VTT P775 Licentiate Thesis ORC Dec2011

VTT PUBLICATIONS 775Otso Cronvall Structural lifetime, reliability and risk analysis approaches for power plant components and systems VTT PUBLICATIONS 775 Structural lifetime, reliability and risk analysis approaches for power plant components and systems Otso Cronvall 2 ISBN 978-951-38-7760-6 (soft back ed.) ISSN 1235-0621 (soft back ed.) ISBN 978-951-38-7761-3 (URL: http://www.vtt.fi/publications/index.jsp) ISSN 1455-0849 (URL: http://www.vtt.fi/publications/index.jsp) Copyright © VTT 2011 JULKAISIJA – UTGIVARE – PUBLISHER VTT, Vuorimiehentie 5, PL 1000, 02044 VTT puh. vaihde 020 722 111, faksi 020 722 4374 VTT, Bergsmansvägen 5, PB 1000, 02044 VTT tel. växel 020 722 111, fax 020 722 4374 VTT Technical Research Centre of Finland, Vuorimiehentie 5, P.O. Box 1000, FI-02044 VTT, Finland phone internat. +358 20 722 111, fax + 358 20 722 4374 Technical editing Marika Leppilahti Kopijyvä Oy, Kuopio 2011 3 Otso Cronvall. Structural lifetime, reliability and risk analysis approaches for power plant components and systems [Rakenteellisen eliniän, varmuuden ja riskien analysoimisen lähestymistapoja voimalai- tosten komponenteille ja järjestelmille]. Espoo 2011. VTT Publications 775. 264 p. Keywords structural mechanics, fracture mechanics, risk analysis, reliability, PFM, RI-ISI Abstract Lifetime, reliability and risk analysis methods and applications for structural systems and components of power plants are discussed in this thesis. These analyses involve many fields of science, such as structural mechanics, fracture mechanics, probability mathematics, material science and fluid mechanics. An overview of power plant environments and a description of the various degradation mechanisms damaging the power plant systems and components are presented first. This is followed with a description of deterministic structural analysis methods, covering e.g. structural mechanics and fracture mechanics based analysis methods as well as the disadvantages of the deterministic analysis approach. Often, physical probabilistic methods are based on deterministic anal- ysis methods with the modification that one or more of the model parameters are considered as probabilistically distributed. Several probabilistic analysis proce- dures are presented, e.g. Monte Carlo Simulation (MCS) and importance sam- pling. Description of probabilistic analysis methods covers both physical and statistical approaches. When the system/component failure probabilities are combined with knowledge of failure consequences, it is possible to assess sys- tem/component risks. Several risk analysis methods are presented as well as some limitations and shortcomings concerning to them. Modelling methods for various degradation (or ageing) mechanisms are pre- sented. These methods are needed in the lifetime analyses of structural systems and components of power plants. In general, the lifetime analyses in question necessitate a thorough knowledge of structural properties, loads, the relevant degradation mechanisms and prevailing environmental conditions. The nature of degradation models of structural systems/components can be deterministic, probabilistic or a combination of these two types. Degradation models of all these kinds are presented here. Some important risk analysis applications are described. These include probabilistic risk/safety assessment (PRA/PSA) and risk informed in-service inspections (RI-ISI). 4 In practise, lifetime and risk analyses are usually performed with a suitable analysis tool, i.e. with analysis software. A selection of probabilistic sys- tem/component degradation and risk analysis software tools is presented in the latter part of this thesis. Computational application of probabilistic failure and lifetime analyses to a representative set of NPP piping components with proba- bilistic codes VTTBESIT and PIFRAP are presented after that. The thesis ends with a summary and suggestions for future research. 5 Otso Cronvall. Structural lifetime, reliability and risk analysis approaches for power plant components and systems [Rakenteellisen eliniän, varmuuden ja riskien analysoimisen lähestymistapoja voima- laitosten komponenteille ja järjestelmille]. Espoo 2011. VTT Publications 775. 264 s. Avainsanat structural mechanics, fracture mechanics, risk analysis, reliability, PFM, RI-ISI Tiivistelmä Tämän lisensiaattityön aiheina ovat voimalaitosten rakennejärjestelmien ja -komponenttien käyttöiän, luotettavuuden ja riskitarkastelujen analyysimenetel- mät ja sovellukset. Näihin analyysimenetelmiin liittyy usea tieteen ala, kuten lujuusoppi, murtumismekaniikka, todennäköisyysmatematiikka, materiaalitiede ja virtausmekaniikka. Ensin esitetään yleiskatsaus voimalaitosympäristöistä sekä kuvaukset erilaisista voimalaitosten rakennejärjestelmiä ja -komponentteja koskevista vaurioitumis- mekanismeista. Sitten käsitellään deterministiset rakenneanalyysimenetelmät, kuten lujuusopin ja murtumismekaniikan menetelmät, sekä eritellään determinis- tisen lähestymistavan puutteita. Usein fysikaaliset probabilistiset menetelmät perustuvat deterministisiin vastaaviin sillä muunnelmalla, että yksi tai useampi malliparametri on asetettu probabilistisesti jakaantuneeksi. Työssä esitetään useita probabilistisia analyysimenetelmiä, kuten Monte Carlo -simulaatio ja tär- keysperusteinen otanta. Todennäköisyysperusteisia analyysimenetelmiä koskeva kuvaus kattaa sekä fysikaaliset että tilastolliset lähestymistavat. Kun rakennejär- jestelmien/-komponenttien todennäköisyydet yhdistetään tietämykseen seuraus- vaikutuksista, voidaan arvioida vastaavat riskit. Työssä esitetään useita riskiana- lyysimenetelmiä sekä eräitä niitä koskevia rajoituksia ja puutteellisuuksia. Työssä esitetään valikoima erilaisia vaurioitumismekanismeja koskevia mallin- nusmenetelmiä. Näitä menetelmiä tarvitaan rakennejärjestelmien ja -komponenttien käyttöikäanalyyseissa. Yleisesti ottaen kyseiset käyttöikäanalyysit edellyttävät perusteellisia tietoja rakenteellisista ominaisuuksista, kuormista, merkittävistä vaurioitumismekanismeista sekä vallitsevista olosuhteista. Rakennejärjestelmien/ -komponenttien vaurioitumista kuvaavat mallit voivat olla tyypiltään determinis- tisiä, probabilistisia tai näiden yhdistelmä. Työssä esitetään kaikkiin näihin tyyppeihin lukeutuvia vaurioitumismalleja. Lisäksi esitetään muutamia merkittä- viä riskianalyysin sovelluksia. Näihin lukeutuvat todennäköisyyspohjaiset turval- lisuus- ja riskianalyysit (PSA ja PRA) ja riskitietoiset tarkastusohjelmat (RI-ISI). 6 Käytännössä käyttöikä- ja riskianalyysit tehdään yleensä jollakin sopivalla analyysityökalulla eli sovellusohjelmalla. Työn jälkipuoliskolla esitetään vali- koima rakennejärjestelmien/-komponenttien vaurioitumisen ja riskien analyysi- sovelluksia. Sen jälkeen edustavalle valikoimalle ydinvoimalan putkistokom- ponentteja esitetään analyysiohjelmilla VTTBESIT ja PIFRAP tehdyt todennä- köisyysperusteiset vikaantumis- ja käyttöikäanalyysit. Lopuksi esitetään yhteenveto sekä jatkotutkimusaiheita koskevia ehdotuksia. 7 Foreword The author expresses his gratitude to VTT for providing both working quarters and altogether nearly three person work months worth of resource support for the preparation of this licentiate thesis. The author expresses his gratitude to Professor Juha Paavola for supervising the thesis and to DrTech Kaisa Simola for being the inspector for this thesis as well as for valuable advice and com- ments. The author also expresses his gratitude to DrTech Heli Talja for encour- agement and preparing the way for this thesis in its earlier phase, to MSc Jouni Alhainen for computational support and music related discussions, and to Niina for general support and caring understanding, in particular during the completion phase of the thesis. Gratitude is also addressed to many fellow researchers from VTT and certain experts from the domestic energy industry for providing useful technical background information and advice during the preparation of this thesis. Espoo, December 2011 Author 8 Contents Abstract ............................................................................................................ 3 Tiivistelmä ......................................................................................................... 5 Foreword........................................................................................................... 7 List of symbols ................................................................................................ 11 List of abbreviations ........................................................................................ 20 1. Objective of the thesis .............................................................................. 23 2. Introduction .............................................................................................. 24 3. Degradation mechanisms concerning power plant components ................ 28 3.1 Introduction ......................................................................................................................... 28 3.2 Overview of conventional and nuclear power plant environments ................................... 31 3.3 Fatigue ................................................................................................................................ 36 3.4 Corrosion ............................................................................................................................ 42 3.5 Creep .................................................................................................................................. 47 3.6 Irradiation embrittlement .................................................................................................... 49 3.7 Thermal ageing .................................................................................................................. 50 3.8 Other degradation mechanisms ........................................................................................ 51 3.9 Interaction of degradation mechanisms ............................................................................ 51 4. Deterministic analysis methods ................................................................ 55 4.1 Introduction ......................................................................................................................... 55 4.2 Analysis methods based on structural mechanics ............................................................ 55 4.3 Fracture mechanics based analysis methods ................................................................... 