Reliability-Based Fatigue Assessment of Existing Steel Bridges

kth royal institute of technology

Licentiate Thesis in Structural Engineering and Bridges Reliability-based fatigue assessment of existing steel bridges

RUOQI WANG

Stockholm, Sweden 2020 Reliability-based fatigue assessment of existing steel bridges

RUOQI WANG

Academic Dissertation which, with due permission of the KTH Royal Institute of Technology, is submitted for public defence for the Degree of Licentiate of Engineering on Friday the 13th November 2020, at 1:00 p.m. in M108, Brinellvägen 23, Stockholm.

Licentiate Thesis in Structural Engineering and Bridges KTH Royal Institute of Technology Stockholm, Sweden 2020 © Ruoqi Wang

ISBN 978-91-7873-663-8 TRITA-ABE-DLT-2031

Printed by: Universitetsservice US-AB, Sweden 2020 Abstract

Fatigue is among the most critical forms of deterioration damage that occurs to steel bridges. It causes a decline of the safety level of bridges over time. Therefore, the performance of steel bridges, which may be seriously aﬀected by fatigue, should be assessed and predicted. There are several levels of uncertainty involved in the crack initiation and propagation process; therefore the probabilistic methods can provide a better estimation of fatigue lives than deterministic methods. When there are recurring similar details which may have correlation with each other and be regarded as a system, there are distinct advantages to analyze them from a system reliability perspective. It allows the engineer to identify the importance of an individual detail or the interaction between details with respect to the overall performance of the system.

The main aim of this licentiate thesis is to evaluate probabilistic methods for reliability assessment of steel bridges, from both a single detail level and a system level. For single details, an eﬃcient simulation technique is desired. The widely applied Monte Carlo simulation method provides accurate estimation, however, is very time-consuming. The Subset simulation method is investigated as an alternative and it shows great feasibility in dealing with a multi-dimensional limit state function and nonlinear crack propagation. For larger systems, the spatial correlation is considered between details. An equicorrelation-based modelling approach has been proposed as supplement to common simulation techniques to estimate the system reliability analytically and signiﬁcantly reduce the simulation time. With correlation considered, the information of one accessible detail could be used to predict the status of the system.

While reliability analysis aims for a specific safety level, risk analysis aims to find the most optimal solution. With consequences considered, a risk-based decision support framework is formulated for the selected system, which is presented as a decision tree. It reveals that the decisions based on reliability assessment can be different from those based on risk analysis, since they have different objective criteria.

Keywords: Fatigue, steel bridges, reliability analysis, spatial correlation, risk analysis, Monte Carlo simulation

Sammanfattning

Utmattning är en av de mest allvarliga nedbrytningsmekanismer som stålbroar utsätts för. Den orsakar en försämrad säkerhet för broar över tid. Därav måste stålbroars tillförlitlighet, som kan påverkas allvarligt på grund av utmattning, bedö- mas och förutsägas. Flera olika nivåer av osäkerheter är involverade i initiering och propagering av utmattningssprickor, varför sannolikhetsbaserade metoder kan ge en bättre uppskattning av utmattningslivslängden än deterministiska metoder. När liknande detaljer återkommer i en konstruktion och med korrelation mellan varandra kan dessa betraktas som ett system, för vilket tillförlitlighetsmetoder på systemnivå kan utnyttjas.. Det gör det möjligt för ingenjören att identiﬁera bety- delsen av en individuell detalj eller interaktionen mellan detaljer med avseende på systemets totala tillförlitlighet.

Det huvudsakliga syftet med denna licentiatuppsats är att utvärdera sannolikhetsbaserade metoder för uppskattning av stålbroars tillförlitlighet, både med avseende på enskilda detaljer och på systemnivå. För enskilda detaljer eftersträvas en tidsef- fektiv simuleringsteknik. Den allmänt tillämpade Monte Carlo-simuleringsmetoden ger en robust uppskattning, men är mycket tidskrävande. Subset-simuleringsmetoden undersöks som ett alternativ och den visar stor potential när det gäller att hantera en ﬂerdimensionell gränsfunktion och en olinjär sprickpropageringsmodell. På sys- temnivå beaktas den rumsliga korrelationen mellan detaljer. En modelleringsmetod baserad på konstant korrelation mellan detaljer har föreslagits som komplement till vanliga simuleringstekniker för att uppskatta tillförlitligheten analytiskt och av- sevärt minska simuleringstiden. Genom att utnyttja korrelationen kan information om en tillgänglig detalj användas för att förutsäga systemets status.

Medan en tillförlitlighetsanalys bedöms mot en speciﬁk säkerhetsnivå används risk- analysen för att hitta den mest optimala åtgärden. Genom att beakta konsekvenser har ett riskbaserat verktyg för beslutsstöd föreslagits och presenterats i form av ett beslutsträd. Resultaten visar att besluten baserade på tillförlitlighet kan skilja sig från de som baseras på en uppskattad risk, eftersom metoderna har olika målfunk- tioner.

Nyckelord: Utmattning, stålbroar, tillförlitlighet, rumslig korrelation, riskanalys, Monte Carlo-simulering

iii

Preface

The research work presented in this thesis was carried out at the Department of Civil and Architectural Engineering, KTH Royal Institute of Technology, Stockholm. It has been appreciatively ﬁnanced by the KTH Railway Group.

I would like to express my most sincere gratitude towards my supervisors, Asso- ciate Professor John Leander and Professor Raid Karoumi for their guidance and continuous support. Special thanks go to Dr. Johan Spross for taking the time to review this thesis and providing valuable comments.

I would also like to thank my colleagues and friends at the Division of Structural Engineering and Bridges. They shared their knowledge with me, oﬀered their support, and make my studies full of joy.

Last but not least, I would like to thank my parents for their endless love and ﬁrm support.

Stockholm, November 2020 Ruoqi Wang

Publications

The current thesis is based on the work presented in the following publications, labelled Paper I-III.

Paper I Wang, R., Leander, J., and Karoumi, R. (2019). Comparison of simulation methods applied to steel bridge reliability evaluations. Proceedings of the 13th International Conference on Applications of Statistics and Probability in Civil Engineering.

Paper II Wang, R., Leander, J., and Karoumi, R. (2020). Fatigue reliability assessment of steel bridges considering spatial correlation in system evaluation. Submitted for review.

Paper III Wang, R., Leander, J., and Karoumi, R. (2020). Risk analysis fordecisionsupport—acasestudy on fatigue assessment of a steel bridge. Proceedings of the 30th European Safety and Relia- bility Conference and the 15th Probabilistic Safety Assessment and Management Conference.

The planning of the papers, the major part of the analyses and of the writing has been performed by Wang. The monitored stress data was supplied by the second co-author and is a result of the work presented in Leander (2018). The co-authors have participated in the planning of the work and contributed to the papers with comments and revisions.

vii

Contents

Preface v

Publications vii

1 Introduction 1 1.1 Background ...... 1 1.2 Aims and scope ...... 3 1.3 Scientiﬁc contribution ...... 4 1.4 Outline of the thesis ...... 5

2 Reliability analysis 7 2.1 Deterministic methods ...... 8 2.2 Probabilistic methods ...... 10 2.3 Risk analysis ...... 15 2.4 System reliability assessment ...... 16

3 Risk analysis 21 3.1 Risk analysis methods ...... 22 3.2 Risk-based decision-making methods ...... 28 3.3 A theoretical model for fatigue assessment ...... 31

4 Summary of appended papers 35

5 Concluding remarks 39 5.1Discussion...... 39 5.2 Conclusions ...... 40 5.3 Further research ...... 41

References 43

Paper I 51

Paper II 61

Paper III 93

Chapter 1

Introduction

Fatigue is among the most critical forms of deterioration damage that occurs to steel bridges. It is usually causing a decline of the safety level over time due to repeated or varying loading. The deterioration process consists of initiation and propagation of cracks, which often accumulate in an invisible way and may eventually trigger diﬀerent levels of consequences. A closure of the bridge for maintenance will lead to inevitable traﬃc disturbance and bring much inconvenience to the travelers and transport operators. More severely, a bridge collapse may induce fatalities as well as economic loss.

Therefore, the performance of steel bridges, which may be seriously aﬀected by fatigue, should be assessed and predicted. An adequate safety level should be guar- anteed by combining theoretical assessment, maintenance and regularly-arranged inspections. It is motivated by economic as well as environmental reasons. An early repair compared to a catastrophic failure will save much material and minimize the construction impact, which is environmentally sustainable.

1.1 Background

Structural reliability is traditionally estimated by deterministic analysis. However, a fatigue life prediction is afflicted by large uncertainties which contribute to the possibility that the structure will not perform as intended. In order to have a good estimation, the uncertainties involved with the traffic induced loads, the deterioration model, the material properties and the detail specific endurance have to be considered and determined. Therefore the probabilistic analysis is applied further as an extension.

Fatigue reliability assessments of a single detail of steel bridges have been accomplished (Kwon and Frangopol (2010); Guo and Chen (2013); Leander and Al-Emrani (2016)). The selected detail is regarded as failed when its reliability condition falls below the target reliability index, a predefined acceptable performance. This criterion can be easily applied when it comes to single details. However, there isn’t such clear regulation for a system or for the whole bridge. The selection of target index is usually the responsibility of the bridge manager and/or its consultant (Casas and Wisniewski (2013)) and generally it should be defined based on cost-benefit analysis (JCSS (2011)).

