a method for assessing risk to engineering...

VILNIUS GEDIMINAS TECHNICAL UNIVERSITY

Virmantas JUOCEVIČIUS

A METHOD FOR ASSESSING RISK TO ENGINEERING STRUCTURES EXPOSED TO ACCIDENTAL ACTIONS

DOCTORAL DISSERTATION

TECHNOLOGICAL SCIENCES, CIVIL ENGINEERING (02T)

Vilnius 2011

Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 2007–2011. Scientific Supervisor Prof Dr Egidijus Rytas VAIDOGAS (Vilnius Gediminas Technical University, Technological Sciences, Civil Engineering – 02T). VGTU leidyklos TECHNIKA 1915-M mokslo literatūros knyga http://leidykla.vgtu.lt ISBN 978-9955-28-998-2 © VGTU leidykla TECHNIKA, 2011 © Virmantas, Juocevičius, 2011 [email protected]

VILNIAUS GEDIMINO TECHNIKOS UNIVERSITETAS

Virmantas JUOCEVIČIUS

YPATINGŲJŲ POVEIKIŲ STATINIAMS SUKELIAMOS RIZIKOS VERTINIMO METODAS

DAKTARO DISERTACIJA

TECHNOLOGIJOS MOKSLAI, STATYBOS INŽINERIJA (02T)

Vilnius 2011

Disertacija rengta 2007–2011 metais Vilniaus Gedimino technikos universitete. Mokslinis vadovas prof. dr. Egidijus Rytas VAIDOGAS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, statybos inžinerija – 02T).

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Abstract The dissertation presents a method developed for assessing the risk to constructed facilities posed by accidental actions. The proposed method is applied to the analysis of structures subjected to accidental actions induced by fire, explosion and extreme wind. The dissertation consists of introduction, five chapters, conclusions, lists of references and papers published by the author of the dissertation, as well as five annexes.

Chapter 1 presents a review of published work on the assessment of risk to structures and non-structural property posed by accidental actions. The review covers general aspects of risk assessment and application of risk analysis tools to the analysis and design of structures. The core of the review is an application of Bayesian methods, which prevail in the field of risk assessment, to a risk-based structural analysis.

Chapter 2 proposes a method developed for the assessment of damage to structures from accidental actions. This method combines structural analysis with Bayesian handling of information related to accidental actions and response of structures to these actions. The method allows to estimate probabilities of the damage to structures caused by accidental actions.

Chapter 3 deals with an application of the proposed method to the estimation of fire risk to structures. It is shown how to apply this method to the estimation of failure probability of a timber beam damaged by fire in an industrial building. This chapter shows how to handle uncertainties in the mathematical model of timber charring and safety margin of the beam.

Chapter 4 describes how to apply the proposed method to a design of a safety barrier, which provides protection against an accidental explosion. The accidental action considered in this study is a blast generated by an explosion of a railway tank car. Bayesian modelling of uncertainties is applied to a mechanical model of barrier elements.

Chapter 5 presents an application of the proposed method to a case of extreme wind risk assessment. The structure under analysis is a reinforced concrete chimney exposed to a hurricane wind action. The analysis of the chimney takes into account uncertainties related to a mechanical model of this structure.

Results of this dissertation were presented in 17 scientific publications: six journal articles included in the Thomson ISI Web of Science data base, two papers published in the conference proceedings included in the Thomson ISI Proceedings data base, one article published in a reviewed scientific journal, five articles published in proceedings of international conferences and three papers included in the proceedings of the Lithuanian conference of young scientists.

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Reziumė Disertacijoje aprašomas metodas, sukurtas vertinti ypatingųjų poveikių

statiniams keliamą riziką. Darbe pateikti trys šio metodo taikymo pavyzdžiai. Disertaciją sudaro įvadas, penki skyriai, išvados, literatūros šaltiniai, autoriaus publikacijų sąrašas ir penki priedai.

Pirmajame skyriuje apžvelgiama mokslinė literatūra, kurioje tiriami rizikos vertinimo metodai ir aptariama jų taikymo konstrukcijoms skaičiuoti galimybė. Pateikiama trumpa ypatingųjų poveikių klasifikacija bei konstrukcijų, kurias jie gali paveikti, pažeidžiamumo modeliavimas.

Antrajame skyriuje aprašomas metodas, sukurtas ypatingųjų poveikių keliamų konstrukcijų pažaidų tikimybėms vertinti. Metodas derina statybinių konstrukcijų patikimumo teorijos ir Bajėso statistinės teorijos priemones. Jis gali būti taikomas tiek projektuojant konstrukcijas, tiek vertinant pramoninių objektų riziką.

Trečiajame skyriuje aprašoma, kaip taikyti siūlomą metodą vertinant medinėms konstrukcijoms keliamą gaisro riziką. Ta rizika išreiškiama gaisro pažeistos konstrukcijos suirimo tikimybe, kuri yra vertinama apriorine ir aposteriorine tankio funkcijomis, skaičiuojamomis Bajėso statistinės teorijos priemonėmis. Šiame skyriuje nagrinėjama medinė gaisro veikiama pramoninio pastato stogo konstrukcijos sija. Gaisro poveikis modeliuojamas maža statistine gaisro charakteristikų imtimi.

Ketvirtajame skyriuje pateiktas siūlomo metodo taikymas projektuojant sprogimo veikiamą apsauginį barjerą. Atsižvelgiama į barjero elementų mechaninio modelio neapibrėžtumus. Ypatingasis poveikis, tirtas šiame skyriuje, yra geležinkelio cisternos sprogimo sukeliama slėgio banga. Šis poveikis nusakomas maža statistine sprogimo bangos viršslėgio reikšmių imtimi.

Penktajame skyriuje pateikiamas siūlomo metodo taikymas uraganinių vėjų atveju. Skyriuje skaičiuojama konstrukcija – gelžbetoninis kaminas, kurį gali pažeisti uraganinis vėjas. Ypatingasis poveikis yra išreiškiamas maža statistine imtimi, sudaryta iš per paskutinį penkiasdešimtmetį matuotų uraganinių vėjų greičių. Skaičiuojant kaminą atsižvelgta į neapibrėžtumus, susijusius su mechaniniu šios konstrukcijos modeliu.

Darbo rezultatai paskelbti septyniolikoje mokslinių straipsnių: šeši – recenzuojamuose mokslo žurnaluose, įtrauktuose į Thomson ISI Web of Science duomenų bazę, du – konferencijų leidiniuose, įtrauktuose į Thomson ISI Proceedings duomenų bazę, vienas – recenzuojamame mokslo žurnale; penki – tarptautinių konferencijų leidiniuose; trys – kituose leidiniuose.

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Notation

Abbreviations AA − accidental action BLEVE − boiling liquid expanding vapour explosion BST − Bayesian statistical theory d.f. − distribution function FF − fragility function MC − Monte Carlo p.d. − probability distribution p.d.f. − probability density function QRA − quantitative risk assessment r.v. − random variable SDOF − single degree of freedom SRA − structural reliability analysis TNT − trinitrotoluene

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Symbols Latin AA − random event of occurrence of AA Ai − the ith accidental event B − bias factor; number of bootstrap samples Bi − the ith basic event C − equivalent mass of TNT Ci − outcome (losses) due to occurrence of the ith damage event CD − drag coefficient d − depth of char Df − a range of variables where the limit state function g(y) becomes

negative Di − the ith damage event DL;T − density of liquid at the temperature and pressure existing in the

container DV;T − density of saturated vapour at the temperature and pressure existing in

the container ed − design value of action effect Ei,j − the jth event in the ith scenario ey,i − correction term of the ith observation of vector iy value ez,i − correction term of the ith observation of vector iz value fX(x) − joint p.d.f. of the ramdom vector X fY(y) − joint p.d.f. of the ramdom vector Y fZ(z) − joint p.d.f. of the ramdom vector Z Fi(y | θ)

iF~ (y | θ)

iF iF

−

− − −

FF with the random parameter vector Θ developed for the damage event Di

predictive fragility of y given θ maximum likelihood estimate of conditional fragility mean estimate of conditional fragility

FX(x) − joint d.f. of the ramdom vector X FΨ(ψ) − joint d.f. of the ramdom vector Ψ F − event of structural failure Fr(AA) − frequency of the random event AA

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Fri − likelihood (usually annual frequency) of the ith damage event g(z| θ) limit state function g of the basic variables z G(θ | M, H) − conditional model, where θ is the vector of input parameters gi(z, y | θ)

ig (z, y | θ)

− −

the ith limit state function with the argument vectors z and y as well as the parameter vector θ the ith candidate limit state function with arguments z and y and parameters θ

Gi(θi | Mi, H) G

− − solution of the ith conditional model weighted average of the solutions of conditional model G(θ | M, H)

H − set of knowledge and beliefs i − incident impulse of the pressure signal I(·) − binary function, which is equal to 1 if the argument is true and 0

otherwise K − hydraulic conductivity k − value of hydraulic conductivity variable K; specific heat of vapour k* − expert estimate of the likelihood function π(k* | k) L(data | θ) − likelihood function used to represent the information in “data”

)|( µµnBL ˆˆ − estimate of the likelihood function related to the damage probability µ Li − likelihood (usually frequency or annual probability) of the ith event M − set of model assumptions that define the model; Mach number Mr − reflected Mach number M~ − epistemic random variable obtained by averaging out aleatory

uncertainty in the model ϕ(X | Ψ) md − design value of safety margin m(·) − function relating epistemic r.v. M~ to Ψ nD − number of foreseeable damage events N − number of elements in the population p − cumulative probabilty bp′ − the bth bootstrap sample obtained by resampling from the original

sample p P(A | B) − conditional probability of A given B P(Ci|E0) − conditional probability of the consequences Ci given an occurrence of

the initiating event E0 P(Di|AA) − conditional probability of the damage event Di given an occurrence of

AA

x

P(Di| y) − probability of the damage event Di given AA with the fixed characteristics y; FF

p(Mi | H) − analyst’s probability that the ith model (set of assumptions) is true Pa − initial atmospheric pressure iP~ − random variable defined as FF with random AA characteristics,

P(Di | Y) iP − upper limit of bootstrap confidence interval

iP − lower limit of bootstrap confidence interval Pf − failure probability Pfl − probability of failure of the lth segment Pmax − peak overpressure pj − element of the fictitious sample p p − fictitious sample consisting of values of the FF P(Di | y); original

sample in terms of boostrap resampling )/2(1 γ−Bp − the (1–γ)/2th empirical percentile of the sample bp (b = 1, 2, … , B) )/2(1 γ+Bp − the (1+γ)/2th empirical percentile of the sample bp (b = 1, 2, … , B)

bp − mean of the bth bootstrap sample bp′ pd(y) − FF developed for the damage event Di and AA characteristics y rp& − rate of reflected overpressure increase Q − TNT charge mass q − specific discharge R − distance from the charge centre rd − design value of resistance Si − significance (magnitude) of the consequences due to the ith damage

event td − positive phase duration of the pressure signal; design working life tf − time to failure treq − required time of fire resistance WL − liquid weight X − vector of random basic variables; vector describing characteristics of

the exposure situation in which an AA can occur; random vector of initial conditions of the fire

x − value of X Y − random vector of characteristic of AA

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Y

iy

−

−

characteristics of AA; vector of variables describing the demands on the structural element measured values of the vector yi

yj − the jth element of the sample y containing values of characteristics of AA

y − statistical sample of action characteristics y′ − experimental sample of AA chatracteristics which are considered

highly relevant to exposure situation Y’ − vector of random variables (y1,Y2,…,Yny) with a fixed y1 y(t) − signal (time history) of fire characteristics z iz

− − value of Z measured values of the vector zi

Greek α − level of a conservative percentile β(⋅) − function relating the rate of charring to the signal y(t) β(y| θ) − conditional generalized reliability index for given θ γ − confidence level (coverage probability) of confidence interval; model

error term ϕ(X | Ψ) − mathematical model of an AA with the vector X describing

characteristics of exposure situation and with the vector Ψ of parameters of ϕ(⋅)

∆ − scaled distance κ(⋅) − kernel function λ − parameter of Poisson p.d. or Poisson process λ − vector of parameters expressing uncertainty in Pf µ − mean value, damage probability

nµ − mean value of the fictitious sample p consisting of n values of the FFP(Di | y) (original sample in terms of boostrap resampling)

π(µ) − prior p.d.f. of µ π(µ | data) − posterior p.d.f. of µ after “data” is obtained π(λ) − p.d.f. expressing the epistemic uncertainty in the estimate of Pf π(θ) − prior p.d.f. for the unknown parameter θ π(θ | data) − posterior p.d.f. of θ after “data” is obtained

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π(θ, M | H) − p.d.f. expressing our beliefs regarding the parameter θ and the validity of model assumptions M, within the entire body of knowledge and beliefs H

πi (θi, Mi | H) − p.d.f of the parameter vector θi of the ith model )|( nµµπ − estimate of a posterior density

Π(p) − d.f. expressing the epistemic uncertainty in the estimate of Pf )|(1 nµαΠ ˆˆ − − estimate of the inverse of the distribution function )(1 ⋅−Π

− vector of parameters − posterior maximum likelihood estimator

θ θ θ − posterior mean vector σ − standard deviation σ′ − Stefan-Boltzmann constant

jv10 − 10-min mean wind velocities at the height of 10.2 m Ψ − the vector of those parameters of the model ϕ(⋅) which are uncertain in

the epistemic sense ψ − value of the random vector Ψ

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Contents

INTRODUCTION ........................................................................................................... 1 Formulation of the problem........................................................................................ 1 Topicality of the problem ........................................................................................... 2 Research object .......................................................................................................... 3 The aim of the work ................................................................................................... 3 Tasks of the work ....................................................................................................... 3 Methodology of research............................................................................................ 3 Scientific novelty........................................................................................................ 4 Practical significance of the work results ................................................................... 5 The statements presented for the defence................................................................... 5 Approval of the results ............................................................................................... 6 Dissertation Structure ................................................................................................. 8

1. ASSESSING THE RISK POSED BY HAZARDOUS PHENOMENA: A REVIEW FROM THE STANDPOINT OF STRUCTURAL ENGINEERING .............. 9 1.1. Quantitative risk assessment as general methodology for dealing with industrial accidents ............................................................................................ 10 1.1.1. General expressions of risk posed by industrial accidents....................... 10 1.1.2. Mathematical tools of dealing with uncertainties related to rare and hazardous phenomena........................................................................................ 12 1.1.3. Utilisation of expert knowledge as potential source of information on rare and hazardous phenomena.......................................................................... 14

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1.2. Attempts to combine quantitative risk assessment and the structural reliability theory ................................................................................................ 17 1.2.1. General possibilities of combining risk assessment and structural analysis .............................................................................................................. 17 1.2.2. Application of structural reliability analysis in the Bayesian frameworks of risk assessment .......................................................................... 18

1.3. Dealing with accidental actions in the structural engineering and structural reliability theory ................................................................................................ 22

1.4. Investigations in assessing fragility of structures to accidental actions............. 24 1.4.1. The concept of structural fragility ........................................................... 24 1.4.2. Mathematical definition of fragility......................................................... 25 1.4.3. Quantification of uncertainties related to fragility functions ................... 26

1.5. First chapter conclusions ................................................................................... 31 2. METHOD DEVELOPED FOR THE ESTIMATION OF RISK TO STRUCTURES POSED BY ACCIDENTAL ACTIONS ............................................. 35 2.1. Assessing risk to structures due to accidental actions: probabilistic problem statement............................................................................................................ 35 2.1.1. The probability and risk of damage ......................................................... 35 2.1.2. Expression of the damage probability through fragility function ............ 37

2.2. Estimation of the probability of damage to structures due to accidental actions: background of the proposed method .................................................... 39 2.2.1. The problem with the amount of statistical data on accidental actions.... 39 2.2.2. The classical statistical approach to an estimation of the damage probability ......................................................................................................... 40 2.2.3. Utilising different sources of information on accidental actions ............. 41

2.3. A proposition of method for a Bayesian estimation of the damage probability 44 2.3.1. Specifying prior for damage probability.................................................. 44 2.3.2. Application of Bayesian bootstrap to the updating of the damage probability ......................................................................................................... 45

2.4. Discussion on the implementation of the proposed method .............................. 48 2.5. Second chapter conclusions............................................................................... 50

3. APPLICATION OF THE PROPOSED METHOD TO ASSESSING THE FIRE DAMAGE TO STRUCTURES ..................................................................................... 53 3.1. Industrial building with fire hazard ................................................................... 53 3.2. Accidental action characteristics obtained by fire simulation ........................... 55 3.3. Developing fragility function for a timber beam............................................... 57 3.4. Failure probability of the beam damaged by fire............................................... 65 3.5. Third chapter conclusions.................................................................................. 71

4. ASSESSING DAMAGE FROM AN ACCIDENTAL EXPLOSION BY MEANS OF THE PROPOSED METHOD .................................................................................. 73 4.1. Situation of explosion accident and safety barrier............................................. 73 4.2. Statistical sample of explosive action values..................................................... 77 4.3. Fragility function of a safety barrier component ............................................... 85

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4.4. Failure probability of barrier component........................................................... 89 4.5. Fourth chapter conclusions................................................................................ 92

5. EXTREME WIND DAMAGE ASSESSMENT BY APPLYING THE PROPOSED METHOD................................................................................................. 95 5.1. Reinforced concrete chimney exposed to the hazard of hurricanes................... 96 5.2. Statistical data on hurricane winds in Lithuania................................................ 98 5.3. Wind fragility function of reinforced concrete chimney ................................. 100 5.4. Estimation of chimney failure probability....................................................... 103 5.5. Fifth chapter conclusions................................................................................. 108

GENERAL CONCLUSIONS ...................................................................................... 109 REFERENCES ............................................................................................................ 113 THE LIST OF SCIENTIFIC AUTHOR'S PUBLICATIONS ON THE SUBJECT OF THE DISSERTATION.......................................................................................... 123 ANNEXES................................................................................................................... 127 Annex A. Spreadsheet file with the results of fire simulation by means of CFAST computer code ........................................................................... 129

Annex B. Results of the analysis of extreme winds recorded in Lithuania in the last 48 years ............................................................................................ 146

Annex C. Computer code “Timber beam” ............................................................. 159 Annex D. Computer code “Barrier” ....................................................................... 173 Annex E. Computer code “Chimney” .................................................................... 185

1

Introduction

Formulation of the problem This work considers in the broad sense the phenomenon of industrial accidents. They are well-known due to their severity and catastrophic losses which sometimes involve the loss of human lives. Industrial accidents can be broadly subdivided into two categories:

� Accidents involving release of hazardous materials, the spread of which can cause harm to people and damage to environment; such accidents occur without damage to the objects of structural and mechanical engineering. An often cited example of the accident belonging to this category is the Bhopal disaster (December 2, 1984).

� Accidents occurring as sudden and unexpected release of large amounts of energy (explosions, catastrophic fires, collisions, landslides, etc.) which can cause damage to structures and non-structural property; clearly, such accidents can also cause release of hazardous substances; however, the main contributor to the consequences of this type of accidents is usually serious mechanical and thermal damage.

The present work is devoted to the assessment of the damage which takes place during the accidents belonging to latter category. The relevance of this

2 INTRODUCTION

work consists in the burning actuality of the accidents involving mechanical damage. Despite the progress in various field of engineering, accidents involving explosions, collisions and fires are still the everyday phenomenon on the worldwide scale, and almost weakly phenomenon on the nationwide scale in most countries. Needless to say that extreme natural phenomena (earthquakes, severe winds, floods, landslides, etc.) are also an often cause of accidents in industrial facilities.

Topicality of the problem The prevention of industrial accidents is unthinkable without assessment of risk posed by hazardous activities in industry. The quantification of risk requires both assessment of accident consequences and likelihood of their occurrence. The risk of an industrial accident can be a relatively complex concept; however, both consequences and their likelihood will in many cases involve damage to constructed facilities. Thus it can be indispensable for the risk assessment to predict damage to specific structures by the estimation of the probability that this damage will occur in the course of an accident. An industrial accident is a natural subject of the quantitative risk assessment (QRA). It is natural to assume that the “structural part” of the risk should also be estimated by means of QRA. From the structural point of view, the industrial accident is a physical process which generates abnormal and unexpected actions and these can be imposed on structures during the accident. In the European design codes these actions are called the accidental actions (AAs). A full scale assessment of the risk related to a potential industrial accident will inevitably involve an analysis of exposed structures for the AAs.

The topicality of the problem considered in the dissertation consists to a large margin in the need to deal with usually very limited statistical data on characteristics of AAs. Experiments imitating the physical phenomena of AAs are usually expensive. The real world observations of AA characteristics obtained from the industrial accidents, which happened in the past, can yield only scarce data, if any at all. This is due to the fact that accidents involving damage to structures are usually rare and highly unexpected events. People as a rule are not prepared to measure characteristics of AAs in the course of such accidents. Consequently, the engineer will usually be faced with the necessity to assess potential damage by means of very limited and usually experimental information on a specific AA. This can be done, at least in some cases, by applying the method proposed in the dissertation.

INTRODUCTION 3

Research object The object of research is damage to constructed facilities caused by AAs which are induced during industrial accidents and extreme natural phenomena. It is investigated how to predict this damage by applying methods of QRA.

The aim of the work The aim of the dissertation was an assessment of the risk to constructed facilities posed by AAs. The core of the risk assessment was an estimation of probabilities of foreseeable damage events. Such estimation required to combine methods of QRA and structural reliability theory.

Tasks of the work 1. To develop a method which allows to assess the damage to structures

due to AAs by applying principles of QRA and structural reliability analysis (SRA). The method should be aimed at an estimation of probabilities of potential damage. Estimates of these probabilities must be suited to the integration into a general framework of QRA.

2. To achieve that the proposed method would allow to combine two sources of information on an AA typically available for the engineer: a mathematical model used for predicting AA and a statistical sample consisting of direct experimental measurements of the AA. This information must be used for choosing a prior density of the damage probability and updating this density to a posterior density.

3. To build an approach which allows to estimate the probability of damage to structure due to an AA in the case when information on this AA is expressed only by a statistical sample which is small from the standpoint of the classical statistics. To apply the method of bootstrap resampling to such an estimation.

4. To apply the method proposed in the dissertation to assessing damage to structures subjected to fires, accidental explosions and extreme winds.

Methodology of research The research methodology was formed by a combination of methods developed and widely used in the fields of QRA and SRA. The methodological core of the

4 INTRODUCTION

research was the Bayesian approach to QRA. This approach underpins most methods of QRA; however, it was only sparsely applied to prediction of AAs and design of structures for AAs. A stochastic simulation (Monte Carlo method) was used as the main computational means for an estimation of probabilities of damage caused by AAs.

Scientific novelty The main element of scientific novelty consists in combining methods of QRA and SRA in an attempt to predict structural damage caused by AAs. As AAs and damage from them are natural subjects of QRA, the damage assessment was carried out by an extensive application of QRA methods to structural analysis. The method of damage assessment proposed in the dissertation can be considered to be an extension of SRA. This method is based on SRA; however, the new element is that results of its application express uncertainties in the values of damage probabilities. These uncertainties can be quantified, for instance, by Bayesian prior and posterior distributions. They match naturally a methodological framework of QRA. Estimates of damage probabilities allow to integrate structural damage into the total spectrum of consequences to be identified in a full-scale QRA.

The previous approaches to damage assessment were based on deterministic methods of conventional structural analysis and classical methods of SRA. They did not allow to utilise specific kinds of knowledge related to individual AAs, first of all, subjective knowledge of experts. Such knowledge can be indispensable for assessing damage from many kinds of AAs. This damage must be often assessed in the situation of limited hard data (post-mortem and experimental data). Subjective knowledge allows to compensate, at least in part, the scarcity of such data. Thus the proposed method, which is based on the Bayesian reasoning has the advantage of an application of different sources of information including expert knowledge. As far as we know such an approach has not been developed until now in the field of the analysis and design of structures for AAs. We think that the results obtained in the dissertation will allow to carry out the damage assessment in a more realistic and methodologically consistent way.

INTRODUCTION 5

Practical significance of the work results The practical significance of the results achieved in this work is a possibility to design structures for AAs by a consistent application of QRA methods. This will allow to increase safety of industrial facilities and transportation objects, which are prone to accidents capable to induce severe AAs. In addition, the method proposed in this work will make the design of structures for AAs more adequate and reliable as compared to the design based on the traditional deterministic approach. The development of the method proposed in this work resulted in several computer codes and these can also contribute to practical implementation of the results.

The statements presented for the defence The main propositions of the dissertation can be expressed by considering the phenomenon of structural damage due to AAs from the standpoint of QRA. The principal proposition is that this phenomenon is a natural subject of QRA. Dealing with this phenomenon by means of the classical approaches prevailing in the today’s structural design has obvious shortcomings. In line with the idea to assess the damage from AAs by applying methodological means of QRA we found that:

1. The phenomenon of AAs and damage to structures caused by AAs are a natural subject of QRA. Dealing with this phenomenon by means of the deterministic approaches prevailing in the today’s structural design practice has obvious shortcomings. The deterministic approaches do not provide adequate means for modelling uncertainties related to AAs and damage due to AAs.

2. Knowledge available about a specific AA and potential damage which can be caused by this AA can have different forms. One part of this knowledge is general information which allows an approximate assessment of AA characteristics and damage probability. Another part of this knowledge can be a small-size sample of AA characteristics. This sample can be highly relevant to the situation of exposure of a structure to a specific AA. A combination of these two sources of information is possible within the scheme of the Bayesian updating.