61 4.4 Analysis methods based on other fields of science .......................................................... 67 5. Probabilistic analysis methods ................................................................. 69 5.1 Introduction ......................................................................................................................... 69 5.2 Failure probability and reliability ........................................................................................ 70 5.3 Uncertainty and sensitivity ................................................................................................. 71 5.4 Response function and performance function .................................................................. 73 5.5 Commonly applied distribution functions ........................................................................... 74 5.6 Review of probabilistic analysis procedures ..................................................................... 75 5.6.1 First-order second moment method (FORM) ..................................................... 75 5.6.2 Second-order reliability method (SORM) ........................................................... 76 5.6.3 Fast probability integration (FPI) ........................................................................ 77 5.6.4 Mean value methods ........................................................................................... 78 5.6.5 Monte Carlo simulation (MCS) ........................................................................... 79 5.6.6 Response surface methods ................................................................................ 81 5.6.7 Importance sampling (IS) .................................................................................... 82 5.6.8 Latin hypercube simulation (LHS) ...................................................................... 84 9 6. Risk analysis methods ............................................................................. 85 6.1 Introduction ......................................................................................................................... 85 6.2 Determination of risk values .............................................................................................. 85 6.3 Qualitative, quantitative and semi-quantitative risk analysis approaches ........................ 86 6.4 Review of risk analysis procedures ................................................................................... 87 6.4.1 Failure mode and effect analysis (FMEA) .......................................................... 87 6.4.2 Fault tree Analysis............................................................................................... 88 6.4.3 Event tree analysis .............................................................................................. 89 6.4.4 Expert opinion ..................................................................................................... 90 6.4.5 Hazard and operability studies (HAZOP) ........................................................... 92 6.5 Risk based regulations, guidelines and standards ........................................................... 92 6.6 Concerns with risk assessment ......................................................................................... 94 7. Deterministic degradation modelling methods for power plant components ....... 97 7.1 Introduction ......................................................................................................................... 97 7.2 Fatigue ................................................................................................................................ 97 7.3 Corrosion .......................................................................................................................... 113 7.3.1 General and pitting corrosion ........................................................................... 113 7.3.2 Stress corrosion cracking (SCC) ...................................................................... 114 7.4 Creep ................................................................................................................................ 121 7.5 Irradiation embrittlement .................................................................................................. 128 7.6 Modelling of interaction of degradation mechanisms...................................................... 131 7.7 Commonly applied component degradation assessment procedures ........................... 139 8. Probabilistic degradation modelling methods for power plant components ...... 143 8.1 Probabilistic structural mechanics based modelling methods ........................................ 144 8.2 Probabilistic fracture mechanics based methods ........................................................... 144 8.2.1 Introduction ........................................................................................................ 144 8.2.2 PFM analysis procedure ................................................................................... 146 8.2.3 Engineering models and PFM .......................................................................... 147 8.2.4 Numerical PFM methods .................................................................................. 149 8.3 Statistical modelling methods .......................................................................................... 152 8.3.1 Short-term ageing models ................................................................................ 152 8.3.2 Long-term ageing models ................................................................................. 159 9. Risk analysis applications for power plant systems ................................. 165 9.1 Introduction ....................................................................................................................... 165 9.2 Probabilistic risk/safety assessment ................................................................................ 165 9.3 Risk informed in-service inspection (RI-ISI) .................................................................... 172 9.3.1 Introduction ........................................................................................................ 172 9.3.2 On principles of RI-ISI ....................................................................................... 174 9.3.3 Process of risk-informed inspection planning .................................................. 174 9.3.4 Qualitative, quantitative and semi-quantitative methods for RI-ISI ................. 175 9.3.5 Brief overview and comparison of two main RI-ISI methodologies ................. 176 9.3.6 Probability of detection of cracks ...................................................................... 179 9.3.7 RI-ISI approach developments by VTT ............................................................ 180 9.3.8 RISMET project – benchmarking of RI-ISI methodologies .............................. 182 10 10. Power plant component ageing analyses ............................................... 183 10.1 Selection of components for analyses............................................................................. 183 10.2 Identification of ageing mechanisms ............................................................................... 185 10.3 Mitigation of ageing effects .............................................................................................. 185 11. Probabilistic ageing and risk analysis tools ............................................. 187 11.1 Component ageing and risk analysis software ............................................................... 187 11.2 System ageing and risk analysis software ...................................................................... 203 11.3 Summary of ageing and risk analysis software............................................................... 208 12. Probabilistic lifetime analysis application for piping components ............. 209 12.1 Introduction ....................................................................................................................... 209 12.2 Examined piping components and associated input data .............................................. 209 12.2.1 Geometry ........................................................................................................... 210 12.2.2 Material properties ............................................................................................ 210 12.2.3 Loads and failure mechanisms ......................................................................... 212 12.2.4 Initial crack size distributions ............................................................................ 216 12.3 Analysis characteristics .................................................................................................... 220 12.3.1 Scope of probabilistic analyses ........................................................................ 220 12.3.2 Details concerning the analysis flow of the two applied analysis codes ......... 221 12.3.3 Sub-critical crack growth ................................................................................... 223 12.4 Results and discussion .................................................................................................... 224 13. Summary and suggestions for future research ....................................... 233 References ................................................................................................... 