1 CHAPTER 1. INTRODUCTION

For steel bridges, it is common to have similar details appear several times within the structure. They might have similar geometries, be built with same materials, and possibly subjected to similar environment conditions and load fluctuations, which leads to a correlation between each other. Correlation is especially relevant when inspection or monitoring are considered (Vrouwenvelder (2004)). Monitoring at one location will provide indirect information about the deterioration progress at another location if the correlation among details is high (Moan and Song (2000); Yang et al. (2004); Maljaars and Vrouwenvelder (2014)). The correlation will also lead to a dependence among failures and further affect the system safety. Therefore it is necessary to consider the bridge as a system and analyze the system reliability (Kang and Song (2010); Roscoe et al. (2015); Schneider et al. (2017)). A general consideration of correlations in the system reliability assessment can be complicated with much variation involved and requires more advanced analyses. In some special cases, however, if the details are regarded as equicorrelated, it is possible to simplify the tedious simulation procedure and evaluate the system reliability efficiently.

Performing a reliability analysis aims to provide more information and support bridge managers or decision makers to decide what is the best option to keep existing bridges in service. Simply using the probability of failure as the criterion will lead to a safe decision, though not necessarily be economic or sustainable. For instance, budget restraints may limit the option for example direct upgrading or replacing the bridge. Instead, the service life of bridges should be extended as far as possible due to sustainability reasons (Kühn et al. (2008); Jensen et al. (2008)). Evaluating the possible interventions and corresponding consequences can facilitate a rational decision (Haimes (2005)), which steps further into risk analysis. An inﬂuence diagram or a decision tree model are usually applied for decision support (Hao (2000); Nielsen and Sørensen (2011); Goyet et al. (2013); Leander et al. (2018)).

Figure 1.1 shows a schematic view of the assessment procedure from reliability level to risk level. The information about one single detail should be obtained from monitoring or inspection. Based on this detail, the reliability of the system consisting of several similar details could be assessed. Furthermore, the possible actions and consequences should be evaluated in order to make the optimal decision.

The Rautasjokk Bridge in north Sweden is selected as a case study to test all theories proposed in this thesis. A photo of the bridge is shown in Figure 1.2. It showed strong indications of an exhausted fatigue life due to high load levels from iron ore trains (Häggström (2015)).

2 1.2. AIMS AND SCOPE

Assessment steps

single detail series system series system

reliability level risk level ିଵ ߚ ൌെȰ ሺ݌୤ሻ ܴ ൌ ݌୤ ȉ ܥ

Paper I Paper II Paper III

Refinement

Figure 1.1: A schematic view of the assessment procedure and the corresponding papers in the thesis.

Figure 1.2: A photo of the Raustasjokk Bridge.

1.2 Aims and scope

The overall aim of the long-term project is to develop a risk-based framework supporting rational decision to extend the service life of steel bridges. And this licentiate thesis is an important part of the project. The thesis focuses on assessing the fatigue life of steel bridges combined with monitoring and inspections on a reliability level and initiating the research on a risk-based decision model on a selected system.

Some speciﬁc objectives with this thesis are:

• Verifying the feasibility of an advanced simulation technique in reliability assessment.

3 CHAPTER 1. INTRODUCTION

• Exploring the connection between details’ spatial correlation and the system reliability. • Establishing a simple risk-based decision support model for maintenance planning considering a combination of theoretical assessments and consequences. • Applying and testing the proposed theories on the fatigue evaluation of the Rautasjokk Bridge.

This thesis is subject to a few limitations and simplifying assumptions. The reliability/risk analysis applied in the thesis is limited to steel bridges and degradation phenomena as fatigue. The stress spectrum used in the case study was obtained from the monitoring of the Rautasjokk Bridge. The characteristic values of stochastic variables are suggested in JCSS (2011) and Leander et al. (2013). However, the proposed method and procedure should be applicable to similar structures. The linear elastic fracture mechanic (LEFM) method has been applied to describe the crack propagation phase and predict the fatigue life in this thesis. It is an established method and the author hasn’t tried to improve it. The separation of the load bearing structure in subsystems consists of 8 details that are built with the same material, have same geometries and are subjected to similar load fluctuations. Therefore those recurrent details are assumed to be equicorrelated. The system was modelled as a series system. It is a reflection of the strategy adhered by the bridge managers in Sweden and also a simplification to enable a thorough study. The costs of different interventions and outcomes in the risk analysis are assigned as tentative values. They should be seen as ratios between costs. Fatalities are not considered here.

1.3 Scientiﬁc contribution

This research project, presented in this thesis together with the appended papers, has resulted in the following scientiﬁc contributions:

1. Presenting a study about a few widely-used methods for reliability and risk analysis and discussing their applicability and limitations. 2. Testing the efficiency and accuracy of the Subset Simulation method in the application of reliability assessment, a multi-dimensional problem considering different levels of uncertainty (study provided in Paper I). 3. Investigating the spatial correlation between similar details and how it influ- ences the system reliability (study provided in Paper II). 4. Exploring the use of a simplified risk-based decision model for maintenance planning of a bridge subjected to fatigue deterioration (study provided in Paper III).

4 1.4. OUTLINE OF THE THESIS

1.4 Outline of the thesis

This thesis is based on three appended papers and additional studies. An overview of various levels of reliability methods is provided in Chapter 2. System reliability is specially illustrated for its complexity. In Chapter3aliterature review of risk analysis is given together with a risk-based decision tree model speciﬁed for the fatigue assessment. Chapter 4 summarizes the main content in the appended papers in which numerical examples are demonstrated. General conclusions based on the research that has been done are presented in Chapter 5. Proposals for further work are given in the same chapter.

Paper I applied the Subset simulation (SS) method in the reliability assessment of a single fatigue-prone detail. Using the Monte Carlo simulation (MCS) method as a reference, SS showed promising feasibility to deal with multi-dimensional limit state functions and the crack propagation task with strong nonlinearity. A sensitivity analysis also implied that the number of samples and the predeﬁned conditional probability for each iteration are important to guarantee the performance of SS.

Paper II investigated the possible impact on system reliability by considering the spatial correlation between details. All details were assumed to be equicorrelated. An efficient modelling approach was specially demonstrated which significantly reduced the simulation time. In this study it was shown that by considering the spatial correlation between similar details, it is possible to estimate the system reliability based on the information of a single accessible detail. Among all parameters, the correlations of material parameters showed more dominating influence on the system reliability, followed by the model uncertainties.

Paper III proposed a risk-based decision tree model specified for fatigue assessment. Demonstrating the model by a case study, a separated load-bearing system, the results of the study showed that probability and consequences both influence on best decision. The decisions based on risk analysis could be different from that based on reliability analysis, with the former one providing a decision that is both economic and safe by evaluating consequences as well.

Chapter 2

Reliability analysis

Generally, methods to measure the reliability of a structure can be divided in four levels, characterized by the extent of information about the structural problem that is used and provided (Madsen et al. (2006)).

Level I The uncertain parameters are modeled by only one characteristic value. Level II The uncertain parameters are modeled by the mean values and the standard deviations, and by the correlation coeﬃcients between the stochastic variables. Level III The uncertain quantities are modeled by their joint distribution functions. The probability of failure is estimated as a measure of the reliability. Level IV The consequences (cost) of failure are also taken into account and the risk (consequence multiplied by the probability of failure) is used as a measure of the reliability.

The classiﬁcation of reliability methods is not exhaustive. For example, a method could employ more information than a level II method and yet not employ the complete distribution information of a level III method; it might even include some of the concepts of economic aspect in level IV.

Another classification from the ISO 2394 (2005) is also widely applied in the field. It describes how the principles of risk and reliability can be utilized to support decisions related to the assessment of existing structures and systems over their service life. Three different but related levels of approach are facilitated as follows:

Risk informed It shall be proven that the total risks, considering loss of lives and injuries, damages to the qualities of the environment, and monetary losses are considered. The sum of all costs/risks should be at a minimum. Reliability-based The structure shall fulﬁll a set of reliability requirements formulated as maximum admissible probabilities of failure or minimum values for the reliability levels. Semi-probabilistic The structure shall fulﬁll a safety format using certain design values of the basic variables.

7 CHAPTER 2. RELIABILITY ANALYSIS

In this chapter, there are many stochastic variables involved in the expressions. For clariﬁcation, the stochastic variables are denoted X = {X1, X2, ..., Xn}. Those n stochastic variables could model uncertainties (e.g. model uncertainty) or physical variables (e.g. load variables). The variables in X are also denoted as basic variables. Realizations of those basic variables are denoted x = {x1, x2, ..., xn}, i.e. x is a point in the n-dimensional basic variable space. More speciﬁc explanations could be found after expressions.

2.1 Deterministic methods

2.1.1 Partial safety factor method In the partial safety factor method single structural components are usually considered (Sørensen (2004)). It has to be verified that the loads acting on the structure or load effects in the structure, are smaller than the resistance of the structure or strength of the materials in the structure. The safety margin M is defined as

M = Rd − Sd (2.1) where Sd is the load eﬀect and Rd is the resistance. If M>0, the component functions safely. If M<0, the assessed component is regarded as failed. If M =0, the component is at the limit state. That’s why the safety margin function can also be called as limit state function (LSF).

