [ exp r $NBay +1]} { i n c r p i e r 1} {

255 s e t elemID [ exp r $N0col + $ l e v e l ∗10 + $ p i e r ]e l eLoad −e l e $elemID −type −beamUniform 0 −$QdlCol ; # COLUMNS}}f o r { s e t l e v e l 2} { $ l e v e l <=[ exp r $NStory +1]} { i n c r l e v e l 1} {

260 f o r { s e t bay 1} { $bay <= $NBay} { i n c r bay 1} {

186

s e t elemID [ e xp r $N0beam + $ l e v e l ∗10 +$bay ]e l eLoad −e l e $elemID −type −beamUniform −$QdlBeam ; # BEAMS}}

265 }# G r a v i t y−a n a l y s i s p a r a m e t e r s −− load−c o n t r o l l e d s t a t i c a n a l y s i ss e t Tol 1 . 0 e−8; # c o n v e r g e n c e t o l e r a n c e f o r t e s tv a r i a b l e c o n s t r a i n t s T y p e G r a v i t y P l a i n ; # d e f a u l t ;i f { [ i n f o e x i s t s Rig idDiaphragm ] == 1} {

270 i f { $RigidDiaphragm =="ON"} {v a r i a b l e c o n s t r a i n t s T y p e G r a v i t y Lagrange ; # l a r g e model : %n

t r y T r a n s f o r m a t i o n} ; # i f r i g i d d iaphragm i s on

} ; # i f r i g i d d iaphragm e x i s t s275 c o n s t r a i n t s $ c o n s t r a i n t s T y p e G r a v i t y ; # how i t h a n d l e s boundary c o n d i t i o n s

numberer RCM; # renumber dof ’ s t o min imize band− %nwid th ( o p t i m i z a t i o n ) , i f you want t o

sys tem BandGenera l ; # how t o s t o r e and s o l v e t h e sys tem %nof e q u a t i o n s i n t h e a n a l y s i s ( l a r g e %n

280 model : t r y UmfPack )t e s t NormDispIncr $Tol 6 ; # d e t e r m i n e i f c o n v e r g e n c e has been %n

a c h i e v e d a t t h e end of an i t e r a t i o n %ns t e p

a l g o r i t h m Newton ; # use Newton ’ s s o l u t i o n a l g o r i t h m : %n285 u p d a t e s t a n g e n t s t i f f n e s s a t e v e r y %n

i t e r a t i o ns e t N s t e p G r a v i t y 1 ; # a p p l y g r a v i t y i n 10 s t e p ss e t DGravi ty [ e xp r 1 . / $ N s t e p G r a v i t y ] ; # f i r s t load i n c r e m e n t ;i n t e g r a t o r LoadCon t ro l $DGravi ty ; # d e t e r m i n e t h e n e x t t ime s t e p f o r %n

290 an a n a l y s i sa n a l y s i s S t a t i c ; # d e f i n e type of a n a l y s i s s t a t i c o r %n

t r a n s i e n ta n a l y z e $ N s t e p G r a v i t y ; # a p p l y g r a v i t y# −−−−−−−−−−−−− m a i n t a i n c o n s t a n t g r a v i t y l o a d s and r e s e t t ime t o z e r o

295 l o a d C o n s t −t i me 0 . 0p u t s " Model B u i l t "# −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# Uniform EQ ground mot ion# e x e c u t e t h i s f i l e a f t e r you have b u i l t t h e model , and a f t e r a p p l i e d g r a v i t y

300 # Uniform E a r t h q u a k e ground mot ion ( un i fo rm acc . input a t a l l s u p p o r t nodes )# s e t f i l e I d [ open " Output−Typ5−NR−N−Res . d a t " w]# f o r { s e t i 1} { $i <321} { i n c r i 1} { ; # Loop f o r ground mot ion s e ts e t GMdirec t ion 1 ; # ground−motion d i r e c t i o ns e t GMfile $ i ; # ground−motion f i l e n a m e s

305 s e t GMfact 1 . 0 ; # ground−motion s c a l i n g f a c t o r# d i s p l a y deformed shape :s e t ViewScale 5 ; # a m p l i f y d i s p l a y o f deformed shape# DisplayModel2D DeformedShape \$ViewScale ; # d i s p l a y deformed shape , %n

310 t h e s c a l i n g f a c t o r needs %nt o be a d j u s t e d f o r each model

# r e c o r d e r p l o t $ d a t a D i r / DFree . o u t D i s p l \10 700 400 400 −columns 1 2 ; # a window t o p l o t t h e n o d a l %n

187

C Software Codes

d i s p l a c e m e n t s v e r s u s t ime315 # s e t up GM−a n a l y s i s p a r a m e t e r s

s e t D t A n a l y s i s [ exp r 0 .01∗ $sec ] ; # t ime−s t e p Dt f o r l a t e r a l a n a l y s i ss e t TmaxAnalys is [ exp r 20 .48∗ $sec ] ; # maximum d u r a t i o n o f GM a n a l y s i s %n

−− s h o u l d be 50∗ $sec# −−−−−−−−−−− s e t up a n a l y s i s p a r a m e t e r s

320 v a r i a b l e c o n s t r a i n t s T y p e D y n a m i c T r a n s f o r m a t i o n ;c o n s t r a i n t s $ c o n s t r a i n t s T y p e D y n a m i c ;v a r i a b l e numbererTypeDynamic RCMnumberer $numbererTypeDynamicv a r i a b l e systemTypeDynamic BandGenera l ; # t r y UmfPack f o r l a r g e p rob lems

325 sys tem $systemTypeDynamicv a r i a b l e TolDynamic 1 . e−8; # Convergence T e s t : t o l e r a n c ev a r i a b l e maxNumIterDynamic 1 0 ; # Convergence T e s t : maximum number %n

of i t e r a t i o n s t h a t w i l l be p e r f o r − %nmed b e f o r e " f a i l u r e t o c o n v e r g e " i s%n

330 r e t u r n e dv a r i a b l e p r i n t F l a g D y n a m i c 0 ; # Convergence T e s t : f l a g used t o %n

p r i n t i n f o r m a t i o n on c o n v e r g e n c e %n( o p t i o n a l ) # 1 : p r i n t i n f o %non each s t e p ;

335 v a r i a b l e t e s tTypeDynamic E n e r g y I n c r ; # Convergence− t e s t typet e s t $ tes tTypeDynamic $TolDynamic $maxNumIterDynamic $ p r i n t F l a g D y n a m i c ;# f o r improved−c o n v e r g e n c e p r o c e d u r e :v a r i a b l e maxNumIterConvergeDynamic 2000 ;v a r i a b l e p r i n t F l a g C o n v e r g e D y n a m i c 0 ;

340 v a r i a b l e a lgor i thmTypeDynamic Modif iedNewtona l g o r i t h m $algor i thmTypeDynamic ;v a r i a b l e NewmarkGamma 0 . 5 ; # Newmark− i n t e g r a t o r gamma p a r a m e t e rv a r i a b l e NewmarkBeta 0 . 2 5 ; # Newmark− i n t e g r a t o r beta p a r a m e t e rv a r i a b l e i n t e g r a t o r T y p e D y n a m i c Newmark ;

345 i n t e g r a t o r $ i n t e g r a t o r T y p e D y n a m i c $NewmarkGamma $NewmarkBetav a r i a b l e ana lys i sTypeDynamic T r a n s i e n ta n a l y s i s $ana lys i sTypeDynamic

# −−−−−−−−−−−− d e f i n e & a p p l y damping350 # RAYLEIGH damping p a r a m e t e r s , Where t o p u t M/K−prop damping , s w i t c h e s

# ( h t t p : / / o p e n s e e s . b e r k e l e y . edu / OpenSees / manuals / u s e r m a n u a l / 1 0 9 9 . htm )# D=$alphaM∗M+ $ b e t a K c u r r ∗K c u r r e n t +$betaKcomm∗KlastCommit+ $ b e a t K i n i t ∗ $ K i n i t i a l

s e t xDamp 0 . 0 2 ; # damping r a t i o355 s e t MpropSwitch 1 . 0 ;

s e t Kcur rSwi t ch 0 . 0 ;s e t KcommSwitch 1 . 0 ;s e t K i n i t S w i t c h 0 . 0 ;s e t n E i g e n I 1 ; # mode 1

360 s e t nEigenJ 3 ; # mode 3s e t lambdaN [ e i g e n [ exp r $nEigenJ ] ] ; # e i g e n v a l u e a n a l y s i s f o r nEigenJ modess e t l ambdaI [ l i n d e x $lambdaN [ exp r $nEigenI −1 ] ] ; # e i g e n v a l u e mode is e t lambdaJ [ l i n d e x $lambdaN [ exp r $nEigenJ −1 ] ] ; # e i g e n v a l u e mode js e t omegaI [ exp r pow ( $lambdaI , 0 . 5 ) ] ;

365 s e t omegaJ [ exp r pow ( $lambdaJ , 0 . 5 ) ] ;# M−prop . damping ; D = alphaM∗M

188

s e t alphaM [ e xp r $MpropSwitch∗$xDamp∗ (2∗ $omegaI∗$omegaJ ) / ( $omegaI+$omegaJ ) ] ;# c u r r e n t −K; + b e a t K c u r r ∗KCurren ts e t b e t a K c u r r [ exp r $Kcur rSwi t ch ∗2 .∗$xDamp / ( $omegaI+$omegaJ ) ] ;

370 # l a s t −commit ted K; +betaKcomm∗Klas tCommi t ts e t betaKcomm [ ex p r $KcommSwitch ∗2 .∗$xDamp / ( $omegaI+$omegaJ ) ] ;# i n i t i a l −K; + b e a t K i n i t ∗Kin is e t b e t a K i n i t [ exp r $ K i n i t S w i t c h ∗2 .∗$xDamp / ( $omegaI+$omegaJ ) ] ;# RAYLEIGH damping

375 r a y l e i g h $alphaM $ b e t a K c u r r $ b e t a K i n i t $betaKcomm ;

# −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− pe r fo rm Dynamic Ground−Motion A n a l y s i s# t h e f o l l o w i n g commands a r e u n i qu e t o t h e Uniform E a r t h q u a k e e x c i t a t i o ns e t IDloadTag [ e xp r (400+ $ i ) ] ; # f o r u n i f o r m S u p p o r t e x c i t a t i o n

380 # Uniform EXCITATION : a c c e l e r a t i o n inputs e t GMfatt [ e xp r $GMfact ] ; # d a t a i n input f i l e i s i n g U n i f t s# t ime s e r i e s i n f o r m a t i o ns e t A c c e l S e r i e s " S e r i e s −d t 0 . 0 1 − f i l e P a t h $GMdir / $GMfile . g3 − f a c t o r $GMfatt " ;# c r e a t e Uniform e x c i t a t i o n

385 p a t t e r n U n i f o r m E x c i t a t i o n $IDloadTag $GMdi rec t ion −a c c e l $ A c c e l S e r i e s ;s e t N ste ps [ e xp r i n t ( $TmaxAnalys is / $ D t A n a l y s i s ) ] ;# a c t u a l l y pe r fo rm a n a l y s i s ; r e t u r n s ok=0 i f a n a l y s i s was s u c c e s s f u l# s e t ok [ a n a l y z e $Ns teps $ D t A n a l y s i s ] ;

390 s e t d r i f t _ m a x 0s e t d r i f t _ j 0

f o r { s e t j 0} { $ j < 2048} { i n c r j 1} { ; # Loop c a l c u l a t i n g MIDR i n each t s t e ps e t ok [ a n a l y z e 1 0 . 0 1 ]

395 s e t d_1 [ nodeDisp 21 1]s e t d_2 [ nodeDisp 31 1]s e t d_3 [ nodeDisp 41 1]s e t d_4 [ nodeDisp 51 1]s e t d_5 [ nodeDisp 61 1]

400 #s e t d r i _ 1 [ exp r ( $d_1 / 3 . 0 ) ]s e t d r i _ 2 [ exp r ( ( $d_2−$d_1 ) / 3 . 0 ) ]s e t d r i _ 3 [ exp r ( ( $d_3−$d_2 ) / 3 . 0 ) ]s e t d r i _ 4 [ exp r ( ( $d_4−$d_3 ) / 3 . 0 ) ]

405 s e t d r i _ 5 [ ex p r ( ( $d_5−$d_4 ) / 3 . 0 ) ]#s e t d r i f t _ 1 [ exp r abs ( $ d r i _ 1 ) ]s e t d r i f t _ 2 [ exp r abs ( $ d r i _ 2 ) ]s e t d r i f t _ 3 [ exp r abs ( $ d r i _ 3 ) ]

410 s e t d r i f t _ 4 [ e xp r abs ( $ d r i _ 4 ) ]s e t d r i f t _ 5 [ exp r abs ( $ d r i _ 5 ) ]#s e t l i s t [ l i s t $ d r i f t _ 1 $ d r i f t _ 2 $ d r i f t _ 3 $ d r i f t _ 4 $ d r i f t _ 5 ]f o r e a c h e l e m e n t [ l r a n g e $ l i s t 1 end ] {

415 i f { $ e l e m e n t > $ d r i f t _ j } { s e t d r i f t _ j $ e l e m e n t } }

i f { ( $ d r i f t _ j >= $ d r i f t _ m a x ) } { s e t d r i f t _ m a x $ d r i f t _ j }

i f { $ok != 0} { ; # a n a l y s i s was n o t s u c c e s s f u l .