239 11 List of symbols Latin symbols A event rate parameter before the first event A rate coefficient in constitutive equation for creep a crack depth a 0 initial crack depth, initial crack length a c critical value of crack depth a CA depth of creep induced cavity/crack a CA,0 threshold depth of creep induced cavity/crack for onset of interac- tion with fatigue induced crack growth a FA depth of fatigue induced crack a i effect of i:th maintenance a n final crack length a 11 , a 12 , a 22 , a 33 coefficients in strain energy density equation B geometry factor b Burgers vector b, k material parameters in S-N curves method c half of the crack length c reduction factor C effective corrosion rate in absence of protective coating C(t) interpolation parameter between small scale creep and extensive creep C t interpolation parameter between small scale creep and extensive creep (C t ) SSC is separately computed small scale creep limit for interpolation parameter between small scale creep and extensive creep C, C F , C S , C J coefficient characterising material and environment in fatigue crack growth rate equation C coefficient characterising material and environment in stress corro- sion crack growth rate equation C 0 , C 1 stress corrosion crack growth rate equation coefficients C1, C2 constants in equation for nominal environmental fatigue correction factor 12 C | coefficient in probabilistic equation for aspect ratio of fabrication cracks CF radiation embrittlement coefficient C i consequence C p retardation parameter C * C * integral C, n, p, q parameters in fatigue crack growth rate equation d critical strength level D total accumulated damage D outer pipe diameter D matrix, which contains the detection probabilities of cracks D I total accumulated damage for crack initiation phase D II total accumulated damage for crack propagation phase D 0 , | material constants in creep crack growth rate equation d(t) general corrosion propagation depth E elastic modulus, Young’s modulus f loading frequency F Faraday’s constant F renewal process f neutron fluence f(t) time dependent probability density function of component failure F(t) time dependent renewal process, i.e. a sequence of independent, identically distributed non-negative random variables, of which not all are zero f x (x) depth probability for existing manufacturing crack f alloy material type dependent coefficient in stress corrosion cracking rate equation F en environmental fatigue correction factor F en,i nominal environmental fatigue correction factor for the stress cycle i F en,nom nominal environmental fatigue correction factor f i0 occurrence rate of stress corrosion cracking f ij dimensionless function of angular coordinate of the polar system having origin at the crack tip F I , F II crack geometry factors for fracture modes I and II f orientation crack propagation direction dependent coefficient in stress corro- sion cracking rate equation 13 f x (x) probability density function of random variable X, PDF F x (x) cumulative distribution function of random variable X, CDF f xy (x, y) joint probability density function of random variables X and Y g performance function, limit state function G shear modulus h pipe wall thickness I n integration constant i 0 initial dissolution current density at bare metal surface i 0 initial value of decay curve of electric current on newly exposed surface I 1 (x) modified Bessel function of order 1 J J-integral, the strain energy rate J 1 Bessel function of first kind and of order 1 J IC mode I fracture toughness J R fracture energy, fracture toughness k constant K General expression for stress intensity factor K I mode I stress intensity factor K II mode II stress intensity factor K III mode III stress intensity factor mean I K mode I stress intensity factor corresponding to mean stress mean II K mode II stress intensity factor corresponding to mean stress mean III K mode III stress intensity factor corresponding to mean stress K Ia lower bound crack arrest value for stress intensity factor K IC mode I fracture toughness K I,max Maximum value of mode I stress intensity factor K I,min Minimum value of mode I stress intensity factor K th crack propagation threshold value for stress intensity factor K * strain hardening coefficient determined near failure L r , K r non-dimensional variables in failure assessment diagram approach l 0 initial crack length m total number of cells in stratified sampling m decay constant of current at newly exposed material surface m dimensionless constant that is approximately 1.0 for plane stress and 2.0 for plane strain 14 m, m y exponent characterising material and environment in fatigue crack growth rate equation M atomic weight of the metal M(t) time dependent renewal function m1 shaping exponent M, N material dependent factors N sample size, number of load cycles n strain hardening exponent n exponent characterising material and environment in stress corro- sion crack growth rate equation n power law exponent for non-linear behaviour in general corrosion n dimensionless work hardening exponent in Ramberg-Osgood equation n numerical constant correlated with the degree of sensitisation, water conductivity, corrosion potential n exponent in constitutive equation for creep n total number of strain increments , , i FS a a N , > the number of times Monte Carlo simulations have reached or exceeded the selected degradation/failure state in terms of crack depth a N(t) time dependent point process describing the occurrence of shocks N I number of load cycles for crack initiation phase N II number of load cycles for crack propagation phase N air,RT fatigue life in air at room temperature n d number of dislocations contributing to formation of slip band N f number of failures, number of load cycles to failure n i number of cycles of operation under certain stress amplitude N i total number of cycles that would produce a failure at certain stress level N j number of samples taken from the j:th cell in stratified sampling N i ´ modified life at some considered loading level N E total accumulated life N slip number of active slip bands N water fatigue life in water at service temperature n * cyclic strain hardening exponent determined near failure O * transformed level of dissolved oxygen P transition probability matrix p internal pressure 15 P, P i probability P f failure probability p f rupture failure probability for pipe section p f0 conditional fracture probability for a given crack length , , t p tj i FS , , failure probability corresponding to selected degradation/failure state p i initial probabilities of various damage states P j probability that an initial crack exists in the j:th cell in stratified sampling P LM Larson-Miller parameter g Q thermal activation energy for crack growth R total risk R reliability R stress ratio R universal gas constant R strength r polar coordinate with origin at the crack tip R(t) time dependent component reliability R eff effective stress ratio R i individual risk R i inner radius R max maximum allowable stress ratio RT NDT reference nil-ductility temperature of the material 0 NDT RT initial NDT temperature for unirradiated material R y extent of current yield zone S stress level, remote stress resulting from the applied load S stress S strain energy density factor S * transformed sulphur content t wall thickness, plate thickness t time in service since start of operation, time since start of corrosion process T temperature T expected total time in service t time in general t 0 short time constant 16 t 0 time constant in pitting corrosion model t 0 starting time of current decay at newly exposed material surface t 1 transition time from short time to long time behaviour t 2 transition time from primary to secondary creep t F time of rupture t i time spent at condition i T i components of traction vector T k maintenance times t r time to rupture t ri time to rupture at condition i T ref absolute reference temperature used to normalise data T * transformed temperature TK 41J 41 J impact energy transition temperature 6J 5 TK 56 J impact energy transition temperature mm 9 . 0 TK 0.9 mm lateral expansion transition temperature T, C, G, k2, k3, s material parameters which are function of temperature, environ- ment and the metallurgical state of the material U factor depending primarily on the stress ratio U en cumulative fatigue usage factor u i components of displacement vector U i partial fatigue usage factor i u displacement rate components V pitting corrosion propagation rate V 0 initial pitting corrosion propagation rate w strain density w stress work rate (power) density x certain value of the random variable x relative initial crack depth, relative initial crack length X ~ Median value X(t) time dependent applied stress X, Y random variable X c continuous load x i exponent parameter related to considered loading level y constant value for strength Y(t) time dependent strength of a component 17 Z response function z number of electrons involved in reaction rate z charge change at electrolysis Z 0 limiting value of response function Z OL plastic zone diameter Greek symbols o ageing acceleration rate o dimensionless constant o crack growth rate coefficient o Parameter describing the degree of interaction between fatigue induced crack growth and creep induced cavity growth | first-order reliability index, geometrical reliability index, Hasofer- Lind reliability index | cyclic hardening rate | i frequency of load cycles at some considered loading level | exponent in stress corrosion cracking rate equation | aspect ratio of fabrication crack |, ì dimensionless constants in stress corrosion cracking rate equation o crack tip opening displacement A total displacement rate t A time dependent creep displacement Ac strain range Ac p Parameter related to plastic strain Ac p current value of plastic strain p c absolute value of plastic strain rate A¸ shear strain range A i damage fraction AJ cyclic J-integral range AK eff effective stress intensity factor range AK eq equivalent stress intensity factor range AK th fatigue threshold in terms of stress intensity factor range AK I (c) strain intensity factor range ART NDT shift in reference nil-ductility temperature of the material 18 AS cyclic strain energy density factor range Ao stress range ATK 41J shift in 41 J impact energy transition temperature ATK 56J shift in 56 J impact energy transition temperature mm 9 . 