3. In the case where Bayesian reasoning is applied to damage assessment, a natural result of this assessment can be a prior or posterior distributions

6 INTRODUCTION

expressing epistemic (state-of-knowledge) uncertainty in the damage probability. Practical decisions concerning the potential damage from an AA can be made by applying conservative percentiles of these distributions.

4. In the specific case where information about an AA is available in the form a small-size sample of its characteristics and it is difficult to develop an adequate prior distribution of the damage probability, this probability can be estimated by confidence intervals computed by applying the method of bootstrap resampling. If necessary, measures of epistemic uncertainty can be introduced in such estimation.

Approval of the results Results of this dissertation were presented in 17 scientific publications: six journal articles included in the Thomson ISI Web of Science data base, two papers published in the conference proceedings and included in the Thomson ISI data base, one article published in a reviewed scientific journal, five articles published in proceedings of international conferences and three papers included in the proceedings of the Lithuanian conference of young scientists.

The main results of the investigation described in this dissertation where reported in the following scientific conferences:

� 10th Conference of Lithuanian Young Scientist „Science – Future of Lithuania“ thematic conference “STATYBA”, Vilnius, Lithuania, 2007;

� 9th International Conference Modern Building Materials, Structures and Techniques, Vilnius, Lithuania, 2007;

� 11th Conference of Lithuanian Young Scientist „Science – Future of Lithuania“ thematic conference “STATYBA”, Vilnius, Lithuania, 2008;

� ESREL 2008 and 17th SRA − Europe Conference, Valencia, Spain, 2008;

� COMPDYN 2009: ECCOMAS thematic conference on computational methods in structural dynamics and earthquake engineering, Rhodes island, Greece, 2009;

� 10th International Conference Modern Building Materials, Structures and Techniques, Vilnius, Lithuania, 2010;

� 16th International Conference “Mechanika 2011”, Kaunas, Lithuania, 2011.

INTRODUCTION 7

Fig. 1. The structure of the dissertation

8 INTRODUCTION

Dissertation Structure The dissertation consists of an introduction, five numbered chapters, conclusions, list of references and author’s publications and five annexes. The structure of the dissertation is presented in Fig. 1.

The volume of the dissertation is 128 pages without annexes, the dissertation includes 117 numbered formulas, 31 figure and 8 tables. The list of references consist of 156 publications.

9

1 Assessing the risk posed by

hazardous phenomena: a review from the standpoint of structural

engineering

The first chapter presents a review and evaluation of the published work on the assessment of risk to structures and non-structural property posed by accidental (abnormal) actions. The review covers general aspects of risk assessment and application of risk analysis tools to the design of structures. The core of the review is an application of Bayesian methods prevailing in the field of risk assessment to a risk-based structural analysis. The principal aim of this review was a preparation of methodological basis for the proposal of a new method described in detail in Chapter 2 and implemented in Chapters 3 to 5. The review was partly presented in the articles published by Juocevičius (2008a), Vaidogas and Juocevičius (2008b, 2009a, 2007a).

10 1. ASSESSING THE RISK POSED BY HAZARDOUS PHENOMENA: …

1.1. Quantitative risk assessment as general methodology for dealing with industrial accidents Risk exists in almost every business activity. A consistent methodological means for expressing and estimating risk are provided by QRA (the list of abbreviations is given in page vii). A principle purpose of QRA is a derivation of risk profiles for a given situations. An industrial accident is usually a highly complex phenomenon as regards its consequences and damage to property, nature, and harm to people. A systematic prediction of these consequences is usually done in the form of risk widely used in the field of QRA. Clearly, this risk will include negative structural events; however, components of risk will cover all types of potential damage.

1.1.1. General expressions of risk posed by industrial accidents The standard expression of risk used in QRA is formulated as follows (e.g., Kumamoto and Henley 1996): Risk = {(Li, Ci, Si), i =1, 2, … , n} (1.1) where Li is the likelihood (usually frequency or annual probability) of the consequences (outcome) Ci; Si is the significance (magnitude) of the consequences. Failure consequences are the results of the action or process of failure (Kumamoto and Henley 1996, Ayyub 2003). They are outcomes or effects of failure as a logical result or conclusion (e.g., gas cloud, fire, explosion, injuries, deaths, environment damages, damage to the facility, etc.). The likelihood Li is expressed in various ways depending on the context, in which the accident may take place (Table 1.1).

Failure significance Si is the quality, condition, strictness, impact, harshness, gravity, or intensity of the failure consequences. The amount of damage that is (or that may be) inflicted by a loss or catastrophe is a measure of significance. The significance cannot be assessed with certainty, but it is preferable to try to define it in monetary terms. Significances Si can be spread as follows (Kumamoto and Henley 1996, Ayyub 2003)

=====

=

losslongevitytimelost

lossesmonetarytial)(consequenindirectinjuriesandfatalitiesofnumber

lossesmonetarydirect

5

4

3

2

1

i

i

i

i

i

i

SSSSS

S (1.2)

1. ASSESSING THE RISK POSED BY HAZARDOUS PHENOMENA: … 11

Таble 1.1. Specific forms of the likelihood Li (Kumamoto and Henley 1996; Vaidogas 2003) Measure Unit Example of hazardous event or activity

Probability Per action (involving an AA hazard)

Transportation of untypically heavy load through the bridge; the burn-in phase of hazardous system which can fail by inducing AA

Frequency Per unit time or during lifetime (of operating facility (carrying out an activity) involving AA hazard)

Rare occurrences of adverse natural phenomena imposing excessively high loads; catastrophic accident capable of causing devastating consequences and making recovery of system un-probable (uneconomical)

Probability at the time t

Per action (taken repeatedly with a changing AA hazard)

Transportation of tank with flammable liquid through the tunnel; Emergency landing of helicopter on the roof of a building design as a landing pad

Frequency associated with a time interval (t1, t2)

During time interval (in which an AA hazard ca be considered to be constant)

Period of very high precipitation which can cause heavy flood or landslide; Repair or upgrading of equipment in the plant running hazardous technology

It is natural to expect that a major industrial accident will cause consequences Ci which can be measured using all types of the significance Si indicated in the expression (1.2). In case where accident is caused by a sudden release of energy, the consequences Ci will include damage to structures and large structural objects which are traditionally assigned to mechanical engineering. This damage will cause direct and indirect monetary losses Si1 and Si3, loss of time Si4, and, what is possible, the loss of longevity Si5 resulting from damage accumulated during the accident.

If accidents in industry, transportation, and exploration and mining are considered from the positions of structural engineering, they can be broadly classified into two groups:

1. Accidents which involve structural failures contributing to the escalation of accident scenarios; however, these failures are only partial contributors to accident consequences Ci. These are more complex than structural failures (e.g., explosion at Nypro factory (1974), explosion at Texas city (2005) (e.g., Broadribb 2008)).


2. Accidents which occur mainly as structural failures; their consequences Ci are almost fully determined by negative structural events (e.g., collisions of vehicles with bridges, explosions of domestic gas in dwelling houses, dam failures during severe floods).

QRA can be used as a general methodology for assessing risk of the aforementioned groups of accidents. Accidents belonging to either group occur as shorter or longer sequences of physical events which are formalised in QRA as random events.

1.1.2. Mathematical tools of dealing with uncertainties related to rare and hazardous phenomena QRA is a method that evolved during the last 40 years (Apostolakis 1990, 2010ab). Its aim is to produce quantitative estimates of the risks associated with complex engineering systems such as nuclear power plants, chemical process facilities, waste repositories etc. The identification of the most likely failure scenarios Eij and the major sources of uncertainty is an essential part of QRA.

The first step in doing QRA is to structure the problem, which means to build a model for the physical situation at hand. It is built on a number of model assumptions and on a number of parameters whose numerical values are required. An essential part of problem structuring in most QRAs is the identification of accident scenarios Eij (event sequences) that lead to the consequence Ci of interest, for example, structural failure, the release of hazardous materials, and so forth.

The development of scenarios introduces model assumptions and model parameters that are based on what is currently known about the physics of the relevant process and the behaviour of systems under given conditions. For example, the failure modes of equipment during given earthquakes, the calculation of heat fluxes in a closed compartment where a fire has started, and the response of plant operators to an abnormal event, are all the results of conceptual models that rely on assumptions about how a real accident will progress. These models include parameters whose numerical values are also assumed to be available (e.g., in the case of fires, the heat of combustion of the burning fuel, the thermal conductivity of the walls of the compartment etc.); that is observable or measurable quantities. A simple example involve the Darcy equation for ground-water flow in saturated media δx

δhKq −= (1.3)


where q is the specific discharge in the x direction, h is the hydraulic head, and K is the hydraulic conductivity.

The parameter K in the model is conditional on assumption that the numerical value of the hydraulic conductivity is known. We generalize the above example and then write the solution for the conditional model as G(θ | M, H), where θ is the vector of input parameters (e.g., the hydraulic conductivity in Eq. (1.3)), M is the set of model assumptions that define the model, and H is the entire body of knowledge and beliefs. This model assumes that the numerical values of its parameters are known and that its model assumptions are true. Since there is usually uncertainty in these condition, we introduce the state-of-knowledge probability density function (p.d.f.) π(θ, M | H) which expresses our beliefs regarding the numerical values of θ and the validity of model assumptions. The lognormal distribution is frequently used in safety studies for the parameters. The p.d.f. for a variable K is given by

−−= 2

2

2)(lnexp2

1)( σµk

kσπσ,µ|kπ (1.4) where K > 0; -∞ < µ < +∞; σ > 0 (K denotes the uncertain variable, and k denotes the value of this variable). Specifying the µ and σ determines the lognormal distribution.

An unconditional model solution of the model is the weighted average of the solution of conditional models where the weights are the probabilities of the parameter values and the assumptions. For a discrete set of n models, we write [ ]∑ ∫

=

=n

iiiiiiiii H|MpθHMθπHMθGG

1)()d,|(),|( (1.5)

where Gi(θi | Mi, H) is the solution of the ith conditional model and πi (θi, Mi | H) is the p.d.f of the parameter vector θi of the ith model. The factor p(Mi | H) is the analyst’s probability that the ith model (set of assumptions is true).

Many important phenomena in safety assessments cannot be modelled by deterministic expression like Eq. (1.3); for example, the occurrence times of earthquakes of given magnitudes cannot be predicted. Various stochastic models have been proposed in the literature to calculate the probability of an event of interest. A simple model that quantifies the probability of ni events occurring in a period of time t uses the Poisson distribution !

)()inevents(i

ntλ

i ntλeH,M,t,λ|tnh i−

= (1.6)


The principal model assumption is that the interval times (that is, the times between successive events) are independent. The constant rate λ of occurrence of the events is the only parameter that may be uncertain, thus requiring a state-of-knowledge distribution π(θ, M | H). Distributions that appear in the model, such as Eq. (1.4), are sometimes called frequency or statistical distributions, so that they can be distinguished from state-of-knowledge distributions.

One of the most fundamental problems when using QRA is related to the way the results of the analyses are expressed and interpreted. Risk is a complex and difficult concept, and there is no consensus on how the risk should be expressed and interpreted. As to quantification of probability the physical situation in practice is characterized by a mixture of classical statistical approach and the Bayesian (subjective) approach (Aven 2003, 2008, 2010, 2011). The quantification of risk requires the quantification of the likelihood of rare accidental events, which normally cannot be done without employing engineering judgement. It has become evident that the problems of risk quantification cannot be handled with the methods of traditional statistics.

The alternative is the Bayesian approach, where the concepts of probability is used as the analyst’s “measure of uncertainty” or “degree of belief” (e.g., Aven and Rettedal 1998). However, this approach has not been commonly accepted; there is still a lot of scepticism among risk analysts when speaking about subjective probabilities.

1.1.3. Utilisation of expert knowledge as potential source of information on rare and hazardous phenomena The role of expert knowledge in QRA The engineering judgement is prevalent in QRA. The events or phenomena of interest are usually very rare, thus lacking significant statistical or experimental support, the opinions of experts acquire great significance. Engineering judgement, which is another, more traditional, name for expert opinion, has always played an important role in engineering work (Apostolakis 1990, 2010ab).

Physical scientists and engineers do not object to the theoretical foundation of Bayesian probability theory, but they are uncomfortable with the extensive use of judgement that QRAs require. The problems related to the elicitation and use of judgement have been recognized and investigated (Mosleh and Apostolakis 1986, Zio and Apostolakis 1996, 2010ab).

The practice of eliciting and using expert opinions became the centre of controversy with the publications of a major risk study of nuclear power plants (Ellingwood 1998, Park et al. 1998). These studies consider explicitly alternate models for physical phenomena that are not well understood and solicits the help


of expert to assess the analyst’s probability p(Mi | H) (Eq. 1.5). Objections have been raised both to the use of expert opinions and to the process of using expert opinions (e.g., the selection of experts). The probabilities p(Mi | H) are an essential part of the decision-making process. A formal mechanism exists to incorporate available evidence into p(Mi | H) (Apostolakis 1990, 2010ab). Unfortunately, many decisions cannot wait until such evidence becomes available, and assessing p(Mi | H) from expert opinions is a necessity (incidentally, such an assessment may lead to the decision to do nothing until experiments are conducted). Application of expert knowledge to the Bayesian estimation According to the theory of subjective probability, Bayes’s theorem is the only way in which a coherent analyst, i.e. an analyst whose probabilities behave according to the laws of probability, can update his state of knowledge. In the case of a single parameter θ, the Bayes’s theorem takes the form )()|(datadata)( 0 θπθL|θπ ∝ (1.7) where π0(θ) is the prior probability density function for the unknown parameter θ (prior to obtaining new information “data”); L(data | θ) is the likelihood function. It is either the conditional probability of observing “data”, given θ, or proportional to that probability. The left-hand side of the Eq. (1.7), π(θ | data), is the posterior probability density function for θ after “data” is obtained. In the denominator, the “const”, is equal to the expectation of L(data | θ) with respect to the prior distribution π0(θ). The most controversial part of a Bayesian analysis is the development of an appropriate prior distribution. It can often be the most resource-intensive work as well (Siu and Kelly 1998). An informative prior distribution is one that reflects the analyst’s beliefs concerning an unknown parameter. The development of an prior distribution can be, in principle, a challenging process, as it requires the analyst to convert his own (and typically qualitative) notions of likelihood into quantitative measures. In practice, however, the process is often less difficult, for reasons discussed in detail by Siu and Kelly (1998).

In cases where updating has not yet been performed, various formal techniques are available to incorporate available information to the prior. Three such are approaches that have been used in QRA and discussed in details by Siu and Kelly (1998) are:

� The two stage Bayes approach; � The empirical Bayes approach; � The method of maximum entropy.


Depending on the particular problem on hand, these approaches can be somewhat difficult to execute. This complication can be viewed as the price to be paid for a more structured and perhaps more robust distribution.

The construction of an appropriate likelihood function requires engineering/scientific knowledge specific to the process being modelled, as well as probabilistic modelling expertise. Once a model has been developed for the process generating the data, it is generally straightforward procedure to construct the likelihood function for a given set of data. The likelihood function, L(data | θ), is proportional to the probability of observing new information “data”, assuming that the parameter θ is actually known. In the case of multiple, conditionally independent sets of data “datai”, the likelihood function is given by L(data | θ) = ∏

ii θ|L )(data (1.8)

The likelihood function for the ith set of “data” can be developed using appropriate distribution functions, e.g., Poisson distribution. The probability that fewer than m failures have occurred, given t and λ, is derived by summing contributions from he different possible values of ni (the random number of failures) L(datai | λ) = ∑−

=

−1

0e!

)(m

n

tλ

i

n

i

i

ntλ (1.9)

where datai = {ni failures in the time interval [0, t]}; this is the standard form of data used in QRA applications for operating failures.

Expert judgement, from the standpoint of Bayesian estimation, simply represents a different form of data. Thus, as long as an appropriate likelihood function can be generated, expert judgement can be used in the estimation process. In realistic situations, when analyst has partial faith in expert, the lognormal distribution can be used as a model for the likelihood function. For the hydraulic conductivity K, given in Eq. (1.3), the likelihood function will have the form

−−= 2

2

2))ln((lnexp2

1)( σBkk

kσπk|kπ*

** (1.10)

where k* denotes the expert’s estimate; B is the bias factor and σ is a dispersion factor. Note that, according to Eq. (1.10), the median value of k* is Bk. The bias factor therefore measures the analyst’s assessment of the tendency of the expert to underestimate (B < 1) or overestimate (B > 1) the true value of k.

The dispersion factor is easiest to interpret if there is no bias (i.e., B = 1); in this case, σ is a direct measure of the expert’s expertise. A small value of σ


implies that the expert is likely to produce an estimate close to the true value (in the limit as σ as approaches 0, the lognormal likelihood function approaches a δ-function about k). A large value of σ implies that the expert may provide a good estimate, but is quite likely to provide a bad one. When the expert is biased, σ still measures the dispersion of the likelihood about the median. Now, a small value of σ implies that, in the judgement of the analyst, the expert is likely to provide an estimate close to the true value modified for bias (Bk).

It is important to observe that in this formalism, both B and σ must be provided by the analyst. Although these may be measureable in principle, they are difficult to assess for the data less, controversial situations in which expert opinion is typically called upon. Perhaps for this reason, many QRAs to date have not used a formal Bayesian approach for incorporating expert opinion; instead, they have used such methods as arithmetic or geometric averaging of the experts’ estimates. However, if expert opinions are to be combined with data, then the Bayesian formalism is needed.

1.2. Attempts to combine quantitative risk assessment and the structural reliability theory 1.2.1. General possibilities of combining risk assessment and structural analysis Methods of SRA are tools for calculating probability. Thus the models used in this type of analysis are standing in line with other reliability models, like lifetime models for mechanic and electronic equipment, reliability models for software, availability models for supply systems and models for calculating the reliability of human actions. All models of this kind can be used to calculate single probabilities that are inputs in different methods used in QRA, such as for the basic events in fault tree analysis and the branching points in event tree analysis. A special feature of methods of SRA is, however, that the influence from several random variables and failure modes may be taken into account in a single analysis. Thus, using methods of SRA, the splitting of events into detailed subevents is often not necessary to the same extent as in fault tree analysis and event tree analysis (Aven and Rettedal 1998). This makes it possible for a whole section of a fault or event tree to be replaced by a single analysis based on SRA methods. Compared to the models traditionally applied in QRA, SRA methods enables the analyst to include more knowledge in the analysis.

The event which normally generates the link between QRA and SRA is the initiating event (Guedes Soares et al. 1995), say accidental action as shown in Fig. 1.1. In this way risk analysis may be the appropriate tool to assess the


probability of a special loading event (an initiating event such as fire, explosion, earthquakes etc.) on the structure. Similarly SRA may be the appropriate tool to assess the probability of, e.g., gas leakage in process components thus causing the initiating event for an explosion or fire.

The Bayesian approach is in our opinion the suitable basis for integrated QRA and SRA modelling. In this modelling, it is necessary to include whatever relevant information is available, and the Bayesian approach provides a consistent tool for combining “hard data” and subjective information (expert opinions, engineering judgements, etc.) The classical statistical approach to risk analysis is not considered suitable for QRA and SRA. There are not sufficient “hard” data available to accurately estimate unknown parameters of the models.

1.2.2. Application of structural reliability analysis in the Bayesian frameworks of risk assessment Classical Bayesian approach This approach is considered attractive by many since it allows a systematic integration of expert opinions and experience data (Aven and Renn 2010, Aven and Vinnem 2007, Aven 2010). Further, it is relatively simple to modify the uncertainty distributions when new data become available. However, there might be practical problems by using classical Bayesian approach. A rigorous Bayesian analysis, with focus on unobservable quantities, is time consuming and difficult, requiring a substantial effort. The coherent assessment of subjective probabilities can pose some formidable difficulties in practice (Selvik and Aven 2011, Flage and Aven 2009). Assessing subjective probabilities requires

Fig. 1.1. Integration of quantitative risk assessment (QRA) and structural

reliability analysis (SRA)


comprehensive and substantial expertise, i.e. a thorough knowledge of the particular issue being addressed. In a large analysis where there may be hundreds of probability distributions to be addressed, it may be difficult to devote the desired level of care and expertise to each individual assessment. Furthermore, such assessments can be subject to a number of biases such as overconfidence.

This classical Bayesian approach deals with analysis/inference related to true, unobservable quantities, which is also the basis of classical statistics (Aven 2011, 2010, 2008, 2003). In QRA such quantities are probabilities of random events. Our attention will be focussed on the random event of structural failure. To estimate the probability of failure, a model is developed (e.g., a limit state function g(z| θ)) with the vector of parameters, θ). Then we can express the failure probability by equations Pf = P(g(Z) ≤ 0) (1.11) for a limit state function g and basic variables Z. Then we can write Pf = ∫

≤ 0})(|{)d(

zz

zzg

f (1.12) where f(z) is the joint probability density function of Z. Assuming the existence of a theoretical, true (but unknown) function f(z) and limit state function g, there will also be a true (unknown) value of Pf. Uncertainty related to the density function f(z) and the limit state function g generates the uncertainty on Pf. Consider first a situation where we ignore uncertainty related to g. And assume that our uncertainty related to f(z) is restricted to specifying a parameter (parameter vector) λ. Thus f(z) = f(z | λ). There exists a true, but unknown, value of λ. We write Pf(λ) to show the dependency on the parameter λ. Hence Pf(λ) = ∫

≤ 0})(|{)d(

zz

zλ|zg

f (1.13) Thus a formula has been established for the uncertainty distribution of Pf based on SRA methods. Here f(z | λ) expresses the aleatory uncertainty, whereas π(λ) expresses epistemic uncertainty.

In the classical Bayesian approach it is assumed that there exists a true value of Pf (e.g., Aven and Rettedal 1998). This value can be interpreted in the classical statistically sense as the relative fraction of times the failure events occur if the situation analysed were hypothetically “repeated” an infinite number of times. The true model g(z) produces the true value of Pf when the input z is true. The true values of Pf, g(⋅), θ and λ are uncertain (unobservable and unknown), and we use probabilities to express this uncertainty. We start with a prior


information about g(⋅), θ and λ, including engineering judgements, that exists before the data is observed. The a priori information is expressed by a prior probability densities π(θ) and π(λ), which reflects our initial knowledge concerning the parameters θ and λ. After having observed the experience data “data”, we apply the Bayes theorem to derive the posterior distributions π(θ | data) and π(λ | data). They express the updated knowledge on the parameters θ and λ after the data have been observed. Due to the functional relationship between Pf and θ and λ we can also establish the posterior distribution π(p | data) of Pf. This uncertainty analysis is very often done by using Monte Carlo simulation. The uncertainty distribution of Pf expressed in the form of a distribution function Π(p) can be written as Π(p) = P(Pf(λ) ≤ p) = ∫

≤ })({ f

)d(pP λ|λ

λλπ (1.14)

where π(λ) denotes the prior or posterior density function of λ. The produced distribution Π(p) reflects our uncertainty related to the true albeit unknown value of Pf. Fully Bayesian approach This approach directly provides the probabilities of the uncertain events that are relevant in the specific situation of decision making (Aven 2003, 2008, 2010). These probabilities are total in the sense that they incorporate all types of uncertainty. Thus the result itself is a total measure of uncertainty, and does not require any further discussion of “uncertainty of the estimates”, given that the model used is considered valid.

Many risk analysts are unfamiliar with this approach, and it often raises strong reaction. Some of the critical questions/comments that can be formulated (Aven and Rettedal 1998) are:

1. How can we ensure that the subjective probabilities are consistently assessed?

2. There could be a large element of arbitrariness when quantifying judgements.

3. How ca we ensure that increased knowledge will “improve the probabilities?”

Most probability judgements, without support of hard empirical data, are subject to a certain degree of inherent imprecision, described as “inexact measurement” of beliefs (Aven and Renn 2010, Aven and Vinnem 2007, Aven 2010). People are often unable to distinguish between different numbers – the different probabilities simply do not “feel” sufficiently different.


The fully Bayesian approach is characterised by a focus on observable quantities, like the occurrence or not of an accidental event, the number of accidental events in a given period of time, lost production in a period of time, etc., and subjective probabilities are used to express the uncertainty in these quantities (Selvik and Aven 2011, Aven and Steen 2010, Flage and Aven 2009). So for example consider above, we focus on the number of accidental events in the given period of time, or simply the occurrence or not occurrence of an accidental event (when it is unlikely that two or more accidental events occur during the time period of interest).

In a fully Bayesian setting all probabilities are quantifying epistemic uncertainty. The probabilities Pf (λ) (when λ is varying) represent alternative models (mathematical expressions) which we consider suitable for expressing our degree of belief concerning the occurrence of the event of structural failure, F. It is a way of standardizing the probability considerations. By introducing the conditional probabilities Pf (λ) we simplify the probability considerations by reducing the dimensions of the background information. It is not essential that the parameter λ has a physical interpretation. Allowing different values of λ is just a way of generating a class of appropriate uncertainty distributions for the failure event F.