189

C Software Codes

420 # −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# change some a n a l y s i s p a r a m e t e r s t o a c h i e v e c o n v e r g e n c e# p e r f o r m a n c e i s s l o w e r i n s i d e t h i s l oop# Time−c o n t r o l l e d a n a l y s i ss e t ok 0 ;

425 s e t c o n t r o l T i m e [ getTime ] ;whi le { $ c o n t r o l T i m e < $TmaxAnalys is && $ok == 0} {

s e t c o n t r o l T i m e [ getTime ]s e t ok [ a n a l y z e 1 $ D t A n a l y s i s ]i f { $ok != 0} {

430 p u t s " T ry i ng Newton wi th I n i t i a l Tangen t . . "t e s t NormDispIncr $Tol 1000 0a l g o r i t h m Newton − i n i t i a ls e t ok [ a n a l y z e 1 $ D t A n a l y s i s ]t e s t $ tes tTypeDynamic $TolDynamic \

435 $maxNumIterDynamic 0a l g o r i t h m $algor i thmTypeDynamic

}i f { $ok != 0} {

p u t s " T ry i ng Broyden . . "440 a l g o r i t h m Broyden 8

s e t ok [ a n a l y z e 1 $ D t A n a l y s i s ]a l g o r i t h m $algor i thmTypeDynamic

}i f { $ok != 0} {

445 p u t s " T ry i ng NewtonWithLineSearch . . "a l g o r i t h m NewtonLineSearch . 8s e t ok [ a n a l y z e 1 $ D t A n a l y s i s ]a l g o r i t h m $algor i thmTypeDynamic

}450 }

} ; # end i f ok !0} ; # Loop f o r c a l c u l a t i n g t h e MIDR f o r each t ime s t e p

# s e t f i l e I d [ open " Outpu t . t x t " w]455 p u t s $ f i l e I d " $GMfile [ exp r $ d r i f t _ m a x ∗100] [ exp r abs ( $d_5 ) ] $ d r i f t _ 5 "

p u t s " Beben $ i "w i p e A n a l y s i s#} ; # Loop f o r ground mot ion s e t# c l o s e $ f i l e I d

460 p u t s " Ground Motion Done . End Time : [ ge tTime ] "

# B u i l d R C r e c t S e c t i o n . t c l −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# based on t h e examples i n# h t t p : / / o p e n s e e s . b e r k e l e y . edu / OpenSees / manuals / ExamplesManual /HTML/p roc B u i l d R C r e c t S e c t i o n { i d HSec BSec coverH coverB core ID cover ID s t e e l I D \

5 numBarsTop barAreaTop numBarsBot ba rAreaBo t numBars In tTo t b a r A r e a I n t nfCoreY \nfCoreZ nfCoverY nfCoverZ } {

################################################# B u i l d R C r e c t S e c t i o n $ i d $HSec $BSec $coverH $coverB $core ID %n$cover ID $ s t e e l I D $numBarsTop $barAreaTop $numBarsBot $ba rAreaBot %n

10 $numBars In tTo t $ b a r A r e a I n t $nfCoreY $nfCoreZ $nfCoverY $nfCoverZ################################################

190

# B u i l d f i b e r r e c t a n g u l a r RC s e c t i o n , 1 s t e e l l a y e r top , 1 bot , %n1 sk in , c o n f i n e d c o r e

# De f i n e a p r o c e d u r e which g e n e r a t e s a r e c t a n g u l a r RC s e c t i o n15 # wi th one l a y e r o f s t e e l a t t o p & bottom , s k i n r e i n f o r c e m e n t and a

# c o n f i n e d c o r e .# by : S i l v i a Mazzoni , 2006# a d a p t e d from Michae l H. S c o t t , 2003# Formal a rgumen t s

20 # i d − t a g f o r t h e s e c t i o n t h a t i s g e n e r a t e d by t h i s p r o c e d u r e# HSec − d e p t h o f s e c t i o n , a l o n g l o c a l −y a x i s# BSec − wid th o f s e c t i o n , a l o n g l o c a l −z a x i s# cH − d i s t a n c e from s e c t i o n boundary t o n e u t r a l a x i s of r e i n f .# cB − d i s t a n c e from s e c t i o n boundary t o s i d e o f r e i n f o r c e m e n t

25 # core ID − m a t e r i a l t a g f o r t h e c o r e patch# cover ID − m a t e r i a l t a g f o r t h e c o v e r p a t c h e s# s t e e l I D − m a t e r i a l t a g f o r t h e r e i n f o r c i n g s t e e l# numBarsTop − number o f r e i n f o r c i n g b a r s i n t h e t o p l a y e r# numBarsBot − number o f r e i n f o r c i n g b a r s i n t h e bot tom l a y e r

30 # numBars In tTo t − TOTAL number o f r e i n f o r c i n g b a r s on t h e %ni n t e r m e d i a t e l a y e r s , symmet r i c a b o u t z a x i s and 2 b a r s %np e r l a y e r −− needs t o be an even i n t e g e r

# barAreaTop − cross−s e c t i o n a l a r e a o f each r e i n f o r c i n g bar %ni n t o p l a y e r

35 # ba rAreaBo t − cross−s e c t i o n a l a r e a o f each r e i n f o r c i n g bar %ni n bot tom l a y e r

# b a r A r e a I n t − cross−s e c t i o n a l a r e a o f each r e i n f o r c i n g bar %ni n i n t e r m e d i a t e l a y e r

# nfCoreY − number o f f i b e r s i n t h e c o r e patch i n t h e y−d i r40 # nfCoreZ − number o f f i b e r s i n t h e c o r e patch i n t h e z−d i r

# nfCoverY − number o f f i b e r s i n t h e c o v e r p a t c h e s wi th %nl ong s i d e s i n t h e y d i r e c t i o n

# nfCoverZ − number o f f i b e r s i n t h e c o v e r p a t c h e s wi th %nl ong s i d e s i n t h e z d i r e c t i o n

45 ## y# ^# |# −−−−−−−−−−−−−−−−−−−−−−−−−

50 # | o o o | | −− coverH# | | |# | o o | |# z <−−− | + | HSec# | o o | |

55 # | | |# | o o o o o o | | −− coverH# −−−−−−−−−−−−−−−−−−−−−−−−−# |−−−−−−−Bsec−−−−−−|# |−−−| coverB |−−−|

60 ## y# ^# |# −−−−−−−−−−−−−−−−−−−−−

191

C Software Codes

65 # | \ c o v e r / |# | \−−−−−−Top−−−−−−/| |# | c | | c |# | o | | o |# z <−−−−−|v | c o r e | v | HSec

70 # | e | | e |# | r | | r |# | /−−−−−−−Bot−−−−−−\ |# | / c o v e r \ |# −−−−−−−−−−−−−−−−−−−−−

75 # Bsec### Notes# The c o r e c o n c r e t e ends a t t h e NA of t h e r e i n f o r c e m e n t

80 # The c e n t e r o f t h e s e c t i o n i s a t ( 0 , 0 ) i n t h e l o c a l a x i s sys tem#s e t coverY [ exp r $HSec / 2 . 0 ] ; # The d i s t a n c e from t h e %n

s e c t i o n z−a x i s t o t h e edge %nof t h e c o v e r c o n c r e t e

85 s e t coverZ [ exp r $BSec / 2 . 0 ] ; # The d i s t a n c e from t h e %ns e c t i o n y−a x i s t o t h e edge %nof t h e c o v e r c o n c r e t e

s e t coreY [ exp r $coverY−$coverH ] ; # The d i s t a n c e from t h e %ns e c t i o n z−a x i s t o t h e edge %n

90 of t h e c o r e c o n c r e t e %n−− edge of t h e c o r e c o n c r .%n/ i n n e r edge of c o v e r c o n c r .

s e t coreZ [ exp r $coverZ−$coverB ] ; # The d i s t a n c e from t h e %ns e c t i o n y−a x i s t o t h e edge %n

95 of t h e c o r e c o n c r e t e −− %nedge of t h e c o r e c o n c r e t e / %ni n n e r edge o f c o v e r c o n c r e t e

s e t numBars In t [ exp r $numBars In tTo t / 2 ] ; # Nr . o f i n t e r m . b a r s p e r s i d e

100 # De f i n e t h e f i b e r s e c t i o ns e c t i o n f i b e r S e c $ i d {# De f i n e t h e c o r e patchpatch quadr $core ID $nfCoreZ $nfCoreY −$coreY $coreZ −$coreY \

−$coreZ $coreY −$coreZ $coreY $coreZ105

# De f i n e t h e f o u r c o v e r p a t c h e spatch quadr $cover ID 2 $nfCoverY −$coverY $coverZ −$coreY \$coreZ $coreY $coreZ $coverY $coverZpatch quadr $cover ID 2 $nfCoverY −$coreY −$coreZ −$coverY \

110 −$coverZ $coverY −$coverZ $coreY −$coreZpatch quadr $cover ID $nfCoverZ 2 −$coverY $coverZ −$coverY \−$coverZ −$coreY −$coreZ −$coreY $coreZpatch quadr $cover ID $nfCoverZ 2 $coreY $coreZ $coreY −$coreZ \$coverY −$coverZ $coverY $coverZ

115

# d e f i n e r e i n f o r c i n g l a y e r sl a y e r s t r a i g h t $ s t e e l I D $numBars In t $ b a r A r e a I n t −$coreY $coreZ \

192

$coreY $coreZ ; # i n t e r m e d i a t e s k i n r e i n f . +zl a y e r s t r a i g h t $ s t e e l I D $numBars In t $ b a r A r e a I n t −$coreY −$coreZ \

120 $coreY −$coreZ ; # i n t e r m e d i a t e s k i n r e i n f . −zl a y e r s t r a i g h t $ s t e e l I D $numBarsTop $barAreaTop $coreY $coreZ \$coreY −$coreZ ; # t o p l a y e r r e i n f o c e m e n tl a y e r s t r a i g h t $ s t e e l I D $numBarsBot $ba rAreaBot −$coreY $coreZ \−$coreY −$coreZ ; # bot tom l a y e r r e i n f o r c e m e n t

125

} ; # end of f i b e r s e c t i o n d e f i n i t i o n} ; # end of p r o c e d u r e

193

C Software Codes

C.4 Risk analysis tools

Matlab files for generating the BPN’s for Example 1

f u n c t i o n BPN_PSHA_Adapazari_RM_Poisson ( T , X1 , Y1 , X2 , Y2 , a , b )



10 %% This script reads a Bayesian probabilistic network for probabilistic% seismic hazard analysis, calculates the probability distribution of the% nodes in the BPN and compiles the BPN with the inference engine of HUGIN.%

15 %%% For each seismic source, Z a set of BPN’s are calculated:f o r Z=1:1

% For controling HUGIN from MATLAB the ActiveX server is loaded and20 % then the available functions in this library are used to alter

% objects created from this library. The HUGIN ActiveX Server is% loaded with the followingcommand and create a HUGIN API object%named bpn:bpn= a c t x s e r v e r ( ’HAPI . G l o b a l s ’ ) ;

25

% An object ’domain’ is created which holds the network:domain= in vo ke ( bpn , ’ LoadDomainFromNet ’ , ’C : \ D i s s e r t a t i o n \ . . .Chap te r4 −1\BPN\ EX1_Typ5_O_Res_single_eps . n e t ’ , 0 , 0 ) ;

30 % The nodes to be manipulated are defined:ndEQ_M= in vo ke ( domain , ’ GetNodeByName ’ , ’ EQ_Magnitude ’ ) ;ndEQ_R= i nv ok e ( domain , ’ GetNodeByName ’ , ’ EQ_Dis tance ’ ) ;ndEps_PGA= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsilon_PGA ’ ) ;ndEps_SD= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsi lon_SD ’ ) ;

35 ndPGA= in vok e ( domain , ’ GetNodeByName ’ , ’PGA ’ ) ;ndSD= i nv oke ( domain , ’ GetNodeByName ’ , ’SD ’ ) ;ndDamage= in vo ke ( domain , ’ GetNodeByName ’ , ’ Damage ’ ) ;

%The number of discrete states of the nodes ’Magnitude’, ’Distance’,40 %’Eps_PGA’, ’Eps_SD’, ’PGA’ and ’SD’ is set:

nM=6;nR =5;nEps_PGA =10;nEps_SD =10;

45 nPGA=7;nSD=7;nDamage =3;s e t (ndEQ_M , ’ NumberOfSta tes ’ ,nM ) ;s e t ( ndEQ_R , ’ NumberOfSta tes ’ , nR ) ;

194

50 s e t ( ndEps_SD , ’ NumberOfSta tes ’ , nEps_SD ) ;s e t ( ndEps_PGA , ’ NumberOfSta tes ’ , nEps_PGA ) ;s e t ( ndSD , ’ NumberOfSta tes ’ , nSD ) ;s e t ( ndPGA , ’ NumberOfSta tes ’ ,nPGA ) ;s e t ( ndDamage , ’ NumberOfSta tes ’ , nDamage ) ;

55

%The probability distribution of node ’Eps_PGA’ is calculated:[ Eps_PGA , dPGA]=EPS_PGA ( nEps_PGA ) ;

%The probability distribution of node ’Eps_SD’ is calculated:60 [ Eps_SD , dSD]=EPS_SD ( nEps_SD , nEps_PGA , T ) ;

%The probability distribution of node ’Distance’ is calculated:[R , P_R , R l i m i t s ]= Line_EQ_R ( 5 0 0 , nR , X1 , Y1 , X2 , Y2 ) ;

65 %The probability distribution of node ’Magnitude’ is calculated:[ Nu_Mmin ,M_EQ, P_M, M l i m i t s ]= EQ_M_NonPoisson ( 5 , a , b , nM, Q, Z ) ;

M=M_EQ;

%The discrete probabilities are set for node ’Magnitude’ in the70 %BPN:

f o r i = 1 : (nM)s e t (ndEQ_M . Table , ’ Data ’ , ( i −1) ,P_M( i ) ) ;s e t (ndEQ_M , ’ S t a t e L a b e l ’ , ( i −1) , [ ’M= ’ num2str (M( i ) ) ] ) ;

end75

%The discrete probabilities are set for node ’Distance’ in the BPN:f o r i = 1 : ( nR )


80 end

%The discrete probabilities are set for node ’Eps_PGA’ in the BPN:f o r i = 1 : ( nEps_PGA )

s e t ( ndEps_PGA . Table , ’ Data ’ , ( i −1) ,Eps_PGA ( i ) ) ;85 s e t ( ndEps_PGA , ’ S t a t e L a b e l ’ , ( i −1) , . . .