0 TK A shift in 0.9 mm lateral expansion transition temperature c strain rate ct c crack tip strain rate c f strain caused by break of protective film at crack tip - f c failure strain as a fraction c i strain accrued at condition i c max maximum strain c min minimum strain c ri rupture strain at condition i - transformed strain rate ij c ~ dimensionless function of work hardening exponent and angular coordinate around crack tip u acceleration factor for fatigue induced crack growth rate equation u(u) standard normal distribution function I contour around crack tip k i principal curvatures of the limit state at the minimum distance point ì, | m coefficients in probabilistic equation for aspect ratio of fabrica- tion cracks ì(t) time dependent component failure rate ì 0 constant random failure rate A(t) time dependent integral of the intensity, cumulative failure function µ i mean value point µ general mean value µ y scale parameter v Poisson’s coefficient t 0 initial crack degradation/size state t k crack degradation/size state at time k u location parameter, angular coordinate around crack tip u angle between the direction of the slip plane and tensile stress 19 u inclination angle of the crack u 0 inclination angle of the crack at the direction of minimum strain energy density µ material density µ d dislocation density µ m density of the metal o standard deviation o 0 reference stress (most often yield strength) o AXIAL axial membrane stress o ij components of stress tensor/matrix ij o ~ dimensionless function of work hardening exponent and angular coordinate around crack tip o y yield strength, average local yield strength o y shape parameter y o local yield strength 20 List of abbreviations AGR advanced gas-cooled reactor AIS adaptive importance sampling AMV advanced mean value method ASME American Society of Mechanical Engineers BWR boiling water reactor CANDU Canadian deuterium uranium reactor CCDP conditional core damage probability CCF common cause failure CDF conditional core damage CDF crack driving force CLERP conditional large early release probability CTOD crack tip opening displacement DFM deterministic fracture mechanics DNV Det Norske Veritas EAC environmentally assisted cracking EDF Electricité de France ENIQ European Network for Inspection and Qualification EPFM elastic-plastic fracture mechanics EPRI Electric Power Research Institute ETA event tree analysis FAC flow accelerated corrosion FAD failure assessment diagram FBR fast breeder reactor FEM finite element method FITNET European Fitness For Service Network FMEA failure mode and effect analysis FMECA failure mode effect and criticality analysis FORM first-order second-moment method FPI fast probability integration FTA fault tree analysis GCR gas cooled reactor GUI graphical user interface HAZ heat affected zone 21 HAZOP hazard and operability study HIDA HIgh temperature Defect Assessment procedure HSE Health and Safety Executive HWR heavy water reactor IAEA International Atomic Energy Agency IASCC irradiation assisted stress corrosion cracking IGSCC intergranular stress corrosion cracking INEEL Idaho National Engineering and Environmental Laboratory IS importance sampling ISI in-service inspection IWM Fraunhofer-Institut für Werkstoffmechanik JRC Joint Research Centre JSME Japan Society of Mechanical Engineers LEFM linear-elastic fracture mechanics LHS Latin hypercube simulation LLNL Lawrence Livermore National Laboratory LOCA loss of coolant accident LWR light water reactor MCS Monte Carlo simulation MIC microbiologically influenced corrosion MPP most probable point MV mean value method NDE non-destructive examination NDT non-destructive testing NDT nil-ductility temperature NPP nuclear power plant OECD Organisation for Economic Co-operation and Development OPDE OECD Piping Failure Data Exchange ORNL Oak Ridge National Laboratory PDF probability density function PFM probabilistic fracture mechanics POD probability of detection PRA Probabilistic Risk Assessment PSA Probabilistic Safety Assessment PTS pressurised thermal shock 22 PWR pressurised water reactor PWROG Pressurized-Water Reactor Owners Group PWSCC primary water stress corrosion cracking QRA quantitative risk analysis RAW risk achievement worth RBI risk based inspection RCM reliability centered maintenance RDF risk reduction factor RIF risk increase factor RI-ISI risk informed in-service inspection RPV reactor pressure vessel RRW risk reduction worth RSM response surface method SA sensitivity analysis SAIC Science Applications International Corporation SCC stress corrosion cracking SINTAP Structural Integrity Assessment Procedures for European Industry SKI Swedish Nuclear Inspectorate SMAW shielded metal arc welding SORM second-order reliability method SSC systems, structures and components SSM Swedish Radiation Safety Authority STUK The Finnish Radiation and Nuclear Safety Authority SwRI Southwest Research Institute TGSCC transgranular stress corrosion cracking TVO Teollisuuden Voima Oyj USNRC U.S. Nuclear Regulatory Commission VTT Technical Research Centre of Finland WRS weld residual stress WWER water-cooled water-moderated energy reactor 1. Objective of the thesis 23 1. Objective of the thesis Several objectives were set for this thesis. One objective is to collect both the methods and background theories for evaluation of operational lifetime, reliabil- ity and risk of structural systems and components of power plants. This was pursued by making an extensive literature survey. Another objective is to collect information concerning the relevant existing probabilistic degradation and risk analysis tools, with which one can assess the remaining lifetime and structural integrity of systems/components of power plants. The scope and characteristics of these tools are described. The last objective is to provide an application ex- ample. This was carried out by making a failure probability analysis to a repre- sentative set of nuclear power plant piping components. 2. Introduction 24 2. Introduction Structural lifetime, reliability and risk analysis methods and applications for structural systems and components of conventional and nuclear power plants are considered in this thesis. The emphasis here is on the latter power plant type. Lifetime analyses of structural systems and components necessitate a thorough knowledge of their structural properties, loads, supports, the relevant ageing mechanisms and prevailing environmental conditions. Due to distributed nature of many of these properties or phenomena, probabilistic modelling methods are deemed suitable for the lifetime analyses. Failure probabilities combined with knowledge of system or component failure consequences allow performing risk analyses. An overview of conventional and nuclear power plant environments and a de- scription of the various degradation mechanisms damaging the power plant sys- tems and components are presented first, see Chapter 3. Ageing degradation mechanisms may be divided into two groups based on the resulting failure modes: (1) those that may cause rupture, and (2) those that may cause cracking. Low-cycle fatigue, high-cycle thermal fatigue, and stress corrosion cracking (SCC) of components may cause cracking. Radiation embrittlement, thermal ageing of cast stainless steel components and vibration fatigue of small diameter piping may cause rupture. The mechanisms that have potential to cause rupture are likely to have significantly more risk impact [1]. A description of deterministic structural analysis methods, where the emphasis is on structural mechanics and fracture mechanics based analysis methods, is presented in Chapter 4. Other associated fields of science are material science, chemistry and radiation physics. The disadvantages of the deterministic analysis methods are discussed. Probabilistic analysis methods are dealt with after that, see Chapter 5. The probabilistic analysis methods are often based on deterministic analysis methods 2. Introduction 25 with the modification that one or more of the model parameters are considered as probabilistically distributed. The presented probabilistic analysis procedures include first-order second moment (FORM) methods, second-order reliability methods (SORM), mean value methods, Monte Carlo simulation (MCS) and importance sampling. A brief description of risk analysis methods is presented then, see Chapter 6. The engineering definition of risk is now accepted as being the product of the likelihood and the consequence of an event [2]. There exist nowadays a number of risk analysis procedures. The considered risk analysis procedures include failure mode and effect analysis (FMEA), fault tree analysis (FTA), event tree analysis (ETA) and expert opinion. Risk based regulations and standards as well as concerns related to risk assessment are considered also. The trend towards a risk based approach in power plants, and especially in nuclear power plants (NPPs), is being supported by extensive plant operating experience, improved understanding of material degradation mechanisms and the availability of fit- ness-for-service assessment procedures [3]. At the same time, developments in non-destructive testing (NDT) technology have increased the scope and efficien- cy of examinations that can be undertaken. Inspection trials have produced a greater appreciation of the limits of NDT performance and reliability. Deterministic and probabilistic computational modelling methods for various ageing degradation mechanisms damaging power plant components are present- ed after this, see Chapters 7 and 8. These methods are needed in the lifetime analyses of structural systems and components of power plants. In general, the lifetime analyses necessitate a thorough knowledge of structural properties, loads, supports, the relevant degradation mechanisms and prevailing environ- mental conditions. The nature of degradation models of structural sys- tems/components can be deterministic, probabilistic or a combination of these two types. Degradation models of all these kinds are presented. Deterministic fracture mechanics is an example of deterministic ageing modelling approaches. Statistic failure models, which are based on observed failure statistics, are an example of probabilistic ageing modelling approach. Probabilistic fracture me- chanics, which is discussed in more detail, is a modelling approach which com- bines both deterministic and probabilistic features. In practise, the degradation analyses are usually performed with suitable analysis tools, i.e. with analysis codes. A brief description of some important risk analysis applications is presented in Chapter 9. This includes e.g. probabilistic risk/safety assessment (PRA/PSA) 2. Introduction 26 and risk informed in-service inspection (RI-ISI). In addition to the energy indus- try, PRA/PSA has emerged as an increasingly popular analysis tool in several other industries as well, especially during the last two decades [1]. PRA/PSA is a systematic and comprehensive methodology to evaluate risks associated with every life-cycle aspect of a complex engineered technological entity or compo- nent from concept definition, through design, construction and operation, and up to removal from service. In most countries, the method is referred to as PSA. In the United States, the method is referred to as PRA. Recently The Finnish Radia- tion and Nuclear Safety Authority (STUK) has adviced/announced that also in Finland this method should be referred to as PRA. Despite having two names, the technique is practically the same. RI-ISI involves the planning of an inspec- tion on the basis of the information obtained from a risk analysis of the equip- ment. The purpose of the risk analysis is to identify the potential degradation mechanisms and threats to the integrity of the equipment and to assess the con- sequences