The conventional fatigue veriﬁcation is a direct comparison between the equivalent stress range ΔσE and the fatigue class (or fatigue strength) σC of the detail,

σC ΔσE · γFf ≤ (2.2) γMf where γFf is the partial factor for equivalent constant amplitude stress ranges, γMf is the partial factor for fatigue strength. The resistance is proven suﬃcient if the condition is fulﬁlled.

The partial factors should be determined such that they take into account both the aleatory and epistemic uncertainties of relevance for the considered failure modes (ISO 2394 (2005)). Some values of the partial factors for fatigue problem can be found in Eurocode 3 Part 1-9 (CEN (2005)) and Eurocode 3 Part 2 (CEN (2006)). Partial safety factors for fatigue assessment of railway bridges in Sweden are given in Traﬁkverket (2019).

Much research has been accomplished to calibrate the partial factors for classes of structures where no code exists beforehand (Sørensen et al. (1994)). They can be determined by minimizing the diﬀerence between the achieved probability of failure and the acceptable probability of failure (Goh et al. (2009); Sørensen et al. (2011);

8 2.1. DETERMINISTIC METHODS

Leander et al. (2015); D’Angelo et al. (2015)). The acceptable probability of failure is usually obtained from the assessment in level II or level III.

2.1.2 Linear damage accumulation In 1945, Miner et al. (1945) popularized a rule that had ﬁrst been proposed by Palmgren (1924), the Palmgren-Miner rule. This rule functions as an extension of the basic partial safety factor method and is mainly applied for fatigue assessment of existing structures. It is typically used when information about the damage equivalent stress range is missing or when measured stresses are used for the assessment. It implies that an analysis of the traﬃc or a measure of the structural response is necessary before applying this method.

The Palmgren-Miner rule assumes that the damage done by each stress repetition at a given stress level is equal. It states that if there are k diﬀerent stress levels and the average number of cycles to failure at the ith stress Δσi is Ni, then the damage fraction D is expressed as

k n i = D (2.3) N i=1 i where ni is the number of cycles accounted at each stress level Δσi, D is the fraction of life consumed by exposure to the cycles at the diﬀerent stress levels. In general, failure occurs when the cumulative damage fraction reaches 1. Considering partial factors, Ni is expressed as m ΔσC/γMf Ni = NE · (2.4) γFf · Δσi where m is the slope of the selected fatigue strength curve, NE the reference endurance typically equal to 2 million cycles together with m equal to 3.

Although the Palmgren-Miner rule may be a useful approximation in many circumstances, it also has major limitations. The sequence in which stress cycles are applied may affect the fatigue life, for which the Palmgren-Miner rule does not account (Szerszen et al. (1999); Eskandari and Kim (2017)). However, for bridges with a natural variation in traffic loads over time, the load sequence effect can be neglected (Leander and Al-Emrani (2016)).

2.1.3 Nonlinear damage accumulation Fracture mechanics is used to study the propagation of deﬁned cracks. Bridges are typically subjected to elastic loading and it is assumed that the size of the plastic zone around the crack tip is very small in comparison to the domain size

9 CHAPTER 2. RELIABILITY ANALYSIS

(Chang (2013)). Therefore the linear elastic fracture mechanics (LEFM) is applicable. LEFM provides a more elaborate description of the crack propagation phase compared to the linear damage accumulation method and is mentioned as a ﬁnal resource for fatigue assessment (Kühn et al. (2008)).

The fatigue assessment based on LEFM can be described by the well-known Paris law (Paris (1961)) formulated as the crack growth rate, which describes the relationship between cyclic crack growth and the stress intensity factor range:

da = AKn (2.5) dN r where a represents the crack size, N is the number of stress cycles, A and n are fatigue growth parameters, Kr is the stress intensity factor range.

The number of cycles to reach a critical crack depth is calculated by integrating the crack growth rate as

−1 acr da Ncr(X)= da (2.6) a0 dN where a0 is the initial crack size and acr is the critical crack size. The vector X contains the basic variables considered in the model.

2.2 Probabilistic methods

The safety margin function as Eq.(2.1) can be regarded as a simplistic limit state function. It is replaced by g(X) in stochastic formulation, where X contains all random variables that are relevant to describe the limit state. The unsatisfactory performance of a component is described as failure, as F = {g(X) ≤ 0}.

Reliability analysis is concerned with the evaluation of the probability of failure, deﬁned as

Pf = P (g (X) ≤ 0) = fX (x) dx (2.7) g(X)≤0 where fX (x) is the joint probability density function of all stochastic variables contained in X, and g (X) ≤ 0 represents the region of failure in the space of x.

The parameter β, often called the ‘Hasofer-Lind reliability index’, is a convenient measure of reliability. It is related to the probability of failure as:

−1 β = −Φ (Pf ) (2.8) where Φ−1() is the inverse of the standardized normal distribution function.

10 2.2. PROBABILISTIC METHODS

Several techniques can be used to estimate the reliability for level II and III methods:

• FORM techniques: In First Order Reliability Methods the limit state function is linearized and the reliability is estimated using level II or III methods.

• SORM techniques: In Second Order Reliability Methods a quadratic approximation to the limit state function is determined and the probability of failure for the quadratic failure surface is estimated.

• Simulation techniques: Samples of the stochastic variables are generated and the relative number of samples corresponding to failure is used to estimate the probability of failure.

2.2.1 Target reliability To ensure acceptable levels of the reliability of structures within the context of structural assessment, acceptance criteria shall be formulated and assessed. Some of these requirements relates to demands on safety for personnel and environment. Others relates to the demands on the functionality of the structures.

−3 For fatigue assessment of existing structures, a target value of Pf =10 is suggested in the ISO 13822 (2010) for non-inspectable details while in combination −2 with inspections, a value of Pf =10 is suggested. The corresponding values expressed in reliability index are β =3.1 and β =2.3, respectively. These values are stated for a reference period equal to the intended remaining service life.

ISO 2394 (2005) provides the tentative target reliabilities related to one-year reference period and ultimate limit states based on the Life Quality index (LQI), by thoroughly considering the economic losses, the social inconvenience, eﬀects to the environment, and the amount of expense and eﬀort required for reliability enhance- ment measures.

NKB (1978) has recommended the maximum probabilities of failure and corresponding reliability index based on a reference period of 1 year. Those numbers are related to the consequences of failure speciﬁed by safety classes and failure types.

Casas and Wisniewski (2013) has summarized the target reliability levels recommended in diﬀerent national and international codes and guidelines (Eurocode, ISO, JCSS, etc.), to be used in the design and assessment of bridges. The values presented correspond to moderate consequences of failure and the reference period of 1 year.

11 CHAPTER 2. RELIABILITY ANALYSIS

The general form of the veriﬁcation in the probabilistic methods is as follows:

β ≥ βt (2.9) where βt represents the deﬁned target reliability.

2.2.2 FORM and SORM FORM is an analytical approximation that solves the probability integral as Eq.(2.7) by simplifying the limit state function g(X) using the ﬁrst-order Taylor series ex- pansion at the Most Probable Point (MPP), also called as the design point. The second-order reliability method (SORM) has been established as an attempt to improve the accuracy of FORM. SORM is obtained by approximating the limit state surface in u-space at the design point by a second-order surface (Fiessler et al. (1979)). The reliability index β is interpreted as the minimum distance from the origin to the limit state surface in standardized normal space (u-space).

The basic framework involves several steps (Der Kiureghian (2000); Ibrahim (2011)):

1. Formulate the reliability problem in terms of the limit state function g(X). 2. Transform original random variables X in x-space to standard uncorrelated normal random variables U in u-space; the limit state function is expressed as gu(U). 3. Find the design point through an iterative process. 4. Obtain the estimation of the probability of failure and the corresponding reliability index β.

If the limit state function is linear, FORM gives an exact solution. For a nonlinear function the solution will be an approximation. The diﬀerence is small for cases with large reliability index (or low probability of failure). For higher probability of failure, the error in the FORM approximation can be signiﬁcant (Zhao and Ono (1999a)).

Some work investigated on the accuracy of the curvature radius at the design point, the number of random variables and the ﬁrst-order reliability index, and proved those parameters help judging when FORM is accurate enough and when SORM is required (Zhao and Ono (1999a,b)).

2.2.3 Simulation techniques There are multiple simulation-based techniques available to perform structural probabilistic studies, such as crude Monte Carlo, importance sampling, Directional

12 2.2. PROBABILISTIC METHODS simulation, etc. They are capable of dealing with complex limit state functions and estimating the probability of failure with several classes of uncertainty taken into account. However, the performance of simulation techniques is sensitive to the number of samples (Sørensen (2004); Melchers and Beck (2018)). A proper sample size is essential for an accurate result. A detailed procedure, application scope and comparison among diﬀerent methods could be found in Faulin et al. (2010). In this thesis only Monte Carlo simulation (MCS) and Subset simulation (SS) are selected as representatives and will be treated henceforth.