[ ’Eps_PGA= ’ num2str (dPGA( i ) ) ] ) ;end

%The discrete probabilities are set for node ’Eps_SD’ in the BPN:90 f o r i = 1 : ( nEps_SD∗nEps_PGA )


%The conditional probability table of the node ’PGA’ is initialized95 %with zeros:

f o r i = 1 : (nM∗nR∗nEps_PGA∗nPGA)s e t ( ndPGA . Table , ’ Data ’ , ( i −1 ) , 0 ) ;

end

100 %The conditional probability table of the node ’SD’ is initialized%with zeros:f o r i = 1 : (nM∗nR∗nSD∗nEps_SD )

195

C Software Codes

s e t ( ndSD . Table , ’ Data ’ , ( i −1 ) , 0 ) ;end

105

%for all combinations of the states in the node ’Magnitude’ and%’Distance’ the peak ground accelerations and spectral%displacements are calculated with the Boore Joyner and Fumal%attenuation model

110 [PGABOORE]=PGA( nEps_PGA ,M, R ) ;[SDBOORE]=SD( nEps_SD , nEps_PGA ,M, R ) ;

%The limits for the discretisation of the nodes ’SD’ and ’PGA’ are%set:

115 CoeffPGA =[0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 max (PGABOORE ) ] ;CoeffSD =[0 0 .005 0 . 0 2 0 . 0 5 0 . 1 0 . 3 0 . 5 max (SDBOORE ) ] ;

f o r i = 1 : ( nSD )S ( i ) = ( CoeffSD ( i )+ CoeffSD ( i + 1 ) ) / 2 ;

120 SDmm( i )=S ( i )∗1 0 0 0 ;end

%The labels of the states are set for node ’Eps_SD’, ’PGA’ and ’SD’%in the BPN:

125 f o r i = 1 : ( nEps_SD )s e t ( ndEps_SD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’ Eps_SD= ’ num2str ( dSD ( i ) ) ] ) ;

endf o r i = 1 : (nPGA)

s e t ( ndPGA , ’ S t a t e L a b e l ’ , ( i −1) , [ ’PGA= ’ num2str ( ( CoeffPGA ( i )+ . . .130 CoeffPGA ( i + 1 ) ) / 2 ) ] ) ;


s e t ( ndSD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’SD= ’ num2str ( ( CoeffSD ( i )+ . . .CoeffSD ( i + 1 ) ) / 2 ) ] ) ;

135 end

%The discrete probabilities are set for node ’PGA’ in the BPN:N=0;f o r i =1 : l e n g t h (PGABOORE)

140 f o r j =1 :nPGAi f PGABOORE( i ) <= CoeffPGA ( j +1) & PGABOORE( i ) > CoeffPGA ( j )

s e t ( ndPGA . Table , ’ Data ’ , ( j +N−1 ) , 1 ) ;N=N+nPGA ;

end145 end

end

%The discrete probabilities are set for node ’SD’ in the BPN:K=0;

150 f o r i =1 : l e n g t h (SDBOORE)f o r j =1 :nSD

i f SDBOORE( i ) <= CoeffSD ( j +1) & SDBOORE( i ) > CoeffSD ( j )s e t ( ndSD . Table , ’ Data ’ , ( j +K−1 ) , 1 ) ;K=K+nSD ;

155 end

196

endend

160 [ P_Damage ]= Fragi l i ty_Typ5_O_Res_RM_1 (SDmm) ;f o r i = 1 : ( nSD∗5∗2∗3)

s e t ( ndDamage . Table , ’ Data ’ , ( i −1) , P_Damage ( i ) ) ;end

165 %The BPN’s are set for each SourceFi l ename =[ ’C : \ D i s s e r t a t i o n \ Chapte r4 −1\BPN\ EX1_Typ5_O_Res_S ’ . . .

num2str ( Z ) ’ _ P o i s s o n _ 1 ’ ’ . n e t ’ ] ;i n vo ke ( domain , ’ SaveAsNet ’ , F i l ename ) ;

end170 end

f u n c t i o n BPN_PSHA_Adapazari_RM ( T , X1 , Y1 , X2 , Y2 , a , b , SS )




15 %%% For each seismic source, Z and each year, Q a set of BPN’s are% calculated:f o r Z=SS : SS

20 f o r Q=1:50% For controling HUGIN from MATLAB the ActiveX server is loaded and% then the available functions in this library are used to alter% objects created from this library. The HUGIN ActiveX Server is% loaded with the followingcommand and create a HUGIN API object

25 % named bpn:bpn= a c t x s e r v e r ( ’HAPI . G l o b a l s ’ ) ;

% An object ’domain’ is created which holds the network:domain= in vo ke ( bpn , ’ LoadDomainFromNet ’ , ’C : \ D i s s e r t a t i o n \ . . .

30 Chapte r4 −1\BPN\ EX1_Typ5_O_Res_single_eps . n e t ’ , 0 , 0 ) ;

% The nodes to be manipulated are defined:ndEQ_M= in vo ke ( domain , ’ GetNodeByName ’ , ’ EQ_Magnitude ’ ) ;ndEQ_R= i nv ok e ( domain , ’ GetNodeByName ’ , ’ EQ_Dis tance ’ ) ;

35 ndEps_PGA= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsilon_PGA ’ ) ;ndEps_SD= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsi lon_SD ’ ) ;ndPGA= in vok e ( domain , ’ GetNodeByName ’ , ’PGA ’ ) ;

197

C Software Codes

ndSD= i nv oke ( domain , ’ GetNodeByName ’ , ’SD ’ ) ;ndDamage= in vo ke ( domain , ’ GetNodeByName ’ , ’ Damage ’ ) ;

40


45 nEps_PGA =10;nEps_SD =10;nPGA=7;nSD=7;nDamage =3;

50 s e t (ndEQ_M , ’ NumberOfSta tes ’ ,nM ) ;s e t ( ndEQ_R , ’ NumberOfSta tes ’ , nR ) ;s e t ( ndEps_SD , ’ NumberOfSta tes ’ , nEps_SD ) ;s e t ( ndEps_PGA , ’ NumberOfSta tes ’ , nEps_PGA ) ;s e t ( ndSD , ’ NumberOfSta tes ’ , nSD ) ;

55 s e t ( ndPGA , ’ NumberOfSta tes ’ ,nPGA ) ;s e t ( ndDamage , ’ NumberOfSta tes ’ , nDamage ) ;


60

%The probability distribution of node ’Eps_SD’ is calculated:[ Eps_SD , dSD]=EPS_SD ( nEps_SD , nEps_PGA , T ) ;

%The probability distribution of node ’Distance’ is calculated:65 [R , P_R , R l i m i t s ]= Line_EQ_R ( 5 0 0 , nR , X1 , Y1 , X2 , Y2 ) ;

%The probability distribution of node ’Magnitude’ is calculated:[ Nu_Mmin ,M_EQ, P_M, M l i m i t s ]= EQ_M_NonPoisson ( 5 , a , b , nM, Q, Z ) ;

M=M_EQ;70


s e t (ndEQ_M . Table , ’ Data ’ , ( i −1) ,P_M( i ) ) ;75 s e t (ndEQ_M , ’ S t a t e L a b e l ’ , ( i −1) , [ ’M= ’ num2str (M( i ) ) ] ) ;

end


80 s e t ( ndEQ_R . Table , ’ Data ’ , ( i −1) ,P_R ( i ) ) ;s e t ( ndEQ_R , ’ S t a t e L a b e l ’ , ( i −1) , [ ’R= ’ num2str (R( i ) ) ] ) ;

end

%The discrete probabilities are set for node ’Eps_PGA’ in the BPN:85 f o r i = 1 : ( nEps_PGA )

s e t ( ndEps_PGA . Table , ’ Data ’ , ( i −1) ,Eps_PGA ( i ) ) ;s e t ( ndEps_PGA , ’ S t a t e L a b e l ’ , ( i −1) , [ ’Eps_PGA= ’ num2str (dPGA( i ) ) ] ) ;

end

90 %The discrete probabilities are set for node ’Eps_SD’ in the BPN:

198

f o r i = 1 : ( nEps_SD∗nEps_PGA )s e t ( ndEps_SD . Table , ’ Data ’ , ( i −1) , Eps_SD ( i ) ) ;

end

95 %The conditional probability table of the node ’PGA’ is initialized%with zeros:f o r i = 1 : (nM∗nR∗nEps_PGA∗nPGA)

s e t ( ndPGA . Table , ’ Data ’ , ( i −1 ) , 0 ) ;end

100


s e t ( ndSD . Table , ’ Data ’ , ( i −1 ) , 0 ) ;105 end

%for all combinations of the states in the node ’Magnitude’ and%’Distance’ the peak ground accelerations and spectral%displacements are calculated with the Boore Joyner and Fumal

110 %attenuation model[PGABOORE]=PGA( nEps_PGA ,M, R ) ;[SDBOORE]=SD( nEps_SD , nEps_PGA ,M, R ) ;

%The limits for the discretisation of the nodes ’SD’ and ’PGA’ are115 %set:

CoeffPGA =[0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 max (PGABOORE ) ] ;CoeffSD =[0 0 .005 0 . 0 2 0 . 0 5 0 . 1 0 . 3 0 . 5 max (SDBOORE ) ] ;

f o r i = 1 : ( nSD )120 S ( i ) = ( CoeffSD ( i )+ CoeffSD ( i + 1 ) ) / 2 ;

SDmm( i )=S ( i )∗1 0 0 0 ;end

%The labels of the states are set for node ’Eps_SD’, ’PGA’ and ’SD’125 %in the BPN:

f o r i = 1 : ( nEps_SD )s e t ( ndEps_SD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’ Eps_SD= ’ num2str ( dSD ( i ) ) ] ) ;


130 s e t ( ndPGA , ’ S t a t e L a b e l ’ , ( i −1) , [ ’PGA= ’ num2str ( ( CoeffPGA ( i )+ . . .CoeffPGA ( i + 1 ) ) / 2 ) ] ) ;


s e t ( ndSD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’SD= ’ num2str ( ( CoeffSD ( i )+ . . .135 CoeffSD ( i + 1 ) ) / 2 ) ] ) ;

end

%The discrete probabilities are set for node ’PGA’ in the BPN:N=0;

140 f o r i =1 : l e n g t h (PGABOORE)f o r j =1 :nPGA

i f PGABOORE( i ) <= CoeffPGA ( j +1) & PGABOORE( i ) > CoeffPGA ( j )s e t ( ndPGA . Table , ’ Data ’ , ( j +N−1 ) , 1 ) ;

199

C Software Codes

N=N+nPGA ;145 end

endend

%The discrete probabilities are set for node ’SD’ in the BPN:150 K=0;

f o r i =1 : l e n g t h (SDBOORE)f o r j =1 :nSD

i f SDBOORE( i ) <= CoeffSD ( j +1) & SDBOORE( i ) > CoeffSD ( j )s e t ( ndSD . Table , ’ Data ’ , ( j +K−1 ) , 1 ) ;

155 K=K+nSD ;end

endend

160 %The discrete probabilities are calculated given the spectral%displacement values are set for node ’Damage’[ P_Damage ]= Fragi l i ty_Typ5_O_Res_RM_1 (SDmm) ;f o r i = 1 : ( nSD∗5∗2∗3)

s e t ( ndDamage . Table , ’ Data ’ , ( i −1) , P_Damage ( i ) ) ;165 end

%The BPN’s are set for each Source and YearFi l ename =[ ’C : \ D i s s e r t a t i o n \ Chapte r4 −1\BPN\ EX1_Typ5_O_Res_S ’ . . .

num2str ( Z ) ’_Y ’ num2str (Q) ’ . n e t ’ ] ;170 i n vo ke ( domain , ’ SaveAsNet ’ , F i l ename ) ;

endend

f u n c t i o n [ P_Damage ]= Fragi l i ty_Typ5_O_Res_RM_1 (SD)

%For the three damage states the parameters of the lognormal%distribution are given in Table X (unretrofitted case).

5 Lambda_Yellow = [ 3 . 6 9 0 3 .7 24 3 .758 3 .792 3 . 8 2 9 ] ;Ze ta_Yel low = [ 0 . 3 4 1 0 .366 0 .390 0 .414 0 . 4 3 9 ] ;Lambda_Red = [ 4 . 1 0 6 4 .160 4 .215 4 .270 4 .3 24 ] ;Zeta_Red = [ 0 . 2 6 6 0 .306 0 .346 0 .386 0 . 4 2 6 ] ;

10 %For each state of the node ’SD’ the probabilities of being in one of the%three damage states are calculated.N=1;f o r i =1 :5

f o r k =1: l e n g t h (SD)15 p1 (N)=1− l o g n c d f (SD( k ) , Lambda_Yellow ( i ) , Ze ta_Yel low ( i ) ) ;

p2 (N)= l o g n c d f (SD( k ) , Lambda_Yellow ( i ) , Ze ta_Yel low ( i ))− . . .l o g n c d f (SD( k ) , Lambda_Red ( i ) , Zeta_Red ( i ) ) ;

p3 (N)= l o g n c d f (SD( k ) , Lambda_Red ( i ) , Zeta_Red ( i ) ) ;N=N+1;

20 endend

%The probabilities of being in one of the three damage states form the

200

%conditional probability table of the node ’Damage’25 DamageNode1 =[ p1 ; p2 ; p3 ] ;

P_Damage1=DamageNode1 . ∗ ( DamageNode1 > 0 ) ;P_Damage1=P_Damage1 ( : ) ;

%For the three damage states the parameters of the lognormal30 %distribution are given in Table X (retrofitted case).