and risks of failure. The inspection plan can then target the high risk equipment and be designed to detect potential degradation before fitness-for- service could be threatened [3]. Component ageing analyses are discussed after this, see Chapter 10. Ageing of various systems, structures and components (SSCs) affect the safety of power plants. The objective of component ageing analyses is to identify the degradation mechanisms of components and the increase in failure occurrences, to assess the remaining lifetime of components and to find suitable means to prevent or miti- gate the effects of ageing. A selection of various probabilistic system/component degradation and risk analysis tools is presented then, see Chapter 11. These are divided into compo- nent and system analysis applications. Another difference in approaches of these programs is that one part of them contains models of one or more specific deg- radation mechanisms, while the other part consists of general structural reliabil- ity analysis codes. Computational application of probabilistic failure and lifetime analyses to a relatively small but representative set of NPP piping components is presented after that, see Chapter 12. In addition to data concerning structural properties, primary and secondary loads, supports and environment, also data concerning relevant degradation mechanisms and failure modes are needed in the probabilis- tic failure and lifetime analyses of NPP piping components. Essential sources for piping degradation and failure data are international and plant specific databases. In the probabilistic failure analysis computations two analysis codes were used, 2. Introduction 27 those being probabilistic VTTBESIT, developed by VTT and Fraunhofer-Institut für Werkstoffmechanik (IWM, Germany), and PIFRAP developed by Det Norske Veritas (DNV), respectively. The thesis ends with a summary and suggestions for further research. 3. Degradation mechanisms concerning power plant components 28 3. Degradation mechanisms concerning power plant components 3.1 Introduction Ageing degradation in power plants should be managed by ensuring that the design functions remain available throughout the service life of the plant. From the safety perspective, this implies that ageing degradation of SSCs important to safety remains within acceptable limits, and that procedures and personnel train- ing remain adapted. Unchecked, ageing degradation has the potential to reduce the safety of operating power plants. The technical definition of ageing given by the International Atomic Energy Agency (IAEA) is the following [4]: Ageing is the continuous time dependent degradation of materials due to normal service conditions, which include normal operation and transient conditions. In this thesis degradation is used as an equiv- alent expression to material ageing. Ageing degradation can be observed in a variety of changes in physical prop- erties of metals, concrete and other materials in a power plant. These materials may undergo changes in their dimensions, ductility, fatigue capacity and me- chanical or dielectric strength. Ageing degradation is caused by a variety of age- ing mechanisms, physical or chemical processes such as fatigue, cracking, em- brittlement, wear, erosion, corrosion and oxidation. These ageing mechanisms act on SSCs due to a challenging environment with relatively high heat and pres- sure, radiation, reactive chemicals and synergistic effects. Some operating prac- tices, such as power plant cycling (i.e., changing power output) and equipment testing, can also create stresses for plant SSCs [5]. Some degradation mecha- nisms can also act simultaneously, like for example fatigue and creep. There is a fairly limited set of degradation mechanisms, a large commonality in used materials, and fairly similar operating conditions. However, due to the 3. Degradation mechanisms concerning power plant components 29 diversity in plant design, construction and used materials, operating conditions and histories as well as maintenance practices, the specific effects of ageing, although similar, are unique for each plant. Even near twin units at the same site can have substantial differences in the remaining life of major SSCs, based on subtle design or material differences and operating histories [5]. In general, ageing in power plants can be taken to mean evolution of person- nel and procedure adequacy and evolution of material or equipment properties, which, after a certain time may not be compatible with the required safety mar- gins, or with an economic functioning of the plant. Repair or replacement of components, as well as change in service conditions for a better compatibility with component reduced capacities are possible [6]. In parallel, safety requirements may increase with time, following the evolu- tion of the public acceptance, and costs of other energy sources may decrease, limiting as a result the economic interest of continued operation. The plant life will then be the result of the consideration of ageing in a changing regulatory, political, technical and economic context. In the context of stable safety requirements, the evolution of the safety mar- gins for a given component can be illustrated as shown in Figure 3.1-1. This figure, which is given for illustrative purpose only and does not take into ac- count the possibility of component replacement, shows that a better evaluation of the applied loading, as well as a better knowledge of component performance can result in a modification of the demonstrated safety margin, and consequently of the potential component life. A reduction of the applied loading or an increase of component performance would result in an increase of the safety margin. Figure 3.1-2 illustrates the combined influence of the above considerations with a possible evolution of safety requirements. 3. Degradation mechanisms concerning power plant components 30 Figure 3.1-1. Illustration of possible component margin evolution during service, from ref. [8]. Figure 3.1-2. Illustration of combined evolution of component margins and safety re- quirements, from ref. [9]. As shown in Figures 3.1-1 and 3.1-2, lifetime extension is not only obtainable through repair or replacement of components, but also through a better use and a better evaluation of the real evolution of component performance, for example a better prediction of material properties evolution, a better evaluation of existing 3. Degradation mechanisms concerning power plant components 31 defects and mechanical behaviour of real defects, or a better knowledge of oper- ating conditions. Power plant structures generally have substantial safety margins when proper- ly designed and constructed. However, the available margins for degraded struc- tures are not well known. In addition, age related degradation may affect the dynamic properties, structural response, structural resistance/capacity, failure mode and location of failure initiation. A better understanding of the effect of ageing degradation on structures and components, especially passive compo- nents, is needed to ensure that the current licensing basis is maintained under all loading conditions [10]. This thesis considers mainly the ageing degradation of metallic SSCs. De- pending of the metal or alloy in question and the intensity of the external loading as well as environmental impact, there may be a high potential for the degrada- tion. The processes of degradation can result in material and geometry changes of which the most important are [4]: - reduced toughness (embrittlement), - cracking, - swelling, - thinning, - denting and - pitting. 3.2 Overview of conventional and nuclear power plant environments The principal route for producing electricity in power plants is the conversion of mechanical energy of rotation into electrical energy using a generator. The large generators used by electric utilities employ a shaft comprising the magnetic field (rotor) which rotates inside a stationary electric field containing conducting wires (stator). Rotation of the shaft is achieved by coupling it to a turbine in which the kinetic energy of a moving fluid is converted into mechanical energy of rotation. The working fluid can be wind, water, steam or combustion gases. The most common turbine type is steam turbine, which employs steam produced from burning fossil fuels in a boiler or from the heat produced by atomic reac- tions inside a nuclear reactor [11]. Due to the nature and requirements of the energy production methods, the power plant environments are characterised by high system temperatures and 3. Degradation mechanisms concerning power plant components 32 pressures. Many power plant components are also exposed to large temperature and pressure variations, which can in certain cases occur quite quickly. The chemistry of the working fluid can affect the structural integrity of the power plant components as well. Besides the operational conditions, the power plants are shutdown at specific intervals for maintenance and inspection outages, after which the operation is started again. During shutdown the system temperature and pressure are lowered, usually to atmospheric conditions, and during start-up they are correspondingly raised to the operational level. Besides the planned and usually yearly outages, the power plants have to be shutdown when certain acci- dents, e.g. malfunctions or structural failures, take place. These accidents are, however, relatively rare, and the power plants are also designed to sustain their safety in the case of their occurrence, e.g. with several auxiliary systems. The pressure, temperature and specific volume of the steam at various stages in coal fired power plants are illustrated in Figure 3.2-1. In general the tempera- tures in conventional power plants vary from atmospheric conditions during shutdown to nearly 1000 °C in the combustion bed of some boiler types under operational conditions. 3. Degradation mechanisms concerning power plant components 33 Figure 3.2-1. Relationships between steam pressure, temperature and specific volume in the various components of a large steam power plant, from ref. [13]. 1000 ksi corre- sponds to approximately 7 MPa and 1000 °F to approximately 540 °C. Two major factors that influence the temperature and pressure levels in conven- tional power plants are the boiler type and used fuel. The most common boiler types of conventional power plants are [12]: - grate-fired furnace, - bubbling fluidised bed combustion, - circulating fluidised bed combustion. 