MCS is straightforward in coding and is robust in practice. However, in the ﬁeld of Civil Engineering, the probability of failure is normally a small number. Therefore a great number of samples are usually required to reach a high level of accuracy (Mooney (1997)), which in turn requires time-consuming calculations. A more advanced method as SS could compensate this shortage. SS is an adaptive Monte Carlo method particularly suitable for handling problems involving a large number of random variables (Au and Wang (2014)), both robust to applications and com- petitive in terms of eﬃciency. Due to its special calculating routine, it consumes less time than MCS for a comparable accuracy. The limitation for this method is that all stochastic variables are recommended to be transferred into the standard space, which means some preparation work is always necessary.

Monte Carlo simulation The MCS was ﬁrstly developed and applied in the 1940s, and has become more and more feasible in the practice of various ﬁelds with the increasing availability of fast computers (Mooney (1997)). It solves a problem by simulating directly the physical process which proceeds with only three steps. These are:

1. Generating samples x = {xi,i=1, ..., n} from the distribution of the input variables X.

2. Evaluating the functions y = g (x) for each sample separately.

3. Evaluating samples y of the function value Y by using statistical method.

Much research has applied MCS in the reliability assessment of bridges. Park et al. (2004) estimated the reliability of a steel bridge member for fatigue and the MCS was performed in conjunction with the Palmgren-Miner rule. Zhang and Cai (2012) established a limit state function of fatigue damage for existing bridges considering traﬃc parameters and MCS was applied for assessment. Salcher et al. (2016) researched on the impact of the seasonal temperature changes on railway bridges subjected to high-speed trains, and MCS and Latin Hypercube were applied to estimate the service life.

13 CHAPTER 2. RELIABILITY ANALYSIS

Subset simulation The SS method was proposed by Au and Beck (2001), and originally developed for solving reliability analysis and risk assessment of civil structures subjected to uncertain earthquake ground motions. Nowadays the SS method has been widely applied overcoming the ineﬃciency of MCS in estimating small probabilities.

A modiﬁed Markov-chain based SS method was proposed by Miao and Ghosn (2011) for performing the reliability analysis of complex structural systems. Tee et al. (2014) proved that SS can provide better estimation for a time-dependent reliability of underground ﬂexible pipelines, which is commonly a rare failure event. Sen et al. (2012) applied the SS method for reliability analysis of a simply supported bridge deck subjected to random moving and seismic loads. An adaptive Markov chain Monte Carlo sampling method from Papaioannou et al. (2015) was referred and applied for sample generation in this thesis.

Before starting the SS routine, it is recommended to transform all the stochastic variables into the probability space u consisting of independent standard normal variables, through a one-to-one mapping U = T (X). The Rosenblatt transformation (Der Kiureghian and Liu (1986)) and Nataf transformation are two of the most widely used methods to deﬁne the mapping, while the Nataf transformation is more applicable when the stochastic variables have correlation between each other (Hohenbichler and Rackwitz (1981)). Then the probability of failure expressed as Eq. (2.7) is transformed into

Pf = fU (u) du (2.10) gu(U)≤0 where f () is the standard normal joint PDF in u-space.

The idea behind SS is to express the event F as an intersection of M intermediate events: M F = Fj (2.11) j=1

It is assumed that the intermediate events are sequentially nested as F1 ⊃ F2 ⊃ ... ⊃ FM, and FM = F is the failure event. Then the probability of failure is estimated as a product of conditional probabilities: ⎛ ⎞ M M ⎝ ⎠ Pf = P Fj = P (Fj | Fj−1) (2.12) j=1 j=1 where F0 is a certain event. The intermediate events are deﬁned as Fj = {g (u)

14 2.3. RISK ANALYSIS

The limit bj is the p0-percentile of the calculated values of g (u) after being as- cended. The iteration only stops when bj ≤ 0. Then the probability of failure is obtained as j−1 Nf Pf = p0 (2.13) NS where Nf is the failure point in the last iteration, NS is the number of samples in each iteration.

2.3 Risk analysis

Risk analysis and risk assessment have been applied to many disciplines (Henley and Kumamoto (1981)). The field of Civil Engineering is one of them. Tetelman and Besuner (1978) described the key elements of structural risk analysis and applied them to evaluate a critical steel suspension bridge component. Ezell et al. (2000) introduced an infrastructure risk analysis model developed for a small community’s water supply and treatment systems and studied the system’s interconnections and interdependence. LeBeau et al. (2000) applied a fault tree model, which offered a a graphical depiction, and presented the element interactions that could contribute to bridge deterioration. Nielsen and Sørensen (2011) evaluated the costs for a single wind turbine with a single component and attempted to make an optimal planning on operation and maintenance based on Bayesian pre-posterior decision theory. Goyet et al. (2013) developed a quantitative fatigue Risk Based Inspection (RBI) framework for offshore structures based on a decision tree model.

Risk is deﬁned as the expected consequences associated with a given activity. Con- sidering an activity with only one event with potential consequences, risk R is thus the probability that this event will occur p multiplied with the consequences given the event occurs C: R = p · C (2.14)

For an activity with n events the risk is deﬁned by n R = pi · Ci (2.15) i=1 where pi and Ci are the probability and consequence of event i.

Typically economic consequences, loss of lives, and adverse eﬀects on people or environment have to be considered in the analysis of consequences. The estimation of consequences requires a thorough understanding of the selected structure and its interrelation with its surroundings. In the analysis of probabilities, the methods for structural reliability in level II and III are usually applied.

More content about risk analysis will be covered in Chapter 3.

15 CHAPTER 2. RELIABILITY ANALYSIS

2.4 System reliability assessment

While many reliability analyses focus on a speciﬁc component, there are distinct advantages to analyze a structure from a system reliability perspective. It allows the engineer to identify the importance of an individual component or the interaction between components with respect to the overall performance of the system.

Estes and Frangopol (2001) proposed to include multiple limit states in the system reliability assessment for a comprehensive understanding. Similarly, Nowak and Cho (2007) proposed an innovative prediction method for the combination of failure modes of an arch bridge, which reveals that the combination of failure modes signiﬁcantly reduced time and eﬀorts in comparison to the conventional system reliability analysis method.

Kang et al. (2008) used a matrix-based system reliability (MSR) method to estimate the probabilities of complex system events by simple matrix calculations. Even when one has incomplete information on the component probabilities and/or the statistical dependence thereof, the matrix-based framework is capable to estimate the narrowest bounds on the system failure probability by linear programming. Two years later, Kang and Song (2010) proposed an eﬃcient method named Se- quential Compounding Method (SCM) in which the reliabilities of components are ﬁrstly computed, and the components are subsequently combined by an iterative procedure, which means that once two components are combined, this new system can be regarded as one component to combine with the next component, until the full system is obtained.

Schneider et al. (2017) enables an integral assessment by proposing two coupled sub-models: a probabilistic system deterioration model for considering stochastic dependence among component deterioration, and a probabilistic structural model for calculating the failure probability of the system.

By considering the correlations between components that form a structural system, a more eﬃcient and accurate evaluation of existing bridges could be obtained.

2.4.1 System model selection Structural systems or their subsystems may be idealized into two simple categories: series and parallel. Series system, typiﬁed by a chain, is also called a ‘weakest link’ system. Any one component’s failure constitutes failure of the structure (Melchers and Beck (2018)). If the failure of all components are needed to obtain the failure of the system, the system is said to be a ‘parallel’ system (Sørensen (2004)). In actual design of bridges, these two types of system are always combined, forming a mixed system. Therefore the probability of failure of a bridge can be very complex and is dependent on how the diﬀerent types of systems are combined.

16 2.4. SYSTEM RELIABILITY ASSESSMENT

In order to draw general conclusions an idealised system model is required. The series system model is supported by the Swedish regulations for bridge inspection. A management system called BaTMan was developed by the Swedish Transport Administration (Trafikverket) to manage their stock of bridges (and tunnels). They collect and store various data from inspections and condition assessments about the bridges required to make decision about their maintenance at operational, tactical and strategic levels (Hallberg and Racutanu (2007)). The physical and functional condition of both the structural components and the entire bridge is determined based on the bridge inspection. Then the bridge inspector will assign a condition class (CC) to the inspected structure. The CC spans from 0 to 3 as listed in Table 2.1 and describes to what extent structural members fulfill the functional requirements at the time of inspection. If a fatigue crack in one of the main load carrying members is detected, the bridge is counted as CC 3 and immediate action is required. From that perspective, the appearance of a crack is a failure of the bridge’s function even though it is not seen as a structural failure in reliability theory. Therefore the definition of failure in the current study is limited as the first presence of damage and a model without redundancy, a series system model, is motivated.

Table 2.1: Condition classes (CC) system used in BaTMan (Traﬁkverket (2015)).

CC Assessment Follow-up 3 Defective function Immediate action is needed 2 Defective function expected within 3yrs Action is needed within 3yrs 1 Defective function expected within 10yrs Action is needed within 10yrs 0 Defective function expected beyond 10yrs No action is needed within 3yrs

2.4.2 Current analytical methods Bounds for system reliability In a series system, two extreme conditions are considered for system failure probability. One is to consider that all details are fully correlated, which means that the failure of a single detail implies a failure of all details simultaneously. Then the system failure probability is equal to the failure probability of single details.