Lambda_Yellow = [ 3 . 7 7 7 3 .811 3 .845 3 .879 3 . 9 1 3 ] ;Ze ta_Yel low = [ 0 . 2 5 0 0 .274 0 .299 0 .323 0 . 3 4 7 ] ;Lambda_Red = [ 4 . 2 1 9 4 .273 4 .328 4 .383 4 . 4 3 7 ] ;Zeta_Red = [ 0 . 1 8 5 0 .225 0 .265 0 .304 0 . 3 4 4 ] ;

35

%For each state of the node ’SD’ the probabilities of being in one of the%three damage states are calculated.N=1;f o r i =1 :5

40 f o r k =1: l e n g t h (SD)p1 (N)=1− l o g n c d f (SD( k ) , Lambda_Yellow ( i ) , Ze ta_Yel low ( i ) ) ;p2 (N)= l o g n c d f (SD( k ) , Lambda_Yellow ( i ) , Ze ta_Yel low ( i ))− . . .

l o g n c d f (SD( k ) , Lambda_Red ( i ) , Zeta_Red ( i ) ) ;p3 (N)= l o g n c d f (SD( k ) , Lambda_Red ( i ) , Zeta_Red ( i ) ) ;

45 N=N+1;end

end

%The probabilities of being in one of the three damage states form the50 %conditional probability table of the node ’Damage’

DamageNode2 =[ p1 ; p2 ; p3 ] ;P_Damage2=DamageNode2 . ∗ ( DamageNode2 > 0 ) ;P_Damage2=P_Damage2 ( : ) ;P_Damage =[ P_Damage1 ; P_Damage2 ] ;

Visual Basic files for evaluating the BPN’s in Example 1 within the GISenvironment

’ OpenShape . bas’ C r e a t e s a shape f i l e − d e v e l o p e d by Adr ienne Gret−Regamey

A t t r i b u t e VB_Name = " OpenShape "5 Opt ion E x p l i c i t

P u b l i c F u n c t i o n O p e n S h a p e f i l e ( s p a t h As S t r i n g , sFi leName As S t r i n g ) _As I F e a t u r e C l a s s

Dim MxDoc As IMxDocument10 S e t MxDoc = ThisDocument

Dim pMap As IMapS e t pMap = MxDoc . FocusMap

’ Get a c c e s s t o F e a t u r e C l a s s15 Dim pWSF As I W o r k s p a c e F a c t o r y

S e t pWSF = New S h a p e f i l e W o r k s p a c e F a c t o r y

201

C Software Codes

Dim pWorkspace As IWorkspaceS e t pWorkspace = pWSF . OpenFromFile ( s p a t h , 0 )

20 Dim pfWorkspace As I F e a t u r e W o r k s p a c eS e t pfWorkspace = pWorkspace

S e t O p e n S h a p e f i l e = pfWorkspace . O p e n F e a t u r e C l a s s ( sFi leName )

25 Dim pFLayer As I F e a t u r e L a y e rS e t pFLayer = New F e a t u r e L a y e r

S e t pFLayer . F e a t u r e C l a s s = O p e n S h a p e f i l e

30 Dim p D a t a s e t As I D a t a s e tS e t p D a t a s e t = O p e n S h a p e f i l e

pFLayer . Name = sFi leName

35 Dim pMxDoc As IMxDocumentS e t pMxDoc = ThisDocumentpMxDoc . AddLayer pFLayerpMxDoc . Act iveView . P a r t i a l R e f r e s h es r iViewGeography , pFLayer , Noth ing

40

End F u n c t i o n

’ Module1 . bas

A t t r i b u t e VB_Name = " Module1 "5 Opt ion E x p l i c i t

P r i v a t e Sub BN_Hugin ( )’ _______________________________________’ A c o l l e c t i o n t o ho ld t h e found p a r s e E r r o r sDim p a r s e E r r o r s As C o l l e c t i o n

10 ’ _______________________________________’ Get MapDim pMxDoc As IMxDocumentDim pMap As IMapS e t pMxDoc = ThisDocument

15 S e t pMap = pMxDoc . FocusMap’ _______________________________________’ Get t h e s h a p e f i l e wi th a l l t h e d a t aDim p F e a t u r e C l a s s As I F e a t u r e C l a s sS e t p F e a t u r e C l a s s = O p e n S h a p e f i l e ( " Z: < Fo lde r >" , " Typ5_O_Res " )

20 Dim pLayer As I F e a t u r e L a y e rS e t pLayer = pMap . Layer ( 0 )Dim p F e a t u r e C l a s s S e l As I F e a t u r e C l a s sS e t p F e a t u r e C l a s s S e l = pLayer . F e a t u r e C l a s s’ _______________________________________

25 ’ c r e a t e c u r s o r t o loop t h r o u g h " b u i l d i n g _ t y p e "Dim IndexFID As I n t e g e rIndexFID = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " FID " )

202

Dim IndexOccupancy As S t r i n gIndexOccupancy = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " Occupancy " )

30 Dim I n d e x S t o r y A r e a As DoubleI n d e x S t o r y A r e a = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " F l o o r A r e a " )Dim IndexEU1 As DoubleIndexEU1 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " EU1 " )Dim IndexEU2 As Double

35 IndexEU2 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " EU2 " )Dim IndexOpt As S t r i n gIndexOpt = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " OptAc t ion " )Dim IndexLiq21 As DoubleIndexLiq21 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " Avg_Liq21 " )

40

’ _______________________________________’ D ef in e c u r s o r f o r s e l e c t e d f e a t u r e sDim pCurso r As I F e a t u r e C u r s o rS e t pCurso r = p F e a t u r e C l a s s S e l . Update ( Nothing , F a l s e )

45 ’ _______________________________________’ C ur s o r t h r o u g h t h e s e l e c t e d rowsDim pRowSel As I F e a t u r eS e t pRowSel = pCurso r . N e x t F e a t u r e

50 Dim pOID As DoubleDim pQuery As I Q u e r y F i l t e r

’ _______________________________________’LOOP THROUGH EACH BUILDING OF THE LAYER

55 Dim M1 As DoubleDim M2 As DoubleDim MTotal1 As DoubleDim MTotal2 As DoubleDim Nu(1 To 9) As Double

60

’ The r a t e o f e x c e e d i n g t h e minimum magni tude o f 5’ f o r each of t h e 9 s e i s m i c s o u r c e s a r eNu ( 1 ) = 0 .0000239883Nu ( 2 ) = 0 .000047863

65 Nu ( 3 ) = 0 .0000891251Nu ( 4 ) = 0 .0000776247Nu ( 5 ) = 0 .0000512861Nu ( 6 ) = 0 .000020893Nu ( 7 ) = 0 .0000354813

70 Nu ( 8 ) = 0 .003019952Nu ( 9 ) = 0 .004073803Dim Source As I n t e g e r

75 Do While Not pRowSel I s Noth ingM1 = 0M2 = 0

For Source = 2 To 280 ’ _______________________________________

203

C Software Codes

’ Im p o r t BPN from HuginDim d As HAPI . DomainDim BN As S t r i n g

85 Dim Netze (1 To 3)Netze ( 1 ) = " EX1_Typ5_O_Res_S "Netze ( 2 ) = CStr ( Source )Netze ( 3 ) = " _ P o i s s o n . n e t "

90 BN = J o i n ( Netze , " " )S e t d = HAPI . LoadDomainFromNet (BN, p a r s e E r r o r s , 10)

’ g e t t h e node wi th l a b e l " . . . " and name " . . . " from t h e domainDim NodeLiq As HAPI . Node

95 S e t NodeLiq = d . GetNodeByName ( " L i q u e f a c t i o n " )Dim NodeLiqTable As HAPI . Tab leS e t NodeLiqTable = NodeLiq . Tab le

Dim NodeCost As HAPI . Node100 S e t NodeCost = d . GetNodeByName ( " Cos t " )

Dim NodeCostTable As HAPI . Tab leS e t NodeCostTable = NodeCost . Tab le

Dim d e c i s i o n R e t r o f i t As Node105 S e t d e c i s i o n R e t r o f i t = d . GetNodeByName ( " R e t r o f i t " )

’ I n i t i a l i z eDim l a u f As I n t e g e rDim M As I n t e g e r

110 M = IndexLiq21For l a u f = 0 To 83 S tep 2NodeLiqTable . Data ( l a u f ) = 1 − pRowSel . Value (M)NodeLiqTable . Data ( l a u f + 1) = pRowSel . Value (M)M = M + 1

115 Next l a u f

Dim F a t a l i t y As DoubleDim R e b u i l d i n g As DoubleDim R e p a i r As Double

120 Dim R e t r o f i t As Double

’ 70%=Occupancy at time of EQ (M2)’80% Occupan t s t r a p p e d (M3)’ 20% Occupants died imediately (M4)

125 ’80% Occupan t s dead a f t e r 10 days (M5)’LSCS=250000 USD ( GDPpc=10000 USD)F a t a l i t y = pRowSel . Value ( IndexOccupancy )∗ _( 0 . 7∗0 . 8∗0 . 2 + 0 . 7∗0 . 8∗ ( 1 −0 . 2 )∗0 . 8 )∗2 5 0 0 0 0

130 ’ Un i t r e b u i l d i n g c o s t = 300USD/ m2 ,’ I m p o r t a n c e f a c t o r f o r h o s p i t a l =10 ,’ N o n s t r u c t u r a l e l e m e n t s 50% of b u i l d i n g va lue ,R e b u i l d i n g = 5 ∗ pRowSel . Value ( I n d e x S t o r y A r e a ) ∗ 300 ∗ 5

204

135 ’ Un i t r e b u i l d i n g c o s t = 300USD/ m2 ,’20%Cost o f r e b u i l d i n g f o r r e p a i r ,’ N o n s t r u c t u r a l e l e m e n t s 50% of building value,’ I m p o r t a n c e f a c t o r f o r h o s p i t a l s =10

140 R e p a i r = 5 ∗ pRowSel . Value ( I n d e x S t o r y A r e a ) ∗ 300 ∗ 0 . 2 ∗ 5

’ Un i t r e t r o f i t c o s t =250USD/ column , a v e r a g e span l e n g t h i s 4mR e t r o f i t = ( ( ( Sqr (5 ∗ pRowSel . Value ( I n d e x S t o r y A r e a ) ) ) / 5 ) ^ 2 ) ∗ 50

145 ’ D i s c o u t r a t e =2%NodeCostTable . Data ( 0 ) = (0 + 0 + 0 + 0) / 0 . 0 2NodeCostTable . Data ( 1 ) = (0 + R e p a i r + 0 + 0) / 0 . 0 2NodeCostTable . Data ( 2 ) = (0 + 0 + R e b u i l d i n g + F a t a l i t y ) / 0 . 0 2NodeCostTable . Data ( 3 ) = ( R e t r o f i t + 0 + 0 + 0) / 0 . 0 2

150 NodeCostTable . Data ( 4 ) = ( R e t r o f i t + R e p a i r + 0 + 0) / 0 . 0 2NodeCostTable . Data ( 5 ) = ( R e t r o f i t + 0 + R e b u i l d i n g + F a t a l i t y ) / 0 . 0 2

d . Compile

155 M1 = M1 + d e c i s i o n R e t r o f i t . E x p e c t e d U t i l i t y ( 0 ) ∗ Nu ( Source )M2 = M2 + d e c i s i o n R e t r o f i t . E x p e c t e d U t i l i t y ( 1 ) ∗ Nu ( Source )Dim FID , Netz (1 To 3 ) , BPNFID = pRowSel . Value ( IndexFID )Netz ( 1 ) = CSt r ( FID )

160 Netz ( 2 ) = "MB_"Netz ( 3 ) = BN ’ " _EX1_Typ5_O_Res_S1_Poisson . n e t "BPN = J o i n ( Netz , " " )d . SaveAsNet (BPN)Next Source

165

MTotal1 = M1MTotal2 = M2pRowSel . Value ( IndexEU1 ) = MTotal1pRowSel . Value ( IndexEU2 ) = MTotal2

170 I f ( MTotal1 <= MTotal2 ) ThenpRowSel . Value ( IndexOpt ) = "No"E l s epRowSel . Value ( IndexOpt ) = " Yes "End I f

175 ’ _______________________________________’ Update c u r s o r o f s e l e c t e d DHM f e a t u r e s’ pCurso r . U p d a t e F e a t u r e pRowSel’ S e t pRowSel = pCurso r . N e x t F e a t u r eLoop

180

’ _______________________________________’END LOOP THROUGH EACH BUILDING OF THE LAYER

End Sub185 ’ _______________________________________

205

C Software Codes

’ Module1−NonPoisson . bas

A t t r i b u t e VB_Name = " Module1 "5 Opt ion E x p l i c i t

P r i v a t e Sub BN_Hugin ( )’ _______________________________________’ A c o l l e c t i o n t o ho ld t h e found p a r s e E r r o r sDim p a r s e E r r o r s As C o l l e c t i o n

10 ’ _______________________________________’ Get MapDim pMxDoc As IMxDocumentDim pMap As IMapS e t pMxDoc = ThisDocument

15 S e t pMap = pMxDoc . FocusMap’ _______________________________________’ Get t h e s h a p e f i l e wi th a l l t h e d a t aDim p F e a t u r e C l a s s As I F e a t u r e C l a s sS e t p F e a t u r e C l a s s = O p e n S h a p e f i l e ( " Z: < Fo lde r >" , " Typ5_O_Res_NonPoissonCopy " )

20 Dim pLayer As I F e a t u r e L a y e rS e t pLayer = pMap . Layer ( 0 )Dim p F e a t u r e C l a s s S e l As I F e a t u r e C l a s sS e t p F e a t u r e C l a s s S e l = pLayer . F e a t u r e C l a s s’ _______________________________________

25 ’ c r e a t e c u r s o r t o loop t h r o u g h " b u i l d i n g _ t y p e "Dim IndexFID As I n t e g e rIndexFID = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " FID " )Dim IndexOccupancy As S t r i n gIndexOccupancy = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " Occupancy " )

30 Dim I n d e x S t o r y A r e a As DoubleI n d e x S t o r y A r e a = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " F l o o r A r e a " )Dim IndexEU1 As DoubleIndexEU1 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " EU1 " )Dim IndexEU2 As Double

35 IndexEU2 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " EU2 " )Dim IndexOpt As S t r i n gIndexOpt = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " OptAc t ion " )Dim IndexLiq21 As DoubleIndexLiq21 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " Avg_Liq21 " )

40

’ _______________________________________’ D ef in e c u r s o r f o r s e l e c t e d f e a t u r e sDim pCurso r As I F e a t u r e C u r s o rS e t pCurso r = p F e a t u r e C l a s s S e l . Update ( Nothing , F a l s e )