Incorporation of SRA methods in the setting of the fully Bayesian approach is rather straightforward. Now P(g(Z) ≤ 0) is a measure of uncertainty, a degree of belief, concerning the occurrence of the event F = g(Z) ≤ 0. The values of quantities Z are uncertain (unknown) and the uncertainty is expressed by the subjective density function f(z), giving the subjective failure probability Pf = ∫

≤ 0})(|{)d(

zz

zzg

f (1.15) If we consider alternative models used to express our belief in the occurrence of the event F, we obtain alternative failure probabilities Pf (λ) = ∫

≤ 0})(|{)d(

zz

zλ|zg

f (1.16) Thus in the fully Bayesian approach the meaning of uncertainty is completely different from uncertainty in the classical Bayesian approach (Aven 2003, 2008, 2010). What is uncertain is the occurrence of the failure event, and the probability Pf expresses this uncertainty. The fact that there could be faults and weaknesses of the models g(z| θ) and f(z | λ) used does not change the interpretation of Pf. There is no sense in speaking about uncertainty of the failure probability Pf, because such a reasoning would presuppose the existence of a true value of Pf.


1.3. Dealing with accidental actions in the structural engineering and structural reliability theory AAs are introduced into the general context of structural design through the so-called design situations (EN 1991 2006, JCSS 2001). A design situation represents a certain time interval with associated hazards, service conditions and relevant limit states. The international standard ISO 2394 (1998) defines the design situation as “set of physical conditions representing a certain time interval for which the design demonstrates that relevant limit states are not exceeded”.

A classification of design situations has been universally accepted in well-known European standards and design codes, such as Eurocode (EN 1990 2002) or JCSS’s probabilistic model code (JCSS 2001). Four types of design situations are distinguished:

� Persistent design situations, which refer to the conditions of normal use. � Transient design situations, which refer to temporary conditions

applicable to the structure, e.g., during execution of repair. � Accidental situations, which refer to exceptional conditions applicable to

the structure or to its exposure, e.g., to fire, explosion, impact or the consequences of localised failure.

� Seismic design situations, which refer to conditions applicable to the structure when subjected to seismic events.

Persistent and transient situations are considered to occur with certainty. Accidental situations by definition act with a low probability during the design working life td. It is obvious that AAs are associated with accidental situations. In most cases, an occurrence of an accidental situation means an occurrence of AA. However, there are subtleties which require distinguishing between accidental situation and AA in some cases. For instance, a local failure under normal service conditions may set the structure into an accidental condition. Clearly, it will take place without a physical occurrence of AA.

The standard ISO 2394 (1998) defines AA as an action which is unlikely to occur with a significant value on a given structure over the design working life td. AA very often implies loading of short duration generating dynamic effects. It can take the structure far beyond its limits of load bearing capacity in terms of static loading (Virdi et al. 2000).

It seems that an exhaustive and universally accepted classification of AAs does not exist. Various standards and codes define the term “Accidental action” and, at the most, give several examples of AAs. For instance, the standard ISO 2394 (1998) provides the following examples in one of its annexes:


� Collisions; � Impacts; � Explosions; � Subsidence of soil; � Tornadoes in regions not normally exposed to them; � Earthquakes under specific conditions; � Fire; � Extreme erosion. The offshore standard DNV-OS-A101 (DNV 2005: 11) calls AAs the

accidental loads and presents the following examples: � Impact loads including dropped objects and collision loads; � Unintended flooding; � Loads caused by extreme weather; � Explosion loads; � Heat loads. One can see that AAs are in most cases of short duration. However,

ISO 2394 (1998) classifies “extreme corrosion” as AA. In addition, the aforementioned local failure, if it is not detected over a longer period, will cause a long lasting accidental situation. In some cases AAs may originate from the same source as variable actions acting in persistent design situations (EN 1991 2006). For example, an impact from a ship which got out of control may be the source of AA, whereas actions from fendering and mooring of ships will be variable actions.

There is yet another line of approach to the definition and hence the classification of AAs. A variable action is classified as an accidental one, if it occurs on the structure which was not designed for it. For instance, structures designed and built in regions not usually prone to earthquakes or tornadoes will be subjected to AAs if these extreme natural phenomena take place. The latter definition of AAs implies that AA can be simply ignored in some cases as they are considered extremely unlikely to occur.

A large part of this dissertation is devoted to the problem of dealing with the scarcity or absence of direct data on AAs. In addition, the problem of a rational utilisation of limited direct (experimental) statistical data is addressed. It is claimed that solving these problems will require an application of probabilistic methods developed in the field of QRA.


1.4. Investigations in assessing fragility of structures to accidental actions 1.4.1. The concept of structural fragility On the level of the design of structures for AAs the concept of fragility is expressed quantitatively as a probability of failure of the structure conditional on specific intensity of accidental action (Der Kiureghian 2000, Sasani et al. 2002, Straub and Der Kiureghian 2008, Celik and Ellingwood 2010). Mathematical expression of structural fragility is called fragility curve or fragility function (FF). FFs have been intensively applied in the field of earthquake engineering for assessing earthquake damage of special types of structures, such as structural parts of nuclear power plants and power transmission systems (Ang et al. 1996, Camensig et al. 1997, Ellingwood 1998, Hwang and Huo 1998, Fong et al. 2009), more specifically in the seismic risk assessment (Ravindra 1995, Yang et al. 2009, Li et al. 2010, Ellingwood et al. 2010). The damage to more conventional structures with respect to seismic events is also assessed using FFs (Ellingwood 2001, Dimova and Elenas 2002, Sarabandi et al. 2003, Kim and Shinozuka 2004, Schotanus et al. 2004). Another field where FFs have been used for risk assessment is prediction of wind induced damage. As regards man-made accidental actions, FFs were applied sparsely; however, recipes proposed in the earthquake and wind engineering can also be applied to developing fragility functions for such actions as: extreme fires, vehicular impacts, explosions. A rare exception of an FF developed for man-made phenomena one can mention the paper published by Low and Hao (2002), who developed FF for an accidental explosion.

AAs induced during man-made accidents are rare events based by relatively sparse data. However, the damage caused by actions of this type can be considerable. FFs developed for the man-made AAs are an indispensable mean of assessing the probability of the damage.

Often fragility is developed with respect to a single demand variable. For example, in earthquake engineering, the fragility of a structure may be developed with respect to a single measure of the ground motion intensity, e.g., the peak ground acceleration (e.g., Vanzi 1996, Lee and Rosowsky 2006, Li and Ellingwood 2007, Pang et al. 2009, Ellingwood and Kinali 2009, Ozel et al. 2011). In that case, the “fragility curve” is a plot of the probability of failure of the structure as a function of the single demand variable. More generally, when several demand variables are specified, one obtains a “fragility surface” in the space of the demand variables.

Having the fragility of a structural element or system and probabilistic descriptions of the demand variables, one can use the total probability law to


compute the failure probability. For example, in earthquake engineering, the fragility curve of a structure is combined with the seismic hazard curve of the site to compute the probability of failure of the structure over its lifetime. Thus, fragility models are essential for risk and reliability assessment of structures.

Clearly, the concept of FF is not specific to earthquake engineering. FFs can be used to relate characteristics of actions induced during industrial accidents to probabilities of the damage events Di (Straub and Der Kiureghian 2008, Sasani and Der Kiureghian 2001). For many of such actions FFs will be functions of several variables (components of the vector y). The difficulty of developing P(Di | y) increases with increasing dimensionality of y. However, the approach proposed in this chapter does not require to construct P(Di | y) in the form of an explicit function. This approach is based on an estimation of P(Di | y) for a relatively small set of argument values yj (components of the sample y).

1.4.2. Mathematical definition of fragility For a structural element, we define the fragility as the conditional probability of the element attaining or exceeding a prescribed limit state for a given set of demand variables at its boundary. Following the conventional notation in structural reliability theory, let gi(z, y | θ) be a mathematical model describing the performance limit state of interest for the structural element, where z denotes a vector of variables describing the capacity of the structural element, e.g., material properties and geometry variables, y is a vector of variables describing the demands on the structural element in terms of a set of boundary forces or displacements, and θ denotes a vector of model parameters. This function is defined such that the event gi(z, y | θ) ≤ 0 denotes the attainment or exceedance of the limit state by the structural element. While z and y, in general, are measurable quantities, i.e., they can be observed and measured in the laboratory or the field, θ are non-measurable parameters. They are introduced in the limit-state model, often as coefficients or powers of the measurable variables, for the purpose of calibration.

According to the above definitions, the fragility of the structural element is given by P(Di | y) = P(gi(Z, y | θ) ≤ 0) (1.17) where P(gi(Z, y | θ) ≤ 0) denotes the conditional probability of event gi(Z, y | θ) ≤ 0 for given y. The uncertainty in the event gi(Z, y | θ) ≤ 0 for given y arises from the inherent randomness in the capacity variables z, the inexact nature of the limit state model gi(Z, y | θ), and the uncertainty inherent in the model parameters θ, as described in the following section.


In detail, the structural fragility can be expressed as the system failure probability ∫=

iDdfP zzZ )(f (1.18)

where the vector Z is given by )( 21 ynZZZ ,...,, ; iD is a range of variables where

the limit state function ),( yzg becomes negative 0}) | ,(,{ ≤= θyz|yz ii gD (1.19) and )(zZf is a density of distribution function of Z.

Methods of SRA are used to calculate the probability of failure Pf, and study the sensitivity of the failure probability to variations of the model parameters (Aven and Rettedal 1998). The Bayesian approach seems to be the dominant probabilistic framework for SRA methods. Updating of probabilities within the classical Bayesian framework is very common, but also elements from the fully Bayesian approach are often seen in SRA.

As mentioned earlier, in some applications a single variable y describes the demand on the structural element. In that case, the fragility curve is a plot of P(Di | y) as a function of y. Furthermore, in this case, provided the model gi(Z, y | θ) is exact and parameters θ are known, P(Di | y) can be interpreted as the cumulative probability distribution function of the capacity of the structural element in terms of the demand variable y. When several variables are used to define the demand on the structural element, then P(Di | y) defines a fragility surface over the space of y.

Assessment of the element fragility using Eq. (1.17) involves two steps: assessment of the limit-state model gi(Z, y | θ) based on the available information, and computation of the probability term with consideration given to the natures of the uncertainties involved. Both steps involve Bayesian thinking.

1.4.3. Quantification of uncertainties related to fragility functions The main types of uncertainty in risk quantification In QRA uncertainties are categorized generally as inherent randomness (aleatory uncertainty) associated with the natural variability in nature and knowledge-based (epistemic uncertainty) due to imperfect modelling related to the assumptions and simplifications in engineering analysis, statistical uncertainty due to small sample size, and measurement errors. All sources of uncertainty must be considered in developing reliability-based decision tools. In contrast to


aleatory uncertainties, these knowledge-based (or epistemic) uncertainties depend on the quality of the analysis and supporting databases, and generally can be reduced, at the expense of more comprehensive (and costly) analysis. Bayesian limit-state models We make use of the Bayesian model assessment technique to develop limit-state models incorporating all types of information and accounting for all relevant uncertainties (Der Kiureghian 2000). The basic framework is the well known Bayesian updating rule π(θ | data) ∝ L(data | θ) π(θ) (1.20) where θ is the vector of parameters to be estimated, π(θ) is the prior distribution representing our knowledge about θ before making observations, L(data | θ) is the likelihood function representing the information contained in the set of observations, and π(θ | data) is the posterior distribution representing our updated state of knowledge about θ.

The Bayesian approach for developing a performance limit state for a structural element begins with the selection of a candidate mathematical model,

ig (Z, y | θ). Ideally, the model should be derived from first principle, e.g., the rules of mechanics applicable to the particular structural element and material. However, one is invariably forced to adopt idealizations and/or simplifications to make the model practical. In extreme cases, when the structural element is too complex to yield itself to any kind of mathematical formulation, one may have no choice but to use a purely empirical model. To account for such imperfections, we write the exact limit-state model in the form gi(z, y, γ | θ) = ig (z, y | θ) + γ (1.21) where γ is a random model correction term. In general, γ accounts for two types of model error. One is the error due to an incorrect model form, e.g., a linear form when the actual relationship is nonlinear. The other is the error that arises when certain variables are missing from the model, either because we are ignorant of their influence or because we drop them for the sake of simplicity. For example, when the intensity of ground motion is represented by the peak ground acceleration alone, the effect of variability of the ground motion time history for the fixed peak acceleration is missing from the model. Likewise, some capacity variables are missing when z contains only a subset of the material or geometry variables that characterize the capacity of the structural element. These effects are contained in γ. In most cases, arguments based on the central limit theorem can be used to justify the normal distribution for γ. In that case, the distribution of γ is completely characterized by its mean and standard deviation. The mean of γ represents the bias in the model. Since usually an


unbiased model is desired, the mean of γ must be set equal to zero. The standard deviation of γ, denoted σ, is a measure of imperfection of the model. Typically, σ is unknown and its estimation is an important part of the model assessment process.

The mathematical model gi(z, y, γ | θ) is assessed by estimating the parameters θ =(θ, σ) on the basis of the available information. Often, this information consists of a set of measured values of variables z and y, denoted zi and yi, i = 1, 2, … , n for a sample of size n, for which the outcomes of gi(z, y, γ | θ) are observed. These measured values, however, could be uncertain due to errors in the measurement devices, or because we make indirect (surrogate) measurements. An example for the latter case is when we use the measured strength of a concrete cylinder to describe the strength of concrete in the actual structural element. To model these errors, we let zi = iz + ez,i and yi = iy + ey,i, where iz and iy are the measured values and ez,i and ey,i are the respective correction terms for the i-th observation. The mean values of these error vectors represent the biases in the measurements, whereas their variances are measures of the respective error magnitudes. These error statistics are usually estimated separately by calibrating the measurement procedures for devices.

The accuracy of estimation of the model parameters θ depends on the observation sample size, n, and the quality of the prior information. The smaller the sample size, the larger the uncertainty in the estimation. This type of uncertainty, commonly known as statistical uncertainty, directly influences the variability present in the model parameters.

In our formulation, the uncertainties in the capacity variables z and load variables y are of aleatory type. Aleatory uncertainty is related also to the model error term γ that reflects the effect of uncertain missing variables. The epistemic uncertainties arise from our lack of knowledge, i.e., that arising from our choice of simplified or idealized models, or that arising from the scarcity of data. This type of uncertainty can be reduced, at least theoretically, by use of more refined models or by collection of additional data. The uncertainties inherent in the model parameters θ and σ are of this type, as is the uncertainty in that element of γ that reflects the effect of an erroneous model form. Estimation of fragility Let θ denote the set of parameters of the limit-state model and π(θ | data) denote its posterior joint probability density function obtained by Bayesian analysis. For given θ, we define the conditional fragility as iF (y | θ) = P( ig (Z, y | θ) + γ ≤ 0) (1.22)


Obviously the above fragility estimate depends on the assumed values of θ. These parameters, however, are unknown due to the epistemic uncertainties. In this section we describe several options for estimating the fragility, depending on the manner of treatment of the epistemic uncertainties. Point estimates of fragility One option for fragility estimation is to use point estimates of the parameters θ. Two natural choices are available: the maximum likelihood estimate θ and the mean estimates θ . We denote the corresponding point estimates of the fragility as iF (y | θ ) ≡ P(Di | y, θ ) (1.23) iF (y |θ ) ≡ P(Di | y, θ ) (1.24) respectively. The random variables involved in computing the above estimates are z and γ. When the distribution π(θ | data) is symmetric or nearly symmetric with respect to mean point, there is no essential difference between iF and iF . Most fragility estimates in the current practice are of this type. This potion clearly does not account for the epistemic uncertainties inherent in the model parameters. Predictive fragility One way to account for the parameter uncertainties in the estimation of fragility is to consider θ as random variables in the same fashion as z and γ. The fragility estimate then is considered a predictive measure. This estimate is obtained by integrating the conditional fragility function over all the possible values of θ with the posterior probability density as a weighing function, i.e., iF~ (y | θ) ≡∫all θ iF (y | θ) π(θ | data)dθ = EθFi(y | θ) (1.25) As is indicated, the predictive fragility is the mean of the conditional fragility with respect to the uncertain parameters θ. Thus, it accounts for the epistemic uncertainties in an average sense. In general, iF (y |θ ) and iF~ (y | θ) deviate only in tail regions. Usually iF (y |θ ) < iF~ (y | θ) for values of fragility near zero, and

iF (y |θ ) > iF~ (y | θ) for values of fragility near unity. The predictive fragility estimate does not distinguish between the natures of

the intrinsic variabilities in z and γ on one hand, and the epistemic uncertainties in θ and σ on the other. As such, it is an appropriate measure for prediction


purposes, e.g., for loss estimation, but not necessarily for the development of design codes and regulations. Bounds on fragility One way to treat the two types of uncertainties separately is to estimate confidence bounds on the fragility that reflect the influence of the epistemic uncertainties. For this we consider iF (y | θ) for a given y a function of random variables θ and determine its distribution. Let iF (y | θ)p denote the fragility value at cumulative probability p, i.e., there is probability p that the fragility

iF (y | θ) is less than iF (y | θ)p. In mathematical form p = P( iF (y | θ) < iF (y | θ)p) (1.26) Then, the interval { iF (y | θ)p, iF (y | θ)1–p} (1.27) denotes the 100)21( ×− p percent confidence interval of the fragility estimate for the given y. The size of this interval is a measure of the influence of the epistemic uncertainties on the fragility estimate.

A simpler but approximate way to compute bounds on the fragility estimate is as follows: Let Θ∇ iF (y | θ) denote the gradient of iF (y | θ) with respect to θ . Using first-order, Taylor-series approximations, the mean and variance of the conditional fragility with respect to the epistemic uncertainties are ΘE iF (y | θ) = iF~ (y | θ)≅ iF (y |θ ) (1.28) ΘVar iF (y | θ)≅ Θ∇ iF (y | θ)T Σ ΘΘ Θ∇ iF (y |Θ ) (1.29) Where the superposed T denotes the vector transpose. With the mean and variance of the fragility estimated, mean ± one standard deviation (1–σ) fragility bounds are obtained from { iF (y | θ)± )((Var θyΘ |Fi } (1.30) The size of this interval, which is twice the fragility standard deviation, is a measure of the influence of the epistemic uncertainties.

The approximations in Eqs. (1.28–1.30) work well for intermediate values of the fragility. In the tail regions, i.e., fragility values near zero or unity, these approximations do not work well due to strong nonlinearity of iF (y | θ) with respect to θ. In such cases, it is best to apply the first-order approximations to a


transformation of the fragility function for which the dependence on θ is less strongly nonlinear. A good choice is the transformation ))(()( 1 θyθy |FΦ|β i

−= (1.31) where )(1 ⋅−Φ is the inverse of the standard normal cumulative probability function. Once the mean plus-minus one standard deviation values of )( θy |β are determined by first-order approximation, the corresponding fragility bounds are computed from the inverse of Eq. (1.30). )( θy |β is the conditional generalized reliability index for given θ. Methods for computation of fragility Computation of the point-estimate fragility functions iF (y | θ ) and iF (y |θ ) requires integration over domains of the random variables z and γ for each given y. Computation of the predictive fragility function, iF~ (y | θ), in addition requires integration over the domain of θ. The confidence bound described in Eqs. (1.26) and (1.27) require nested probabilistic analysis, since the argument of the probability function itself is a probability term. Finally, the variance in Eq. (1.29) requires computation of probability sensitivities. Methods for such computations are well developed in the theory of structural reliability.

1.5. First chapter conclusions The review of published work presented above allows to make several

statements about the current state of dealing with AAs in the field of structural design:

I. There are two general approaches to the design of structures for AAs, namely, deterministic one and probabilistic one. Design codes containing descriptions of AAs remain deterministic; however, the general climate starts to be more probabilistic. The recognition of this fact can be illustrated by the seventh part of Eurocode 1 (EN 1991-2-7 2006). The main text of this code describes accidental actions in the traditional, deterministic way whereas annexes of this document contain probabilistic elements of modelling AA occurrences.

II. Journal articles and other publications devoted to assessing damage from AAs are very unspecific as regards practical recipes of calculation. A special notice should be made about the fact that most articles devoted to assessing damage due to AAs consider this damage on element level.


Damage to the system is barely covered. These statements refer to AAs induced during man-made accidents. The situation is better in the field which covers extreme natural phenomena, mainly by earthquake engineering and assessing damage induced by extreme winds. Methodological recipes developed in the fields of earthquake and wind engineering can also be applied to the case of man-made AAs.

III. The fact that AAs and damage induced by them are natural subjects of QRA is recognized in a relatively small number of scientific articles. Another small number of articles seek to combine QRA and SRA. The reader will not found too many of such articles in the leading journals of SRA: “Structural Safety”, “Probabilistic Engineering Mechanics”, “Reliability Engineering and System Safety”. The published articles do not provide too many recipes for the practical structural design for AAs.

IV. A prediction of damage to structures due to AAs will often face the problem of scarce statistical information on AA characteristics and incomplete knowledge about modelling the behaviour of structures subjected to AAs. Thus the traditional probabilistic models used in SRA, which are fitted to large amounts of data by means of classical statistics, may not be applicable to the case of AAs. A natural alternative, which can be very helpful in case of design for AAs, are methods based on the Bayesian statistical theory (BST). This theory is recognized as indispensable in some articles on AAs; however, beyond this recognition little was done to implement Bayesian methods for the damage assessment on practical level.

The assessment of damage to structures from AAs generates the need for a

method which allows a structural design for AAs on detailed, practical level. This method should concentrate available knowledge related to the phenomenon of AAs and implement computational tools developed to date. A development of such a method requires to achieve the following objectives:

1. To express the problem of damage assessment in the form of risk containing probabilities of foreseeable (identifiable) damage events.

2. To model AAs and mechanical events of damage due to AAs in probabilistic way and, especially, by applying elements of BST.

3. To apply the format of BST to an extensive use of expert judgements and other sources of knowledge relevant to the damage assessment.

4. To consider the behaviour of structure subjected to an AA on a highly detailed level, the level on which the structural engineer works.


5. To formulate the results obtained by the method in a simplified form understandable to decision-makers who have limited knowledge in the fields of QRA and structural engineering.

It is highly probable that the engineer having to assess risk posed by an AA will have at hand only a small-size statistical sample of AA characteristics. This is a highly realistic situation and the proposed method should allow to estimate the risk by utilising this small-size sample. In what follows it will be demonstrated that the small-size sample can be combined with other information on the AA within the framework of Bayesian updating.

35

2 Method developed for the estimation

of risk to structures posed by accidental actions

The second chapter proposes a method developed for the assessment of damage to structures due to accidental actions. This method combines structural analysis with Bayesian handling of information related to accidental actions and response of structures to these actions. The proposed method can be used for both structural design and assessment of risk to constructed facilities. The present chapter summarises results of investigations published by Juocevičius et al. (2010), Vaidogas and Juocevičius (2009ab, 2007ab).

2.1. Assessing risk to structures due to accidental actions: probabilistic problem statement 2.1.1. The probability and risk of damage As damage induced by AAs is a natural subject of QRA, the result of assessment of such damage can be expressed in the form of risk

36 2. METHOD DEVELOPED FOR THE ESTIMATION OF RISK…

)},(,..),,(,..),,{(Risk 11 DD nnii LFrLFrLFr= (2.1) where Fri is the likelihood (usually frequency, say, annual probability) of the damage event Di; Li outcome (losses) due to occurrence of Di; nD is the number of foreseeable damage events. In principle, the set of damage events, Di (i = 1, 2, … , nD), may include the event of no damage or negligible damage. This will allow to form a set of collectively exhaustive events (Lefebre 2009). Clearly, it can be difficult to identify all possible damage events Di or to represent different degree of damage by a discrete set of such events.

The frequency Fri can be expressed in several ways; the simplest and most natural one is an expression through the conditional damage probability, namely )()()( AADPAAFrDFrFr iii |=≡ (2.2) where Fr(AA) is the frequency of the random event AA and P(Di | AA) is the conditional probability of the damage event Di given an occurrence of AA.

The expression of risk (2.1) is consistent with the general definition of risk presently accepted in the field of QRA (e.g., Kumamoto and Henley 1996, Ayyub 2003). Eq. (3.2) is based on the assumption that the frequency of AA is constant over the lifetime of structure subjected to this AA. This assumption is difficult to verify in case that AA is a rear phenomenon. Further considerations of the subject can be found in the book (Vaidogas 2007: 34). We think that the structural engineer will be more interested in the estimation of the conditional probability P(Di | AA) rather then in the assessment of Fr(AA). Occurrences of AA are usually rear, specific types of AAs are generally backed by scarce historic and experimental data, and a lot of speculation is usually needed for specifying the value of Fr(AA). An estimation of the probability P(Di | AA) is, to the contrary, a very concrete task solved by standard means of the structural reliability analysis (SRA). Such an estimation is a probabilistic counterpart of the deterministic structural design for AAs.

The problem of the estimation of the damage probability P(Di | AA) depends on the complexity of the damage events Di. Roughly, these events can be classified in the events of direct damage and indirect damage. The former occur due to a direct impingement of AA; the latter may occur due to impact of adjacent structures which fail due to local action of AA and these failures occur one after another.