3. Degradation mechanisms concerning power plant components 34 Some typical fuels used in conventional power plants are [12]: coal, oil, gas, waste (industrial, municipal), black liquor, peat, biomass and a combination of biomass and e.g. coal, oil or peat. In grate-fired boilers, the fuel is loaded on the fire grate and combustion air is fed from below the grate to the primary combustion chamber. In the secondary chamber gases evaporated from the fuel bed are incinerated. Pyrolysis of volatile mat- ters starts at 150–200 °C, is fastest at 250–500 °C and finishes around 800 °C [12]. In fluidised bed combustion, the bed is a mixture of inert material, like sand, and coarse fuel particles. A continuous stream of air is fed through the bed to produce turbulence in the bed due to which particles in the bed start to move and the bed becomes fluidised. In bubbling fluidised bed furnaces the fluidising par- ticles remain under a defined level. The bed temperature is maintained relatively low, between 800–900 °C, in order to keep the ash solid and the bed material in the fluidised state. In the circulating system higher gas stream and finer fluidised particles are used and solid fluidised particles drift throughout the combustor with the gas stream. Cyclone is used to separate flue gas from escaped particles, which are continuously circulated back to the bed. Fuel is normally mixed with matter returning from a cyclone [12]. The quality of the working fluid in conventional power plants is monitored and it is attempted to maintain such that the concentrations of various impurities stay low enough. Iron and copper concentrations in the water/steam are an indi- cator of the efficiency of conditioning. Their values give information about the corrosion/deposition processes in the water/steam cycle, and respectively in the boiler and turbine. Silica concentrations in some boilers may not exceed certain values due to the specification for turbine operation. When operating with fully demineralized water, the silica content will be far below the specified level [14]. The typical pressure boundary materials in conventional power plants are fer- ritic, martensitic, austenitic and Ni-based alloys. For instance, the material of the steam tubes is typically austenitic stainless steel [11]. Most NPPs use steam cycles that differ little from those of conventional power plants except for the source of heat for the steam generator and for steam supply conditions [18]. According to the reactor type the NPPs are divided to two categories: - light water reactor (LWR), - heavy water reactor (HWR). 3. Degradation mechanisms concerning power plant components 35 The LWRs are further divided to the following categories [21]: - advanced gas-cooled reactor (AGR), - boiling water reactor (BWR), - fast breeder reactor (FBR), - gas cooled reactor (GCR), - pressurised water reactor (PWR), - water-cooled water-moderated energy reactor (WWER). The LWRs are designed so that the core is both moderated and cooled by highly purified water and the fuel is enriched uranium that fissions with thermal neu- trons [18]. The most common LWR types are the BWR and PWR. The BWRs operate at a pressure that allows boiling of the coolant water adja- cent to the fuel elements inside the reactor core, producing slightly radioactive steam that passes directly to the steam turbines. The radioactivity in the steam, however, has a half-life of only a few seconds. The propagation of radioactivity to the turbine-feedwater system is virtually non-existent [18]. BWR plant operating pressures are typically from 6.90 to 7.24 MPa (1000 to 1050 psi), whereas the operating temperatures range from 282 to 288 °C (540 to 550 °F), respectively [19]. The PWR is currently the predominant nuclear reactor type in the world. The water in the PWR is in slightly higher temperature as in the BWR but is at a higher pressure, so that the reactor coolant remains liquid through the entire reactor coolant loop. Another difference between these two reactor types is that PWRs have an additional separate water loop that isolates the turbine steam loop from the reactor coolant [18]. The Table 3.2-1 are shows typical operational temperatures and pressures for common PWR plant types. Table 3.2-1. Typical operational temperatures for common PWR plant types, from ref. [20]. Parameter/ Reactor type Westinghouse (4-loop plant) Framatome Siemens WWER 440 WWER 1000 EPR Cold leg temperature, °C 292.5 292 291.3 270 290 295.5 Hot leg temperature, °C 325.5 329.5 326.1 300 325 328 Primary circuit pressure, MPa 15.51 15.51 15.2–15.8 12.2–12.5 15.7 15.5 3. Degradation mechanisms concerning power plant components 36 The best known HWR is the Canadian Deuterium Uranium (CANDU) reactor. It utilises natural uranium as a fuel and heavy water as a moderator and coolant. These reactors produce a substantial saving due to the absence of fuel enrich- ment costs, but a large chemical plant is required to supply the quantities of heavy water required [18]. As the vast majority of the operating NPPs are LWRs, the description of fur- ther details of HWRs is omitted here. As for local loads, one important issue is welding process induced residual stresses. They are locally confined to the weld regions and can be relatively high, e.g. of the scale of yield stress in tension on and near the inner surface. Thus they have to be taken into account in the structural integrity/safety anal- yses. The resulting residual stress state in a welded component is determined by welding related parameters, material properties and geometrical constraints. The first issue refers to the local shrinkage, quench and phase transformations caused by the localised thermal cycle. While the latter issue concerns the constraining effect of the surrounding structure, and in case of dissimilar metal welds the unbalance in material properties. It is generally known that there are three prima- ry sources for weld residual stress redistribution or relaxation, which are the following [22]: transient mechanical loads, thermal treatments and irradiation exposure. Mainly weld residual stresses have to be taken into account in NPP environments, whereas due to relatively high operational temperatures of con- ventional power plants, they can be considered as mostly relieved and thus not necessary to be included in the structural integrity/safety considerations. Even though the LWRs are moderated and cooled by highly purified water, the water still contains chemicals, like boric acid, and impurities, like sulphate, chloride and peroxides that can cause corrosion. Suspended particulate material can also accelerate erosion processes, which can lead to wall thinning. In addi- tion to this the reactor pressure vessels (RPVs) are exposed to degrading effect of the products from the fission reactions [23, 24]. The typical pressure boundary materials in the BWRs and the PWRs are car- bon steels and stainless steels. For instance, the material of the steam tubes is typically austenitic stainless steel [24]. 3.3 Fatigue Fatigue is caused by periodic application of stresses by mechanical or thermal loading. The metal subjected to fluctuating stress will fail at stresses much lower 3. Degradation mechanisms concerning power plant components 37 than those required to cause fracture in a single application of load [11]. The key parameters are the range of stress variation and the number of its occurrences [6]. The fatigue process can be roughly divided into four stages: cyclic harden- ing/softening, crack nucleation, crack propagation, and fracture [7]. Low-cycle fatigue, usually induced by mechanical and thermal loads is distin- guished from high-cycle fatigue, mainly associated with vibration or high num- ber of small thermal fluctuations. Although, there is no distinctive limit between these two types of fatigue, the traditional approach is to classify failures occur- ring above 10 000 cycles as high-cycle fatigue and those occurring below that value as low-cycle fatigue. An important distinction between low-cycle fatigue and high-cycle fatigue is that in high-cycle fatigue most of the fatigue life is spent in crack initiation, whereas in low-cycle fatigue most of the life is spent in crack propagation, because cracks are found to initiate within 3 to 10 % of the fatigue life [11]. An example of fatigue fracture of a structural component is presented in Figure 3.3-1. The cyclic nature of the fatigue crack growth through the component section can be recognised in the patterned fracture surface shown in the figure. Figure 3.3-1. Fatigue damage of a shaft component, from ref. [3]. 3. Degradation mechanisms concerning power plant components 38 Fatigue failures occur in many different forms. Mere fluctuations in externally applied stresses or strains result in mechanical fatigue. Cyclic loads acting in association with high temperatures cause creep-fatigue. When the temperature of the cyclically loaded component also fluctuates, thermo-mechanical fatigue is induced. Recurring loads imposed in the presence of a chemically aggressive or embrittling environment give rise to corrosion-fatigue. The repeated application of loads in conjunction with rolling contact between materials produces rolling contact fatigue, while fretting fatigue occurs as a result of pulsating stresses along with oscillatory relative motion and frictional sliding between surfaces [25]. In NPP environments fatigue due to temperature fluctuation only, i.e. ther- mal fatigue, can also in certain cases be identified. All in all, the majority of damage/failures in power plant components can be attributed to one of the above fatigue processes. Such failures take place under the influence of cyclic loads with peak values being considerably lower than the allowed loads according to static fracture analyses. One characteristic of fatigue in metals is work hardening under reversed load- ing conditions. With continued cyclic loading, the rate of hardening progressive- ly diminishes and a quasi-steady state of deformation, known as saturation, is reached. Once saturation occurs, the variation of the resolved shear stress with the resolved shear strain is not altered by further load cycles and the stress-strain hysteresis loop develops a stable configuration. Obviously work hardening be- haviour is a metal specific phenomenon [25]. The lowering of the yield strength upon reloading that follows unloading, even if the reloading is in the same direction as the original loading, is called the Bauschinger effect. This phenomenon decreases with continued cycling. Un- symmetric cycles of stress between prescribed limits will cause progressive