45 ’ _______________________________________’ Cu r s o r t h r o u g h t h e s e l e c t e d rowsDim pRowSel As I F e a t u r eS e t pRowSel = pCurso r . N e x t F e a t u r e

50 Dim pOID As DoubleDim pQuery As I Q u e r y F i l t e r

’ _______________________________________

206

’LOOP THROUGH EACH BUILDING OF THE LAYER55 Dim M1 As Double

Dim M2 As DoubleDim MTotal1 As DoubleDim MTotal2 As DoubleDim Nu(1 To 9) As Double

60 Nu ( 1 ) = 0 .0000239883Nu ( 2 ) = 0 .000047863Nu ( 3 ) = 0 .0000891251Nu ( 4 ) = 0 .0000776247Nu ( 5 ) = 0 .0000512861

65 Nu ( 6 ) = 0 .000020893Nu ( 7 ) = 0 .0000354813Nu ( 8 ) = 0 .003019952Nu ( 9 ) = 0 .004073803Dim Source As I n t e g e r

70 Dim Year As I n t e g e r

Do While Not pRowSel I s Noth ingM1 = 0M2 = 0

75

’ For each s e i s m i c s o u r c e and each y e a r t h e e x p e c t e d c o s t s a r e a g g r e g a t e dFor Source = 1 To 9For Year = 1 To 50’ _______________________________________

80 ’ Im po r t BPN from Hugin

Dim d As HAPI . DomainDim BN As S t r i n g

85 Dim Netze (1 To 5)Netze ( 1 ) = " EX1_Typ5_O_Res_S "Netze ( 2 ) = CStr ( Source )Netze ( 3 ) = "_Y"Netze ( 4 ) = CStr ( Year )

90 Netze ( 5 ) = " . n e t "

BN = J o i n ( Netze , " " )S e t d = HAPI . LoadDomainFromNet (BN, p a r s e E r r o r s , 10)

95 ’ g e t t h e node wi th l a b e l " . . . " and name " . . . " from t h e domainDim NodeLiq As HAPI . NodeS e t NodeLiq = d . GetNodeByName ( " L i q u e f a c t i o n " )Dim NodeLiqTable As HAPI . Tab leS e t NodeLiqTable = NodeLiq . Tab le

100

Dim NodeCost As HAPI . NodeS e t NodeCost = d . GetNodeByName ( " Cos t " )Dim NodeCostTable As HAPI . Tab leS e t NodeCostTable = NodeCost . Tab le

105

Dim d e c i s i o n R e t r o f i t As Node

207

C Software Codes

S e t d e c i s i o n R e t r o f i t = d . GetNodeByName ( " R e t r o f i t " )

’ I n i t i a l i z e110 Dim l a u f As I n t e g e r

Dim M As I n t e g e rM = IndexLiq21For l a u f = 0 To 83 S tep 2NodeLiqTable . Data ( l a u f ) = 1 − pRowSel . Value (M)

115 NodeLiqTable . Data ( l a u f + 1) = pRowSel . Value (M)M = M + 1Next l a u f

Dim F a t a l i t y As Double120 Dim R e b u i l d i n g As Double

Dim R e p a i r As DoubleDim R e t r o f i t As Double

125 ’70%=Occupancy a t t ime of EQ (M2)’ 80% Occupants trapped (M3)’20% Occupan t s d i e d i m e d i a t e l y (M4)’ 80% Occupants dead after 10 days(M5)’LSCS=250000 USD ( GDPpc=10000 USD)

130 F a t a l i t y = pRowSel . Value ( IndexOccupancy )∗ _( 0 . 7∗0 . 8∗0 . 2 + 0 . 7∗0 . 8∗ ( 1 −0 . 2 )∗0 . 8 )∗2 5 0 0 0 0

’ Un i t r e b u i l d i n g c o s t = 300USD/ m2135 ’ I m p o r t a n c e f a c t o r f o r h o s p i t a l =10

’ N o n s t r u c t u r a l e l e m e n t s 50% of building value,R e b u i l d i n g = 5 ∗ pRowSel . Value ( I n d e x S t o r y A r e a ) ∗ 300 ∗ 5

’ Un i t r e b u i l d i n g c o s t = 300USD/ m2140 ’ 20%Cost of rebuilding for repair

’ N o n s t r u c t u r a l e l e m e n t s 50% of b u i l d i n g v a l u e’ I m p o r t a n c e f a c t o r f o r h o s p i t a l s =10 D i s c o u t r a t e =2%R e p a i r = 5 ∗ pRowSel . Value ( I n d e x S t o r y A r e a ) ∗ 300 ∗ 0 . 2 ∗ 5

145

’ Un i t r e t r o f i t c o s t =250USD/ column , a v e r a g e span l e n g t h i s 4mR e t r o f i t = ( ( ( Sqr (5 ∗ pRowSel . Value ( I n d e x S t o r y A r e a ) ) ) / 5 ) ^ 2 ) ∗ 50

NodeCostTable . Data ( 0 ) = (0+0+0+0)∗Exp (−0.02∗Year )150 NodeCostTable . Data ( 1 ) = (0+ R e p a i r +0+0)∗Exp (−0.02∗Year )

NodeCostTable . Data ( 2 ) = (0+0+ R e b u i l d i n g + F a t a l i t y ) ∗ Exp (−0.02∗Year )NodeCostTable . Data ( 3 ) = ( R e t r o f i t +0+0+0)∗Exp (−0.02∗Year )NodeCostTable . Data ( 4 ) = ( R e t r o f i t + R e p a i r +0+0)∗Exp (−0.02∗Year )NodeCostTable . Data ( 5 ) = ( R e t r o f i t +0+ R e b u i l d i n g + F a t a l i t y )∗Exp (−0.02∗Year )

155

d . Compile

M1 = M1 + d e c i s i o n R e t r o f i t . E x p e c t e d U t i l i t y ( 0 ) ∗ Nu ( Source )M2 = M2 + d e c i s i o n R e t r o f i t . E x p e c t e d U t i l i t y ( 1 ) ∗ Nu ( Source )

208

160

Next YearNext Source

MTotal1 = M1165 MTotal2 = M2

pRowSel . Value ( IndexEU1 ) = MTotal1pRowSel . Value ( IndexEU2 ) = MTotal2I f ( MTotal1 <= MTotal2 ) ThenpRowSel . Value ( IndexOpt ) = "No"

170 E l s epRowSel . Value ( IndexOpt ) = " Yes "End I f’ _______________________________________’ Update c u r s o r o f s e l e c t e d DHM f e a t u r e s

175 pCurso r . U p d a t e F e a t u r e pRowSelS e t pRowSel = pCurso r . N e x t F e a t u r e

Loop

’ _______________________________________180 ’END LOOP THROUGH EACH BUILDING OF THE LAYER

End Sub’ _______________________________________

209

C Software Codes


f u n c t i o n BPN_PSHA_Adapazari_RA_Poisson_Res ( T , X1 , Y1 , X2 , Y2 , a , b )




15 %%% For each seismic source, Z a set of BPN’s are calculated:f o r Z=2:2 %e.g. for seismic source S2

% For controling HUGIN from MATLAB the ActiveX server is loaded and20 % then the available functions in this library are used to alter

% objects created from this library. The HUGIN ActiveX Server is% loaded with the following command and create a HUGIN API object% named bpn:bpn= a c t x s e r v e r ( ’HAPI . G l o b a l s ’ ) ;

25

% An object ’domain’ is created which holds the network:domain= in vo ke ( bpn , ’ LoadDomainFromNet ’ , ’C : \ D i s s e r t a t i o n \ . . .Chap te r4 −1\BPN\ EX2_Typ5_O_Res_single_eps . n e t ’ , 0 , 0 ) ;

30 % The nodes to be manipulated are defined:ndEQ_M= in vo ke ( domain , ’ GetNodeByName ’ , ’ EQ_Magnitude ’ ) ;ndEQ_R= i nv ok e ( domain , ’ GetNodeByName ’ , ’ EQ_Dis tance ’ ) ;ndEps_PGA= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsilon_PGA ’ ) ;ndEps_SD= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsi lon_SD ’ ) ;

35 ndPGA= in vok e ( domain , ’ GetNodeByName ’ , ’PGA ’ ) ;ndSD= i nv oke ( domain , ’ GetNodeByName ’ , ’SD ’ ) ;ndDamage= in vo ke ( domain , ’ GetNodeByName ’ , ’ Damage ’ ) ;

%The number of discrete states of the nodes ’Magnitude’, ’Distance’,40 %’Eps_PGA’, ’Eps_SD’, ’PGA’ and ’SD’ is set:

nM=6;nR =5;nEps_PGA =10;nEps_SD =10;

45 nPGA=7;nSD=7;nDamage =3;s e t (ndEQ_M , ’ NumberOfSta tes ’ ,nM ) ;s e t ( ndEQ_R , ’ NumberOfSta tes ’ , nR ) ;

50 s e t ( ndEps_SD , ’ NumberOfSta tes ’ , nEps_SD ) ;s e t ( ndEps_PGA , ’ NumberOfSta tes ’ , nEps_PGA ) ;

210

s e t ( ndSD , ’ NumberOfSta tes ’ , nSD ) ;s e t ( ndPGA , ’ NumberOfSta tes ’ ,nPGA ) ;s e t ( ndDamage , ’ NumberOfSta tes ’ , nDamage ) ;

55


%The probability distribution of node ’Eps_SD’ is calculated:60 [ Eps_SD , dSD]=EPS_SD ( nEps_SD , nEps_PGA , T ) ;

%The probability distribution of node ’Distance’ is calculated:[R , P_R , R l i m i t s ]= Line_EQ_R ( 5 0 0 , nR , X1 , Y1 , X2 , Y2 ) ;

65 %The probability distribution of node ’Magnitude’ is calculated:[ Nu_Mmin ,M_EQ, P_M, M l i m i t s ]= EQ_M_NonPoisson ( 5 , a , b , nM, Q, Z ) ;

M=M_EQ;

%The discrete probabilities are set for node ’Magnitude’ in the70 %BPN:

f o r i = 1 : (nM)s e t (ndEQ_M . Table , ’ Data ’ , ( i −1) ,P_M( i ) ) ;s e t (ndEQ_M , ’ S t a t e L a b e l ’ , ( i −1) , [ ’M= ’ num2str (M( i ) ) ] ) ;

end75



80 end

%The discrete probabilities are set for node ’Eps_PGA’ in the BPN:f o r i = 1 : ( nEps_PGA )

s e t ( ndEps_PGA . Table , ’ Data ’ , ( i −1) ,Eps_PGA ( i ) ) ;85 s e t ( ndEps_PGA , ’ S t a t e L a b e l ’ , ( i −1) , . . .


%The discrete probabilities are set for node ’Eps_SD’ in the BPN:90 f o r i = 1 : ( nEps_SD∗nEps_PGA )


%The conditional probability table of the node ’PGA’ is initialized95 %with zeros:

f o r i = 1 : (nM∗nR∗nEps_PGA∗nPGA)s e t ( ndPGA . Table , ’ Data ’ , ( i −1 ) , 0 ) ;

end

100 %The conditional probability table of the node ’SD’ is initialized%with zeros:f o r i = 1 : (nM∗nR∗nSD∗nEps_SD )

s e t ( ndSD . Table , ’ Data ’ , ( i −1 ) , 0 ) ;end

211

C Software Codes

105

%for all combinations of the states in the node ’Magnitude’ and%’Distance’ the peak ground accelerations and spectral displace-%ments are calculated with the Boore Joyner and Fumal attenuation%model

110 [PGABOORE]=PGA( nEps_PGA ,M, R ) ;[SDBOORE]=SD( nEps_SD , nEps_PGA ,M, R ) ;

%The limits for the discretisation of the nodes ’SD’ and ’PGA’ are%set:

115 CoeffPGA =[0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 max (PGABOORE ) ] ;CoeffSD =[0 0 .005 0 . 0 2 0 . 0 5 0 . 1 0 . 3 0 . 5 max (SDBOORE ) ] ;

f o r i = 1 : ( nSD )S ( i ) = ( CoeffSD ( i )+ CoeffSD ( i + 1 ) ) / 2 ;

120 SDmm( i )=S ( i )∗1 0 0 0 ;end

%The labels of the states are set for node ’Eps_SD’, ’PGA’ and ’SD’%in the BPN:

125 f o r i = 1 : ( nEps_SD )s e t ( ndEps_SD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’ Eps_SD= ’ num2str ( dSD ( i ) ) ] ) ;


s e t ( ndPGA , ’ S t a t e L a b e l ’ , ( i −1) , . . .130 [ ’PGA= ’ num2str ( ( CoeffPGA ( i )+ CoeffPGA ( i + 1 ) ) / 2 ) ] ) ;


s e t ( ndSD , ’ S t a t e L a b e l ’ , ( i −1) , . . .[ ’SD= ’ num2str ( ( CoeffSD ( i )+ CoeffSD ( i + 1 ) ) / 2 ) ] ) ;

135 end

%The discrete probabilities are set for node ’PGA’ in the BPN:N=0;f o r i =1 : l e n g t h (PGABOORE)

140 f o r j =1 :nPGAi f PGABOORE( i ) <= CoeffPGA ( j +1) & PGABOORE( i ) > CoeffPGA ( j )

s e t ( ndPGA . Table , ’ Data ’ , ( j +N−1 ) , 1 ) ;N=N+nPGA ;

end145 end

end

%The discrete probabilities are set for node ’SD’ in the BPN:K=0;

150 f o r i =1 : l e n g t h (SDBOORE)f o r j =1 :nSD

i f SDBOORE( i ) <= CoeffSD ( j +1) & SDBOORE( i ) > CoeffSD ( j )s e t ( ndSD . Table , ’ Data ’ , ( j +K−1 ) , 1 ) ;K=K+nSD ;

155 endend

end

212

%The discrete probabilities are calculated given the spectral160 %displacement values are set for node ’Damage’

[ P_Damage ]= Fragi l i ty_Typ5_O_Res_RA_1 (SDmm) ;f o r i = 1 : ( nSD∗5∗3)

s e t ( ndDamage . Table , ’ Data ’ , ( i −1) , P_Damage ( i ) ) ;end

165

%The BPN’s are set for each SourceFi l ename =[ ’C : \ D i s s e r t a t i o n \ Chapte r4 −1\BPN\ EX2_Typ5_O_Res_S ’ . . .

num2str ( Z ) ’ _ P o i s s o n _ 1 ’ ’ . n e t ’ ] ;i n vo ke ( domain , ’ SaveAsNet ’ , F i l ename ) ;

170 endend

f u n c t i o n [ P_Damage ]= Fragi l i ty_Typ5_O_Res_RA_1 (SD)

%For the three damage states the parameters of the lognormal%distribution are given in Table X.