The estimation of the damage probability P(Di | AA) will highly influenced by the complexity of the event Di considered from the mechanical standpoint. This estimation will be a relatively simple task when Di represents a failure of simple, statically determined structure, for instance, flexural failure or shear failure of a facade slab idealized as simple reinforced concrete beam (Low and Hao 2002). In this case the mechanical model (limit state function) g(·) will have

2. METHOD DEVELOPED FOR THE ESTIMATION OF RISK... 37

simple, explicit form. In case where Di represents substantial damage resulting from progressive collapse, the mechanical model g(·) may not exist in an explicit form. What is more, a precise definition of Di can be difficult. In order to develop a rule which defines occurrence or non-occurrence of Di, a numerical analysis of structure and possibly physical experimentation may be necessary. They will yield highly case-specific results which can be applied to the estimation of the damage probability P(Di | AA) (Luccioni et al. 2004, Anwarul and Yazdaru 2008).

2.1.2. Expression of the damage probability through fragility function In the standard form of SRA the probability P(Di | AA) is expressed as ∫=

yyyYy||

all)d()()( fDPAADP ii (2.3)

where P(Di | y) is the probability of the damage event Di given an AA with fixed characteristics y; fY(y) is the joint probability density function (p.d.f.) of the random vector Y which models the aleatory (stochastic) uncertainty in the values of AA characteristics y.

The probability P(Di | y) is by its nature an FF which relates the damage probability to specific characteristics. In general the characteristics y will have more than one component and thus P(Di | y) will be a function having several arguments.

Strictly speaking, the function P(Di | y) will be time-depended one. This statement follows naturally from the fact that the resistance of structures decreases with time and so does the reliability. In the simplest, time-independent case values of this FF are expressed through the standard formula of SRA. It relates the occurrence of Di to mechanical model of exposed structure and its random characteristics represented, say, by the random vector Z:

{ } ∫∫ ==

≤ zZ

,|Z zzyzzzy|

all0)()d(),()d()( fIfDP

yzgzi (2.4)

where z is the value of the random vector Z with the joint p.d.f. fZ(z) quantifying aleatory uncertainty in the characteristics of exposed structure; g(z, y) is the mathematical model (limit state function, say) quantifying an occurrence of the damage event Di ; I(z, y) is the binary function which takes on the value of 1 if g(z, y) ≤ 0 and the value of 0 otherwise.

Eqs. (2.3) and (2.4) are standard expressions of SRA. The damage probability P(Di | AA) can be calculated using these expressions with relative ease. Such a calculation of the damage probability will as long remain standard


problem of SRA as is possible to fit the action model, namely the p.d.f. fY(y) to the data on AA in question. Such data must have the form of a single-dimensional or multi-dimensional statistical sample y = {y1, y2, … ,yj, … , yn} (2.5) where the elements of y are vectors containing characteristics of AA measured in an experiment.

The sample y can also be represented in another format, namely, in the format of FF values. To do this, one can notice that Eq. (2.3) is nothing more as an expression of a mean value ))(()( Y|| Y ii DPEAADPµ == (2.6) The above expression implies that the damage probability P(Di | AA) is a mean of fragility function values.

Consequently, an estimation of the damage probability amounts to an estimation of the mean ))(( Y|Y iDPE . As shown in Fig. 2.1 the sample for the estimation of this mean follows from the transformation )( jij DPp y|= (2.7) The scalar values pj can serve as elements of a univariate sample p = {p1, p2, … pj, … , pn} (2.8)

y(1) y(2) … y(j) … y(n-1) y(n)

z=P(Di | y)

Density of Y=φ(X | Ψ=ψ)

Range of the sample yy

z

00

1

π(µ)

π(µ|data)

µ=P(Di | AA)

Rang

e of th

e sam

ple p

µ=P(Di | y(n))

µ=P(Di | y(1))

Fig. 2.1. Schematic representation of the densities related to the damage

probability estimation by means of Bayesian bootstrap (Vaidogas and Juocevičius 2009a)


This represents statistical material for the estimation of P(Di | AA). Eq. (2.7) allows to transform the problem of probability estimation from a

generally multi-dimensional case expressed by the sample components yj to one-dimensional case represented by sample components pj. Which of the two cases is preferable may depend on the size of the samples y and p. It is natural to expect that in most cases the sample size n will be relatively small.

2.2. Estimation of the probability of damage to structures due to accidental actions: background of the proposed method 2.2.1. The problem with the amount of statistical data on accidental actions In case that the sample size n is sufficiently large and the sample itself possesses the property of representativeness, fitting the p.d.f. fY(y) is practicable task. However, such a case is far from typical in case of AAs. These are rare and unexpected events. Data from post-mortem investigations of AAs can be sparse or not available at all. In addition, experiments imitating specific AAs can be expensive and yield sample y with relatively low number of elements n.

A further factor, which can negatively influence the sample size n, can be non-positive attitude of the concerned persons towards experiments carried out under conditions which differ only in statistical sense. For these persons carrying out the experiment under n statistically different conditions can appear as a repetition of the same experiment n times in a row and thus as a needless wasting of time and resources. However, if we want to assess uncertainties related to the action characteristics y by the way of experiment we have no choice as to carry out a series of n repetitive experiments even if this number is relatively small.

One can expect that the series of experiments yielding the sample y will in most cases be small in sense of classical statistics. Consequently, the engineer in most cases will face the problem of estimating the damage probability P(Di | AA) using a small-size sample y.

In case that the sample elements yi have two or more dimensions and the sample size n is small, fitting the p.d.f. fY(y) to the sample can be highly problematic. Then a practical alternative can be an estimation of the damage probability P(Di | AA) by a direct application of the sample y, that is, by side stepping the problem of model fitting. To tell the truth, in case of the decision not to use the model fY(y), the sample p is a more preferable choice to estimate


P(Di | AA) than the small-size sample yi. With p the damage probability can be estimated by applying the Bootstrap method, that is, by resampling components of p (Vaidogas 2005ab, Vaidogas and Juocevičius 2007ab, 2009ab). This can be done in the classical statistical format or in line with the Bayesian statistical approach.

2.2.2. The classical statistical approach to an estimation of the damage probability The possibility to express the damage probability P(Di | AA) as a mean value of a fictitious population of FF values opens the possibility to calculate interval estimates, one-sided or two-sided, for this mean value. The fictitious sample serving for such an estimation is the sample of FF values p (Vaidogas Juocevičius 2007a, 2009a). Formally, the sample p can be used for calculation of interval estimates within the classical statistics. However the traditional calculation of confidence interval can encounter two problems:

� The size of the sample p can be relatively small. � The type of fictitious population from which p is drawn will generally be

unknown. To bypass these problems one can replace the calculation of the classical

statistical confidence interval by a approximate confidence interval calculated by means of Bootstrap resampling (Efron 1993, Davison 1998).

The two-sided and one-sided Bootstrap confidence intervals are given by the respective expressions [] ii P,P and []0 iP, . As the large values of the damage probability are of interest, the upper limit of the one-sided interval, iP can be used for the decision making concerning the damage event Di. The two-sided interval [] ii P,P can also be applied especially in the case when the accuracy of the estimation of the P(Di | AA) is of interest. The theory of Bootstrap resampling suggest several specific expressions of the confidence intervals

[] ii P,P and []0 iP, , each of them with somewhat different estimation accuracy (Efron 1993, Davison 1998). As an example, one can calculate the so-called percentile intervals. In the two-sided case, the percentile interval will have the form [,][,] )/2(1)/2(1 γγ +−= BBii ppPP (2.9) where )/2(1 γ−Bp and )/2(1 γ+Bp are the (1–γ)/2th and (1+γ)/2th empirical percentiles of a sample consisting of the mean values bp of B bootstrap samples


drawn from p, respectively (b = 1, 2, … , B). The two-sided 100γ% percentile interval for the damage probability P(Di | AA) is given by the two values which encompass the central 100γ% of the distribution of the mean values bp .

The present approach presumes that the FF P(Di | y) and sample components pj express the aleatory uncertainty only. Such an representation of the involved quantities is in line with the classical statistical approach. Any epistemic uncertainties are excluded from the analysis when the damage probability is estimated by means of Bootstrap confidence intervals [,] ii PP .

In addition, the confidence intervals are calculated by applying only direct information which comes from the sample y; this means that the values of AA characteristics beyond the range of data values (potential zones of distribution tales) are excluded from the analysis. Be the sample size n large such an exclusion would be unacceptable and could be avoided by fitting the p.d.f. fY(y). In case that n is small the direct use of data seems to be a natural approach.

In many practical situations the knowledge available for the engineer will not be restricted by the sample y. The estimation of the probability P(Di | AA) can also utilise a less formalised knowledge indirectly related to action characteristics. An application of this knowledge along with the sample y can be applied to damage assessment in line with BST.

2.2.3. Utilising different sources of information on accidental actions Prior knowledge related to the situation of exposure to hazardous effects In a pure Bayesian analysis, the prior distribution of µ should be specified subjectively. However, the purely subjective specification does not utilise prior knowledge about many types of AAs. Such knowledge, more or less relevant to the exposure situation under investigation, is usually available for the engineer. Therefore one can make a compromise between the frequentist’s and Bayesian statistical analysis and develop priors for damage probabilities from the prior knowledge.

The development of priors is considered to be one of the most controversial and resource-consuming problems of BST (Siu and Kelly 1998). A large number of formal means for specifying priors have been suggested in the literature on BST (e.g., Siu and Kelly 1998, Bernardo and Smith 1994). This section proposes a simple heuristic approach to specifying priors from existing knowledge. The approach is based on the knowledge which is specific to a particular AA and is expressed in the form of a mathematical model ϕ(⋅) relating characteristics of exposure situation to characteristics of AA, namely


y = ϕ(x | ψ) (2.10) where x is the vector describing characteristics of the exposure situation in which an AA can occur; ψ is the vector of those parameters of ϕ(⋅) which are uncertain in epistemic sense. The exposure situation represented by x may be uncertain in the aleatory sense and this can be modelled by a random vector X with an aleatory d.f. FX(x).

A part of prior knowledge should be represented by the FF pd(y) which can be established for an exposed structure by using methods of SRA. Thus the FF pd(⋅) together with the model ϕ(⋅) form the main part of the prior knowledge, the structure of which is shown in Fig. 2.2. Acquisition and processing of the new knowledge relevant to the exposure situation The need to apply Bayesian inference to estimating the damage probability P(Di | AA) may stem mainly from a partial irrelevance of the prior knowledge to a particular exposure situation. The configuration of a structure exposed to an AA as well as the accident capable of inducing the AA may be unique by a large margin and so may not fit in the prior knowledge. The source of the partial irrelevance may lie in

� Structure of the model ϕ(⋅). � Data used to fit the d.f. FX(x) and estimate parameters of ϕ(⋅), that is,

components of ψ. The partial irrelevance may require a correction of AA prediction by

experimental data which can be considered highly relevant to the exposure situation under investigation. Clearly, experiments can be used also for improving the model ϕ(⋅) by, say, increasing its relevance to the exposure situation. However, the highly relevant (case-specific) data on AA characteristics y and, possibly, interaction of AA with the exposed structure can be used directly to estimating P(Di | AA).

In theory, the amount of the case-specific data may be such that the model ϕ(⋅) will no longer be needed. In practice, however, the amount of the data may be limited because experiments on AAs, especially full-scale ones, are often expensive. This may require combining the new, case-specific data with the prior knowledge behind ϕ(⋅).

The case-specific data necessary for estimating P(Di | AA) should be gathered and represented in the form of a sample y.

It contains experimental observations of the AA characteristics. Clearly, each experiment in the series yielding the sample y′ should imitate a potential


accident and the sample y′ itself should posses the property termed by statisticians the representativeness (e.g., Barnett 1991). This property of y′ is very important for the estimation of P(Di | AA). A discussion about how to ensure the representativeness is given in Sec (2.4). In the subsequent text, it is assumed that the sample y′ is a representative one.

Given the sample y′ and an FF of interest, pi(y), one can simplify estimating )|( AADP i by introducing a fictitious sample p.

Analogously to the sample defined in Eq. (2.7) Components of p are calculated by )|()( jijij DPpp yy ′=′= (2.11) Expensiveness of experiments on AAs may cause that the size n of the sample y′ will be too small to apply methods of classical freequentist’s statistics to estimating P(Di | AA). In addition, experiments on an AA may be unique and a series of them resulting in y′ may be carried out only once. This implies that the procedure of Bayesian updating using y′ will be a single act, rather than a more or less constant process.

In this way the problem of estimating P(Di | AA) is made less complicated by switching from a multi-dimensional analysis to a one-dimensional case. Then the components pj of p can be treated as realisations of the r.v. iP~ defined by

Fig. 2.2. The role of the Bayesian bootstrap in estimating probabilities of

damage to structures due to accidental actions (Vaidogas and Juocevicius 2009a)


)( Y|~ii DPP = (2.12)

Where the function P(Di | Y) is used in the same mean as in Eq. (2.6).

2.3. A proposition of method for a Bayesian estimation of the damage probability 2.3.1. Specifying prior for damage probability In many cases characteristics of AAs can be roughly predicted by applying the mathematical model ϕ(X | Ψ) related to the physical phenomenon of AA and information on its input. This model can be applied even if it is not fully relevant to the situation of exposure to a specific AA.

Within the scheme represented by Eqs. (2.2) … (2.4), the mathematical model ϕ(X | Ψ) can be used to specify the distribution of epistemic uncertainty related to P(Di | AA).

Epistemic uncertainties related to parameters ψ of the model ϕ(x | ψ) can be expressed by introducing a random vector Ψ with a d.f. FΨ(ψ). Then averaging out the aleatory uncertainty expressed by X in the random function ϕ(X | Ψ) will yield an epistemic r.v. xxxX~

Xx

X )d())|(|())|(()(all

)( fDPpEmM ii ΨΨΨ ϕϕ ∫=== (2.13)

where m(⋅) denotes some function relating M~ to Ψ. A value µ of M~ is the damage probability at givenψ, namely, ( )))|(|( ψXX ϕiDPE . A density of M~ denoted, say, by π(µ) can be used as prior quantifying epistemic uncertainty in the damage probability P(Di | AA) (Fig. 2.3).

The source of the epistemic uncertainty do not necessarily may be only on the side of the model ϕ(x | ψ) used to predict AA characteristics. Epistemic uncertainty can also be related to values of the FF pi(y) ≡ P(Di | y) at given y. However, this part of epistemic uncertainty can be handled in the framework of (2.13).

Thus in the Bayesian format the damage probability can also be conservatively estimated with the small size sample y. However, the scheme represented by Eqs. (2.11) and (2.12) is based on three principal assumptions:

� Elements of the sample y are “crisp”, components of each observation yj are fixed values and do not involve any uncertainties.


� The FF P(Di | y) expresses the aleatory uncertainty only; values of P(Di | yj) can be calculated as fixed numbers with the necessary accuracy.

� The fictitious sample p follows from the FF P(Di | y) and the sample y; elements of p are fixed numbers expressing the aleatory uncertainty only.

Unfortunately, in most cases values of the FF P(Di | y) will be uncertain in the epistemic sense and the damage probabilities P(Di | y) not always can be calculated with sufficient accuracy (the problem of evaluation of multiple integrals over the p.d.f. fZ(z)). In addition, some or all elements of the original data y can be inaccurate, say, stem from post-mortem investigations of accidents. Even if the above problems can be solved the estimation of P(Di | AA) can be complicated by a high dimensionality of the argument vector y, difficulties with calculating FF values P(Di | y) for mechanically complex damage events Di and other intricacies.

2.3.2. Application of Bayesian bootstrap to the updating of the damage probability The usual Bayesian posterior (e.g., Congdon 2006) has the following form π(µ | data) ∝ π(µ) L(data | µ) (2.14)

Dens

ity

Fig. 2.3. Schematic representation of densities related to probabilistic damage assessment in Bayesian context: )|( ψˆ µPf = density of the r.v.

))|(( ψXˆ ϕii pP = with the mean µψ; π(µ) = prior distribution of µ; )|( nµµπ ˆ = posterior distribution of µ


where “data” is represented eventually by the samples p or y′ . The main idea followed in this chapter is to replace the usual Bayesian posterior π(µ | data) by an estimated posterior data)|(µπ ∝ )|(data)( µLµπ B (2.15) where )|(data µLB is an estimate of the likelihood function based on bootstrap estimation of the density of the pivotal quantity M~µn − with nµ = ∑ =

− nj jpn 1

1 (2.16) A possibility of such a replacement was suggested by Boos and Monahan (1986).

The first step is to estimate the d.f. of the data p using the empirical d.f. nF of the pjs. In the second step, a set of B random samples of the size n is generated from nF , namely ),...,,...,,( 21 nbjbbbb ppppp ′′′′=′ (b = 1, 2, … , B) (2.17) Then a mean bnµ′ is calculated for each sample bp′ (b = 1, 2, … , B). From the B simulated estimates 1nµ′ , 2nµ′ , … , Bnµ′ , one can compute a kernel density estimate ∑

=

−′−= B

b

nbnB w

µµuκwBuk1

)(1)( (2.18)

where w is a bandwidth (window width, smoothing parameter) and κ(⋅) is a kernel function. Such functions are used for constructing smooth estimates of continuous p.d.f.s (e.g., Davison and Hinkley 1998: 79). Kernel functions are also applied to developing approximate likelihood functions (Efron and Tibshirani 1993: Chapter 24).

Since the function )( µukB − is an estimate of the sampling density of nµ given µ, the likelihood function of nµ can be estimated by ∑

=

′−−=−= B

b

bnnnBnB w

µµµκwBµµkµµL1

21)()|( ˆˆˆˆˆˆ (2.19)

The resulting estimate of posterior of the damage probability is )|()()()|( µµLµπµCµµπ nBnn = (2.20)


where the normalising constant )( nµC can be found by numerical integration. Practical implementation of the bootstrap-based updating procedure is

relatively simple, as the estimates )|( µµL nB and )|( nµµπ can be computed almost automatically. The estimate )|( µµL nB is relatively insensitive to the choice of the kernel function κ(⋅) (Davison and Hinkley 1998). The only implementation problem is associated with the choice of the bandwidth w which can have considerable influence on )|( µµL nB .

A simple recipe for choosing the “best” value of w seems not to be available and many authors sidestep this problem by presenting estimates like )|( µµL nB and )|( nµµπ for different values of w (e.g., Davison et al. 1992). A criterion for the choice of w should be smoothness of the estimated likelihood curve

)|( µµL nB . Davison and Hinkley (1998) suggest that w should be proportional to B. Shao and Tu (1995) provide a review of approaches to choosing w by means of a cross validation and the bootstrap. In the present context, one can only state that a further investigation is necessary to develop an algorithm for an automatic choice of w. Without such an algorithm, updating via the Bayesian bootstrap can hardly be attractive to applying it in the field of practical structural engineering.

The estimate of the posterior density, )|( nµµπ , will express the epistemic uncertainty of the true value of the failure probability P(Di | AA). The result having the form of the density )|( nµµπ is well-understandable for the persons who understand the methodology of QRAs. However, most decision-makers have a pretty vague understanding of peculiarities of QRA and SRA. Therefore the final result of the analysis can be a single value expressing the likelihood of the damage Di and defined as a conservative percentile of the probability distribution given by the density )|( nµµπ , for instance, α,iP . Such percentile can be computed by )|(1, niP µαΠα

ˆˆ −= (2.21)

where )|(1 nµαΠ ˆˆ − is the estimate of the inverse of the distribution function )(⋅Π obtained by integrating the density )|( nµµπ ; and α is the level of the

conservative percentile (α = 0.9, say). We think that the single value α,iP will be easier to understand to decision-

makers than the density )|( nµµπ .


2.4. Discussion on the implementation of the proposed method The key requirement for the application of the method proposed in Sec 2.3 is the representativeness of the sample y. This sample should contain characteristics of AA under study obtained by experiment or, in rare cases, from post-mortem investigation of accidents. In theory, the property of representativeness is achieved by following the principles of statistical sampling theory (e.g., Tillé 2006). However, practical assurance of this property may require a great deal of judgement. It is well known among statisticians and practitioners that there is no yardstick for representativeness.

We think that it is possible to organize experiments on AAs in such a manner that they could yield representative samples y. In addition, the problem of obtaining representative samples has a general character and is not specific to the physical phenomena of AAs. Therefore numerous recipes of obtaining representative samples used in the various fields of experimentation can be utilised for composing y. The reminder of this section presents a short discussion on organising experiments which can be considered to be once yielding the representative sample y.

The representativeness of y should be achieved by careful arranging the series of n repetitive experiments on specific exposure situation. Apart from technical details, an organisation of these experiments should follow the principles of statistical sampling. “Representativeness” of y is understood in terms of generating this sample. Methods of sampling imply that y should be considered a sample from certain population, finite or infinite. The choice of a sampling method depends, among other things, on the structure of this population. In addition, the sampling method applied determines also estimators used to assessing population parameters, say, confidence intervals of the population mean (Fuller 2009).

It is stated somewhat pessimistically that “there is no one “best” method of drawing a sample from the population” (Mason 1982) and “there is no yardstick against which to measure “representativeness” ” (Barnett 1991). However, it is generally agreed that schemes of probability sampling allow tackling the problem of representativeness. As long as the relatively small size of the sample y is considered economical and/or practical, the simple random sampling seems to be the best recipe to draw y. In principle, further schemes of probability sampling, say, cluster or stratified sampling can be applied to drawing y (see, e.g., Fuller 2009). However, if the sample size n is, say, up to 20 elements, these schemes may not be attractive.

The fact that the sample y is small aggravates the problem of assuring the representativeness of y. In case of large samples, the problem of not


representative elements, say, outliers is alleviated by presence of a large number of “representative” elements (e.g., Barnett 1991). The small-size sample y will be always vulnerable to criticism in terms of its statistical reliability and so will be estimates of the damage probabilities P(Di | AA). However, in case of non-negligible uncertainties in AA characteristics and absence or irrelevance of knowledge about a specific exposure situation, one has little choice but to draw y according to principles of random sampling.

In practice, textbook recipes provided by the schemes of probability sampling are to be adapted to specific exposure situation under investigation. A systematic investigation of how to achieve the statistical representativeness of samples containing measurements of AA characteristics requires a separate study. It is beyond the scope of the present work. However, one can mention several arrangements of the series of n experiments which naturally suggest themselves:

1. If elements in the population of the known size N are nominally identical (their properties and conditions at occurrence of AA differ only in the statistical sense), the scheme of simple random sampling can be applied to choosing the sample of n experimental units from the population of N units (e.g., Fuller 2009: 100).

2. If elements in the population of the known size N are nominally not identical but can be divided into non-overlapping groups of similar units, stratified random sampling can be applied to drawing n experimental units and applying them in the experiments which yields y (e.g., Fuller 2009: 189). In general, the larger is the number of these groups (strata), the larger size n of y may be necessary.

3. If elements in the population of the known size N can be divided into groups (clusters), a sample of these groups can be selected and the required elements of the sample y collected from the elements within each cluster . This may be done for every element in these clusters or a subsample of elements may be selected within each of the clusters. Such a collection of elements of y is known as random cluster sampling (e.g., Rao 2000: Chapter 7).

An example of an application of the simple random sampling can be a selection of nominally identical pressure vessels for a series of experiments, in which the vessels are exploded and the resulting blast and fragmentation recorded. To form the sample y, experiments on the total of n vessels should be carried out.

An application of the stratified sampling scheme in the context of an investigation of an AA can be illustrated by considering tank cars (wagons) used for the transportation of liquefied gases. Such tanks can undergo BLEVE


explosions. Strata can be formed from tanks with nominally identical design and/or with the same type of gas being transported. The elements of the sample y can then be obtained by carrying out experiments on tanks selected from each stratum.

An example of application of the cluster sampling can be related to spatial clustering of occurrences of an AA. Specific fires in industry, say, tank fire or jet fires can be divided into groups (clusters) related to specific administrative units of a country. Then the sample y can be formed by selecting several clusters (units) and carrying post mortem investigations of all fires in each selected cluster.

The theory of statistical sampling provides other schemes of obtaining representative samples (Rao 2000, Tillé 2006, Fuller, 2009). In principle, they can be applied to obtaining the sample y; however, a detailed analysis of these sampling schemes is beyond the scope of our investigation.

2.5. Second chapter conclusions This chapter presented a method for the assessment of damage to structures

due to accidental actions in terms of Bayesian estimation of damage probability. Therefore conclusions of the present chapter will summarise the main features and potential fields of application of the proposed method:

1. The proposed method allows to predict damage to structures due to accidental actions in terms of damage probabilities.

2. Estimates of the damage probabilities obtained by means of the proposed method can be integrated into the general expression of risk posed by hazardous phenomena inducing the accidental actions.

3. The proposed method is based on elements of the Bayesian statistical theory and so allows to utilise different sources of information on accidental actions.

4. A particular feature of the proposed method is that it allows the estimation of the damage probability with a small-size statistical sample containing measurements (observations) of an accidental action under analysis.

5. The mechanical aspect of the damage probability estimation problem is expressed through a fragility function which must be developed for the damage event under analysis.

6. The key operation of the proposed method is quantification and propagation of aleatory and epistemic uncertainties related to accidental actions and response of structures to these actions. This operation yields failure probability estimates.


7. The proposed method is suitable to dealing with accidental actions induced by man-made and natural hazardous phenomena. The applicability of the method is illustrated in the three subsequent chapters of the present dissertation.

53

3 Application of the proposed method

to assessing the fire damage to structures

This chapter deals with an application of the method proposed in Chapter 2 to the estimation of fire risk to structures. The main reason for the choice of this application area was that accidental fires are highly uncertain phenomena posing serous risk to structural property. The thermal actions posed on structures during fires are undoubtedly accidental ones. In addition, uncertainties related to such actions are considerable and can be handled at its best within the Bayesian format. The study described in this chapter considers a timber beam bearing a roof of an industrial building. The fire action is expressed by a small-size sample containing time-histories of simulated fire characteristics. The present chapter presents results of investigations published by Juocevičius (2008b), Vaidogas and Juocevičius (2008a), Vaidogas et al. (2009), Linkutė et al. (2010).