in- crease in the strain response or “ratchetting” in the direction of the mean stress. Typically, transient ratchetting is followed by stabilisation (zero ratchet strain) for low mean stresses, while a constant increase in the accumulated ratchet strain is observed at high mean stresses [25]. The damage process leading to the formation of fatigue cracks is initiated by locally limited plastic deformation in the micro scale range of individiual crys- tallites at low nominant net stresses. Depending on the lattice orientation and on the dislocation configuration, slip bands can be activated causing irriversible slip processes, see Figure 3.3-2. The repeated loading leads to micro-scale damage accumulation. After a saturation of energy accumulation has occurred, at which state no further slip processes can take place in the crystallite, small micro- 3. Degradation mechanisms concerning power plant components 39 cracks form. With an increasing number of load cycles these micro-cracks start to grow and additional regions undergo the process of micro-scale damage. As the crack grows to macroscopic size the stress intensity at the crack tip usually increases so that the crack growth is accelerated until the remaining ligament can no longer bear the stress and complete failure occurs [4]. Three fundamental aspects related to the fatigue crack initiation are: the sig- nificance of the free material surface, the irreversibility of cyclic slip, and envi- ronmental effects on micro-crack initiation. The micro-cracks usually initiate at the free surface of the material. The restraint on cyclic slip is lower than at inside the material because of the free surface at one side of the surface material. Fur- thermore, micro-cracks initiate more easily in slip bands with slip displacements perpendicular to the material surface [26], which seems to be logical when look- ing at Figure 3.3-2. There are two reasons for irreversibility. One argument is that (cyclic) strain hardening occurs which implies that all dislocations do not return to their original position. Another important aspect is the interaction with the environment [27]. Figure 3.3-2. Geometry of slip at the material surface, from ref. [28]. The cyclic growth of a fatigue crack can be seen in the formation of striations, with one striation forming per load cycle, see Figure 3.3-3. The striations are supposed to be remainders of microplastic deformations at the crack tip, but the mechanism can be different for different materials. Because of micro-plasticity at the crack tip and the crack extension mechanism in a cycle, it should be ex- pected that the profile of striations depends on the material type. Striations have not been observed in all materials, at least not equally clearly. Moreover, the visibility of striations also depends on the severity of load cycles. Striations have also shown that the crack front is a curved line and the crack tip is rounded [27]. 3. Degradation mechanisms concerning power plant components 40 Figure 3.3-3. Correspondence between striations and load cycles during fatigue crack growth in an Al-alloy specimen, from ref. [27]. The fatigue life of a structural component under cyclic loading can be consid- ered to consist of two phases, which are the crack initiation life followed by a crack growth period until failure. This can be represented in a block diagram, see Figure 3.3-4. Work hardening and other continuum type fatigue phenomena affect the component throughout the fatigue life. Figure 3.3-4. The different phases of the formation and growth of a fatigue crack, from ref. [27]. The fatigue limit is an important material property from an engineering point of view. It can be formally defined as a stress amplitude for which the fatigue life becomes infinite in view of the asymptotic character of the stress versus the number of load cycles (S-N) curve, see Figure 3.3-5. As fatigue tests must be terminated after a long testing time, it appears to be more logical to define a fatigue limit as the highest stress amplitude for which failure does not occur after 3. Degradation mechanisms concerning power plant components 41 high numbers of load cycles. A more physically based definition of the fatigue limit could be defined as the threshold stress amplitude to take care of micro- crack nucleation and subsequent growth to a macro-crack [27]. Figure 3.3-5. Outlook of a typical stress versus the number of load cycles (S-N) fatigue testing curve, from ref. [25]. Components/locations in conventional power plants which have experienced fatigue degradation include [29]: headers, downcomers, main steam piping, hot reheater piping, tubing, ductwork, precipitator, drums and geometry discontinui- ties, such as nozzles, reducers, transitions and small fillet radii. Components/locations in NPPs which have experienced fatigue degradation include [1, 4]: - BWRs; RPV, feed water piping, main steam piping and recirculation pump, - PWRs; safe-ends, reactor coolant piping, feedwater piping and nozzles, - LWRs in general; RPV internals, - HWR RPVs; calandria vessel, pressure tubes and calandria tubes, - steam generator; vessel shell, tubes and grid, 3. Degradation mechanisms concerning power plant components 42 - steam and water piping vessels valves, - main coolant pump; casing and shaft impeller, - turbine; casing, blades and shaft, - condensor tubes, - geometry discontinuities in general; nozzles, reducers, transitions and small fillet radii. 3.4 Corrosion Corrosion is the degradation of a material by chemical or electrochemical reac- tion with its environment. Most of the technical corrosion processes, however, are electrochemical in nature and are accompanied with hydrogen production. Corrosion reduces the component wall thickness, either uniformly or locally. The corrosion phenomenon is governed by the so called corrosion system which consists of the metallic material and the corrosive medium (the environment) with all the participating matter that can influence the chemical and physical behaviour and the corrosion parameters. The variety of chemical and physical variables of environments and materials leads to a large number of types and appearance of corrosion [4, 15]. In the case of electrochemical processes, corrosion occurs in several steps. Metal ions dissolve in liquid electrolyte (anodic dissolution) and hydrogen is produced in the process. This is the process of material loss from the corroded component and of the creation of the corrosion products. The free hydrogen can metallurgically interact with the metal (by adsorption, absorption or interstitial bonding) and can produce secondary damage, e.g. hydrogen embrittlement [15, 16]. When mechanical stresses or strains are acting in addition to the corrosion im- pact, the anodic dissolution of the metal can be stimulated, protective oxide lay- ers can rupture or hydrogen absorption can be promoted. The combined action of a corrosive environment and mechanical loading can cause cracking even when no material degradation would occur under either the chemical or mechanical conditions alone [4, 15]. Corrosion can manifest itself in a number of forms, and there are generally ac- cepted categories of corrosion based on the appearance and electrochemical pro- cesses. The types of corrosion include (but are not limited to) [30]: 3. Degradation mechanisms concerning power plant components 43 - uniform attack, - galvanic (two-metal) corrosion, - crevice corrosion, - pitting, - intergranular corrosion, - erosion corrosion, and - environmental cracking. Corrosion without mechanical loading Corrosion without mechanical loading can be divided to uniform corrosion, local corrosion and selective corrosion. Uniform corrosion refers to a uniform attack over surfaces of the material and results in thinning of the material. Uniform corrosion rates vary with fluid oxy- gen content, temperature and flow rate. When a protective Magnetite layer forms to the metal surface under uniform corrosion, which occurs e.g. for all hot water and steam pipes of unalloyed or low alloyed ferritic steel, the corrosion rate de- creases considerably or may even stop. This process is not considered as an age- ing mechanism, since the related material loss is very small [4, 15]. In case of inhomogeneities at the metal surface and/or local differences in the electrochemical reactivity of the environment, the creation of local cells is pos- sible which results mostly in local corrosion attack. Localised corrosion includes pitting, crevice corrosion, etc. Pitting is commonly caused by the breakdown of the passive film on a metal in local areas, by species such as chlorides. Crevice corrosion results from local environment conditions in the restricted region of a crevice being different and more aggressive than the bulk environment [15, 16]. In selective corrosion, the attack is concentrated on distinct material phases, regions adjacent to the grain boundaries or specific alloying elements. The cor- rosion proceeds into the depth of the material without changing the overall ge- ometry of the component. Since the corrosion attack propagates along the grain boundaries, the material damage proceeds from the surface into the bulk accord- ing to a grain disintegration process. An example of selective corrosion is the intergranular corrosion of sensitised austenitic stainless steels and nickel alloys. Environments with liquid that can cause intergranular corrosion in practice can include, among others, oxygen, chlorides, sulphates and hydroxides [4, 15]. 3. Degradation mechanisms concerning power plant components 44 Stress corrosion cracking (SCC) SCC is a localised non-ductile failure which occurs only under the combination of three factors: tensile stress, aggressive environment and susceptible material. The SCC failure mode can be intergranular, IGSCC, or transgranular, TGSCC. In NPP, primary water stress corrosion cracking, PWSCC, and irradiation assist- ed stress corrosion cracking, IASCC, are also identified [4, 17]. An example of SCC is presented in Figure 3.4-1. For some combinations of metals and corro- sive environments even very low stresses are sufficient to cause SCC. The phase of first macroscopic cracking is often preceded by a long incubation phase [4]. Actual SCC mechanisms are not clearly known yet [31]. However, many dif- ferent mechanisms have been proposed to explain the synergistic interactions of stress and corrosion that occur at the crack tip, and there may be more than one process that causes SCC [32]. Generally, the mechanisms can be divided to an- odic dissolution mechanisms and cathodic hydrogen induced cracking mecha- nisms. In case of hydrogen induced SCC, the additional effect beyond typical SCC exists in an interaction of hydrogen with the metal lattice. In this case, the level of mechanical stresses is of minor importance compared with the reaction of hydrogen. A prerequisite for hydrogen induced SCC is a small uniform or local corrosion attack of the material or a reaction between reaction layers and the electrolyte with subsequent hydrogen production [4]. Figure 3.4-1. Microscopic detail of SCC in a steel component, from ref. [3]. 