5 Lambda_Yellow = [ 3 . 6 9 0 3 .724 3 .758 3 .792 3 . 8 2 9 ] ;Ze ta_Yel low = [ 0 . 3 4 1 0 .366 0 .390 0 .414 0 . 4 3 9 ] ;Lambda_Red = [ 4 . 1 0 6 4 .160 4 .215 4 .270 4 . 324 ] ;Zeta_Red = [ 0 . 2 6 6 0 .306 0 .346 0 .386 0 . 4 2 6 ] ;

10 %For each state of the node ’SD’ the probabilities of being in one of the%three damage states are calculated.N=1;f o r i =1 :5

f o r k =1: l e n g t h (SD)15 p1 (N)=1− l o g n c d f (SD( k ) , Lambda_Yellow ( i ) , Ze ta_Yel low ( i ) ) ;

p2 (N)= l o g n c d f (SD( k ) , Lambda_Yellow ( i ) , Ze ta_Yel low ( i ))− . . .l o g n c d f (SD( k ) , Lambda_Red ( i ) , Zeta_Red ( i ) ) ;

p3 (N)= l o g n c d f (SD( k ) , Lambda_Red ( i ) , Zeta_Red ( i ) ) ;N=N+1;

20 endend

%The probabilities of being in one of the three damage states form the%conditional probability table of the node ’Damage’

25 DamageNode1 =[ p1 ; p2 ; p3 ] ;P_Damage1=DamageNode1 . ∗ ( DamageNode1 > 0 ) ;P_Damage=P_Damage1 ( : ) ;

f u n c t i o n [ P_C]= C o s t D i s t S 2g lob = a c t x s e r v e r ( ’HAPI . G l o b a l s ’ ) ;N=1;f o r i =1 :6 %over all states of the magnitude node

5 f o r j =1 :5 %over all states of the distance nodef o r k =1:5 %over all states of the epsilon node

f o r m=1:534f i l e n a m e =[ ’Z : \ D i s s e r t a t i o n a n a l y s i s \ . . .

S e i s m i c h a z a r d a n a l y s i s \ PhD \ GIS−BPN Outpu t \ . . .10 RA_Typ5_O_Res_Poisson_Proper ty \ . . .

213

C Software Codes

’ num2str (m−1) ’ _aEX2_Typ5_O_Res_S1_Poisson . n e t ’ ] ;domain= in vo ke ( glob , ’ LoadDomainFromNet ’ , f i l e n a m e , 0 , 0 ) ;ndEQ_M= in vo ke ( domain , ’ GetNodeByName ’ , ’ EQ_Magnitude ’ ) ;ndEQ_R= i nv ok e ( domain , ’ GetNodeByName ’ , ’ EQ_Dis tance ’ ) ;

15 ndEps_Frag= in vo ke ( domain , ’ GetNodeByName ’ , ’ Eps_Frag ’ ) ;s e t ( ndEQ_R . Table , ’ Data ’ , 0 , 0 . 0 0 0 0 0 1 ) ;s e t ( ndEQ_R . Table , ’ Data ’ , 1 , 0 . 0 0 0 0 0 1 ) ;s e t ( ndEQ_R . Table , ’ Data ’ , 4 , 0 . 0 0 0 0 0 1 ) ;ndCost = i nv ok e ( domain , ’ GetNodeByName ’ , ’ Cos t ’ ) ;

20 i n vo ke ( domain , ’ Compile ’ ) ;i n vo ke (ndEQ_M , ’ S e l e c t S t a t e ’ , ( i −1 ) ) ;i n vo ke ( ndEQ_R , ’ S e l e c t S t a t e ’ , ( j −1 ) ) ;i n vo ke ( ndEps_Frag , ’ S e l e c t S t a t e ’ , ( k −1 ) ) ;i n vo ke ( domain , ’ P r o p a g a t e ’ , ’ hEqui l ib r iumSum ’ , ’ hModeNormal ’ ) ;

25 f o r l =1:10P_Cost1 (m, l )= g e t ( ndCost , ’ B e l i e f ’ , ( l −1 ) ) ;

endendf o r m=1:712

30 f i l e n a m e =[ ’Z : \ D i s s e r t a t i o n a n a l y s i s \ . . .S e i s m i c h a z a r d a n a l y s i s \ PhD \ GIS−BPN Outpu t \ . . .RA_Typ5_N_Res_Poisson_Proper ty \ . . .’ num2str (m−1) ’ _aEX2_Typ5_N_Res_S1_Y1 . n e t ’ ] ;

domain= in vo ke ( glob , ’ LoadDomainFromNet ’ , f i l e n a m e , 0 , 0 ) ;35 ndEQ_M= in vo ke ( domain , ’ GetNodeByName ’ , ’ EQ_Magnitude ’ ) ;

ndEQ_R= i nv ok e ( domain , ’ GetNodeByName ’ , ’ EQ_Dis tance ’ ) ;ndEps_Frag= in vo ke ( domain , ’ GetNodeByName ’ , ’ Eps_Frag ’ ) ;s e t ( ndEQ_R . Table , ’ Data ’ , 0 , 0 . 0 0 0 0 0 1 ) ;s e t ( ndEQ_R . Table , ’ Data ’ , 1 , 0 . 0 0 0 0 0 1 ) ;

40 s e t ( ndEQ_R . Table , ’ Data ’ , 4 , 0 . 0 0 0 0 0 1 ) ;ndCost = i nv ok e ( domain , ’ GetNodeByName ’ , ’ Cos t ’ ) ;i n vo ke ( domain , ’ Compile ’ ) ;i n vo ke (ndEQ_M , ’ S e l e c t S t a t e ’ , ( i −1 ) ) ;i n vo ke ( ndEQ_R , ’ S e l e c t S t a t e ’ , ( j −1 ) ) ;

45 i n vo ke ( ndEps_Frag , ’ S e l e c t S t a t e ’ , ( k −1 ) ) ;i n vo ke ( domain , ’ P r o p a g a t e ’ , ’ hEqui l ib r iumSum ’ , ’ hModeNormal ’ ) ;f o r l =1:10

P_Cost2 (m, l )= g e t ( ndCost , ’ B e l i e f ’ , ( l −1 ) ) ;end

50 endP_Cost =[ P_Cost1 ; P_Cost2 ] ;P_C (N ) . P_C=P_Cost ;N=N+1;c l e a r d

55 endend

endsave 1P_S1M6R5 P_C

f u n c t i o n [ P_C]= C o s t D i s t S 2 _ I n t e g r a t e dg lob = a c t x s e r v e r ( ’HAPI . G l o b a l s ’ ) ;N=1;f o r m=1:534

214

5 f i l e n a m e =[ ’Z : \ D i s s e r t a t i o n a n a l y s i s \ S e i s m i c h a z a r d a n a l y s i s \ PhD \ . . .GIS−BPN Outpu t \ RA_Typ5_O_Res_Poisson \ . . .’ num2str (m−1) ’ _EX2_Typ5_O_Res_S1_Poisson . n e t ’ ] ;

domain= in vo ke ( glob , ’ LoadDomainFromNet ’ , f i l e n a m e , 0 , 0 ) ;ndCost = i nv oke ( domain , ’ GetNodeByName ’ , ’ Cos t ’ ) ;

10 i n vo ke ( domain , ’ Compile ’ ) ;f o r l =1:10


end15 f o r m=1:712

f i l e n a m e =[ ’Z : \ D i s s e r t a t i o n a n a l y s i s \ S e i s m i c h a z a r d a n a l y s i s \ PhD \ . . .GIS−BPN Outpu t \ RA_Typ5_N_Res_Poisson \ . . .’ num2str (m−1) ’ _EX2_Typ5_N_Res_S1_Poisson . n e t ’ ] ;

domain= in vo ke ( glob , ’ LoadDomainFromNet ’ , f i l e n a m e , 0 , 0 ) ;20 ndCost = i n v o ke ( domain , ’ GetNodeByName ’ , ’ Cos t ’ ) ;

i n vo ke ( domain , ’ Compile ’ ) ;f o r l =1:10


25 endP_Cost =[ P_Cost1 ; P_Cost2 ] ;P_C (N ) . P_C=P_Cost ;save 1_S1MR P_C

f u n c t i o n [EX,M,MVERT, EXVERT, Risk ,CX_MEAN]= A g g r e g a t i o n _ P o i s s o n _ S 2

%For each combination of the 150 states (6 magnitude states*5 distance%states*5 epsilon states) the discrete probabilities for the states of the

5 %node ’Cost’ were stored in PC_Poisson_S2load PC_Poisson_S2

%Total number of buildingsN=1246;

10 %States of the node ’Cost’c = [ 0 ; 9 9 ; 1 6 8 ; 2 6 6 ; 4 4 2 ; 1 1 9 6 ; 2 5 1 9 ; 3 9 9 3 ; 6 2 8 7 ; 1 4 4 5 4 ; 1 4 4 5 4 ] ;%The temporary number of states for the node ’Cost S2’ are set to 10000n d i s =10000;

15 %The state boundaries CXLIM and representative value for each state CX_MEAN%are setCmax=max ( c )∗N;Cmin =0;f o r i =1 : n d i s

20 CXLIM( i )= i ∗ (Cmax−Cmin ) / n d i s ;endCXLIM=[0 CXLIM ] ;f o r i =1 : n d i s

CX_MEAN( i ) = (CXLIM( i )+CXLIM( i + 1 ) ) / 2 ;25 end

CX_MEAN=CX_MEAN’ ;

%50000 samples of random numbers are drawn for each of the 1246 buildings

215

C Software Codes

%and depending on the probability of the node ’cost’, the corresponding30 %cost terms are added

f o r m=1:150d i s p l a y (m)A=PC_Poisson_S2 ( 1 ,m) . PC_Poisson_S2 ;A=A’ ;

35 p=cumsum (A ) ;p_temp= z e r o s ( 1 ,N ) ;p =[ p_temp ; p ] ;

K=50000;40 CT=0;

f o r j =1 :Kr =rand ( 1 ,N ) ;f o r i =1 :N

CT=CT+c ( sum ( ( r ( i ) > p ( : , i ) ) ) ) ;45 end

CT_T ( j )=CT ;CT=0;

end

50 M= h i s t ( CT_T ,CX_MEAN) ;M=M’ ;M=M/K;MVERT(m) .m=M;

end55

%The marginal distribution of each of the 150 combinations are calculatedT=1;P_M=[0 .724894 0 .199652 0 .054984 0 .015145 0 .004171 0 . 0 0 1 1 4 9 ] ;P_R =[0 .000001 0 .000001 0 .626 0 .374 0 . 0 0 0 0 0 1 ] ;

60 P_Eps = [ 0 . 0 2 2 8 0 .1359 0 .6826 0 .1359 0 . 0 2 2 8 ] ;f o r i =1 :6

f o r j =1 :5f o r k =1:5

P_Marg ina l ( T)=P_M( i )∗P_R ( j )∗ P_Eps ( k ) ;65 T=T+1;

endend

end

70 Risk = z e r o s ( nd i s , 1 ) ;

%The distribution of risk is calculatedf o r T=1:150

Risk = Risk +MVERT( 1 , T ) .m∗P_Marg ina l ( T ) ;75 end

save Risk_S2 Risk

f u n c t i o n [EX,M,MVERT,CX_MEAN]= A g g r e g a t i o n _ P o i s s o n _ S 2 _ I n t e g r a t e d

%For each combination of the 150 states (6 magnitude states*5 distance%states*5 epsilon states) the discrete probabilities for the states of the

216

5 %node ’Cost’ were stored in PC_Poisson_S2load 1_S1MR

%Total number of buildingsN=1246;

10 %States of the node ’Cost’c = [ 0 ; 9 9 ; 1 6 8 ; 2 6 6 ; 4 4 2 ; 1 1 9 6 ; 2 5 1 9 ; 3 9 9 3 ; 6 2 8 7 ; 1 4 4 5 4 ; 1 4 4 5 4 ] ;%The temporary number of states for the node ’Cost S2’ are set to 10000n d i s =10000;

15 %The state boundaries CXLIM and representative value for each state CX_MEAN%are setCmax=max ( c )∗N;Cmin =0;f o r i =1 : n d i s

20 CXLIM( i )= i ∗ (Cmax−Cmin ) / n d i s ;endCXLIM=[0 CXLIM ] ;f o r i =1 : n d i s

CX_MEAN( i ) = (CXLIM( i )+CXLIM( i + 1 ) ) / 2 ;25 end

CX_MEAN=CX_MEAN’ ;

%50000 samples of random numbers are drawn for each of the 1246 buildings%and depending on the probability of the node ’cost’, the corresponding

30 %cost terms are addedA=P_C . P_C ;A=A’ ;p=cumsum (A ) ;p_temp= z e r o s ( 1 ,N ) ;

35 p =[ p_temp ; p ] ;

K=50000;CT=0;f o r j =1 :K

40 r =rand ( 1 ,N ) ;f o r i =1 :N

CT=CT+c ( sum ( ( r ( i ) > p ( : , i ) ) ) ) ;endCT_T ( j )=CT ;

45 CT=0;end

%The distribution of risk is calculatedM= h i s t ( CT_T ,CX_MEAN) ;

50 M=M’ ;M=M/K;Risk =M;

save R i s k _ S 2 _ I n t e g r a t e d Risk

217

C Software Codes

















40

End F u n c t i o n

’ Module1−Ex2 . basA t t r i b u t e VB_Name = " Module1 "Opt ion E x p l i c i tP r i v a t e Sub BN_Hugin ( )