Psys = max {pi} (2.16) i where i counts the number of details in the system. The other case is to consider that all details are independent and every detail should function to keep the system safe. Therefore the system failure probability is expressed as:

m Psys =1− (1 − pi) (2.17) i=1

17 CHAPTER 2. RELIABILITY ANALYSIS where m is the total number of the details within the selected system. Assuming all the details having the same probability of failure p1 = ... = pi = p, this results in the following bounds on the probability of failure of series systems:

m p ≤ Psys ≤ 1 − (1 − p) (2.18) 1

In real applications, those two extreme cases are relatively rare. Two similar details at diﬀerent locations are usually partially correlated. Then the probability of system failure can be expressed as follows, suggested by Ditlevsen (1979): m i−1 ≥ P (F1)+ =2 max Pi − =1 P (Fi Fj) , 0 Psys i j (2.19) ≤ m − m { } i=1 P (Fi) i=2 maxj≤i P (Fi Fj) where F1,Fi,Fj represent the failure events of the corresponding details.

A conversion from probability of failure to reliability index can be attained by setting

P (Fi)=Φ(−βi) (2.20) and P Fi Fj =Φ2 (−βi, −βj; ρij) (2.21) where βi and βj are the corresponding reliability indices of the failure events of the ith and jth detail; the function Φ2 (−βi, −βj; ρij) is the distribution function of the two-dimensional normal distribution with mean values (0, 0), variances (1, 1) and correlation coeﬃcient ρij.

The probability of the intersection of Fi and Fj for positive correlation (ρij > 0) can be estimated as (Ditlevsen and Madsen (1996)): ≥ max Φ(−βi)Φ −βi|j , Φ(−βj )Φ −βi|j Φ2 (−βi, −βj ; ρij ) (2.22) ≤ Φ(−βi)Φ −βi|j +Φ(−βj )Φ −βi|j

The conditional reliability index βi|j is expressed as (Ditlevsen (1979)): − βi ρijβj βi|j = (2.23) − 2 1 ρij

Assuming every detail has the same reliability level and are equicorrelated with each other, 1 − ρ β | = β (2.24) i j 1+ρ

18 2.4. SYSTEM RELIABILITY ASSESSMENT then the probability of failure of a multi-details system expressed by Eq.(2.19) can be written as: ⎧ ⎨≥ Φ(−β) 1+ m 1 − 2(i − 1)Φ −β 1−ρ , 0 i=2 1+ρ Psys (2.25) ⎩ ≤ Φ(−β) m − (m − 1) Φ β 1−ρ 1+ρ

Exact solution for equicorrelated details Ditlevsen (1983) considers the safety margin of the selected system as the intersection of safety margin of each detail, given by the conditions M1 > 0, ..., Mm > 0, where Mi is the linear safety margin of the ith detail. Mi is expressed as √ Mi = βi + ρX + 1 − ρXi,i=1, ..., m (2.26) where X, X1, ..., Xm are mutually independent standardized normally distributed random variables.

Assume m safety margins are equicorrelated with the correlation coeﬃcient ρ. They all have the variance of 1 and the reliability index βi as the mean value. Then the system safety margin is expressed as (Ditlevsen (1983)):

P (M1 > 0, ..., Mm > 0) ∞

= P (M1 > 0, ..., Mm > 0 | X = x) ϕ (x) dx −∞ ∞ m = P (Mi > 0 | X = x) ϕ (x) dx −∞ i=1 √ ∞ m β + ρx = Φ √i ϕ (x) dx (2.27) −∞ 1 − ρ i=1 where ϕ (x) represents the one-dimensional standardized normal density of stochastic variables x. Assuming β1 = ... = βm = β, Eq.(2.27)becomes √ ∞ β + ρx m P (M1 > 0, ..., Mm > 0) = Φ √ ϕ (x) dx (2.28) −∞ 1 − ρ where β is the reliability level of the monitored/inspected detail with accessible information.

Therefore the probability of the system failure is:

√ ∞ β + ρx m Psys =1− Φ √ ϕ (x) dx (2.29) −∞ 1 − ρ

19 CHAPTER 2. RELIABILITY ANALYSIS

2.4.3 Equicorrelation-based modelling approach Based on the derived equation Eq.(2.29), an eﬃcient modelling approach is proposed. The so-called Equicorrelation-based modelling approach (ECM) is based on the assumptions that every detail within one system is equicorrelated with each other and they have the same individual reliability level. It facilitates an estimation of the system reliability analytically with high accuracy.

To distinguish the diﬀerence, in the rest of the thesis ρv is applied to represent the correlation between underlying stochastic variables, while ρeq represents the equivalent correlation between similar details within the same system.

The main procedure is as follows:

1. Perform a prior reliability assessment of the known detail and a two-details series system considering the correlated stochastic variables. The conditional correlated samples are generated in the normal space by applying Cholesky factorization ﬁrstly, then transformed to the required space. Then the prior reliability of the known details β1 and the probability of failure for the two- details series system Psys are obtained respectively from simulations.

2. Apply Eq.(2.29) inversely to derive the equicorrelation ρeq between these two details.

3. Since all details are assumed to be equicorrelated, ρeq also applies to other details in this system. By using Eq.(2.29) with the known values of β and ρeq, the probability of failure for the m-details series system can be computed analytically instead of performing simulations.

More details about this modelling approach and a case study can be found in Paper II in this thesis.

20 Chapter 3

Risk analysis

Kaplan and Garrick (1981) deﬁned risk as the combined answer to the three questions of:

1. What can go wrong? To answer this question, the possible hazards and threats must be identiﬁed, which may lead to harm to some assets that should be protected. These assets can extensively be people, animals, the environment, infrastructures, etc.

2. What is the likelihood of that happening? A causal analysis should be carried out to identify the basic causes that may lead to the hazardous event. The frequency of the hazardous event can be estimated based on experience, data and/or expert judgments. The answer can be given as a qualitative statement or as probabilities.

3. If it does happen, what are the consequences? For each hazardous event, the potential harm or adverse consequences to the assets mentioned in the ﬁrst question must be identiﬁed.

Therefore risk can be expressed as a set of triplet (Kaplan and Garrick (1981)):

R = {sk,pk,Ck} (3.1) where k counts for the events involved in; sk is a scenario identiﬁcation or description; pk is the probability of that scenario; and Ck is the consequence of that scenario.

ISO 31000 (2018) defines risk as effects of uncertainty on objectives. The effects are deviations from the expected, which can be positive, negative or both. Risk is usually expresses in terms of:

• risk resources: element which alone or in combination has the potential to give rise to risk;

• potential events: occurrence or change of a particular set of circumstances;

21 CHAPTER 3. RISK ANALYSIS

• consequences: outcome of an event aﬀecting objectives;

• likelihood: chance of something happening.

According to IEC 60300-3-9 (1995) a risk analysis is usually carried out to provide answers to the three questions above, raised by Kaplan and Garrick (1981). It is a systematic use of available information to identify hazards and to estimate the risk to individuals, property, and the environment. ISO 31000 (2018) addressed that the purpose of risk analysis is to comprehend the nature of risk and its characteristics. It involves a detailed consideration of factors such as the likelihood of events, the nature and magnitude of consequences, the connectivity of events, confidence levels, etc. The main difference in these two definitions is whether risk identification is included in the risk analysis. IEC 60300-3-9 (1995) involves risk identification within risk analysis, while ISO 31000 (2018) treats risk identification as a separate step to be carried out before risk analysis. In this thesis, risk analysis will be defined as ISO 31000 (2018) and only likelihoods and consequences are evaluated.

3.1 Risk analysis methods

Risk analysis can be undertaken with varying degree of detail and complexity, depending on the purpose of the analysis, the availability and reliability of information, the availability of resources and how the results are to be used. There are mainly two categorises of risk analysis methods: qualitative analysis and quantitative analysis.

A qualitative risk analysis uses words and/or descriptive scales to describe the frequency of the hazardous events identified and the severity of the potential consequences that may result from these events (Rausand (2013)). The scales may be adapted or adjusted to suit the circumstances, and different descriptions may be used for different categories of the risk. A quantitative risk analysis is a analysis that provides numerical estimates for probabilities, consequences, and severities. Sometimes the associated uncertainties are considered along with the analysis.

The term semi-quantitative risk analysis is sometimes used to denote analyses where the qualitative scales are given values. The numbers allocated to each terms do not have to have any accurate relationship to the actual value of the probability or the severity. The objective is to produce a more detailed risk picture than that achieved in a qualitative analysis; but not to suggest any realistic values for the risk, as attempted in a quantitative analysis (Rausand (2013)).

Aven (2015) provides a different classification of the risk analysis methods as simplified risk analysis, standard risk analysis and model-based risk analysis. They are

22 3.1. RISK ANALYSIS METHODS also associated with either qualitative or quantitative. The descriptions are listed in Table 3.1.

Table 3.1: Main categories of risk analysis methods (Aven (2015)).

Category Type Description An informal procedure that establishes the risk picture using brainstorming and Simpliﬁed Qualitative group discussions. The risk might be risk analysis presented on a coarse scale, e.g. low, moderate or high. A more formalised procedure in which Standard Qualitative recognised risk analysis methods are risk analysis or quantitative used. Risk matrices are often used to present the results. Techniques such as Event tree analysis Model-based Primarily and Fault tree analysis are applied to risk analysis quantitative calculate the risk.