3.1. Industrial building with fire hazard In this chapter, the method proposed for the estimation of structural failure probability in Chapter 2 will be applied to an assessment of risk posed by a

54 3. APPLICATION OF THE PROPOSED METHOD TO ASSESSING THE FIRE …

potential fire in industrial building now under construction near Lithuania city Kaunas (Fig. 3.1). The proposed method will be used to estimate the probability of failure of heavy timber beams subjected to fire in the hall used for storage of industrial goods. The plan and section of the hall in which the fire will be simulated and failure probability of roof structure estimated are shown in Fig. 3.2.

The simply supported beams of the roof structure were made from glue-laminated timber. The cross-sectional dimensions of the beams are 0.14 m × 1.6 m and so they exceed 0.05 m and the beams must be considered elements of a heavy timber structure (Purkiss 2006). As compared to steel and concrete structures, heavy timber structures are recognized as having good fire resistance. There are many examples of such structures surviving fire exposure without collapse (e.g., Buchanan 2002: 274). However, the relatively high fire resistance of heavy timber structures does not automatically mean that they are safe from failures caused by fires.

Fig. 3.1. The storage hall used for fire simulation: general view on the

timber roof structure and materials stored

11.62m

3.82

m4.02

3.78

Fig. 3.2. The plan and section of the room (hall) in which the fire will be

simulated

3. APPLICATION OF THE PROPOSED METHOD TO ASSESSING THE FIRE … 55

3.2. Accidental action characteristics obtained by fire simulation The accidental action considered in this case study will be a thermal action induced by fire in the building under study. As an experimental assessing of the thermal action in the building is hardly possible, the action will be predicted by means of deterministic computer fire models. Reviews of these models are presented by Rasbach et al. (2004: 294) and Karlsson and Quintiere (2000: 255), among others. The deterministic models rely on the basic assumption that for a given vector of initial conditions, x, the outcome y at time t is entirely determined. This can be reflected by the function y(x, t | ψ), where x is the vector representing the input of a computer fire model and ψ is the vector of parameters of this model. The model y(x, t | ψ) is an special case of the general model y = ϕ(x | ψ) defined in Chapter 2 by Eq. (2.10).

In general, the vector y can express fire development, its characteristic features, and its consequences. However, it is possible to simulate by means of y(x, t | ψ) thermal actions applied to a specific structure exposed to a particular fire situation. Computer fire simulators and theoretical models underlying them can differ in accuracy and the time required for simulation. The signal y(t) can be obtained by applying competitive models, say, zone models or field models (Rasbach et al. 2004: 256).

In the case under study, the fire simulation will be carried out by applying a two-zone fire model implemented in the code CFAST (Peacock et al. 2008). This model and its implementation in the CFAST code determine the structure of the input vectors x and ψ. Components of these vectors are listed in Table 3.1.

The simulation carried out with CFAST can yield the simulated realizations yj of y(x, t | ψ) or, briefly, y(t) having the form of the time sequences yj = {yj(tτ), τ = 1, 2, … , Nτ} (3.1) where the times tτ are obtained by dividing the period of required time to failure, [0, treq], into a relatively large number Nτ of small intervals ∆t. The sequences yj can be generated by a MC simulation of fire scenarios as described by Hostikka and Kesti-Rahkonen (2003) and Hietaniemi (2005, 2007).

We will apply a charring model suggested by Hietaniemi (2005). According to this model, components of y must be fire room temperature on the level of the beam, y1 (°C), and oxygen content on this level, y2 (%). The CFAST code allows to simulate and store the time sequences y1(tτ) and y2(tτ). Running this code N times within a Monte Carlo simulation will yield N realisations yj(tτ) which can be used as values of a demand function developed for the failure probability estimation.


Table 3.1. Aleatory and epistemic random variables used in the two-zone simulation of fire in building shown in Fig. 3.2 Description and notation Mean/coeff.

of variation Probability distribution

Aleatory random quantities (components of X) Initial interior temperature, X1 (°C) 10/0.3 Normal(1) Exterior exterior temperature, X2 (°C) -7/0.36 Normal(1) Fire growth time, X3 (s) 500/0.115 Uniform on

[400, 600](2) Steady burning time, X4 (s) 750/0.077 Uniform on

[650, 850](2) Decay time parameter, X5 (s–1) 0.0091/0.105 Uniform on

[0.00769, 0.0111](2) Maximum heat release rate, X6 (kW) 7100/0.2 Normal(2) Number of fire objects, X7 (dimensionless) 10/0.26 Binomial(0.323, 31)(2) Height of the individual fire object, X8 (m) 0.388/1.73 Discrete two-weight

distribution over 0 m and 1.55 m(2)

Time of window breakage and fall out, X9 (s) 500/0.1 Normal(3) Opening fraction of the supply gates, X10 (-) 0.1/3 Binomial(0.1, 1)(3) Opening fraction of the supply gates, X11 (-) 0.03/4.36 Binomial(0.05, 1)(3) (1) Toratti et al. (2007) (2) According to Hostikka and Rahkonen (2003) (3) According to Li et al. (2007)

In the context of the notation used in Chapter 2 for the description of the

proposed method of failure probability estimation, the realisations yj can be grouped into a simulated sample y = {y1, y2, … , yj, … , yn} (3.2)

The sample size n will not be large as the fire simulation for a sufficiently large building and storage of simulation results in the form suitable for further MC simulation can be a time-consuming process. This is especially true for fire simulation which are carried out by applying the field models (Yang et al. 2010).

We applied the fire simulation by means of the CFAST code to the building shown in Figs. 3.1 and 3.2. A total of 30 simulations were carried out. Results of


the simulations are presented in Annex A. CFAST code runs the programme Smokeview used to visualise fire simulation results. An example of such visualisation is shown in Fig. 3.3. Fig. 3.4 contained the visualisation of the five sequences y1(tτ) and y2(tτ) computed with ∆t = 1 min and Nτ = 60.

Fig. 3.3. Visualisation of fire simulation by means of two zone model (results of computation by means of CFAST computer code visualized by

means of SMOKEVIEW code)

The simulated realisations {yj(tτ), τ = 1, 2, … , Nτ} were used as values of a time-dependent demand variable y in a fragility function P(Di | y) developed for the timber from Fig. 3.1 subjected to fire and undergoing the process of charring.

3.3. Developing fragility function for a timber beam General expression of the fragility function In the deterministic structural analysis for fire safety, the verification of structure is carried out in the time domain or strength domain by checking the respective inequalities (e.g., Purkiss 2006: 16, Buchanan 2002: 227): tf – treq ≥ 0 (3.3) md = rd– ed ≥ 0 (3.4)


where tf = calculated time to failure; treq = required time of fire resistance (required time to failure); and md, rd, ed = design values of safety margin, resistance and action effect of the structure under fire situation, respectively. Although the condition (3.4) does not contain the time treq specified in the building/design codes, it is stated that the inequalities (3.3) and (3.4) give equivalent result as the positive value of the difference tf – treq corresponds to the positive value of rd(t) – ed(t) at the moment t = treq. In order to take into account the uncertainties related to fire severity (destructive potential) of natural (real, not nominal) and uncertainties inherent in

0 10 20 30 40 50 60

tτ (m in)

0

200

400

600

800

1000

Temp

eratur

e of g

ases

y 1(t τ

), oC

0 10 20 30 40 50 60

tτ (min)

4

8

12

16

20

Oxyg

en co

ntent y 2

(t τ), %

Fig. 3.4. Five simulated realizations yj = {(y1j(tτ), y2j(tτ)), τ = 1, 2, … , 60}

of the signal y(t) (∆t = 1 min) obtained with the CFAST code


the response of a structure to fire, it makes sense to measure the fire safety of a structural element by a probability that the element will fail during the time [0, treq] (the failure event Di will occur during this period). This probability will be denoted by Pf (treq). With the function y(x, t | ψ) introduced in Sec. 3.2 and used to express the time history of fire, the failure probability Pf (treq) can be expressed as follows (Vaidogas and Juocevičius 2008a): P(Di | AA) ≡ Pf(treq) = P(mi(Z, y(X, t | ψ)) ≤ 0 | ∀t ∈ [0, treq]) (3.5) where mi(⋅) is the deterministic model of the time-dependent safety margin related to the limit state i; Z is the vector representing time-independent random characteristics of structure; and X is the random vector quantifying uncertainties related to the values of x. For a timber structure, components of Z will represent original dimensions, mechanical properties of residual cross section, and time-independent loads applied to structure during fire. Uncertainty in values of Z can be expressed by a joint probability density function fZ(z). The time-dependent formation of char and so gradual reduction of resistance of the structure can be expressed through components of y(X, t | ψ). For the sake of brevity, the function y(X, t | ψ) will be denoted, where appropriate, by y(t).

The conventional approach to developing fragility functions can be adapted to the case of fire by subdividing the general problem of the failure probability estimation into two tasks:

� A computer simulation of fire, which imitates the exposure of the structure under analysis to fire, or an imitation of fire by full-scale or large-scale experiments (Task 1) (see also Sec. 3.2).

� An estimation of the failure probability Pf (treq) by applying results of the previous task (Task 2). The connecting link between Task 1 and Task 2 can be the signal (time-

history) of fire actions, y(t), generated by simulation or recorded in experiment. The simulated signals y(t) can be generated by means of MC method (e.g., Hostikka and Kesti-Rahkonen 2003). The reason for the subdivision in the two tasks is that it simplifies the estimation of Pf (treq) and opens several theoretical possibilities (Vaidogas and Juocevičius 2008a; Vaidogas et al. 2009c).

Components of y(t) are time-dependent demand variables. With y(t), the fragility function used for the estimation of Pf (treq) can be formulated as a probability of the failure event Di conditioned on the given signal y(t): tftmPttDP

reqt

ireqi d)d()0))(,(())(|(0

1 ∫ ∫

≤= −

zZ zzyzy (3.6)


The above expression of the failure probability ))(|( tDP i y applies averaging over required time treq of the instantaneous probabilities )0))(,(( ≤tmP i yz . Such an operation is a standard approach to the time-dependent reliability estimation (Melchers 1999: 184) Then the failure probability Pf (treq) can be expressed as a mean calculated over all signals y(t): )))(|(()( )(f tDPEtP itreq YY==µ (3.7) where Y(t) = time-dependent random vector modelling the aleatory uncertainty related to the signals y(t); EY(t)(⋅) = mean value with respect to Y(t). Peculiarities of assessing the fire fragility of timber structures A calculation of the estimates Pe(Di | yj) of the failure probability ))(|( tDP i y for timber structures can be substantially facilitated by utilizing specific features of fire damage to these structures and making several simplifying assumptions (Vaidogas and Juocevičius 2008a):

1. The interaction of fire with structure occurs as a gradual process of charring over the time [0, treq]. 2. Charring leads to a gradual reduction of cross-section and monotonic decrease in time of the section resistance ri(t) (i.e., ri(t) is a monotonically decreasing function of t, Fig. 3.4). The monotonic decrease of ri(t) is caused by monotonic growth of the depth of char front (char depth, in short) d(t).

3. Loads applied to a structure during fire can be assumed to be time-independent random variables. This leads to a time-independence of random action effect ei Fig. 3.4. This assumption will not be valid in cases where ei can be influenced by such processes as evacuation of people and goods during fire and concentration of people trapped by fire in small areas. The rapid melting of snow cover due to the thermal effect of fire will lead to a monotonic decrease of ei. 4. The reduction of mechanical properties of residual cross section during fire is low (Purkiss 2006: 113). This allows to make the assumption that these properties can be modeled as time-independent random variables.

5. The monotonic decrease of section resistance ri(t) over the time interval [0, treq] and the time-independence or monotonic decrease of the action effect ei will result in a monotonic decrease of safety margin mi given by the difference ri(t) – ei (Fig. 3.4).

6. The monotonic decrease of safety margin mi allows to express the probability of failure during the time [0, treq] as the probability of failure at the required time treq.


The facts and assumptions listed above allow to replace the time-dependent safety margin mi(z, y(x, t | ψ)) by a time-independent one, namely )()),(,()),(,( zzzzz ireqireqi etdrtdm −= (3.8) where ri(z, d(treq, z)) is the resistance of section at the time treq given the char depth d(treq, z) reached at treq; ei(z) is the action effect in the section. The notations ri(z, ⋅ ) and d( ⋅ , z) mean that some components of z are arguments of the resistance and some are ones of the char depth. However, ri(⋅) and d(⋅) do not have common arguments in our study.

The char depth d(treq, z) reached at treq is introduced into the expression of ri(⋅) through a simple modification of original cross-sectional dimensions (e.g., Purkiss 2006: 232). If, for instance, the original depth and width of a timber beam section exposed to fire on all four sides are the first two components of the vector z, the resistance of the section at the time treq should be calculated using the cross-sectional depth h – 2 d(treq, z) and width b – 2 d(treq, z). Thus the damage due to the fire with the given signal y(t) accumulated over [0, treq] can be unambiguously expressed by the single char depth d(treq). Consequently, the estimation of the fragility function ))(|( tDP i y for the given signal y(t) can be replaced by its estimation for d(treq), namely, =≤−=≡ 0))()),(,(())(|())(|( ZZZy ireqireqii etdrPtdDPtDP ∫ ≤−=

zZ zzzzz )d(0))()),(,(( fetdrP ireqi (3.9)

The form of the function (3.9) is very close to the one of traditional fragility functions used in the structural reliability assessment (e.g., Ellingwood et al. 2004, Lee and Rosowsky 2006). The only difference is that the geometrical quantity d(treq) and not an external action is used as a demand variable. A developing of the fragility function ))(|( reqi tdDP is a standard problem of structural reliability analysis (e.g., Melchers 1999). For the double-sloped timber beam introduced in Sec. 3.1 (Fig. 3.2), the probability ))(|( reqi tdDP of the flexural failure (the damage event) Di can be expressed through the functions ri(z, d(treq), z) and ei(z) in the following form:

−−×−= 2528121 ))(2())(2(0.1667((),),(|( reqreqreqi tdZtdZZPtdDP ΘΘΘ

0)))(795)(9.92( 3421726321 ≤−+++− ZZZZZZZZZ .Θ (3.10) The above expression means that the vector of the random basic variables, Z, contains eight components Z1 to Z8 expressing aleatory uncertainty related to the


beam structure. The parameter Θ1 is used to model the epistemic uncertainty in the resistance of the beam. The parameter Θ2 expresses the epistemic uncertainty in the ratio of snow loads on roof of the building under study and adjacent ground. The probability distributions of Z1 to Z8 as well as Θ1 and Θ2 are presented in Table 3.2. The remaining components of Z, namely, Z9 and Z10 will be used to simulate the random processes of fire and charring of the beam induced by this fire. Modelling the charring of timber structures subjected to fire For a given signal y(t), the char depth d(treq) is obtained from the expression ∫=

reqt

reqj ttd0

d)|,),(,()|,,( ττβ θθ zxyzx (3.11) where β(⋅) is the function relating the rate of charring to the signal y(t). The expression )|,,( θz⋅⋅β means that some components of the vectors z and θ used above as arguments and parameters of the mechanical model (safety margin) mi⋅(⋅) serve also as arguments and parameters of β(⋅). In addition, arguments of the model β(⋅) include components of the vector x used in this work for modelling the situation of exposure to an accidental action (or fire in the present case) (Sec. 2.2). Components of x were applied to the fire simulation described in Sec. 3.2. Probability distributions of these components are listed in Table 3.1. The components of the vectors z and θ are defined in Table 3.2.

t

efi, rfi(t)

0

density of rfi(0)

treq

density of efi

tf

m fi(t) = rfi(t) − efi

0

1

2

12

Fig. 3.5. Monotonic decrease of resistance ri(t) and safety margin mi(t) over the time [0, treq] (1 = realizations of ei and ri(t) resulting in survival over [0, treq]; 2 = realizations of ei and ri(t) leading to a failure at tf)


Buchanan (2002: 295) provides a short review of literature on the charring rate modelling. The state-of-the-art form of the function β(t, y(t), x, z | θ) is presented by Hietaniemi (2005; 2007). The components of the vector z influencing the charring rate are wood density z1 and moisture content of the wood, z9 (Table 3.1). The empirical expression of β(⋅) used in this study was adopted from Hietaniemi (2005, 2007):

( )( )

1998

1101

411011116

54

.73702233

)()()())((

}/exp{))/()((1)|,),(,(

7

−− ++×)(′+−×

×−+=

zztyzxty

txtytt

θθθσθθ

θθθθβ

θ

θzxy

(3.12)

where σ ′ is the Stefan-Boltzmann constant (e.g., Buchanan 2002: 52). With the sequence yj defined by Eq. (3.1), the jth simulated value of the char

depth d(treq) was calculated by ∑

1=∆≈

τΝ

τττβ ))|,),(,(()|,,( txttxtd jreqj θθ zyz (3.13)

For given x, z and θ, the value of dj(treq, x, z | θ) yields a single value of the fragility function P(Di | ⋅), namely, P(Di | dj(treq, x, z | θ)). This value must be estimated by evaluating the integral in Eq. (3.10). This integral by its definition is used to average out only the aleatory uncertainties expressed by the vector z. However, the structure of uncertainties in the problem of estimating the failure probability Pf (treq) is more complicated, because the parameter vector θ is uncertain in the epistemic sense. Modelling aleatory and epistemic uncertainties The failure probability Pf (treq) can be treated as a measure of the aleatory uncertainty in the occurrence of the failure event Di. The aleatory uncertainty is also expressed by the vector Z used to model characteristics of structure subjected to fire. Probability distributions of components of Z are listed in Table 3.2.

The key models y(x, t | ψ) and β(t, y(t), z | θ) used to estimate Pf (treq) contain parameters (components of ψ and θ) which can be uncertain in the epistemic sense. In principle, measures of epistemic uncertainty can also be assigned to outputs of these models, namely, the simulated signal y(t) and the charring rate β(t).

The presence of the two different sources of uncertainty will be denoted by the functions y(x, t | Ψ) and β(t, y(t), z | Θ), where Ψ and Θ are the random


vectors expressing the epistemic uncertainty in ψ and θ. Components of Ψ are present in the computer models used to the fire simulation. Components of Ψ were listed in Table 3.1. These components were determined by the fire simulation model implemented in the computer code CFAST.

Components of Θ are constituents of the model of the charring rate and mechanical model of the timber beam. The components of Θ depend on the specific form of β(t). In our study we used the model β(t) developed by Hietaniemi (2005; 2007). The random vector Θ is defined in Table 3.2. Epistemic probability distributions of the parameters grouped into Θ were suggested by Hietaniemi (2005; 2007).

Table 3.2. Aleatory and epistemic random variables used in the analysis of timber beam Description and notation Mean/coeff.

of variation Probability distribution

Aleatory random quantities (components of Z) Wood density Z1 (kN/m3) 3.5/0.05 Lognormal(1) Width Z2 (m) 0.14/0.0357 Lognormal(1) Height at the supports Z3 (m) 0.47/0.017 Lognormal(1) Height at the mid-span Z4 (m) 1.24/0.0081 Lognormal(1) Critical section height Z5 (m) 0.50/0.01 Lognormal(1) Self weight of roof cover Z6 (kN/m) 1.53/0.05 Lognormal(1) Snow load Z7 (kN/m) 0.78/0.5 Gumbel(1) Timber strength Z8 (MPa) 37.4/0.13 Lognormal(2) Wood moisture content Z9 (%) 8.1/0.06 Normal(2) Beam surface emissivity Z10 ≡ ε (4) (-) 0.75/0.05 Normal(2) Epistemic random quantities (components of Θ) Dimensionless factor expressing uncertainty in the resistance of the beam, Θ1

1.0/0.05 Lognormal(2)

Ground to roof snow conversion factor, Θ2 0.8/0.2 Lognormal(2) Parameter of the function relating charring to oxygen concentration, Θ3 ≡ ξ (%)(4)

0.575/0.0753 U(0.5, 0.65) (3,5)

Parameter in the charring rate model, Θ4 ≡ ψ0(4) (mm/min)

3.77/0.1256 T(2.7, 3.6, 5.0) (3,5)


Parameter (time constant) in the charring rate model, Θ5 ≡ τ(4) (min)

100/0.041 T(90, 100, 110) (3,5)

Parameter in the heat flux model, Θ6 ≡ ϑ(4), (kW/m2)

1.192/0.0624 T(1.026, 1.162, 1.387) (3,5)

Dimensionless parameter in the heat flux model, Θ7 ≡ p(4)

0.5/0.04 Normal(3)

Parameter in the charring rate model, Θ8 ≡ A(4), (kJkg)

800/0.213 U(505, 1095) (3,5)

Parameter in the charring rate model, Θ9 ≡ B(4), (kJkg)

2490/0.0139 U(2430, 2550) (3,5)

Parameter relating charring rate to wood density, Θ10 ≡ ρ0(4), (kN/m3)

4.65/0.93 Normal (3)

Heat transfer coefficient Θ11 ≡ hs(4) 13/0.0628 T(11, 13, 15) (3,5) (1) According to Spaethe (1986) (2) According to Toratti et al. (2007) (3) According to Hietaniemi (2005, 2007) (4) The original notation used by Hietaniemi (2005, 2007) (5) T = triangular probability distribution; U = uniform probability distribution

3.4. Failure probability of the beam damaged by fire The beam failure probability Pf(treq) will be estimated by applying input information given in Table 3.2 for the required time to failure treq equal to 60 min. The estimation will be carried out by specifying a prior distribution of Pf(treq) from a prior developed for the char depth d. Then the prior of Pf(treq) will be updated and a posterior of Pf(treq) obtained by applying new data. This data is expressed through the simulated realisations of the fire time-history, {yj(tτ), τ = 1, 2, … , Nτ} with Nτ = 60, introduced in Sec 3.2.

The influence of the char depth d on the mechanical behaviour of the beam is simple and unambiguous (Buchanan 2002, Purkiss 2006). This implies that the estimation of the failure probability Pf (treq) in line with the Bayesian procedure proposed in Chapter 2 should start from a specification of a probability density fD(d | θd) of d related to the required time to failure treq. As the data for fitting the density fD(d | θd) will hardly be available in most practical cases, the estimation of the parameters θd will have to rely on the development of a prior density π(θd). This density will express the epistemic uncertainty and can be updated using new data in the form of char depth values obtained by


experiment or by means of the computer fire simulation (Sec. 3.2). The models fD(d | θd) and π(θd) can be used for developing a prior distribution for the failure probability π(µ) (Sec. 2.3.1). Specifying the prior for the failure probability The simple model suitable to express the aleatory uncertainty in the char depth is a normal distribution. Although this model is not bounded at the zero value (zero char depth), the tail over the negative region of d values can be ignored at a proper specification of the mean and standard deviation, θ1d and θ2d. The priors of θ1d and θ2d will be specified as informative prior distributions for normal distribution parameters (see Siu and Kelly (1998) for the interpretation of the term “informative”).

The normal distribution of d can be easily handled within the Bayesian procedure (Congdon 2006). If d ~N (θ1d, θ2d), then the priors for the parameters θ1d and θ2d can be specified using standard recipes of Bayesian statistics. The priors for the uncertain mean Θ1d and variance Θ2d are usually specified by considering these two epistemic variables independent. Thus the joint prior distribution will be expressed by the product of two prior densities π(θ1d)× π(θ2d). The recommended prior for Θ1d is a normal distribution, whereas the standard prior for the dispersion parameter is specified not for the variance Θ2d but for the precision 1

2−

dΘ . A gamma distribution is used as a standard prior of 1

2−

dΘ (e.g., Congdon 2006). The following priors were assumed for parameters of the normal distribution of the char depth: )mm0.0016m,N(0.04),N( 22

001 =σµΘ ~d (3.14a) 0.01)78,2Gamma(),Gamma( 00

12 =− βαΘ ~d (3.14b)

With the above assumptions one can rewrite the prior densities π(θ1d) and π(θ2d) in the form ),|( 2

001 σµθπ d and ),|( 002 βαθπ d . The mean value 0µ of d1Θ was assumed to be equal to 40 mm on the basis

of the deterministic models used to predict fixed values of the char depth d. Two models were used to the deterministic estimation of d. In line with the linear model presented by Purkiss (2006), the depth d is estimated by the product β×treq, where β is the fixed and constant charring rate equal to 0.667 mm/min for the glue-laminated timber considered in the present case study. In the case where treq = 60 min, the linear model yields the result d = 40.02 mm. A further, non-linear model is presented by Buchanan (2002). This model allows to estimate d by means of the expression 0.813

n2.58 reqtβ× , where βn is the nominal charring rate


obtained from the char depth measured after 1 hour of fire exposure (βn = 0.635 mm/min). The latter model gave the following result: d = 45.7 mm. As most deterministic models used in the structural engineering tend to yield conservative results, the smaller value of 0.04 m was assumed as the mean value 0µ of the prior of Θ1d. The value of 2

0σ was taken to be equal to 0.0016 m2 by assuming that the coefficient of variation of Θ1d is equal to 10%. The latter assumption was made subjectively.