5 ’ _______________________________________’ A c o l l e c t i o n t o ho ld t h e found p a r s e E r r o r sDim p a r s e E r r o r s As C o l l e c t i o n’ _______________________________________

218

’ Get Map10 Dim pMxDoc As IMxDocument

Dim pMap As IMapS e t pMxDoc = ThisDocumentS e t pMap = pMxDoc . FocusMap

’ _______________________________________15 ’ Get t h e s h a p e f i l e wi th a l l t h e d a t a

Dim p F e a t u r e C l a s s As I F e a t u r e C l a s sS e t p F e a t u r e C l a s s = O p e n S h a p e f i l e ( " Z: < Fo lde r >" , " Typ5_O_Res " )Dim pLayer As I F e a t u r e L a y e rS e t pLayer = pMap . Layer ( 0 )

20 Dim p F e a t u r e C l a s s S e l As I F e a t u r e C l a s sS e t p F e a t u r e C l a s s S e l = pLayer . F e a t u r e C l a s s’ _______________________________________’ c r e a t e c u r s o r t o loop t h r o u g h " b u i l d i n g _ t y p e "Dim IndexFID As I n t e g e r

25 IndexFID = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " FID " )Dim IndexOccupancy As S t r i n gIndexOccupancy = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " Occupancy " )Dim I n d e x S t o r y A r e a As DoubleI n d e x S t o r y A r e a = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " F l o o r A r e a " )

30 Dim IndexLiq21 As DoubleIndexLiq21 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " Avg_Liq21 " )Dim I n d e x C o s t T o t a l 1 As DoubleI n d e x C o s t T o t a l 1 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " C o s t T o t a l 1 " )Dim I n d e x C o s t T o t a l 2 As Double

35 I n d e x C o s t T o t a l 2 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " C o s t T o t a l 2 " )Dim I n d e x C o s t T o t a l 3 As DoubleI n d e x C o s t T o t a l 3 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " C o s t T o t a l 3 " )

’ _______________________________________40 ’ D e f in e c u r s o r f o r s e l e c t e d f e a t u r e s

Dim pCurso r As I F e a t u r e C u r s o rS e t pCurso r = p F e a t u r e C l a s s S e l . Update ( Nothing , F a l s e )’ _______________________________________’ Cu r s o r t h r o u g h t h e s e l e c t e d rows

45 Dim pRowSel As I F e a t u r eS e t pRowSel = pCurso r . N e x t F e a t u r e

Dim pOID As DoubleDim pQuery As I Q u e r y F i l t e r

50

Dim Nu(1 To 9) As DoubleNu ( 1 ) = 0 .0000239883Nu ( 2 ) = 0 .000047863Nu ( 3 ) = 0 .0000891251

55 Nu ( 4 ) = 0 .0000776247Nu ( 5 ) = 0 .0000512861Nu ( 6 ) = 0 .000020893Nu ( 7 ) = 0 .0000354813Nu ( 8 ) = 0 .003019952

60 Nu ( 9 ) = 0 .004073803

219

C Software Codes

’ _______________________________________’LOOP THROUGH EACH BUILDING OF THE LAYERDo While Not pRowSel I s Noth ing

65 Dim Source As I n t e g e rFor Source = 1 To 9’ _______________________________________’ Im p o r t BPN from Hugin

70 Dim d As HAPI . DomainDim BN As S t r i n gDim Netze (1 To 3)Netze ( 1 ) = " EX2_Typ5_O_Res_S "Netze ( 2 ) = CStr ( Source )

75 Netze ( 3 ) = " _ P o i s s o n . n e t "BN = J o i n ( Netze , " " )S e t d = HAPI . LoadDomainFromNet (BN, p a r s e E r r o r s , 10)

80 ’ g e t t h e node wi th l a b e l " . . . " and name " . . . " from t h e domainDim NodeLiq As HAPI . NodeS e t NodeLiq = d . GetNodeByName ( " L i q u e f a c t i o n " )Dim NodeLiqTable As HAPI . Tab leS e t NodeLiqTable = NodeLiq . Tab le

85

Dim NodeCost As HAPI . NodeS e t NodeCost = d . GetNodeByName ( " Cos t " )Dim NodeCostTable As HAPI . Tab leS e t NodeCostTable = NodeCost . Tab le

90

’ I n i t i a l i z eDim l a u f As I n t e g e rDim M As I n t e g e r

95 M = IndexLiq21For l a u f = 0 To 83 S tep 2NodeLiqTable . Data ( l a u f ) = 1 − pRowSel . Value (M)NodeLiqTable . Data ( l a u f + 1) = pRowSel . Value (M)M = M + 1

100 Next l a u f

Dim F a t a l i t y As DoubleDim R e b u i l d i n g As DoubleDim R e p a i r As Double

105 Dim C o s t T o t a l 1 As DoubleDim C o s t T o t a l 2 As DoubleDim C o s t T o t a l 3 As Double

110 ’ I n i t i a l i z eDim i n i t 1 As S i n g l eFor i n i t 1 = 0 To 29NodeCostTable . Data ( i n i t 1 ) = 0Next i n i t 1

220

115

’70%=Occupancy a t t ime of EQ (M2)’ 80% Occupants trapped (M3)’20% t r a p p e d Occupan t s d i e d i m e d i a t e l y (M4)’ 80% Occupants dead after 10 days(M5)

120 ’LSCS=250000 USD ( GDPpc=10000 USD)F a t a l i t y = pRowSel . Value ( IndexOccupancy ) ∗ _

( 0 . 7 ∗ 0 . 8 ∗ 0 . 2 + 0 . 7 ∗ 0 . 8 ∗ (1 − 0 . 2 ) ∗ 0 . 8 ) ∗ 250000

125 ’ Un i t r e b u i l d i n g c o s t = 200USD/ m2 ,’ I m p o r t a n c e f a c t o r f o r h o s p i t a l =10’ 2% discountingR e b u i l d i n g = 5 ∗ pRowSel . Value ( I n d e x S t o r y A r e a ) ∗ 300 ∗ 5

130 ’ Un i t r e b u i l d i n g c o s t = 200USD/ m2 ,’ 15%Cost of rebuilding for repair,’ I m p o r t a n c e f a c t o r f o r h o s p i t a l s =10’ D i s c o u t r a t e =2%R e p a i r = 5 ∗ pRowSel . Value ( I n d e x S t o r y A r e a ) ∗ 300 ∗ 0 . 2 ∗ 5

135

C o s t T o t a l 1 = 0 ∗ Nu ( Source )C o s t T o t a l 2 = R e p a i r ∗ Nu ( Source )C o s t T o t a l 3 = ( R e b u i l d i n g ) ∗ Nu ( Source )

140 I f ( C o s t T o t a l 1 <= 62 And C o s t T o t a l 1 >= 0) ThenNodeCostTable . Data ( 0 ) = 1

E l s e I f ( C o s t T o t a l 1 < 134 And C o s t T o t a l 1 > 62) ThenNodeCostTable . Data ( 1 ) = 1

E l s e I f ( C o s t T o t a l 1 < 212 And C o s t T o t a l 1 >= 134) Then145 NodeCostTable . Data ( 2 ) = 1

E l s e I f ( C o s t T o t a l 1 < 329 And C o s t T o t a l 1 >= 212) ThenNodeCostTable . Data ( 3 ) = 1


150 E l s e I f ( C o s t T o t a l 1 < 1863 And C o s t T o t a l 1 >= 557) ThenNodeCostTable . Data ( 5 ) = 1


E l s e I f ( C o s t T o t a l 1 < 4930 And C o s t T o t a l 1 >= 3073) Then155 NodeCostTable . Data ( 7 ) = 1



160 End I f

I f ( C o s t T o t a l 2 <= 62 And C o s t T o t a l 2 >= 0) ThenNodeCostTable . Data ( 1 0 ) = 1

E l s e I f ( C o s t T o t a l 2 < 134 And C o s t T o t a l 2 > 62) Then165 NodeCostTable . Data ( 1 1 ) = 1

E l s e I f ( C o s t T o t a l 2 < 212 And C o s t T o t a l 2 >= 134) ThenNodeCostTable . Data ( 1 2 ) = 1

221

C Software Codes


170 E l s e I f ( C o s t T o t a l 2 < 557 And C o s t T o t a l 2 >= 329) ThenNodeCostTable . Data ( 1 4 ) = 1


E l s e I f ( C o s t T o t a l 2 < 3073 And C o s t T o t a l 2 >= 1863) Then175 NodeCostTable . Data ( 1 6 ) = 1




End I f

I f ( C o s t T o t a l 3 <= 62 And C o s t T o t a l 3 >= 0) Then185 NodeCostTable . Data ( 2 0 ) = 1

E l s e I f ( C o s t T o t a l 3 < 134 And C o s t T o t a l 3 > 62) ThenNodeCostTable . Data ( 2 1 ) = 1




E l s e I f ( C o s t T o t a l 3 < 1863 And C o s t T o t a l 3 >= 557) Then195 NodeCostTable . Data ( 2 5 ) = 1





End I f205

Dim FID , Netz (1 To 3 ) , BPNFID = pRowSel . Value ( IndexFID )Netz ( 1 ) = CStr ( FID )Netz ( 2 ) = " _a "

210 Netz ( 3 ) = BNBPN = J o i n ( Netz , " " )d . SaveAsNet (BPN)Next SourcepRowSel . Value ( I n d e x C o s t T o t a l 1 ) = C o s t T o t a l 1

215 pRowSel . Value ( I n d e x C o s t T o t a l 2 ) = C o s t T o t a l 2pRowSel . Value ( I n d e x C o s t T o t a l 3 ) = C o s t T o t a l 3

’ _______________________________________220 ’ Update c u r s o r o f s e l e c t e d DHM f e a t u r e s

222

pCurso r . U p d a t e F e a t u r e pRowSelS e t pRowSel = pCurso r . N e x t F e a t u r e

Loop’ _______________________________________

225 ’END LOOP THROUGH EACH BUILDING OF THE LAYER

End Sub’ _______________________________________

223

C Software Codes


f u n c t i o n BPN_PSHA_Adapazari_BU ( T , X1 , Y1 , X2 , Y2 , a , b , SS )




15 %%% For each seismic source, Z and each year, Q a set of BPN’s are% calculated:f o r Z=SS : SS

20 f o r Q=1:50% For controling HUGIN from MATLAB the ActiveX server is loaded and% then the available functions in this library are used to alter% objects created from this library. The HUGIN ActiveX Server is% loaded with the following command and create a HUGIN API object

25 % named bpn:bpn= a c t x s e r v e r ( ’HAPI . G l o b a l s ’ ) ;

% An object ’domain’ is created which holds the network:domain= in vo ke ( bpn , ’ LoadDomainFromNet ’ , ’C : \ D i s s e r t a t i o n . . .

30 Chapte r4 −1\BPN\ EX3_Typ5_O_Res_single_eps . n e t ’ , 0 , 0 ) ;

% The nodes to be manipulated are defined:ndEQ_M= in vo ke ( domain , ’ GetNodeByName ’ , ’ EQ_Magnitude ’ ) ;ndEQ_R= i nv ok e ( domain , ’ GetNodeByName ’ , ’ EQ_Dis tance ’ ) ;

35 ndEps_PGA= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsilon_PGA ’ ) ;ndEps_SD= in vo ke ( domain , ’ GetNodeByName ’ , ’ Epsi lon_SD ’ ) ;ndPGA= inv ok e ( domain , ’ GetNodeByName ’ , ’PGA ’ ) ;ndSD= i nv oke ( domain , ’ GetNodeByName ’ , ’SD ’ ) ;ndDamage= in vo ke ( domain , ’ GetNodeByName ’ , ’ Damage ’ ) ;

40


45 nEps_PGA =10;nEps_SD =10;nPGA=7;nSD=7;nDamage =3;

50 s e t (ndEQ_M , ’ NumberOfSta tes ’ ,nM ) ;s e t ( ndEQ_R , ’ NumberOfSta tes ’ , nR ) ;

224

s e t ( ndEps_SD , ’ NumberOfSta tes ’ , nEps_SD ) ;s e t ( ndEps_PGA , ’ NumberOfSta tes ’ , nEps_PGA ) ;s e t ( ndSD , ’ NumberOfSta tes ’ , nSD ) ;

55 s e t ( ndPGA , ’ NumberOfSta tes ’ ,nPGA ) ;s e t ( ndDamage , ’ NumberOfSta tes ’ , nDamage ) ;


60

%The probability distribution of node ’Eps_SD’ is calculated:[ Eps_SD , dSD]=EPS_SD ( nEps_SD , nEps_PGA , T ) ;

%The probability distribution of node ’Distance’ is calculated:65 [R , P_R , R l i m i t s ]= Line_EQ_R ( 5 0 0 , nR , X1 , Y1 , X2 , Y2 ) ;

%The probability distribution of node ’Magnitude’ is calculated:[ Nu_Mmin ,M_EQ, P_M, M l i m i t s ]= EQ_M_NonPoisson ( 5 , a , b , nM, Q, Z ) ;

M=M_EQ;70


s e t (ndEQ_M . Table , ’ Data ’ , ( i −1) ,P_M( i ) ) ;75 s e t (ndEQ_M , ’ S t a t e L a b e l ’ , ( i −1) , [ ’M= ’ num2str (M( i ) ) ] ) ;

end


80 s e t ( ndEQ_R . Table , ’ Data ’ , ( i −1) ,P_R ( i ) ) ;s e t ( ndEQ_R , ’ S t a t e L a b e l ’ , ( i −1) , [ ’R= ’ num2str (R( i ) ) ] ) ;

end

%The discrete probabilities are set for node ’Eps_PGA’ in the BPN:85 f o r i = 1 : ( nEps_PGA )

s e t ( ndEps_PGA . Table , ’ Data ’ , ( i −1) ,Eps_PGA ( i ) ) ;s e t ( ndEps_PGA , ’ S t a t e L a b e l ’ , ( i −1) , . . .