Risk in the ﬁeld of Civil Engineering are associated with low-probability and high- consequence events. Qualitative methods by using coarse scale cannot reveal the exact diﬀerence for various hazardous events or scenarios. The qualitative methods or semi-quantitative methods are better suited for quantifying risk in such applications. The reliability methods illustrated in Chapter 2 can provide a good evaluation of the probability. The consequences can be presented either by accurate results based on comprehensive researches or certain assigned values. Both could provide a more detailed picture.

In the following sections, the Fault tree analysis method is illustrated for causal and frequency analysis; and the Event tree analysis method is introduced to develop the accident scenarios and evaluate the consequences.

3.1.1 Fault tree analysis (FTA) The fault tree concept was originally developed at the Bell Telephone Labora- tories in 1962 to evaluate the safety of the Minuteman Launch Control System (Dhillon and Singh (1979)). Since the 1970s Fault tree analysis (FTA) has become widespread and is today one of the most used reliability and risk analysis methods. Its applications are found in most industries (NUREG-75/014 (1975); CCPS (2000); Stamatelatos et al. (2002)).

FTA is a technique for acquiring and representing information about a system,

23 CHAPTER 3. RISK ANALYSIS accordingly the method requires a thorough understanding of the system (Barlow and Chatterjee (1973)). The fault tree is a top-down logic diagram that comprises symbols that show the interrelationships between a potential hazardous event in a system and the causes of this event. The potential hazardous event is called the TOP event. The causes at the lowest level are called basic events. The graphical symbols that show the relation are called logical gates. The output from a logical gate is determined by the input states. Figure 3.1 shows the most important symbols in a fault tree together with the interpretations. Figure 3.2 shows a simple fault tree model.

Figure 3.1: Fault tree symbols (Aven (2015)).

A fault tree provides valuable information about possible combinations of basic events that can result in the TOP event. Such a combination of basic events is called a cut set, which is deﬁned as a set of basic events whose (simultaneous) occurrence ensures that the TOP event occurs. The most interesting cut sets are those minimal cut sets, deﬁned as the shortest combination of events leading to the top event. It cannot be reduced and still insure the occurrence of the TOP event. These combinations are found using Boolean algebra (Davis-Mcdaniel (2011)). FTA is a binary analysis. All events, from the TOP event down to the basic events, are assumed to be binary events that either occur or do not occur. No intermediate states are therefore allowed in the fault tree model.

FTA is suitable for both qualitative and quantitative analysis of complex systems, depending on the scope of the analysis. Qualitative evaluation produces a graphical Boolean depiction of the events leading to the TOP event (Davis-Mcdaniel (2011)). Multiple cut sets are developed and the minimal cut sets should be found. Quanti- tative evaluation of a fault tree transforms the logical structure into an equivalent

24 3.1. RISK ANALYSIS METHODS

4OP VN

/R !ND /R

!ND "ASICºEVENTS

Figure 3.2: A simple fault tree model. probabilistic form so that the probability of the TOP event can be computed from the probabilities of the basic events (Sianipar and Adams (1997)).

The main objectives of FTA are (Barlow and Chatterjee (1973); Rausand (2013):

• To identify all possible combinations of basic events that may result in a critical event in the system.

• To find the probability that the critical event will occur during a specified time interval or at a specified time.

• To identify the weak points of the system that need to be improved to reduce the probability of the critical event and how they lead to undesired events.

There are many advantages of FTA in applications (Rausand (2013)). This technique is easy to use and employs a clear and logical form of presentation. Besides, it is suitable for many diﬀerent hazardous events, though each event needs a separate fault tree model. Both technical faults and human errors can be analyzed by FTA. It also exposes the shortage of the system and provides a chance to rethink and further eliminate many potential hazards. However, FTA also has its limitations. FTA treats only foreseen events, therefore a comprehensive understanding of the hazardous event is required. For large systems, FTA possibly becomes too complicated, time-consuming, and diﬃcult to follow. FTA gives a static picture of the combinations of failures and events that can cause the TOP event to occur.

25 CHAPTER 3. RISK ANALYSIS

However, FTA is not well suited to handle dynamic systems, e.g. when systems are subjected to complex maintenance strategies

Plenty of research related with bridges have also applied FTA as a strong analysis tool (Sianipar and Adams (1997); LeBeau and Wadia-Fascetti (2007); Davis- McDaniel et al. (2013)). While other available methods and tools in bridge assessment, such as visual inspections, computerized simulations and structural health monitoring, have some limitation or downfalls, FTA has its special advantages as resolution (Davis-Mcdaniel (2011); Davis-McDaniel et al. (2013)). FTA assesses the condition of individual components and identifies the relationships between the different components to assess the failure risk of the whole bridge system. The fault tree model is developed using the chain of events; therefore, the events leading to failure can be identified through analysis. When it is difficult to estimate the exact failure possibilities of vague events, an educated guess or probable range can be input. Some research integrate the fuzzy set theory with FTA (Singer (1990); Pan and Wang (2007); Jafarian and Rezvani (2012)). The fuzzy fault tree model is not considered in this thesis.

3.1.2 Event tree analysis Development of potential accident scenarios is an essential element in a risk analysis, and the Event tree analysis (ETA) is usually the preferred method for this purpose. The concept of the ETA was ﬁrst used in the United States to evaluate the risk in nuclear energy programs (Srivastava (2008)). It is by far the most commonly used method for the development of accident scenarios and has been applied within many diﬀerent application areas.

Event tree is a graphical tool indicating all the possible scenarios resulting from a specific event occurrence. This starts with the hazardous event, continues through the splits at certain stages in the structure and ends in outcome identification. The splitting takes place when specified pivotal events occur.

The pivotal events are listed as headings above the tree model. It is crucial that the pivotal events are modeled to correspond to the timing and occurrence of events in the accident scenario. It is recommended that each pivotal event be formulated as a ‘negative’ statement, such as ‘Barriers X fails on demand’. For each pivotal event, at least one branch splits into two new branches: the upper branch signiﬁes that the event description is ‘true’, and a lower branch signiﬁes that it is ‘false’. By this approach, the most serious accident scenarios will come highest up in the event tree diagram. The branches can be categorized to be related to either physical phenomena or barriers in the system. Often, ETA covers both categories.

Event trees may be developed independently or combined with FTA. FTA is used to study the causes of a hazardous event, while ETA is used to study the possible

26 3.1. RISK ANALYSIS METHODS accident scenarios and consequences following the same event. In this case, FTA and ETA are like the two sides of a bow-tie diagram, as shown in Figure 3.3. Fault trees can also be nicely integrated into the event tree to analyze barrier failures (pivotal events), as shown in Figure 3.4. Consequences

Fauůt tree analysis Hazardous Eventtree analysis event Hazards/threats

Figure 3.3: A simpliﬁed bow-tie diagram combined with FTA and ETA.

Hazardous Pivotal Pivotal Pivotal End event No. event event 1 event 2 event 3 description

TRUE D 1

C FALSE B 2 D'

A 3 C'

4 B'

Figure 3.4: Pivotal events of an event tree analyzed by fault trees.

The main objectives of an ETA are (Rausand (2013)):

• To identify the accident scenarios that may follow the hazardous event.

27 CHAPTER 3. RISK ANALYSIS

• To identify the barriers that are (or are planned to be) provided to prevent or mitigate the harmful eﬀects of the accident scenarios. • To determine the probability of each accident scenario. • To determine and assess the consequences of each accident scenario.

The ETA may be qualitative, quantitative, or both, depending on the objectives of the analysis and the availability of relevant data. In the former case, the method provides a picture of the possible scenarios. In the latter case, probabilities are linked to the various event sequences and their consequences.

Due to limited access to information in the traditional approach of ETA, the assessment of potential scenarios resulting from the occurrence of a speciﬁc event is diﬃcult and sometimes even impossible. In such a condition, combining ETA with fuzzy logic may enable construction industry specialists to assess the risk event by using linguistic terms instead of numerical values and use fuzzy arithmetic operations to perform Event trees analysis (Abdelgawad (2011)). Some research have used this fuzzy approach during ETA (Kenarangui (1991); Huang et al. (2001); Baraldi and Zio (2008); Abdelgawad and Fayek (2012)). However, this method has not been widely used in the construction industry including bridge construction projects.

The main advantages of ETA are that it is well documented and simple to use. It clearly presents the event sequences following a hazardous event and the corresponding consequences for each scenario. It can also identify the weaknesses of the system and/or barriers. A limitation is that ETA requires that the sequence of pivotal events has to be foreseen. Besides, all the probabilities in an event tree are, in principle, conditional probabilities, indicating that there are various dependencies between the pivotal events. It can be diﬃcult to handle and quantify the dependency in a quantitative analysis.

3.2 Risk-based decision-making methods

Although scientific description or explanation may be an intermediate goal, in engineering practice the use of any analysis is ultimately in situations where the engineer must make a decision (Benjamin and Cornell (2014)). The aim for conducting a risk analysis is to support decision-making. Risk-based decision making can provide an important basis for finding the right balance between different concerns, such as safety and costs (Aven (2015)).