The prior distribution of the second parameter Θ2d was developed by assuming that the coefficient of variation of the normally distributed char depth d is equal to 15%. This means that the standard deviation of d at the most likely mean of 0.04 m is equal to 0.006 m. Then the corresponding value of the precision will be 27 778 m–2. This value was taken as an approximate mode of a gamma distribution used to express the epistemic uncertainty in the parameter Θ2d. The latter assumption yielded the distribution parameters α0 = 278 and β0 = 0.01 given in Eq. (3.14b). The ratio α0/β0 is the mode of a gamma distribution. The values of α0 and β0 give the ratio α0/β0 equal to 27 800 m–2. This value is approximately equal to 27 778 m–2. The prior densities

)m.00160m,.040|( 21dθπ and )m0.0178,2|( 2

2dθπ are shown in Fig 3.6.

Densi

ty

0.02 0.03 0.04 0.05 0.06Θ1d, m

0

20

40

60

80

100

Densi

ty x 0

.0000

1

19 22 25 28 31 34 37(X 1000.0)

0

48

12

16

20

24

Θ , m-222−

d Fig. 3.6. Graphs showing prior densities of the uncertain parameters Θ1d

and Θ2d of the char depth distribution

With the prior distributions of the char depth parameters Θd = (Θ1d, Θ2d) and the parameters Θ1 and Θ2, which are present in the mechanical model and are uncertain in the epistemic sense, the prior distribution of the failure probability will be a distribution of the epistemic random variable =),,|( 21 ΘΘdiDP Θ

ddffedrP dDd

ii )d|()d(0))()|,(( 21 Θzz|zzz

Z∫ ∫

<−= ΘΘ (3.15)


where 1Θ and 2Θ are the epistemic random variables used to model uncertainty in the accuracy of the resistance of the beam and uncertainty related to conversion of ground snow load to roof snow load, respectively (Table 3.2). The distribution of ),,|( 21 ΘΘdiDP Θ was obtained by means of a Monte Carlo simulation. The simulation started from sampling the values θdk, θ1k and θ2k of Θd, Θ1 and Θ2 from the probability distributions given by Eqs. (3.14) and in Table 3.2. Then the integral in Eq.(3.15) was evaluated for the sampled values by computing the following estimate of ),,|( 21 kkdkiDP θθθ : ∑

=

−=

N

lkkdklk N

121

1 ),,|( θθµ θz1 (3.16a)

≤= otherwise0

)( ),,( if1),,|( 2121

kk |zzz θθθθ lillikkdkl

edrθ1 (3.16b) where zl is the vector of values of stochastic random variables sampled in the simulation loop l from the probability distributions given in Table 3.2; dl is the char depth sampled from the normal distribution with the simulated parameters θdk; ),,( 1kz θlli dr and )( 2k|z θlie are the values of the beam section resistance and action effect computed in repetition l of the nested loop and repetition k of the outer loop, respectively.

A total of 1000 estimates µk of ),,|( 21 kkdkiDP θθθ was computed by applying N = 1×105 repetitions of the stochastic (Monte Carlo) simulation. The histogram of the simulated sample {µ1, µ2, … , µk, … , µ1000} is shown in Fig. 3.7. The prior distribution π(µ) of Pf(treq) was obtained by fitting a Weibull distribution W(0.0951, 1.106) to this sample. Form of new data The initial new information which can be used for updating the prior π(µ) of Pf(treq) is expressed in the form of 30 simulated fire time-histories {yj(tτ), τ = 1, 2, … , 60} grouped in the sample y = {y1, y2, … , yj, … , y30} (Sec. 3.2). As the estimation of the damage probability Pf(treq) is based on the fragility function developed for the char depth d used as a demand variable, namely,

))(|( reqi tdDP , the initial new information represented by y was transformed into information expressed through the char depths corresponding to the fire time-histories {yj(tτ), τ = 1, 2, … , 60}. The transformation was not straightforward due to epistemic uncertainties inherent in the mechanical model of the timber beam, )|),(,( Θzz reqi tdm , and the char model dj(treq, x, z | Θ). These uncertainties are expressed by the components of the vector Θ. The safety


margin mi(⋅) contains two uncertain parameters Θ1 and Θ2, whereas the char depth dj(⋅) depends on nine uncertain parameters Θ3 to Θ11 (Table 3.2). This means that the char depth dj(treq, yj(tτ), z) and so the value of the fragility function ))(|( τtDP ji y corresponding to the time history yj(tτ) will be uncertain in the epistemic sense. In other words, the fragility function related to the time-history j will be an epistemic random variable )),(|( ΘτtDPP jij y~

= (3.17) With the random variables jP

~ (j = 1, 2, … , 30), the Bayesian updating of the prior π(µ) becomes a problem of an updating with imprecise data (e.g., Siu and Kelly 1998). This means that it is impossible to form the sample of new fixed data, p, defined by Eq. (2.8). As suggested by Vaidogas and Juocevičius (2009ab), this sample can be replaced by an alternative sample expressed as p′ = {(pjk, k = 1, 2, … , m), j = 1, 2, … , 30} (3.18)

0.0 0.1 0.2 0.3 0.4 0.5Failure probability

0

40

80

120

160

200

240

Num

ber o

f obs

erva

tions

Kolmogorov-Smiornov D = 0.031, p = 0.297

π(p | 0.0951, 1.106)

Fig. 3.7. Graph illustrating the choice of the prior π(p): histogram of the sample { µ1, µ2, … , µ1000} and the density of the Weibull distribution

W(0.0951, 1.106) fitted to this sample

where pjk (k = 1, 2, … , m) are estimates of the fragility function value )),(|( kji tDP θτy related to the time-history j and the value θk of the vector Θ

sampled from the probability distributions given in Table 3.2. The sample p′ was formed by computing 5 estimates of )),(|( kji tDP θτy , that is, m = 5.


Consequently, the sample p′ contains 150 elements (Annex A). This sample was used instead of p as new data for the updating of the prior density π(µ) expressed as a density of the Weibull distribution π (µ | 0.0951, 1.106). Bayesian updating and posterior estimate of failure probability The sample p′ containing 150 elements was used to calculate the likelihood function estimate )|( 150 µµBL by means of Eq. (2.19). Then Eq. (2.20) was used to obtain the approximation of the posterior, )|( 150µµπ ˆˆ . The normalizing constant )( 150µC obtained by a numerical integration was equal to 2.699297. Fig. 3.8 shows the graphs of the functions π(µ), )|( 150 µµBL , and )|( 150µµπ ˆˆ .

0.00 0.05 0.10 0.15 0.20 0.25 0.30Failure probability µ

0

2

4

6

8

10

12

14

Dens

ity

Fig. 3.8. Likelihood function estimate )|( 150 µµL (solid line), prior π(µ) (dash and line) and estimate of the posterior, )|( 150µµπ ˆˆ , (dotted line)

obtained with the bandwidth w = 0.1

The number of bootstrap replications, B, necessary to generate the sample { 1nµ ′ˆ , 2nµ ′ˆ , … , Bnµ′ˆ } was taken to be equal to 1000. The choice of B was based on the rules of thumb suggested by Efron and Tibshirani (1993: 39) and Davison and Hinkley (1998: 21). The estimate of the likelihood function,

)|( 150 µµBL was obtained by applying the Gaussian kernel function κ(.) (e.g., Davison and Hinkley 1998: 168). The approximations of the posterior,


)|( 150µµπ ˆˆ , were computed at the bandwidth w = 0.1. This value was chosen using the rule w ∝ B–1/3 proposed by Davison and Henley (1998: 227).

The approximation of the posterior, )|( 150µµπ ˆˆ , expresses the updated epistemic uncertainty in the failure probability Pf(treq). Fig. 3.8 indicates that

)|( 150µµπ ˆˆ is more accurate (narrow) than the prior π(µ). The degree of “accuracy” can be expressed by the ranges of non-conservative and conservative percentiles given in Table 3.3. The 30 simulated fire histories represented by the sample y decreased the uncertainty expressed by the prior π(µ). One can anticipate that the decision-maker will understand the conservative percentiles shown in Table 3.3 better than the entire density )|( 150µµπ ˆˆ . Thus the decision concerning the potential failure event Di can be made by applying these percentiles. Table 3.3. Pairs of approximate percentiles derived from the prior density π(µ) and the approximation of the posterior densities, )|( 150µµπ ˆˆ Characteristic of density Prior π(µ) Posterior estimate )|( 150µµπ ˆˆ

10th percentile 0.0125 0.0090 90th percentile 0.2035 0.1155 Range 0.191 0.1065

3.5. Third chapter conclusions 1. Building fires are severe and highly random phenomena capable to

cause substantial damage to built property. An assessment of fire risk to structures requires to quantify uncertainties related to fire actions and fire damage. These uncertainties can be effectively handled within the Bayesian format.

2. Statistical information on accidental actions caused by fires is expensive and can be obtained by means of computer fire simulation. This information can be expressed in the form of a small-size sample of time-histories of fire characteristics. The method proposed in Chapter 2 allows to integrate this small-size sample into an estimation of fire risk to timber structures.

3. The small-size sample of time-histories of fire characteristics can be transformed into a set of char depths. These can be used for an estimation of a failure probability of a timber beam subjected to fire.


4. The small-size sample of time-histories of fire characteristics (temperature and oxygen content, say) can be integrated into fire risk assessment by developing a fragility function with a demand variable computed from these time-histories. For timber structures, such a demand variable will be a depth of char caused by fire.

5. Mathematical models used for predicting the char depth of timber structures contain a large number of uncertain parameters which can be modelled within Bayesian format. Quantitative measures expressing such uncertainties can be propagated and expressed in the form of uncertainty related to failure probability of a timber structure subjected to fire.

6. The method proposed in Chapter 2 is suitable for assessment of risk to timber structures having to sustain the action of fire.

73

4 Assessing damage from an

accidental explosion by means of the proposed method

Chapter 4 describes how to apply the method from Chapter 2 to a design of a safety barrier intended to protect against an accidental explosion. This field of application of the proposed method was chosen due to a highly random nature of accidental explosions. The behaviour of structures built to protect against them carries also a great deal of uncertainty. The Bayesian modelling is used in this chapter mainly for quantifying uncertainties inherent in the mechanical model of the barrier. The accidental action considered in this study was a blast generated by an explosion of a railway tank car. The action is represented by a small-size sample of pressure signal characteristics. This chapter summarises the results of investigations published by Juocevičius (2009), Juocevičius et al. (2010) Juocevičius and Vaidogas (2010), Linkutė et al. (2011)

4.1. Situation of explosion accident and safety barrier In the present chapter, the failure probability estimation method proposed in Chapter 2 will be applied to a design of a key element in a barrier intended for a

74 4. ASSESSING DAMAGE FROM AN ACCIDENTAL EXPLOSION BY MEANS …

Fig. 4.1. The train formation station near Vilnius (Lithuania) (obtained by

using the Google Earth software)

(a) (b)

(c) Fig. 4.2. The aftermath of a tank car explosion which happen on 8

November 2010: (a) oil tank explosion; (b) failed tank; (c) remains after explosion and fire

4. ASSESSING DAMAGE FROM AN ACCIDENTAL EXPLOSION BY MEANS … 75

Fig. 4.3. A tank car used for the transportation of liquefied propane

Fig. 4.4. The areal view on the diesel fuel tanks exposed to the danger of a

potential BLEVE on rail

protection against an accidental explosion on railway (briefly, explosion barrier). The accident capable to escalate into an explosion can happen in a train formation station in the vicinity of Lithuanian capitol Vilnius (Fig. 4.1). Such an accident occurred on 8 November 2010 in Polish town Bialystok which is located relatively close to Vilnius. Photos taken in the aftermath of this accident are given in Fig. 4.3.

The explosion can happen as a boiling liquid expanding vapour explosion (BLEVE) of a tank car used for the transportation of liquefied propane gas (Fig. 4.3). Such explosions generate effects of three kinds: blast, thermal radiation and projectiles (fragments of the tank projected during the explosion). The present study will consider only the mechanical effect induced by the blast.


Albeit the explosion barrier will be designed for the effect of blast, it can be useful for the protection against thermal radiation and impact by projectiles.

The accident situation analysed in the present case study involves a diesel fuel tank (”target”) located 63 m form the external track of the station (Figs. 4.1 and 4.4, 4.5). The worst case scenario is considered, according to which the tank car will be in the least favourable position in terms of the blast intensity. The exploding tank car will be directed perpendicularly to the fragment the explosion barrier. Consequently, the angle of incidence of the shock front generated by the tank BLEVE will be close to the mostly adverse 90 degrees (Fig. 4.6).

Fig. 4.5. The elevation of the accident situation (the section A-A is

indicated in Fig. 4.6)

R 1

Fig. 4.6. The plan of the potential accident site

The fuel tank is surrounded by a protective embankment which will be used as a foundation of the explosion barrier (Fig. 4.5). In this way the height of the future barrier will be enhanced by the height of the barrier. The barrier is to be


built from double-tee steel posts serving as the main support structure and profile steel sheets spanning horizontally between the posts (Fig. 4.7). This arrangement of structural elements in typical to walls and barriers used for explosion isolation (Boh et al. 2004; Langdon and Schleyer 2005, 2006).

h

s

Fig. 4.7. Structural details of the steel explosion barrier: (a) vertical section of the protective barrier; (b) plan of one segment of protective barrier;

(c) cross-section of profiled sheet

4.2. Statistical sample of explosive action values The form of sample The AA considered in the present case study will be the blast induced in the course of BLEVE of a tank car. Data on this AA will have the form a two-dimensional statistical sample y = {y1, y2, … , yj, … , yn} (4.1) where the elements of y are vectors yj = (y1j, y2j) containing the overpressure y1j and positive phase duration y2j of the incident shock front generated by BLEVE. The model used to estimate characteristics of the blast The overpressure y1 can be determined by applying methods used to relate the energy of BLEVE to explosion of TNT mass with an equivalent energy (Prugh 1991). The equivalent mass of TNT in kg can be calculated for a container with liquefied vapour by ))101(1(1

1042 1)-(4

kkPk

VPC −−

⋅⋅=

− *. (4.2)


where P is the pressure of liquefied vapour in kPa; k = 1.13 is the specific heat of propane vapour; the quantity V* is expressed using the following equation ))(1/)/(( TLTVLT DDfWVV ;;

* −+= (4.3) where VT is the total volume of the container in m3; WL is the weight of liquid in the container in kg; f is the flashing fraction; DL;T and DV;T are densities of the liquid and the saturated vapour at the temperature and pressure existing in the container at the moment of rupture, respectively. Densities DV;T and DL;T can be taken from the thermodynamic tables given in Perry et al. (1997) or using fitted equations expressing densities DV;T and DL;T through the pressure of liquefied vapour P

294

26

1046511025211006402100940

PPPPD TV −−

−

⋅+⋅−

⋅−+=

..

...

; (4.4)

)ln(165102530630296663 39 PPPPD TL ....; −⋅−+−= − (4.5) The effects of the equivalent TNT mass explosion Peak overpressure y1 ≡ Pmax, positive phase duration y2 ≡ td and incident impulse i can be expressed in the sense of equivalent TNT mass and distance to the centre of explosion using following equations (Korenev, 1981)

Fig. 4.8. Functions of liquid and vapour densities DL;T and DV;T fitted to the

data given by Perry et al. (1997)


32

3 234143010 RC

RC

RCP ...max ++= (4.6)

RCtd 631071 −⋅= . (4.7)

RCi

3 236.= (4.8)

where C is the equivalent mass of TNT in kg; R is the distance between the barrier and explosion centre.

Using Eqs. (4.6–4.9) positive phase duration td can be expressed through the incident overpressure Pmax (as shown in Fig. 4.9) [ ] 2)0.88ln(2.68 maxd Pt += (4.9)

Peak reflected overpressure in terms of incident angle α can be expressed through the reflection ratio Cr, introducing Mach number M and reflected Mach number Mr (Bangash, 2009): )7)((7

25 22

amaxamax

max

PPPPPM

++= (4.10)

t d, m

s

Fig. 4.9. Function expressing positive phase duration td through incident

overpressure Pmax


αMM r sin= (4.11)

1)42(361))(717(

2

22

−

−−−=

MMM

C Rr (4.12)

where Pa is initial atmospheric pressure. Then the peak reflected pressure can be expressed in terms of reflection ratio Cr by aamaxrr PPPCP +−= )( (4.13) Pressure signal of TNT explosion In the case of an impulsive explosive loading, the response of the protective barrier shown in Fig. 4.7 will occur in so short a time that no viscous damping can be invoked (Tedesco et al. 1999). For a structure subjected to such a loading, the first displacement peak will be the most severe. Subsequent cycles will decrease significantly in magnitude and the oscillation will die down rapidly. Moreover, under severe loading, the structure is likely to undergo excessive permanent deformation during its first displacement, and it is very unlikely for the structure to fail during its second displacement peak. Therefore, in most cases, only the first displacement peak is considered in analyzing structural response to explosive loading (Low and Hao 2001).

)/atan()atan( risemax tPp =&

Fig. 4.10. Pressure signal of the shock fronts, incident or reflected, resulting

from an explosion

A loading of a structure directly exposed to an incident shock front generated by an above ground explosion takes place during the reflection of this front. The typical pressure signal p(t) of incident and reflected shock fronts of a large and distant free field explosion is characterised by the peak overpressure maxP and the positive phase duration trise + tdecay (Fig. 3.10, e.g., (Bulson 1997)).

The negative phase is usually ignored in the explosive damage assessment. In order to distinguish between the pressure signals of incident and reflected shock


fronts, characteristics of these signals will be denoted by the subscripts “i” and “r”, respectively.

Records of the pressure time-history of distant explosions allow to assume a linear form of the function pi(t) within the rise time ti,rise (Fig. 3.10 (Wu and Hao 2005, 2007)). This linear part of pi(t) is expressed as i,risei

rise,i

i,maxi tttptt

Ptp <== ,)( & (4.14)

where Pi,max is the peak overpressure of the incident shock front; ip& is the rate of pressure increase. The assumption of the linear form of pi(t) within ti,rise implies that the pressure increases at a constant rate ip& . During the reflection time tr,rise + tr,decay, the structure will be subjected to a varying straining rate. However, the traditional approach is to assume this rate to be constant for within the time tr,rise + tr,decay. It is stated that this assumption gives good results (Krauthamer et al. 1994, Krauthamer 2008).

The rise time ti,rise can be roughly assumed as maximum 25% of the decay time ti,decay (Zhou and Hao 2008). Low and Hao used for their calculations the value ti,rise = 0.1ms (Low and Hao 2002). However, for TNT (trinitrotoluene) explosions, a more accurate modelling exists and allows to assess ti,rise of the incident shock front from the following empirical relation (Krauthammer et al. 1994) 2

1κ

rise,i κt ∆= (4.15) with the scaled distance 31/QR∆ = (4.16) where R is the distance from the charge centre; Q is the TNT charge mass; 1κ and 2κ are parameters, namely, regression coefficients (Juocevičius and Vaidogas 2010).

In the course of the reflection, the overpressure on the face of the structure rises to r,maxP at the instant tr,rise and then decreases to the ambient pressure after the time tr,decay. It is reasonable to suppose that the enhancement of mechanical properties of concrete and steel is affected by loading rate Pr,max / tr,rise (hereafter denoted by rp& ) within the rise phase (0, tr,rise). The question how the enhancement might be influenced by sudden stop of pressure increase at tr,rise and a decrease in the early part of the decay phase (tr,rise, tr+) remains unanswered, to the best of our knowledge.


One can roughly estimate the reflected specific impulse ir by assuming a similarity between the time-histories of the overpressure in the incident and reflected shock fronts. This assumption yields the following relation (Baker 1983)

i

r

i,max

r,max

ii

PP

≈ (4.17) where ii is the specific impulse of the incident shock front.

The similarity of pressure signals allows to make also an assumption about the ratio of the rising times of the incident and reflected shock fronts:

i,rise

r,rise

i,max

r,maxtt

PP

= (4.18) The above expression means that the rates of pressure increase in the incident and reflected fronts, ip& and rp& , are equal. To make this assumption less stringent one can assume the linear relationship. The assumption (4.18) implies that the rate rp& can be estimated by the parameters of the incident pressure signal:

rise

i,maxr t

Pp =& (4.19) The incident overpressure i,maxP can be estimated by means of an empirical function of the scaled distance ∆ (Krauthammer et al. 1994):

≤<≤≤−=101if110if

7

4

6

53

∆∆κ∆κ∆κP κ

κ

i,max. (4.20)

where 3κ to 7κ are regression coefficients (Juocevičius and Vaidogas 2010). As the empirical relations (4.15) and (4.20) depend on ∆ , the rate rp& can be expressed as a function of the incident overpressure i,maxP alone:

,

72

6

11

κκi,max

i,maxr κPPκp

−

−

=&

MPa1.008MPa0.0098 << i,maxP (4.21)


,

42

3

511

κκi,max

i,maxr κκPPκp

−

−

+=&

MPa384.5MPa1.008 << i,maxP (4.22) Eqs. (4.19), (4.21) and (4.22) imply that the loading rate rp& can be predicted with the peak overpressure of the incident (not the reflected) shock front i,maxP . The simulated sample of the accidental action Experiments which could yield the elements yj of the sample y should imitate a real-world BLEVE of a propane tank. We know that such experiments were carried out in previous years. For instance, in 1999 The Federal Institute for Materials Research and Testing (BAM) in Germany has carried out a full-scale BLEVE test with a tank car (see a footage of the test at http://www.bam.de/filme/bam_047/film_047_tank_waggon.htm). However, such tests are hazardous and extremely expensive. Therefore, the sample y used in the present case study was obtained by calculation and not by a direct recording the sample components yj. Instead of measuring the overpressure y1j and the duration y2j, values of y1j and y2j were calculated by the respective Eqs. (4.6) and (4.7). The key input value of these equations, the TNT mass C, was calculated by applying the equivalent TNT energy method proposed by Prugh (1991) and represented by Eq. (4.2). The real-world statistical sample used in the present case study was compiled from 30 pairs (Wj, Pj), where Wj and Pj is the weight and pressure of liquefied propane measured in the tank car j. The sequence of calculations starting with the values Wj and Pj and ending with the values y1j and y2j for all 30 tank cars in presented in Table 4.1.

Thus the bivariate sample y defined by Eq. (4.1) will consist of 30 pairs y1j and y2j and have the form

=

4444444 34444444 21

elements sample30ms73.24kPa13.50,...,ms08.25

kPa14.04 ,ms74.24kPa13.51

y (4.23)


Table 4.1. Information on the sample containing characteristics of the blast, Pmax and td

No. Liquid weight WL, kg

Container pressure P, kPa

Liquid density DL;T, kg/m3

Vapour density DV;T, kg/m3

Equivalent mass of TNT, kg

Peak incident overpressure y1j ≡ Pmax, kPa

Positive phase duration Y2j ≡ td, ms

1 60939 2575.78 409.9 74.50 83.29 13.51 24.74 2 57566 2462.15 417.2 69.12 90.30 14.04 25.08 3 57419 2395.92 421.2 66.19 77.14 13.04 24.43 4 59472 2602.83 408.1 75.85 99.41 14.69 25.48 5 54108 2453.13 417.7 68.71 73.21 12.72 24.21 6 56751 2312.08 426.0 62.69 66.63 12.18 23.84 7 61307 2615.38 407.3 76.50 71.69 12.60 24.13 8 59950 2264.70 428.7 60.79 89.97 14.01 25.06 9 55176 2572.24 410.2 74.32 74.78 12.85 24.30 10 58094 2531.86 412.8 72.36 73.30 12.73 24.22 11 57839 2446.98 418.1 68.43 83.50 13.53 24.75 12 58116 2270.15 428.4 61.01 52.10 13.42 24.68 13 57777 2424.60 419.4 67.44 83.45 13.52 24.75 14 60724 2457.83 417.4 68.92 79.33 13.21 24.54 15 56333 2411.36 420.2 66.86 77.83 13.09 24.46 16 55878 2193.22 432.6 58.03 71.71 12.60 24.13 17 59339 1922.22 446.2 48.51 64.05 11.96 23.68 18 52549 2301.81 426.6 62.27 64.18 11.97 23.69 19 59697 2364.44 423.0 64.85 82.32 13.44 24.69 20 59215 2406.56 420.5 66.65 74.52 12.83 24.29 21 60088 2492.00 415.3 70.48 86.58 13.76 24.90 22 55379 2581.78 409.5 74.79 78.12 13.11 24.48 23 58567 2502.49 414.6 70.97 71.68 12.60 24.13 24 53204 2613.84 407.4 76.42 73.93 12.78 24.25 25 57594 2204.21 432.0 58.45 81.13 13.35 24.63 26 58586 2355.46 423.5 64.48 70.36 12.49 24.05 27 53499 2461.10 417.2 69.07 78.41 13.14 24.49 28 51802 2508.66 414.3 71.26 79.44 13.22 24.55 29 57106 2351.95 423.7 64.33 68.00 12.30 23.92 30 55286 2471.80 416.6 69.56 83.16 13.50 24.73


4.3. Fragility function of a safety barrier component General expression of the fragility function The fragility function P(Di | y) introduced in Chapter 2 will be developed for the profiled steel sheets of the barrier. The key structural elements of the barrier are vertical posts fixed in the foundation and horizontal profiled sheets spanning between posts. The posts will not be considered in this study. The characteristics of the incident shock front y discussed in Sec. 4.2 will be used as demand variables.