90

%The discrete probabilities are set for node ’Eps_SD’ in the BPN:f o r i = 1 : ( nEps_SD∗nEps_PGA )


95

%The conditional probability table of the node ’PGA’ is initialized%with zeros:f o r i = 1 : (nM∗nR∗nEps_PGA∗nPGA)

s e t ( ndPGA . Table , ’ Data ’ , ( i −1 ) , 0 ) ;100 end


225

C Software Codes


%for all combinations of the states in the node ’Magnitude’ and%’Distance’ the peak ground accelerations and spectral

110 %displacements are calculated with the Boore Joyner and Fumal%attenuation model[PGABOORE]=PGA( nEps_PGA ,M, R ) ;[SDBOORE]=SD( nEps_SD , nEps_PGA ,M, R ) ;

115 %The limits for the discretisation of the nodes ’SD’ and ’PGA’ are%set:CoeffPGA =[0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 max (PGABOORE ) ] ;CoeffSD =[0 0 .005 0 . 0 2 0 . 0 5 0 . 1 0 . 3 0 . 5 max (SDBOORE ) ] ;

120 f o r i = 1 : ( nSD )S ( i ) = ( CoeffSD ( i )+ CoeffSD ( i + 1 ) ) / 2 ;SDmm( i )=S ( i )∗1 0 0 0 ;

end

125 %The labels of the states are set for node ’Eps_SD’, ’PGA’ and ’SD’%in the BPN:f o r i = 1 : ( nEps_SD )

s e t ( ndEps_SD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’ Eps_SD= ’ num2str ( dSD ( i ) ) ] ) ;end

130 f o r i = 1 : (nPGA)s e t ( ndPGA , ’ S t a t e L a b e l ’ , ( i −1) , [ ’PGA= ’ num2str ( ( CoeffPGA ( i )+ . . .

CoeffPGA ( i + 1 ) ) / 2 ) ] ) ;endf o r i = 1 : ( nSD )

135 s e t ( ndSD , ’ S t a t e L a b e l ’ , ( i −1) , [ ’SD= ’ num2str ( ( CoeffSD ( i )+ . . .CoeffSD ( i + 1 ) ) / 2 ) ] ) ;

end

%The discrete probabilities are set for node ’PGA’ in the BPN:140 N=0;

f o r i =1 : l e n g t h (PGABOORE)f o r j =1 :nPGA

i f PGABOORE( i ) <= CoeffPGA ( j +1) & PGABOORE( i ) > CoeffPGA ( j )s e t ( ndPGA . Table , ’ Data ’ , ( j +N−1 ) , 1 ) ;

145 N=N+nPGA ;end

endend

150 %The discrete probabilities are set for node ’SD’ in the BPN:K=0;f o r i =1 : l e n g t h (SDBOORE)

f o r j =1 :nSDi f SDBOORE( i ) <= CoeffSD ( j +1) & SDBOORE( i ) > CoeffSD ( j )

155 s e t ( ndSD . Table , ’ Data ’ , ( j +K−1 ) , 1 ) ;K=K+nSD ;

end

226

endend

160

[ P_Damage ]= Fragi l i ty_Typ5_O_Res_RA_2 (SDmm) ;f o r i = 1 : ( nSD∗5∗2)

s e t ( ndDamage . Table , ’ Data ’ , ( i −1) , P_Damage ( i ) ) ;165 end

%The BPN’s are set for each Source and YearFi l ename =[ ’C : \ D i s s e r t a t i o n \ Chapte r4 −1\BPN\ EX3_Typ5_O_Res_S ’ . . .

num2str ( Z ) ’_Y ’ num2str (Q) ’ . n e t ’ ] ;170 i n v o k e ( domain , ’ SaveAsNet ’ , F i l ename ) ;

endend

f u n c t i o n [ P_Damage ]= Fragi l i ty_Typ5_O_Res_RA_2 (SD)

%For the two damage states the parameters of the lognormal%distribution are given in Table X.

5 Lambda_Red = [ 4 . 1 6 0 4 .215 4 . 2 7 0 ] ;Zeta_Red = [ 0 . 3 0 6 0 .346 0 . 3 8 6 ] ;

%For each state of the node ’SD’ the probabilities of being in one of the%two damage states are calculated.

10 N=1;f o r i =1 :5

f o r k =1: l e n g t h (SD)p1 (N)=1− l o g n c d f (SD( k ) , Lambda_Red ( i ) , Zeta_Red ( i ) ) ;p2 (N)= l o g n c d f (SD( k ) , Lambda_Red ( i ) , Zeta_Red ( i ) ) ;

15 N=N+1;end

end

%The probabilities of being in one of the two damage states form the20 %conditional probability table of the node ’Damage’

DamageNode1 =[ p1 ; p2 ] ;P_Damage1=DamageNode1 . ∗ ( DamageNode1 > 0 ) ;P_Damage=P_Damage1 ( : ) ;





227

C Software Codes













40

End F u n c t i o n

’ Module1−BU−Typ5_O_Res . basA t t r i b u t e VB_Name = " Module1 "Opt ion E x p l i c i tP r i v a t e Sub BN_Hugin ( )

5 ’ _______________________________________’ A c o l l e c t i o n t o ho ld t h e found p a r s e E r r o r sDim p a r s e E r r o r s As C o l l e c t i o n’ _______________________________________’ Get Map

10 Dim pMxDoc As IMxDocumentDim pMap As IMapS e t pMxDoc = ThisDocumentS e t pMap = pMxDoc . FocusMap

’ _______________________________________15 ’ Get t h e s h a p e f i l e wi th a l l t h e d a t a

Dim p F e a t u r e C l a s s As I F e a t u r e C l a s sS e t p F e a t u r e C l a s s = O p e n S h a p e f i l e ( " Z: < Fo lde r >" , " Typ5_O_Res " )Dim pLayer As I F e a t u r e L a y e rS e t pLayer = pMap . Layer ( 0 )

228

20 Dim p F e a t u r e C l a s s S e l As I F e a t u r e C l a s sS e t p F e a t u r e C l a s s S e l = pLayer . F e a t u r e C l a s s’ _______________________________________’ c r e a t e c u r s o r t o loop t h r o u g h " b u i l d i n g _ t y p e "Dim IndexFID As I n t e g e r

25 IndexFID = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " FID " )Dim IndexOccupancy As S t r i n gIndexOccupancy = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " Occupancy " )Dim I n d e x S t o r y A r e a As DoubleI n d e x S t o r y A r e a = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " F l o o r A r e a " )

30 Dim IndexLiq21 As DoubleIndexLiq21 = p F e a t u r e C l a s s S e l . F i n d F i e l d ( " Avg_Liq21 " )

’ _______________________________________35 ’ D e f in e c u r s o r f o r s e l e c t e d f e a t u r e s

Dim pCurso r As I F e a t u r e C u r s o rS e t pCurso r = p F e a t u r e C l a s s S e l . Update ( Nothing , F a l s e )’ _______________________________________’ C ur s o r t h r o u g h t h e s e l e c t e d rows

40 Dim pRowSel As I F e a t u r eS e t pRowSel = pCurso r . N e x t F e a t u r e

Dim pOID As DoubleDim pQuery As I Q u e r y F i l t e r

45

’ _______________________________________’LOOP THROUGH EACH BUILDING OF THE LAYERDo While Not pRowSel I s Noth ingDim Source As I n t e g e r

50 For Source = 1 To 9’ _______________________________________’ Im po r t BPN from Hugin

Dim d As HAPI . Domain55 Dim BN As S t r i n g

Dim Netze (1 To 3)Netze ( 1 ) = " EX3_Typ5_O_Res_S "Netze ( 2 ) = CSt r ( Source )Netze ( 3 ) = " _ P o i s s o n . n e t "

60 BN = J o i n ( Netze , " " )S e t d = HAPI . LoadDomainFromNet (BN, p a r s e E r r o r s , 10)

’ g e t t h e node wi th l a b e l " . . . " and name " . . . " from t h e domainDim NodeLiq As HAPI . Node

65 S e t NodeLiq = d . GetNodeByName ( " L i q u e f a c t i o n " )Dim NodeLiqTable As HAPI . Tab leS e t NodeLiqTable = NodeLiq . Tab le

’ I n i t i a l i z e70 Dim l a u f As I n t e g e r

Dim M As I n t e g e rM = IndexLiq21

229

C Software Codes

For l a u f = 0 To 83 S tep 2NodeLiqTable . Data ( l a u f ) = 1 − pRowSel . Value (M)

75 NodeLiqTable . Data ( l a u f + 1) = pRowSel . Value (M)M = M + 1Next l a u f

Dim FID , Netz (1 To 3 ) , BPN80 FID = pRowSel . Value ( IndexFID )

Netz ( 1 ) = CSt r ( FID )Netz ( 2 ) = " _ "Netz ( 3 ) = BNBPN = J o i n ( Netz , " " )

85 d . SaveAsNet (BPN)Next Source

’ _______________________________________’ Update c u r s o r o f s e l e c t e d DHM f e a t u r e s

90 pCurso r . U p d a t e F e a t u r e pRowSelS e t pRowSel = pCurso r . N e x t F e a t u r e

Loop’ _______________________________________’END LOOP THROUGH EACH BUILDING OF THE LAYER

95

End Sub’ _______________________________________

f u n c t i o n UpdateBPN

% From GIS Map the ID of the buildings with observed damage state along% with the damage state is loaded

5 load BU_BuildingID_Damage

bpn= a c t x s e r v e r ( ’HAPI . G l o b a l s ’ ) ;

%Initially the nodes for modeling uncertainty are normal distributed10 P_Lambda_Yellow = [ 0 . 1 5 9 0 .682 0 . 1 5 9 ] ;

P_Zeta_Yel low = [ 0 . 1 5 9 0 .682 0 . 1 5 9 ] ;P_Lambda_Red = [ 0 . 1 5 9 0 .68 2 0 . 1 5 9 ] ;P_Zeta_Red = [ 0 . 1 5 9 0 .682 0 . 1 5 9 ] ;

15 f o r i =1:203m= f ( i , 1 ) ;%ID of the building with observed damage staten= f ( i , 2 ) ;%damage state of the building 0=No damage, 1=Damagei f n==1

n=n +1;%State number for red is 2 and not 1, hence the addition of 120 end

Fi l ename =[ ’Z : \ D i s s e r t a t i o n a n a l y s i s \ S e i s m i c h a z a r d a n a l y s i s \ PhD \ . . .GIS−BPN Outpu t \ BU_O_Res \ ’ num2str (m) ’ _BU_O_Res_S3 . n e t ’ ] ;

domain= in vo ke ( bpn , ’ LoadDomainFromNet ’ , F i lename , 0 , 0 ) ;

25

ndLambda_Yellow= in vo ke ( domain , ’ GetNodeByName ’ , ’ Lambda_Yellow ’ ) ;ndZeta_Yel low = in vo ke ( domain , ’ GetNodeByName ’ , ’ Ze ta_Yel low ’ ) ;

230

ndLambda_Red= i nv ok e ( domain , ’ GetNodeByName ’ , ’ Lambda_Red ’ ) ;ndZeta_Red= in vok e ( domain , ’ GetNodeByName ’ , ’ Zeta_Red ’ ) ;

30 ndDamage= i n v ok e ( domain , ’ GetNodeByName ’ , ’ Damage ’ ) ;

% Initially the nodes for modeling uncertainty are normal distributed,% recursively given the damage state of the i-th building they are% updated

35 f o r i =1 :3s e t ( ndLambda_Yellow . Table , ’ Data ’ , ( i −1) , P_Lambda_Yellow ( i ) ) ;s e t ( ndZeta_Yel low . Table , ’ Data ’ , ( i −1) , P_Zeta_Yel low ( i ) ) ;s e t ( ndLambda_Red . Table , ’ Data ’ , ( i −1) , P_Lambda_Red ( i ) ) ;s e t ( ndZeta_Red . Table , ’ Data ’ , ( i −1) , P_Zeta_Red ( i ) ) ;

40 end

i n vo ke ( domain , ’ Compile ’ ) ;i n vo ke ( ndDamage , ’ S e l e c t S t a t e ’ , n ) ;i n vo ke ( domain , ’ P r o p a g a t e ’ , ’ hEqui l ib r iumSum ’ , ’ hModeNormal ’ ) ;

45 f o r l =1 :3P_Lambda_Yellow ( l )= g e t ( ndLambda_Yellow , ’ B e l i e f ’ , ( l −1 ) ) ;P_Zeta_Yel low ( l )= g e t ( ndZeta_Yel low , ’ B e l i e f ’ , ( l −1 ) ) ;P_Lambda_Red ( l )= g e t ( ndLambda_Red , ’ B e l i e f ’ , ( l −1 ) ) ;P_Zeta_Red ( l )= g e t ( ndZeta_Red , ’ B e l i e f ’ , ( l −1 ) ) ;

50 endend

f i l e n a m e =[ ’Z : \ D i s s e r t a t i o n a n a l y s i s \ S e i s m i c h a z a r d a n a l y s i s \ PhD \ . . .GIS−BPN Outpu t \ BU_O_Res \ ’ num2str (m) ’ _BU_O_Res_S3_FINAL . n e t ’ ] ;

55 i n vo ke ( domain , ’ SaveAsNet ’ , f i l e n a m e ) ;P_Lambda_YellowP_Zeta_Yel lowP_Lambda_RedP_Zeta_Red

231

Curriculum Vitae

Yahya Y. BayraktarliBorn 3 February 1972 in Mannheim (D)Citizen of Germany

1995 Bachelor of Science in Civil EngineeringBogazici University, Istanbul, Turkey

1995 - 1999 Diploma in Structural EngineeringUniversity of Karlsruhe, Karlsruhe, Germany

1999 - 2000 Student exchangeINSA Lyon, Lyon, France

2000 Diploma ThesisUniversity of Karlsruhe, Karlsruhe, Germany

2000 - 2002 Research and teaching assistantCollaborative Research Center 461University of Karlsruhe, Karlsruhe, Germany

2002 - 2008 Research and teaching assistantETH Zurich, Zurich, Switzerland

2008 - Technical scientific staffRisk analyses of existing nuclear power plantMühleberg, Bern, SwitzerlandBernische Kraftwerke AG

233

construction and application of bayesian probabilistic network for earthquake management

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