An important aspect of decision analysis is to define a utility function u.Itis usually a numerical assessment which reflects the decision makers’ preferences over the possible outcomes: a higher utility reflects preference for one possible outcome

28 3.2. RISK-BASED DECISION-MAKING METHODS over another with lower utility (Björnsson (2017)). The value of the utility function for different outcomes may include benefits as well as costs or drawbacks. In the context of bridge management, it may be difficult to directly determine the benefit of a structure while the consequences of various damage states can be more readily determined and compared (Wong et al. (2005)).

3.2.1 Decision tree Decision trees have their genesis in the work of Von Neumann and Morgenstern (2007) on extensive-form games. They are in some way an extension of an event tree so that the full decision problem could be expressed (Smith (2010)).

When problems are well deﬁned and their dependencies are reasonably known, then they can be put into a hierarchical structure known as a decision tree. A decision tree has three basic nodes: a decision node presenting alternative actions or options a, a chance node representing the random outcome of action or option called the state of nature θ and a terminal node where the ﬁnal outcomes (the utility) are given with respect to the decision u(a, θ). Therefore, the decision-making process can be formulated as the process of choosing an action a from among the available alternative actions a1, a2,..., am, which are the members of an action space A. Once the decision has been made, the decision-maker can see the corresponding states of nature θ. As a result of having taken action a and having found the true state θ, the decision-maker will receive the utility value u(a, θ), a numerical measure of the consequences of this action-state pair (Benjamin and Cornell (2014); Björnsson et al. (2019)). A sketch of a decision tree model is shown as Figure 3.5.

Although decision trees are expressive and computationally eﬃcient, they indeed have several drawbacks. First, since decision trees explicitly represent acts and events, the trees grow too fast in some problems. For example, an n-stage decision problem with m choices or events will lead to at least mn endpoints. Second, probabilities of events may not be available in the form that decision tree requires. When several stages are considered, the probability of a following stage is usually a conditional probability depending on the previous stage. Some calculations based on probability theory are needed.

3.2.2 Influence diagram Influence diagram (ID) is a Bayesian probabilistic network augmented with decision variables and utility functions of a decision problem which can be used for solving sequential interleaved decision problems (Kjaerulff and Madsen (2008)). It was initially proposed as an alternative to decision trees for representing decision problem as a more compact and computationally more efficient approach (Miller III et al. (1976); Varis (1997)). Olmsted (1985) and Shachter (1986) devised methods

29 CHAPTER 3. RISK ANALYSIS

ݑሺܽ ǡ ߠ ሻ ߠଵ ଵ ଵ

ߠଶ ݑሺܽଵǡ ߠଶሻ ... ߠ௠

ݑሺܽଵǡ ߠ௠ሻ ܽଵ

ߠଵ ݑሺܽଶǡ ߠଵሻ

ܽଶ ߠଶ ݑሺܽଶǡ ߠଶሻ ... ߠ௠

ݑሺܽଶǡ ߠ௠ሻ ܽ௠ ݑሺܽ ǡ ߠ ሻ ߠଵ ௠ ଵ

ߠଶ ݑሺܽ௠ǡ ߠଶሻ

ߠ௠

ݑሺܽ௠ǡ ߠ௠ሻ

Figure 3.5: The sketch of a decision tree (Benjamin and Cornell (2014)). for solving inﬂuence diagram directly. Nowadays the inﬂuence diagram has gained increasing popularity within decision analysis (Oliver (1990)).

The influence diagram provides a graphical representation of the decision problem and a framework to discover optimal decision rules. It contains nodes related to the different aspects of the decision problem. The node types are (Kjaerulff and Madsen (2008)):

• Decision nodes corresponding to actions (rectangles); • Uncertainty nodes corresponding to uncertain states (ovals); • Value nodes which contain the utilities (diamonds).

Directed paths between each node reﬂect dependence and (usually) causality between nodes. Arrows between node pairs indicate inﬂuences of two types (Howard and Matheson (2005)):

1. Informational inﬂuences, represented by arrows leading into a decision node. These show exactly which variables will be known by the decision maker at the time that the decision is made.

30 3.3. A THEORETICAL MODEL FOR FATIGUE ASSESSMENT

2. Conditioning inﬂuences, represented by arrows leading into a chance node. These show the variables on which the probability assignment to the chance node variable will be conditioned.

Figure 3.6 shows a sketch of the inﬂuence diagram with focus on bridge condition assessment (Björnsson et al. (2019)). In this case, the methods can extensively be inspections, on-site testing or monitoring, etc., and the corresponding method outcomes indicate if there is any damage detected. According to the outcome, several actions could be considered, such as ‘Do nothing’, ‘Repair’, ‘Replace’, etc.

utilities͕ Action͕ ܽ ݑሺܽǡ ߠǡݖሻ

Method Method͕ ݁ outcome͕ ݖ

Method State of cost͕ ݑሺ݁ሻ nature͕ ߠ

Figure 3.6: The sketch of an inﬂuence diagram (Björnsson et al. (2019)).

An influence diagram cannot be effectively used to represent all decision problems since they depend to a significant degree on a certain type of symmetry being present (Smith (2010)). However the conditions in which they are a useful tool are met quite often in practice and Gómez (2004) catalogues over 250 documented practical application of the framework before 2003.

An influence diagram can also be used for decision problems where more than one person’s decision affects the final utility. This can be useful related to bridge management in situations where multiple stakeholders are involved in the process and where a decision taken by one stakeholder may affect the subsequent decisions of others involved (Koller and Milch (2003)).

3.3 A theoretical model for fatigue assessment

To avoid failure and limit the potential cost, decision makers should consider that there are a number of possible interventions based on the current condition of the selected structure and be able to determine which intervention is suitable in a

31 CHAPTER 3. RISK ANALYSIS speciﬁc case. In this section a decision tree model, which has been speciﬁed for the fatigue assessment, was established with several possible scenarios as shown in Figure 3.7.

Prior assessment

ܽ଴

ܽଵ Status of nature

ܽଶ Decision

Alternatives

Chance node

Posterior assessment

Figure 3.7: A decision tree model.

The decision tree is based on a reference time frame. Based on the status of the current time point, the so-called initial condition, diﬀerent scenarios can be followed and an optimal decision will be made for a future point in time. The three intervention alternatives are explained as follows:

• a0: do nothing. The prior reliability assessment of the selected structure should be performed, which estimates the probability of failure at the studied point in time. This scenario is justiﬁed if the initial condition determines that the structure does not, within a reference time frame, require any more attention and that no immediate action is required.

32 3.3. A THEORETICAL MODEL FOR FATIGUE ASSESSMENT

For instance, if the prior reliability assessment estimates that the reliability of the selected structure still has some margin above the target value at the studied point in time, no extra intervention is needed. However, it would be diﬃcult to apply this criterion for a system assessment, since there is no deﬁned target system reliability.

• a1: direct repair. Here refers to implement strengthening to the selected structure without any inspection performed. This scenario could be an option if the prior assessment shows an exhausted fatigue life since a direct repair could eﬃciently increase the fatigue resistance. For example, for welded structures increasing the cross-sectional dimensions or improving the weld geometry will eﬃciently reduce the stress range at the critical details, which will further improve the fatigue resistance.

• a2: inspection. This scenario is applied if the prior assessment indicates an exhausted fatigue life, but more information about the damage state is required before making any decisions. An inspection brings new knowledge and it corresponds to the posterior reliability assessment. There are various methods of performing an inspection, and they achieve different accuracy level. In the case study in appended papers, visual inspection is selected due to its simplicity and low-cost in practice.

Chapter 4

Summary of appended papers

To avoid unduly conservatism and due to the various levels of uncertainty involved in the problem considering fatigue, a probabilistic model was established. It is based on linear elastic fracture mechanics (LEFM) and Paris law for crack growth. The Rautasjokk Bridge, located along the iron ore railway line in Northern Swe- den, is selected as a case study to test all theories proposed in this thesis. It is representative for many similar railway bridges in Sweden and globally. Based on the deterministic code based assessment by Häggström (2015), the gusset plate on the stringer beam is shown to be the most critical section. A crack is assumed to initiate and propagate into the ﬂange at the end of a welded on in-plane gusset plate.

Paper I: Comparison of simulation methods applied to steel bridge reliability evaluations

The initiation and propagation of a fatigue crack is a strongly nonlinear phe- nomenon due to the exponential relation between crack growth and stress range, varying crack growth rates, and threshold levels. Besides, the limit state function (LSF) constructed for a fatigue problem is multi-dimensional considering diﬀer- ent levels of uncertainty. Both aspects impose more demands on the simulation techniques.

There are multiple simulation-based procedures available to estimate the structural reliability. In Paper I, Monte Carlo simulation (MCS) and Subset simulation (SS) are selected. Their performances are evaluated and compared against fatigue deterioration for the gusset plate detail. The performance of SS mainly depends on two parameters: NS, the sample size of each iteration; and p0, the conditional probability of each iteration (presented in detail in Section 2.2.3).

The prior reliability assessment and updated reliability assessment combined with a parametric study were performed respectively. Three values of NS and p0 were applied in the SS setting. In general, the assessments obtained by SS are consistent with MCS. Fatigue life is represented by the number of cycles when the reliability of a single detail falls to 3.1. Regarding the fatigue lives obtained by MCS as true, the results using SS are all within acceptable deviation, though the deviations are due to inevitable randomness and cannot be eliminated by simply increasing the sample numbers NS. The elapsed time of SS is much shorter than MCS for all

35 CHAPTER 4. SUMMARY OF APPENDED PAPERS settings. The robustness of SS has been tested and proved by 400 independent runs with a speciﬁc setting.