The damage event Di will consist in attaining or exceedance of two limit states of the profiled sheets: (i) ultimate limit state when peak reflected overpressure exceeds maximum dynamic capacity of the barrier; (i) serviceability limit state when dynamic plastic deflection exceeds maximum tolerable value. The fragility function P(Di | y) will be expressed through the random safety margins related to these limit states: 0)))((0))((()|()|( 2121 ≤≤== yyyy MMPDDPDP i UU (4.24) where D1 is the event of exceeding the dynamic elastic flexural capacity of the profiled sheet; D2 is the event of exceeding the ultimate dynamic plastic deflection; M1(y) and M2(y) are the safety margins correlating to the failure events D1 and D2 )()|()|,(1 yΘZΘyZ rR ppM −= (4.25) )|,()|,()|,(2 ΘyZΘyZΘyZ

,, dynplupl uuM −= (4.26) with ) , , ,( 4321 ZZZZ=Z (4.27) ) , ... , ,( 521 ΘΘΘ=Θ (4.28) where Z and Θ are the vectors of random variables used to model aleatory and epistemic uncertainties, respectively (components of Z and Θ are explained Table 4.2); )(⋅Rp is the deterministic model used to compute resistance (dynamic pressure capacity) of profiled sheet; )(⋅rp is the deterministic model used to compute the reflected overpressure (see Eq. (4.13)); upl,u(·) is the deterministic model used to compute the ultimate dynamic plastic deflection; upl,dyn(·) is the dynamic plastic deflection due to the external load applied during the reflection of the shock front. The cross-sectional dimensions are considered to be fixed (deterministic quantities).


Profiled steel sheets in the barrier shown in Fig. 4.7 will be idealised as one-way slabs and analysed as a single degree of freedom system (SDOF) (Louca et al. 2004; Biggs 1964). The profiled sheets are fixed on both ends; however,

Table 4.2. Aleatory and epistemic random variables used in the analysis of the explosion barrier shown in Fig. 4.7

Description and notation (notation used this dissertation ≡ notation from the original text by Louca et al. (2004))

Mean/coeff. of variation

Probability distribution

Aleatory random quantities (components of Z) Span Z1 ≡ L (m) 2.0/0.005(1) Lognormal Static yield strength of steel, Z2 ≡ py (MPa) 554/0.11(1) Lognormal Modulus of elasticity of steel, Z3 ≡ E (GPa) 200/0.06(1) Normal Natural period of elastic vibration, Z4 ≡ T (ms) 3.4/0.05 Normal

Epistemic random quantities (components of Θ) Enhancement factor for steel strength, Θ1 ≡ γ; the uncertainty in Θ1 was modelled by the expression 1 + ∆×ξ(2) (∆ = 0.12)

1.012/0.011 Beta, ξ ~ Be(1, 9)

The factor of uncertainty related to the model of ductility ratio µ, Θ2 1/0.04 Normal

N(1, 0.04) Reduction factor for stiffness of profiled sheet, Θ3 ≡ fK; the uncertainty in Θ3 was modelled by the expression 1 – ∆×ξ(3) (∆ = 0.3); the mode of Θ3 is equal to 0.85

0.85/0.05 Beta, ξ ~ Be(6, 6)

Reduction factor for transverse stress effect, Θ4 ≡ fC 0.99/0.085 Beta

Be(70, 1) Reduction factor for flattering of cross-section, Θ5 ≡ fF; the uncertainty in Θ5 was modelled by the expression 1 – ∆×ξ(3) (∆ = 0.2); the mode of Θ5 is equal to 0.952

0.933/0.0382 Beta, ξ ~ Be(2, 4)

(1) Spaethe (1987) (2) This linear transformation is used to obtain a Beta distribution defined on the interval ]1, 1.12 [ which covers potential values of the strength enhancement factor (Juocevičius and Vaidogas 2010) (3) This linear transformation is used to obtain a Beta distribution defined on the interval [∆, 1]


the mechanical model, which will be used for an analysis of the sheets, requires considering the ends to be partially fixed. The mechanical model was proposed by Louca et al. (2004) and is adjusted to steel sheets regulated by the British standard (BS EN 10088-1 2005). The standard requires to produce the profiled sheets from the steel with a minimum 0.2% proof stress of 460 MPa. The deterministic modulus of elasticity is equal 200 GPa. Geometry and material properties of profiled steel section under analysis is presented in Table 3.1. Natural frequency T of the profiled steel sheet section is approximately 3.4ms (Louca et al. 2004).

The dynamic pressure capacity of profiled sheet is given by 54

312

12

),(8)|( ΘΘlΘZl

wΘZpE

elR =ΘZ (4.29)

where lE(·) is the effective span; wel is the deterministic elastic section modulus depending on the cross-sectional dimensions t, l3, s and h shown in Fig. 4.7 with numerical values given in Table 4.3; l is the cross-sectional width (Fig. 4.7 and Table 4.3).

The reflected overpressure is given by )()( 1yppr ≡y (4.30) where p(y1) is the reflected overpressure defined by Eq. (4.13). In the present study, the characteristic pr(·) is considered as a deterministic function of incident overpressure (demand variable) y1 and calculated by Eqs. (4.10) to (4.13) with Pa = 98.07 kPa and α = 90°.

The maximum plastic dynamic deflection capacity is given by ×⋅⋅=

5

54

3

312

12

0.5),(

485)|,(

ΘΘΘ

hZΘZlΘZu E

upl ΘyZ,

),,,|,,,,( 543121421 ΘΘΘΘyyZZZµ× (4.31) where µ(·) is the function used to compute the ductility ratio and given by

= )(

),,,|Z,(,)|,(1

543121

4

22 yp

ΘΘΘΘZpZyφΘµ

r

RΘYZ (4.32) where φ(·,·) is the function fitted to the graphs developed by Biggs (1964: 74) and used for retrieving values of µ(·).

The dynamic plastic deflection due to the external load applied during the reflection of the shock front is computed using the following expression


)(384),()()|,( 2

3

114

1 yδZΘZlypu Er

dynpl ⋅=ΘyZ,

(4.33) where δ(y2) is the dynamic loading factor computed by

≤≤−+−= 2

242

442 0y2

)2sin()2cos(1max)( ytyt

ZπZπtZπtyδ

t (4.34)

Modelling aleatory and epistemic uncertainties The mechanical model presented in the previous subsection contains aleatory random variables (components of the vector Z) and epistemic random variables (components of the vector Θ) (see also Table 4.2). Probability distributions of the components of Z and Θ were chosen partly on the basis of information on natural variability of the quantities used in the analysis and partly on the basis of subjective reasoning.

The probability distributions of the aleatory random variables Z1 to Z3 can be easily specified from information on random properties of steel structures (e.g., Spaethe 1987). The natural period of elastic vibration, Z4 (or T according to Louca et al. 2004), is considered to be an aleatory quantity because this dynamic characteristic of profiled steel sheets can be measured experimentally. We assumed the nominal value of T given by Louca et al. (2004) to be a mean value of a normal distribution and used this value, along with the coefficient of variation equal to 0.05, as a parameter of a normal distribution expressing the uncertainty in Z4. The probability distributions of the epistemic variables grouped into the vector Θ were used to express uncertainty related to parameters of the models pR(·), upl,u(·) and upl,dyn(·). These distributions quantify the doubts about quantities represented by Θ expressed by Louca et al. (2004) and Juocevičius and Vaidogas (2010). We interpret the epistemic density π(θ) of the components of Θ as a prior distribution which can be updated, at least in theory, given a new data. Then the posterior density will have the form π(θ | data). However, new statistical evidence (“data”) necessary for such an updating is not available at present.

With the random parameter vector Θ, the fragility function P(Di | y) becomes an epistemic random variable defined as === ),|()()( 21 ΘΘ y|y~y~ DDPPP ii U 0)))|,((0))|,((( 21 ≤≤= ΘyZΘyZ MMP U (4.35)


The density of )( y~iP is illustrated in Fig. 4.11 for three elements of the

ordered sample y . This illustration assumes that the vector y has only one component and this is valid for the structures, the response of which is determined by the overpressure only (Watson 1994). The typical approach to dealing with epistemic uncertainties in fragility functions is establishing confidence bounds around the point estimates of fragility curve or median fragilities (e.g., Der Kiureghian 2000, Sasani et al. 2001, Straub et al. 2008). Most authors consider the confidence bounds the final result of analysis. However, a further propagation of the epistemic uncertainty quantified by Θ is necessary to estimate the failure probability Pf. In case where the explosion demand y is represented by the small-size sample y , the estimation of Pf can be expressed as an estimation of a mean of fragility function values with uncertain data )(f y~P (Vaidogas and Juocevičius 2009b).

4.4. Failure probability of barrier component The standard developing of the fragility function )(f y~P starts from specifying a set of values of the demand variable, {y, m = 1, 2, … , nm}, with elements usually distributed at equal intervals, i.e., my1 – 1,1 −my = ∆1 = const and my2 –

1,2 −my = ∆2 = const for m = 1, 2, … , nm – 1. Then a set of, say, Nk values θk of the random parameter vector Θ is sampled from π(θ) or π(θ | data) and the probabilities )|(f kjP θy (k = 1, 2, … , Nk) are estimated for each jy . This

)1( −jy y)( jy )1( +jy

)( )(f jyP~ )(f yP~

0

1

0

} , ... 2, 1, ,{ kk Nkp =

1)0.9]([ +⋅kNp

)1(y )(nyL L

kjp )(ˆ

kjp )1( −ˆ

kjp )1( +ˆ

Fig. 4.11. A schematic illustration of the epistemic uncertainty in the value

of the fragility function )(f y~P


yields nm samples, each consisting of Nk estimates of the fragility function. These samples are used to fit the median line and confidence bounds characterising the uncertain fragility function )(f y~P (e.g., (Sasani et al. 2001, Straub et al. 2008)). The ranges of the specified demand variable values, [ 11y ,

mny1 ] and [ 21y ,

mny2 ], is usually chosen to cover the range of possible

fragility function values, that is, [0, 1]. In the present case study, these ranges took the following values [ 11y ,

mny1 ] = [11.6 kPa, 14.4 kPa] and [ 21y ,

mny2 ] =

[23.0 ms, 25.8 ms]. An illustration of the fragility function )|(f kP θy estimated for one realisation kθ of Θ is shown in Fig. 4.12. Values of the failure

0.3 0.29 0.28 0.27 0.26 0.25 0.24

11.6 12.0 12.4 12.8 13.2 13.6 14.0 14.4Pmax, kPa

23.0

23.4

23.8

24.2

24.6

25.0

25.4

25.8

t d, m

s

Fig. 4.12. The surface of the fragility function )(f y~P with demand variables y1 (peak overpressure) and y2 (positive phase duration)

probability Pf shown in Fig. 4.12 were determined by the limit state of excessive ultimate dynamic plastic deflection (an occurrence of the event D2). The values of Pf were deliberately kept large because Pf is proportional to the percentage of the profiled sheets which will be destroyed during an accidental explosion (BLEVE). We think that this percentage does not need to be very small because BLEVE is a low-likelihood event and failures of the profiled sheets will not cause any harm to people.


Create the bootstrap samples pbk (b = 1, … , B) by sampling from the empirical p.d. of pk) , ... 2, 1, ( Bbbk =p kp

k < NkkNk ≤

End

no

Store the value pkkp

Sample θk from the p.d. of Θkθ Θ

Compute estimates pik of Pfl(v10, i ) and form the sample pk = {pik, i = 1, … , n}} , ... 2, 1, ,{ nipikk == ˆp)|(f kjP θyikp

Compute the bootstrap confidence interval ]0, pk[ for the samples pbkbkp[]0, kp

Form the sample {pk, k = 1, 2, … , Nk} } , ... 2, 1, ,{ kk Nkp =

yes

1 += kk

Compute the conservative percentile p([N 0.9]+1) )10.9]([ +⋅kNp

Fig. 4.13. The flowchart of estimating the failure probability of the profiled

barrier sheet, Pf

We think that the estimation of the fragility function )(f y~P can be carried out in a more effective way. After all, the final aim of the analysis is an estimation of the failure probability Pf and not the fragility function itself. An estimate of Pf can be obtained by estimating fragility function values )|(f kjP θy for the elements jy of the sample y . This will require to estimate n×Nk values of the random function )( Θ|y~

iP defined in Eq. (4.37). For each k, the estimates ikp of )|(f kjP θy can be grouped into the sample } , ... 2, 1, i ,{ npikk == ˆp . An

illustration of the sample elements ikp is given in Fig. 4.11. This sample can be used to calculate a one-sided bootstrap confidence interval []0, kp for Pf as described in Sec. 2.2.2 (see also Vaidogas and Juocevičius 2009b). A repetition of this process Nk times will yield the sample { ,kp k = 1, 2, … , Nk }. This sample will express the epistemic uncertainty related to the upper limit of this interval (Fig. 4.13). A conservative percentile of this sample, say, 1)0.9]([ +⋅kNp can


be used as the final result of the conservative estimation of the failure probability Pf (Fig. 4.11).

The failure probability Pf was estimated by computing the estimates jkp of the probabilities )|(f kjP θy with k = 1, 2, … , 500 (Nk = 500). A total of 15 000 of such estimates were computed for 30 elements jy of the sample y given in Table 4.1. Fig. 4.14 shows the histogram of the estimates kp . A conservative 90th percentile of this sample, 1)0.9]([ +⋅kNp , was equal to 0.263. This percentile can be used as the final result of the conservative estimation of the failure probability Pf. This percentile shows that 26.3% of profiled sheets will suffer a considerable damage (will be sacrificed). The value of Pf can be considered acceptable as the accidental explosions are infrequent events. If necessary, profiled sheets with larger moment of resistance or smaller spacing between vertical posts of the barrier can be selected in order to decrease the value of the conservative percentile 1)0.9]([ +⋅kNp .

0.257 0.258 0.259 0.260 0.261 0.262 0.263 0.264 0.265 0.266Failure probability

0

20

40

60

80

100

120

140

160

180

Frequ

ency

Fig. 4.14. Histogram of the simulated sample 00}5 , ... 2, 1, ,{ =kpk

4.5. Fourth chapter conclusions 1. Accidental explosions are hazardous phenomena with a great potential

of mechanical damage. The risk posed by accidental explosions should be assessed by taking into account considerable uncertainties related to explosive loading and response of structures subjected to this loading. A


proper way of quantification and propagation of these uncertainties is provided by Bayesian methods.

2. Accidental explosions are rare and unexpected events. Therefore statistical information on actions imposed by explosions will be sparse. In case where this information can be expressed in the form of a small-size statistical sample of explosive action characteristics, the method proposed in Chapter 2 can be applied assessing risk of explosive damage.

3. The small-size sample of explosive action characteristics (peak incident overpressure and positive phase duration) can be integrated into the risk assessment through a fragility function developed for an explosive action.

4. Uncertainties related to explosive loading and response of structure to this loading will be of both stochastic and epistemic nature. It is possible to propagate these uncertainties through the fragility function and develop a probability distribution for the probability of foreseeable explosive damage.

5. The method proposed in Chapter 2 is suitable for the design of structures built to protect against accidental explosions.

95

5 Extreme wind damage assessment by applying the proposed method

Chapter 5 presents an application of the risk assessment method from Chapter 2 to a case of extreme wind risk. The potential damage due to hurricane winds observed in Lithuania is studied. The structure under analysis is a reinforced concrete chimney. The hurricane wind pressures are treated as an accidental action. The statistical information on this action is expressed in the form of a small-size sample containing hurricane wind speeds recorded in the last half-century. Uncertainties related to a mechanical model of concrete chimney are expressed in the Bayesian format. The main motivation of the study presented in this chapter was that wind sensitive structures built in Lithuania were designed for annual maxima winds and not hurricane winds. Therefore there exists a need to reassess the safety of wind sensitive structures by taking into account hurricane winds. This chapter summarises the results published by Vaidogas and Juocevičius (2011), Vadlūga et al. (2011).

96 5. EXTREME WIND DAMAGE ASSESSMENT BY APPLYING…

5.1. Reinforced concrete chimney exposed to the hazard of hurricanes This chapter contains an application of the method proposed in Chapter 2 to an assessment of risk posed by an extreme natural phenomenon, namely, hurricane winds. Unlike man-made accidental actions induced during industrial accidents, extreme wind actions are created by nature. However, extreme winds are relatively rare and heavy-to-predict phenomena and so they are similar to man-made accidental actions. This is especially true for Lithuania, where hurricanes occur not every year, albeit damage to built property caused by them is far from being negligible.

Lithuania, is affected by hurricanes of moderate intensity. Local winds such as thunderstorms, downbursts, and tornadoes are infrequent and not considered a serous hazard to such tall structures as chimneys. The prediction of safety of tall structures during hurricanes requires the assessment of various uncertainties in the design procedure and then the estimation of the probability of failure.

The present chapter will consider the estimation of the failure probability of a reinforced-concrete chimney subjected to hurricane winds. A 250 m tall reinforced-concrete chimney, designed for non-hurricane winds and built in the 1980s in accordance with the deterministic structural design codes used in Lithuania at that time, will be analysed (Fig. 5.1). The chimney is supposed to be built again at a site in the coastal region of Lithuania which is prone to the most severe hurricanes.

The aim of the analysis is to estimate the probability of failure due to a hurricane wind. The simplified analysis will be carried out by considering only one horizontal section of the chimney and only the first mode of vibration.

The fragility analysis of tall reinforced-concrete chimneys under hurricane wind loads is presented. The analysis is carried out for the situation of limited data on hurricane wind speeds. This situation is characteristic of hurricane prone regions of Lithuania. The fragility is estimated for small-size samples of hurricane wind records. The standard mean wind velocity recorded at the standard height is used as a demand variable of a wind fragility function. It is proposed not to develop the fragility function for the entire range of probability values but to compute estimates of this function only for elements of a small-size sample of the hurricane wind records. These estimates can be used for the estimation of a chimney failure probability. The estimation of the values of chimney fragility must take into account both aleatory and epistemic uncertainties inherent in the problem. A tall reinforced concrete chimney is used to illustrate the proposed procedure of fragility analysis. Extreme winds

5. EXTREME WIND DAMAGE ASSESSMENT BY APPLYING… 97

21.80

± 0.00 m5.00

250.009 . 0 8

23.46

11A

1-1

8.03

ttop= 0.2tbottom= 0.7

187.50

H

h l

2-2

t = 0.222

A

segment l

2.5

h k

segment k

Fig. 5.1. Chimney built in 1988 in Vilnius (Lithuania) and analysed for

hurricane winds: (a) photo; (b) schematic sketch

constitute a hazardous loading condition to such tall and slender structures as chimneys. Probabilistic methods and statistical analysis of extreme wind data are fundamental to evaluation of structural performance of chimneys subject to wind effects.

Extreme wind hazards are generally categorised according to causative meteorological system as hurricanes (cyclonic winds), tornadoes, thunderstorms, and downbursts (Holmes 2003, Twisdale and Vickery 1995). The prevailing wind hazard to tall structures built in Lithuania is hurricanes. The prediction of the structural safety of tall chimneys during hurricanes requires the assessment of various uncertainties in the design procedure of chimney and then the estimation of the probability of failure. The assessment of chimney fragility to extreme winds is an essential ingredient of the failure probability estimation. This study presents a procedure for the computation of estimates of a chimney fragility function and further use of these estimates for the failure probability estimation.


5.2. Statistical data on hurricane winds in Lithuania The form of sample of accidental action records The accidental action considered in this case study will be induced by a hurricane. In Eurocodes, wind loads are generally classified as variable, not accidental loads. However, the load induced by a natural phenomenon and not taken into account due to, say, very low likelihood of occurrence is considered to be accidental one (EN 1991 2006, JCSS 2001, ISO 2394 1998). In the Lithuanian situation, wind actions induced by hurricanes must be treated as accidental ones. Structures in the hurricane prone regions of our country have been designed for annual wind maxima and not the winds which occurred as hurricanes. Hurricanes in Lithuania are not annual events and thus design of structures for loads specified using annual wind speed maxima is insufficiently conservative. Consequently, the chimney shown in Fig. 5.1 will be analysed for the sample y = {y1, y2, … , yj, … , yn} (5.1) where the elements yj are hurricane wind speeds recorded by means of a standard measurement procedure. The sample size n will be equal to the number of hurricanes in a particular region of Lithuania. The wind speeds yj are the so-called 10-min mean wind velocities recorded at the standard height of 10.2 m. Therefore the wind speeds yj will be denoted by the symbols jv10 specific to wind engineering. The sample y will take the form 10v = ), ... , , ... ,( ,10,10,110 ni vvv (5.2)

An in-depth analysis of extreme winds recorded in Lithuania in the last 48 years was carried out by Vaidogas and Juocevičius (2011). Results of this analysis are presented in Annex B. Lithuania is affected by moderate cyclonic winds which are classified as hurricanes. Local winds such as thunderstorms, downbursts, and tornadoes are infrequent and not considered a serous hazard to such tall structures as chimneys. Annual frequency of hurricanes in coastal region of Lithuania is 0.39 per year and in the south-western part is 0.28 per year (Table 5.1) (Vaidogas and Juocevičius 2011).

The recorded hurricane wind speeds 10v ranged between 33 m/s and 40 m/s. In the period 1962 to 2005, the numbers of hurricanes in these two regions were equal to 17 and 10 respectively. We think that the risk to wind sensitive structures posed by hurricanes in Lithuania is non-negligible. The design wind speeds specified in the current Lithuanian loading code for three regions I, II and III are 27.4 m/s, 31.9 m/s, and 36.5 m/s, respectively. These values are insufficiently conservative and were exceeded in all three regions during


hurricanes (Vaidogas and Juocevičius 2011). Design wind speeds specified in the preceding loading codes used in Lithuania from the late 1980s till 2003 were even less conservative.

The chimney shown in Fig. 5.1 will be analysed for the sample 10v defined by Eq. (5.2). The sample size n will be equal to 17 and 10 for the hurricane prone regions A and B, respectively. For the brevity sake, results will be presented only for the sample consisting of 17 observations: 10v = (34 m/s, 40 m/s, … , 40 m/s, 37 m/s) (5.3)

Table 5.1. Statistical samples of hurricane wind speeds (m/s) in the hurricane wind regions A, and B (Fig 5.2) (Vaidogas and Juocevičius 2011)

10-min 10 m wind speed No of sample element Region A Region B

1 34 (1962/1) 34 (1967/5) 2 40 (1967/5) 38 (1969/7) 3 34 (1969/7) 34 (1970/10) 4 36 (1969/8) 34 (1971/14) 5 33 (1969/9) 38 (1972/17) 6 34 (1970/10) 34 (1973/18) 7 35 (1970/11) 35 (1976/21) 8 40 (1970/12) 35 (1981/24) 9 35 (1971/15) 35 (1993/30) 10 37 (1972/17) 34 (2002/32) 11 35 (1973/18) 12 34 (1981/24) 13 35 (1982/25) 14 35 (1983/26) 15 36 (1993/30) 16 40 (1999/31) 17 37 (2005/33)

Mean v (m/s) 35.9 35.1 Std.dev. s (m/s) 2.23 1.60 v50 (m/s) 43.3 40.4 v100 (m/s) 44.9 41.5 Period of record 44 years 36 years Annual frequency 0.39 per year 0.28 per year


The statistical quality of this sample is impaired by two facts. Firstly, the size of the sample is too small from the standpoint of the classical statistics. Secondly, the sample contains a relatively large number of repeated values. This makes a reliable fitting of a probability distribution to this sample problematic. However, we think that this sample is still suitable to a probabilistic wind risk assessment by applying method proposed in Chapter 2.

5.3. Wind fragility function of reinforced concrete chimney General expression of the fragility function The fragility function P(Di | y) introduced in Chapter 2 will be developed for the chimney considered in the present case study. In general, the failure probability for a chimney exposed to a hazard of an extreme wind is expressed as ∫==

vVii vvfvDPDPP

allf )d()()(

10| (5.4)

where Di is the random event of chimney failure in the mode i (attainment or exceedance of the limit state i), v is the wind demand on chimney, P(Di | v) is the conditional failure probability, conditional on v and )(

10vfV is the probability

density function of the random demand 10V . The density )(10

vfV defines the wind hazard and the conditional probability is the fragility of chimney. Eq. (5.4) makes clear that a structural fragility analysis is an essential ingredient of a

≥33 m/s (hurricane)≥35 m/s (catastrophic wind)

Fig. 5.2. Weather stations with the records of hurricane and catastrophic winds as well as proposed hurricane wind regions A, B, and C (Vaidogas

and Juocevičius 2011)


failure probability estimation. A simple example of wind fragility curves can be found in the article (Kareem and Hseih 1986).

In the case where the random demand variable 10V expresses uncertainty related to hurricane wind speeds, the density )(

10 vV vf θ| with the parameter vector θv will have to be fitted to the aforementioned samples 10v consisting of 17 or 10 elements. Unfortunately, such fitting is problematic because the size of the samples is small and this will lead to estimates of θv which are insufficiently precise. However, an estimate of the chimney failure probability Pf defined by Eq. (5.4) can be obtained without fitting of the model )(

10 vV vf θ| , that is, by a direct application of the samples 10v as explained in Chapter 2.