SS shows its feasibility to deal with fatigue problem with high nonlinearity and multi-dimensional LSF, and shows outstanding time eﬃciency compared to MCS for a comparable accuracy. Recommended values on SS setting in dealing with fatigue problems are provided.

Paper II: Fatigue reliability assessment of steel bridges considering spatial correlation in system evaluation

Components of the structure might share similar characteristics which may bring correlations in their fatigue deterioration processes. Therefore it is necessary to assess the structural reliability from a system perspective. In Paper II, the system reliability of steel bridges is addressed with focus on the inﬂuence of spatial correlations between fatigue sensitive details. Their spatial correlation is considered among the details which are built with the same material, have similar geometries and are subjected to similar load ﬂuctuations.

Eight gusset plate details, which are prone to failure, are idealized into a series system. An eﬃcient modeling approach has been developed under the assumption that each detail within the same series system has the same individual reliability level and those details are equicorrelated with each other. This approach, referred as Equicorrelation-based modelling approach (ECM), supplements the common simulation techniques and facilitates a quantiﬁcation of the correlation and help estimate the system reliability analytically.

The prior reliability of the 8 details system was assessed by MCS with supplement ECM. A sample size of 106 can guarantee a quick convergence of the equivalent correlation and the estimated system reliability is consistent with the simulated result. However, SS is not the best option dealing with system reliability assessment. SS has its inevitable randomness (presented in Paper I) which makes it diﬃcult to have a converged equivalent correlation. Besides, SS needs to assess details one by one and cannot run simulations in parallel. ECM shows high eﬃciency and accuracy in system reliability estimation as supplement to MCS, and its feasibility to assess large structural systems without inducing extra calculation time.

The value of the correlation between details depends on the correlations of underlying random variables contributing to the failure. A parametric study was performed to identify the decisive variables and how they influence the system reliability. Five cases were created and the results show that material parameters have more dominating influence on the system reliability, followed by the model uncertainties, while the correlation of initial crack size has only a modest influence on the reliability.

36 Paper III: Risk analysis for decision support — a case study on fatigue assessment of a steel bridge

Many structures have been assessed on the reliability level, which focuses on finding the reliability index and comparing it with the target reliability, a predefined acceptable performance level. If the selected structure falls below the target reliability, it is regarded as deficiency and requires further actions. The optimal intervention and the corresponding consequences are of great importance.

In Paper III, a risk-based decision support framework is formulated aiming for rational decisions, which is presented as a decision tree. Based on the evaluation in Paper II, an eight details series system was further analyzed on the risk level. Three possible intervention actions are presented, ‘Do nothing’, ‘Repair’ and ‘Inspection’. The corresponding consequences are evaluated based on cost with tentative values.

Reliability assessment and risk analysis have different criteria in decision-making. When reliability assessment tries to find the safest option — the option with the lowest probability of failure, risk analysis tries to find an option that is both safe and economic. Therefore they may impose different decisions. For instance when a crack is detected, ‘Replace’ the whole subsystem will lead to the lowest probability of failure, while ‘Repair’ will bring a relatively safe and economic decision. The values of the consequences matter when it comes to making an optimal decision. A tiny change in the values could directly alter the decision. When the cost of ‘Repair’ increases, the risk of ‘Repair’ will also increase, which could possibly make ‘Do nothing’ as the option with lowest risk. Therefore an accurate estimation of consequences will increase the credibility of the decision.

Chapter 5

Concluding remarks

5.1 Discussion

The first damage appearance in a detail is defined as failure in this thesis. Therefore the procedure of risk identification has not been carried out. The definition of risk analysis is consistent with ISO 31000 (2018), which includes analysis on frequency and consequence. However, in real applications especially for larger systems there could be multiple hazardous events and they are difficult to identify. From this perspective the risk identification is very important and should be the first step before further analysis on frequency or consequences. The review of methods for risk analysis is not exhaustive. In Chapter 3 only two representative methods have been presented: FTA for frequency analysis and ETA for consequence analysis. There are more methods available covering a broad variation of applications. The selection of methods should be made according to the aim and focus of the study.

Three papers are built up with an order as:

1. reliability assessment of a single detail;

2. reliability assessment of a series system consisting of multiple similar details;

3. risk analysis of the system.

However, those connections are valid under several assumptions. For instance, Paper II provided a comprehensive case study based on the idealization of series system. However, in actual design of bridges mixed systems are usually applied. Then the system reliability will become very specific for the selected bridge, depending on how the subsystems are combined. The parametric study has shown how the correlations influence on the system reliability. However, it is difficult to provide or justify a reasonable range of correlations. In Paper III, the consequences are evaluated by cost which is provided by tentative values. The study came to the conclusion that a small change in consequences could alter the optimal decision. In reality, bridges always have a very low value in probability of failure and a high value in consequences, which makes the risk of bridge failure become moderate. If there is a small change in consequence (the actual value), it is possible that it won’t have much impact on risk or the selection of optimal decision.

39 CHAPTER 5. CONCLUDING REMARKS

This thesis lays the foundation for the long-term project which aims to develop a risk-based framework supporting rational decision to extend the service life of steel bridges. The reliability assessment can be regarded as the foundation of the frequency analysis, while a comprehensive consequence analysis should be accomplished in the later stage. Several decision-making models are of interest and need to be tried out.

5.2 Conclusions

The main conclusions based on each paper appended to this thesis are summarized below:

Paper I

• For reliability assessment of single details against fatigue deterioration, Subset simulation (SS) shows promising feasibility to deal with multi-dimensional limit state functions and the crack propagation task with strong nonlinearity. Using Monte Carlo simulation (MCS) as a reference, SS shows reassuring consistency at low time consumption. The sample number NS for each subset iteration can provide a good accuracy when it is around thousand or higher. An empirical value p0 =0.1 is recommended for this application.

• The robustness of SS was tested by 400 independent simulation runs both for prior reliability and updated reliability. The scattered estimated fatigue lives are quite centered with limited standard deviation. The robustness of SS is proved and it is believed that the result from one single run, though with inevitable randomness, is acceptable and reliable.

• Inspection provides a noticeable increase in single details’ reliability and fatigue life, even the visual inspection which has the lowest accuracy. SS is also feasible for updated reliability application.

Paper II

• An equicorrelation-based modelling approach (ECM) was outlined and it showed high eﬃciency and accuracy in system reliability estimation as a supplement to common simulation techniques. It is able to deal with large structural systems without inducing extra calculation time.

• Diﬀerent from the single detail assessment, SS is not the best choice for system reliability assessment. Its randomness makes it diﬃcult to derive a converged equivalent correlation and further implement ECM. A direct SS routine will

40 5.3. FURTHER RESEARCH

be very time-consuming since the simulation procedure cannot be shortened. In this aspect MCS shows better feasibility.

• The correlations of material parameters have a more dominating inﬂuence on the system reliability, followed by model uncertainties. The inﬂuence of the initial crack size could be noticed only when all other variables are fully correlated. A higher correlations of material parameters or model uncertainties will induce a higher value of the equivalent correlation between two detail, as well as an increase in system reliability.

• The inspected detail is able to provide information about the updated reliability of other similar details depending on the level of correlation. However, not much information can be expected for the actual crack size at other locations even with high spatial correlations.

Paper III

• Three possible interventions and their corresponding consequences have been discussed. The values of consequences (presented by cost) matter when it comes to making an optimal decision. A small change of the consequence could directly alter the decision. An accurate estimation of the consequence will increase the reliability of the decision.

• The decisions based on reliability assessment can be diﬀerent from those based on risk analysis. The former one only considers the safest option, which could induce unnecessary waste of money and resources, while the latter one evaluates the intervention and consequences together, which could provide a decision that is both economic and safe.

• Visual inspection brought in new information and provided a noticeable im- provement in the system reliability, which has been proved in Paper I and Paper II. However, it didn’t show much beneﬁt on the value of risk when compared to other alternatives. This is due to that the probability of non- failure and the cost for repairing/replacement are dominating, making the value of risk relatively high.

5.3 Further research

This licentiate thesis has investigated reliability-based assessment of an existing steel bridge, and a simpliﬁed risk-based decision tree model has been established for a selected subsystem. Additional work could be done to improve and move further in the research of this topic. Some suggestions for further studies are:

41 CHAPTER 5. CONCLUDING REMARKS

• The actual probability of failure of a bridge can be very complex and be the result of a combination of different types of systems. In this thesis only one series system is considered. Additional work could be done to combine several systems of different types, as either series or parallel systems. • There are defined target reliability levels for single details; however, no such clear criteria have been proposed for systems. Target values of system reliability under various conditions are needed, so as to further help engineers making good judgement about steel bridges’ status. • The costs of different interventions and outcomes in the risk analysis are assigned as tentative values. These values are essential and can for some conditions alter the decision. Therefor a comprehensive research on the real costs should be done. • A risk-based analysis framework for existing steel bridges is expected to be developed. It is expected to well combine the information from inspection, monitoring, etc., and be applicable in general for steel bridges in fatigue evaluation.

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