In general, the random event Di stands for limit states associated with bending moment exceeding the moment capacity of the chimney cross-section at any level and excessive wind-induced deflection at the top are the most dangerous. The deflection at the chimney top is problematic from the serviceability standpoint. In many cases chimneys are built with a lining to prevent corrosion due to flue gases. A large lateral deflection could cause damage to the lining. Therefore, a limit state of deflection is established. For reasons of simplicity and brevity, the present study will consider only the limit state of excessive bending moment. In what follows, the random event D will stand for exceedance the ultimate limit state of excessive bending moment. The analysis of chimney reliability for both ultimate and serviceability limit states was carried out by Kareem and Hseih (1986).

Failure due to due to excessive bending moment may occur at any level under the action of wind loading which fluctuates in time and space. This requires considering the chimney a multiple-failure mode system. Such system is defined by dividing the chimney into segments, the number of which will be denoted by nl. With these segments, the fragility function is expressed as

≤−=

==

==

IIll n

lll

n

ll vERPvDPvDPvP

11010

11010f )0)(()()( | (5.5)

where Dl is the random event of exceedance of the ultimate moment capacity of the chimney at the level l (bottom level of segment l, say); Rl is the random resistance; )( 10vEl is the random action effect. In Eq. (5.5) the action effect is expressed as a function of a fixed (non-random) demand variable 10v ; nevertheless, El is a random variable because its value is influenced by many random factors which not necessarily depend on 10v . Safety margins Rl – )( 10vEl at different levels are correlated and this correlation increases as the distance between the chimney levels decreases. The correlation among the safety


margins follows from the correlations between resistances Rl and action effects )( 10vEl at different levels (Bottenbruch et al. 1989).

The action effect El is usually decomposed into its average (static) component and the fluctuation around the mean (dynamic component). The dynamic turbulent fluctuations are transformed into alongwind and acrosswind load effects. The total maximum bending moment at the level l is calculated as the square root of the sum of the squares of alongwind and acrosswind load effects (Kareem and Hseih 1986): == ),()( 1010 XvevE ll 1/22

10,2

10,10, )),(()),(),(( XXX veveve aclallstl ++= (5.6) where el(·) is the deterministic model of the total maximum bending moment at the level l; el,st(·) is the deterministic model of the static bending moment; el,al(·) and el,ac(·) are deterministic models of the maximum values of the dynamic bending moments in the alongwind and acrosswind directions, respectively; and X is the vector of random variables that describes fluctuation component, geometric parameters, random mechanical properties of concrete and steel, etc.). Some components of the vector X will appear also on the side of the resistance Rl and this can be denoted by the random function rl(X). With the expressions of the action effect and resistance as functions of X, the probability of chimney failure at the level l will take the form 0)),()(()()( 101010f ≤−== XX| verPvDPvP llll (5.7) The probabilities related to individual levels, )( 10f vP l , can be used to calculate upper and lower bounds of the chimney failure probability Pf (e.g., Kareem and Hseih 1986) Alternatively, the probability Pf can be estimated by means of Monte Carlo methods. The simplest method is called the direct Monte Carlo simulation. This method yields the following estimate of Pf: ∑

==

−=

N

jjl

n

lINp l

1 11

f )}({max Xˆ (5.8) where N is the number of Monte Carlo trials; Il(·) is the binary (zero/one) function indicating the survival or failure of chimney at the level l; Xj is the value of X simulated in the trial j.

There is a vast literature devoted to the mathematical modelling of the dynamic alongwind and acrosswind response to wind action, el,al(·) and el,ac(·). However, many authors note that the models el,al(·) and el,ac(·) include uncertainties related to both model parameters and model output (Pagnini 2010). The uncertainties in el,al(·) and el,ac(·) lead to uncertainties in the fragility


function )( 10f vP . The standard way to quantify these uncertainties is an application of the Bayesian approach to developing )( 10f vP . The Bayesian approach to modelling fragility is widely applied in the field of earthquake risk assessment (Choe et al. 2007, Der Kiureghian 2000, Sasani et al. 2001, 2002, Straub et al. 2008). Let θ be a vector which includes parameters of the models el,st(·), el,al(·), el,ac(·), and rl(·) and distribution parameters of X. In line with the Bayesian approach, the parameters θ are treated as uncertain quantities and uncertainty in them is quantified by means of prior and posterior densities, π(θ) and π(θ | data). To stress the role of θ in the problem, the fragility function can be reformulated as follows: 0))|,()|((),()|( 101010f ≤−== θθθθ XX| verPvDPvP llll (5.9) Modelling aleatory and epistemic uncertainties The prior and posterior densities π(θ) and π(θ | data) define a random vector Θ which expresses the epistemic (state-of knowledge) uncertainty, whereas the vector X express the aleatory (stochastic) uncertainty (e.g., Garrick 2008, Kumamoto 2007). With the random parameter vector Θ, the fragility function becomes an epistemic random variable )|()( 10f10f ΘvPvP ll =

~ (5.10) The density of )( 10f vPl

~ is illustrated in Fig. 5.3 for three elements of the ordered sample 10v . The typical approach to dealing with epistemic uncertainties in fragility functions is establishing confidence bounds around the point estimates of fragility curve or median fragilities (e.g., Der Kiureghian 2000, Sasani et al. 2001, Straub et al. 2008). Most authors consider the confidence bounds the final result of analysis. However, a further propagation of the epistemic uncertainty quantified by Θ is necessary to estimate the failure probability Pf. In case where the wind demand 10v is represented by the small-size sample 10v , the estimation of Pfl and Pf can be expressed as an estimation of a mean of fragility function values with uncertain data )( 10f vPl

~ (Vaidogas and Juocevičius 2009b).

5.4. Estimation of chimney failure probability The standard developing of the fragility function )( 10f vPl

~ starts from specifying a set of values of the demand variable, { mv ,10 , m = 1, 2, … , nm}, with elements usually distributed at equal intervals, i.e., mv ,10 – 1,10 −mv = ∆ = const for m = 1, 2,


… , nm – 1. Then a set of, say, Nk values θk of the random parameter vector Θ is sampled from π(θ) or π(θ | data) and the probabilities )|( 10f kml vP θ

, (k = 1, 2, …

, Nk) are estimated for each mv ,10 . This yields nm samples, each consisting of Nk estimates of the fragility function. These samples are used to fit the median line and confidence bounds characterising the uncertain fragility function )( 10f vPl

~ (e.g., Sasani et al. 2001, Straub et al. 2008). The range of the specified demand variable values, [ 1,10v ,

mnv ,10 ], is usually chosen to cover the range of possible fragility function values, that is, [0, 1].

We think that the estimation of the fragility function can be carried out in a more effective way. After all, the final aim of the analysis is an estimation of the failure probabilities Pfl and Pf and not the fragility function itself. Estimates of the failure probabilities can be obtained by estimating fragility function values

)|( 10f kil vP θ,

for the components iv,10 of the sample 10v . This will require to

estimate n× Nk values of the random function )|( 10f ΘvPl . For each k, the estimates ikp of )|( 10 kifl vP θ

, can be grouped into the sample

} , ... 2, 1, ,{ nipikk == ˆp . An illustration of the sample elements ikp is given in Fig. 5.3. This sample can be used to calculate a one-sided bootstrap confidence interval []0, kp for Pfl (Vaidogas and Juocevičius 2009b). A repetition of this process Nk times will yield the sample { ,kp k = 1, 2, … , Nk }. This sample will express the epistemic uncertainty related to the upper limit of this interval (Fig. 5.3). A conservative percentile of this sample, say, 1)0.9]([ +⋅kNp can be used

)1(,10 −iv 10v)(,10 iv )1(10, +iv

)( )(,10f il vP~ )( 10f vPl~

0

1

0

} , ... 2, 1, ,{ kk Nkp =

1)0.9]([ +⋅kNp

)1(,10v )(10, nvL L

kip )(ˆ

kip )1( −ˆ

kip )1( +ˆ

Fig. 5.3. A graphical illustration of the epistemic uncertainty in the value

of the fragility function )( 10f vP l~


as the final result of the conservative estimation of the failure probability Pfl (Fig. 5.4). The limit state of excessive bending moment was considered in this

example. For brevity, only the first mode of vibration was taken into account in the analysis. The natural frequency for the first mode was 0.31 Hz. The logarithmic decrement of damping was assumed to be equal to 0.15. The gust load factor G based on the displacement response of the chimney was used to obtain equivalent static wind loading (Zhou and Kareem 2001). The acrosswind vibration was ignored. The fluctuating part of the wind action is not modelled explicitly in this example.

To carry out the computation of the wind response, the chimney was divided into 100 segments along the height. The section at the level of 20 m (l = 8) was analysed and thus the probability Pf8 was the final aim of the analysis. This section was found in a pilot analysis to have the smallest safety margin M8 among Ml (l = 1, 2 … , 100).

The maximum bending moment )|,( 108 θXve was computed as a sum of the moment due to the horizontal wind action, )|,( 10,8 θXve st , and additional moment due to a displacement, )|,( 10,,8 θXve addst : =+= )|,()|,()|,( 10,,810,8108 θθθ XveXveXve addstst ∑ ∑

= =

+−=100

8

100

81010 )|,()|())(|,(

l lllll XveXwhHXvq θθθ (5.11)

where the )(⋅lq , )(⋅lw , and )(⋅le are the deterministic functions used to calculate equivalent static wind load applied to the segment l, the weight of the segment l and the horizontal displacement of the load )(⋅lw , respectively. The heights H and hl are explained in Fig. 5.1. The force )(⋅lq was computed by the standard formula

} , ... 2, 1, ,{ nipikk == ˆp

)|( ,10f kil vP θ ) , ... 2, 1, ( Bbbk =pkp bkp

[]0, kpikp

kθΘ kpkNk ≤1 += kk

)10.9]([ +⋅kNp } , ... 2, 1, ,{ kk Nkp = Fig. 5.4. The flowchart of the estimating the chimney failure probability Pfl


Table 5.2. Aleatory and epistemic random variables used in the analysis of the chimney shown in Fig. 5.4

Description Mean/coeff. of var.

Probability distribution

Aleatory random quantities (components of X) External segment diameters X1l (l = 1, 2, … , 100) (m)

Nominal value /(0.4…0.5)×10–2

Wall thickness of segments X2l (l = 1, 2, … , 100) (m)

Nominal value/0.03

Concrete density X3l (l = 1, 2, … , 100) (kN/m3) 25/0.06 Brick lining thickness X4l (l = 1, 2, … , 100) (m) 0.12/0.05 Brick lining density X5l (l = 1, 2, … , 100) (kN/m3) 20/0.05 Concrete secant modulus X5l (l = 1, 2, … , 100) (MPa) 31 000/0.124 Mu

ltivari

ate no

rmal

with

the co

rrelat

ion

coeff

icien

ts ρkl (se

e Note

)

Concrete strength in the base of segment 8, X7 (MPa) 33/0.15 UN (a) Reinforcement strength in the base of segment 8, X8 (MPa) 490/0.11 UL (b) Additional reinforcement strength at openings, X9 (MPa) 490/0.11 UL (b) External concrete cover in the base of segment 8, X10 (m) 0.05/0.025 UN (a) Internal concrete cover in the base of segment 8, X11 (m) 0.05/0.025 UN (a) Epistemic random quantities (components of Θ)

The coefficients θm used to calculate the correlation coefficients ρkl

0.15/δ, 0.2/δ, 0.4/δ, 0.1/δ, 0.1/δ, 0.2/δ; δ = 0.05

UN (a)

The drag coefficient θ7 ≡ Cd [18] 0.6/0.12 UN (a) The power law coefficient θ8 ≡ α [1: 49, 19] 0.133/0.114 UL (b) The dynamic coefficient θ9 [18] 2.18/0.1 Beta over

[0, 2.4] The coefficient of fluctuation of wind pressure θ10 [18] 1.9/0.1 UN (a) Note: ρkl = exp{–θm|hk – hl|}, k = 1, 2, … , 100; m = 1, 2, … , 6 [5] (the heights hk and hl are explained in Fig. 5.1)

(a) UN – Univariate normal; (b) UL – Univariate lognormal


≡= GvhdCXvq llDl210

210 0)1/(0.5)|,( αρθ

)(10)1/(0.613 109210

217

8 θθvhXθ θll +≡ (5.12)

in which Cd is the drag coefficient; dl diameter of the segment l; α is the power law coefficient; θ9 is the dynamic coefficient depending on the natural frequency of the first vibration mode and logarithmic decrement of vibration; θ10 is the coefficient of fluctuation of the wind pressure at the top of the chimney. The segment displacements )|,( 10 θXvel were calculated by applying the linear elastic analysis.

0.000 0.005 0.010 0.015 0.020 0.025 0.030Failure probability

0

40

80

120

160

200

240

Frequency

Fig. 5.5. Histogram of the simulated sample 00}5 , ... 2, 1, ,{ =kpk

Components of the vectors X and Θ in Eqs. (5.9) and (5.10) are defined in Table 5.2. The failure probability Pf8 was estimated by computing estimates ikp of the probabilities )|( 108f kivP θ

, with k = 1, 2, … , 500 (Nk = 500). A total of

8500 of such estimates were computed for 17 elements of the sample 10v given in Table 5.1. Fig. 5.5 shows the histogram of the estimates kp . A conservative 90th percentile of this sample, 1)0.9]([ +⋅kNp , was equal to 0.009451. This percentile can be used as the final result of the conservative estimation of the failure probability Pf8. This percentile is larger than the tolerable failure probability 0.724×10–4 recommended in the European code (EN 1990 2002). However, the tolerable values are specified for point estimates of failure probabilities and not conservative percentiles. The true value of Pf8 will be lower than 0.009451 with


a fairly high probability and the value 0.009451 can be considered acceptable. If necessary, the reinforcement area of the 8th segment of chimney can be increased to obtain the value 1)0.9]([ +⋅kNp which is less than 0.724×10–4.

5.5. Fifth chapter conclusions 1. Extreme cyclonic winds known as hurricanes can cause serious damage

to constructed facilities. Loads imposed by hurricanes must be considered accidental ones in the regions where structures are designed for annual maxima winds and not specifically for hurricane winds. Such a region is Lithuania.

2. Assessment of risk induced by hurricanes in Lithuania will face the problem of limited statistical information on hurricane wind speeds. Hurricanes in Lithuania are relatively rare events. The size of statistical samples of hurricane wind records will not exceed 10 to 20 elements depending on a regionalisation of Lithuanian territory. In such a case the risk to structures posed by hurricanes can be assessed by means of the methods proposed in Chapter 2.

3. The small-size sample of hurricane wind records will consist of 10 min mean wind velocities measured at 10.2 m height. It is possible to introduce two hurricane wind regions in Lithuania, a costal one and a south-western one. For these regions, two small-size samples can be established. The samples will consist of 17 and 10 elements, respectively. The method proposed in Chapter 2 can be applied to assessing extreme wind risk with these samples.

4. The small-size samples of hurricane wind records can be integrated into the risk assessment through a fragility function developed for 10 min and 10.2 m wind velocities used as a demand variable.

5. The method proposed in Chapter 2 is suitable to an assessment of risk to wind sensitive structures posed by hurricane winds. A reinforced concrete chimney is a typical example of such a structure. The mechanical model of chimney response to extreme winds contains a relatively large number of parameters which can be uncertain in the epistemic sense. A propagation of these uncertainties through a fragility function will yield a distribution of epistemic uncertainty in chimney failure probability. Such probability is an essential element of hurricane wind assessment.

109

General conclusions

The results presented in the dissertation were obtained by solving the problem of a design of structures which can be damaged by accidental actions. Such a design differs considerably from the structural design which takes into account actions to be resisted during normal service of structures. Solving the problem considered in the dissertation led to the following conclusions:

1. The analysis of methods used in the fields of quantitative risk assessment and structural reliability allows to conclude that these two methodologies are not fully suited to the design of structures which can be damaged by accidental actions. A prediction of damage to structures due to accidental actions requires to combine methods of quantitative risk assessment and structural reliability theory. Predicting accidental actions is a connecting link between these two fields.

2. An analysis of mathematical expressions of structural failure probability led to a discovery of a possibility to express the probability of damage due to an accidental action in the form of a mean of fragility function values. This allows to estimate the damage probability by applying various statistical methods used for an mean estimation. It was found that the probability of damage caused by accidental actions can be estimated by means of a statistical sample consisting of fragility function values. This function can be obtained from a sample of accidental action

110 GENERAL CONCLUSIONS

characteristics recorded in experiment. The sample of fragility function values can be applied to compute an interval estimate of the damage probability.

3. An analysis of the general procedure of estimating a mean value of random variable used in the Bayesian statistical theory led to a finding that the probability of structural damage caused by accidental actions can be estimated using two sources of information: an approximate mathematical model of the accidental action and a sample containing records of characteristics of this action. The first source of information can be applied to developing a prior distribution for the damage probability. The second source of information can be used for updating the prior distribution and obtaining a posterior distribution of this probability. In the case where the information on the accidental action is available only in the form of the sample of action characteristics, the damage probability can be estimated by means of a confidence interval.

4. The method proposed in the dissertation can be used to assessing fire damage to timber structures. The sample containing characteristics of accidental action induced by fire can be obtained by means of computer simulations of the fire. Such a sample can be transformed into a sample of containing values of the depth of char layer caused by fire. The latter sample can be used for generating a sample of fragility function values and consequently to estimating the failure probability of a timber structure damaged by fire.

5. An analysis of the hazard posed by a potential accidental explosion of a railway tank car allows to conclude that the method proposed in the dissertation is suitable to a design of a safety barrier aimed at protecting against this explosion. The probability of damage to components of this barrier can be estimated by applying a sample consisting of 30 values of characteristics of the blast generated by the explosion. It was found that the fragility function developed for barrier is uncertain in the epistemic sense. The probability of explosive damage to the barrier should be estimated by taking into account this uncertainty. The damage probability can be estimated by means of a confidence interval obtained using bootstrap resampling.

6. A collection and processing of data on hurricane winds recorded in Lithuania during the past 48 years led to a proposition of two hurricane wind regions represented by two statistical samples of wind speed records including 17 and 10 elements. It was proved that the method proposed in the dissertation allows to apply these small-size samples to estimating the probability of wind-induced damage to a wind sensitive structure. Such estimating required to compute values of wind fragility


function for all elements of wind speed samples by taking into account uncertainties related to mechanical model of structure. The method proposed in the dissertation can be used for a design of a reinforced concrete chimney subjected to the hazard of hurricane winds.

113

References

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123

The List of Scientific Author's

Publications on the Subject of the Dissertation

In the reviewed scientific periodical publications Vaidogas, E. R.; Juocevičius, V. 2011. A critical estimation of data on extreme winds in Lithuania. Journal of Environmental Engineer and Landscape Management 19(2): 178−188. ISSN 1648-6897. (Thomson ISI Web of Science) Juocevičius, V.; Vaidogas, E. R. 2010. Effect of explosive loading on mechanical properties of concrete and reinforcing steel: towards developing a predictive model. Mechanika 1(81): 5−12. ISSN 1392-1207. (Thomson ISI Web of Science) Vaidogas, E. R.; Juocevičius, V. 2009a. Assessment of structures subjected to accidental actions using crisp and uncertain fragility functions. Journal of Civil Engineering and Management 15(2): 95−104. ISSN 1392-3730. (Thomson ISI Web of Science) Juocevičius, V. 2009. Dependence of strength enhacement on shock front overpressure: the case of RC structures. Science – Future of Lithuania 1(5): 25−29. ISSN 2029-2341 Vaidogas, E. R.; Juocevičius, V. 2008a. Reliability of a timber structure exposed to fire: estimation using fragility function. Mechanika 5(73): 35−42. ISSN 1392-1207. (Thomson ISI Web of Science)

124 THE LIST OF SCIENTIFIC AUTHOR'S PUBLICATIONS…

Vaidogas, E. R.; Juocevičius, V. 2008b. Sustainable development and major industrial accidents: the beneficial role of risk−oriented structural engineering. Technological and Economic Development of Economy: Baltic Journal on Sustainability 14(4): 612−627. ISSN 1392-8619. (Thomson ISI Web of Science) Vaidogas, E. R.; Juocevičius, V. 2007a. Assessing external threats to structures using limited statistical data: an approach based on data resampling. Technological and Economic Development of Economy: Baltic Journal on Sustainability 13(2): 170−175. ISSN 1392-8619. (Thomson ISI Web of Science) In other editions Vaidogas, E. R.; Juocevičius, Virm.; Juocevičius, Virg. 2009. Developing fragility function for a timber structure subjected to fire, in ESREL 2008 and 17th SRA−Europe Conference, held in Valencia, Spain on 22−25 September. 2009. Universidad Politécnica de Valencia. London: Taylor & Francis Group. Selected papers 1−4: 1641−1649. ISBN 978-0-415-48513-5. (Thomson ISI Proceedings) Vaidogas, E. R.; Juocevičius, V. 2007b. Utilising fragility functions in assessing damage to structures due to accidental actions, in the 9th International Conference Modern Building Materials, Structures and Techniques, held in Vilnius, Lithuania on 16−18 May. 2007. Vilnius: Technika. Selected papers 3: 1090−1099. ISBN 978-9955-28-200-6. (Thomson ISI Proceedings) Vadlūga, R.; Juocevičius, V.; Vaidogas, E. R. 2011. Developing fragility functions for the estimation of extreme wind damage to chimneys, in the 16th International Conference “Mechanika 2011”, held in Kaunas, Lithuania on 7−8 April. 2011. Kaunas: Technologija. Selected papers: 311−316. ISSN 1822-2951. Linkutė, L.; Juocevičius, V.; Vaidogas, E. R. 2011. Numerical evaluation of concrete and steel strain rate in the RC section subjected to high rate loading, in the 16th International Conference “Mechanika 2011”, held in Kaunas, Lithuania on 7−8 April. 2011. Kaunas: Technologija. Selected papers: 209−214. ISSN 1822-2951. Juocevičius, V.; Linkutė, L.; Vaidogas, E. R. 2010. Assessment of potential mechanical damage to tanks of flammable liquids, in the 10th International Conference Modern Building Materials, Structures and Techniques, held in Vilnius, Lithuania on 19−21 May. 2010. Vilnius: Technika. Selected papers 2: 1237−1245. ISBN 978-9955-28-594-6. Linkutė, L.; Juocevičius, V.; Vaidogas, E. R. 2010. On reliability of timber structures subjected to fire, in the 10th International Conference Modern Building Materials, Structures and Techniques, held in Vilnius, Lithuania on 19−21 May. 2010. Vilnius: Technika. Selected papers 2: 1266−1273. ISBN 978-9955-28-594-6. Vaidogas, E. R.; Juocevičius, V. 2009b. Propagation of epistemic uncertainty in the fragility function used for bayesian estimation of failure probability, in the COMPDYN 2009: ECCOMAS thematic conference on computational methods in structural dynamics and earthquake engineering, held in island of Rhodes, Greece on 22−24 June. 2009.

THE LIST OF SCIENTIFIC AUTHOR'S PUBLICATIONS… 125

Athens: National Technical University of Athens. In the volume of abstracts and accompanying CD−ROM: 1−19. ISBN 978-9602-54-682-6. Juocevičius, V. 2008a. Assessment of Damage due to accidental actions: discussion on problematic issues, in 11th Conference of Lithuanian Young Scientist „Science – Future of Lithuania“ thematic conference “STATYBA”, held in Vilnius, Lithuania on April 2−4. 2008. Vilnius: Technika. Selected papers: 93−102. ISBN 978-9955-28-319-5. Juocevičius, V. 2008b. Developing fragility function for a timber structure subjected to fire, in 11th Conference of Lithuanian Young Scientist „Science – Future of Lithuania“ thematic conference “STATYBA”, held in Vilnius, Lithuania on April 2−4. 2008. Vilnius: Technika. Selected papers: 103−112. ISBN 978-9955-28-319-5. Juocevičius, V. 2007. Probabilistic prediction of prestress losses in concrete members, in 10th Conference of Lithuanian Young Scientist „Science – Future of Lithuania“ thematic conference “STATYBA”, held in Vilnius, Lithuania on March 29−30. 2007. Vilnius: Technika. Selected papers: 160−165. ISBN 978-9955-28-163-4.

127

ANNEXES

Annexes are given in the enclosed compact disc. They include textual information, computation results and three computer codes used for the implementation of the damage assessment method proposed in this dissertation. Annexes are arranged in five folders entitled as follows:

� Annex A. Spreadsheet file with the results of fire simulation by means of CFAST computer code

� Annex B. Results of the analysis of extreme winds recorded in Lithuania in the last 48 years

� Annex C. Computer code “Timber beam” � Annex D. Computer code “Barrier” � Annex E. Computer code “Chimney”

Virmantas JUOCEVIČIUS A METHOD FOR ASSESSING RISK TO STRUCTURES EXPOSED TO ACCIDENTAL ACTIONS Doctoral Dissertation Technological Sciences, Civil Engineering (02T) Virmantas JUOCEVIČIUS YPATINGŲJŲ POVEIKIŲ STATINIAMS SUKELIAMOS RIZIKOS VERTINIMO METODAS Daktaro disertacija Technologijos mokslai, Statybos inžinerija (02T)

2011 10 26. 12,0 sp. l. Tiražas 20 egz. Vilniaus Gedimino technikos universiteto leidykla „Technika“, Saulėtekio al. 11, 10223 Vilnius, http://leidykla.vgtu.lt Spausdino UAB „Ciklonas“ J. Jasinskio g. 15, 01111 Vilnius.

a method for assessing risk to engineering...

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