3. Degradation mechanisms concerning power plant components 45 SCC is a delayed failure process. That is, cracks initiate and propagate at a slow rate (for instance, 10E-9 to 10E-6 m/s) until the stresses in the remaining liga- ment of metal exceed the fracture strength. The sequence of events involved in the SCC process is usually divided into three stages [33]: 1. crack initiation, 2. steady state crack propagation, and 3. final failure. IGSCC is associated with a sensitised material, e.g. sensitised austenitic stainless steels are susceptible to IGSCC in an oxidising environment. Sensitisation of unstabilised austenitic stainless steels is characterised by a precipitation of a network of chromium carbides with depletion of chromium at the grain bounda- ries, making these boundaries vulnerable to corrosive attack. TGSCC is caused by aggressive chemical species, especially if coupled with oxygen and combined with high stresses. PWSCC is a form of IGSCC and is defined as intergranular cracking in primary water within specification limits. So PWSCC can occur without the presence of additional aggressive species. An example of PWSCC is IGSCC of Inconel 600 alloy in primary water. IASCC refers to intergranular cracking of materials exposed to ionising radiation. As with SCC, IASCC re- quires stress, aggressive environment and a susceptible material. However, in the case of IASCC, a normally non-susceptible material is rendered susceptible by exposure to neutron irradiation [4, 15]. Flow accelerated corrosion (FAC) FAC, also called corrosion/erosion mechanism, refers to the combined action of corrosion and erosion (i.e. the mechanical action of a fluid on a metal surface) [1, 6]. The corrosion resistance of unalloyed or low alloyed ferritic steels in wa- ter and steam depends on the formation of protective oxide layers. When these layers are removed, the corrosion attack on the metal surface can occur in the form of metal dissolution. The flow conditions shift the equilibrium between protection layer removal and formation which determines the material consump- tion. The severity of erosion varies with the material type, the fluid temperature, the fluid velocity, the oxygen content in the fluid and the component geometry. The result of FAC is an increased rate of attack on metal because of the relative movement between a corrosive environment and the metal surface. Carbon and low-alloy steels are susceptible to FAC [4, 15]. 3. Degradation mechanisms concerning power plant components 46 The FAC process is an extension of the generalised carbon steel corrosion process in stagnant water. In stagnant water, the carbon steel corrosion rate is low and decreases parabolically with time due to the formation of a protective oxide film at the surface. FAC takes place at low flow velocities and the corro- sion rate is constant. The difference between generalised corrosion and FAC is the effect of water flow at the oxide-feedwater interface [1]. Consequences of the corrosion/erosion mechanism are thinning of component walls until leakage or rupture occurs, depending on the total stress as well as the deposition of the eroded particles at locations where there is a low flow velocity (wastage) [4]. Microbiologically influenced corrosion (MIC) MIC is the accelerated corrosion of materials resulting from surface microbio- logical activity. MIC is characterised by the formation of microbial colonies and associated scale and debris on the surface of the metal. MIC affects carbon steels and, to a lesser extent, stainless steels and nickel alloys. Any buried system or system using untreated water is susceptible to MIC. The major factors increasing the growth of MIC are temperature, pressure, pH, water content and oxygen [15]. Components in conventional power plants which have experienced corrosion degradation in one or more of its various forms include [29]: headers, downcomers, tubing, ductwork, precipitator and drums. Components in NPPs which have experienced corrosion degradation in one or more of its various forms include [1, 4]: - BWRs; main steam pipe, safe ends, recirculation pipe and recirculation pumps, - PWRs; pressurizer, feedwater piping, nozzles and reactor coolant pump, - LWRs; RPV, RPV internals, control rod drive mechanisms, feedwater pipe and emergency diesel generators, - HWR RPVs; calandria vessel, - steam generator; vessel shell, tubes and grid, - steam and water piping vessels valves, - turbine; casing and blades, - condensor tubes. 3. Degradation mechanisms concerning power plant components 47 3.5 Creep Creep is the deformation that occurs over a period of time in a material subjected to a stress, even below the elastic limit. Creep reaches a significant level above 0.4 times the melting point of the material [4, 6]. Due to thermal activation, ma- terials can slowly and continuously deform even under constant stress and even- tually fail. Depending on the component, the final failure may be limited either by deformation or by fracture [11]. An example of creep crack in a piping com- ponent is presented in Figure 3.5-1. Figure 3.5-1. Creep induced crack that has propagated through a pipe wall, from ref. [3]. The propagation of creep is determined by several competing reactions, includ- ing [11]: - strain hardening, - softening processes; recovery, recrystallisation, strain softening and precipitate overageing, - damage processes; cavitation and cracking. Of the above mentioned reactions, strain hardening tends to decrease the creep rate, whereas the other reactions tend to increase the creep rate [11]. Basically, two processes occur in the micro-structure that affect the material properties [11, 4]: plastic deformation processes and changes in the microstructure. 3. Degradation mechanisms concerning power plant components 48 Both of these processes are time dependent and they show certain interactions. The changes in the micro-structure can be caused by high temperatures alone, since they are the consequence of thermally activated mechanisms [4]. Deformation due to creep is usually divided into three regimes; primary, sec- ondary and tertiary, see Figure 3.5-2. These regimes are controlled by different mechanisms. Components are usually designed to be operated only in the prima- ry and secondary regimes. The time dependent creep damage processes occur according to the following steps [11, 4]: - Creep damage in the primary and secondary regimes, in which the oc- curred damage is not irreversible. - Nucleation of creep pores near the end of the second regime. - Coagulation of the creep pores to form micro-cracks. - Crack growth due to creep. In this regime spontaneous failure can oc- cur when large sections of the material have already suffered from this process. - Creep rupture. With increasing exposure time and accumulation of mi- cro-structural damage the ability for creep deformation decreases and thus ruptures occur in the part that very low ductility. An aggressive environment can have an influence on the creep behaviour and the degree of damage resulting from it. In metallic materials, an internal oxida- tion process can occur and accelerate the creep process [4]. The initial creep strength depends on the initial strength of the material at the considered tempera- ture. If the thermally induced process (thermal ageing) leads to a decrease in strength, then more unfavourable conditions have to be expected concerning the long range creep behaviour [4, 11]. 3. Degradation mechanisms concerning power plant components 49 Figure 3.5-2. Schematic illustration of creep damage curve showing the three different creep regimes; primary, secondary and tertiary, from ref. [11]. Here c r is rupture strain and t r is rupture time, respectively. Components in conventional power plants which have experienced creep degra- dation include [29]: superheater and reheater headers, piping as well as super- heater and reheater tubing. Components in NPPs in which creep degradation has occurred include [1, 4]: RPV internals of PWRs and in HWR RPVs calandria vessel, pressure tubes and calandria tubes. 3.6 Irradiation embrittlement Materials exposed to neutron irradiation undergo changes in microstructure and properties. The extent of the changes depends on the material, neutron flux, flux spectrum, fluence and irradiation temperature. In addition to the lattice defects caused by the interaction of fast neutrons with the material atoms, helium can be produced as a result of nuclear reactions. This damage mechanism occurs at neutron fluences above 10E+22 1/cm 2 and temperatures usually above 400 °C. Helium diffuses easily under elevated temperatures and forms voids [4, 6]. The effect of neutron irradiation on metals is mainly to increase the yield and ultimate strength and to reduce the fracture toughness. Helium production does 3. Degradation mechanisms concerning power plant components 50 not only lead to changes in material properties but is also combined with an in- crease in volume (swelling) [4]. Fracture toughness is the property that governs the material resistance to fast fracture, which for ferritic steels is small at low temperature where the material behaves in a brittle manner, and increases with the temperature until the material becomes ductile. It is generally accepted that the effect of irradiation is to shift this fracture toughness curve to higher temper- atures, the shape of the curve being only slightly affected with a small decrease of the upper shelf toughness in the ductile regime [4, 6]. The irradiation degradation sensitivity of the materials depends among other things on [4]: material type, chemical composition, heat treatment and initial mechanical properties. For steels the neutron irradiation is considered to cause significant ageing degradation only for neutron fluences above 10E+17 1/cm 2 . Irradiation embrittlement is the major degradation mechanism associated with the ageing of RPVs of PWRs. It is less important for BWR RPVs because of the lower fluence experienced in the BWR vessel environment [32]. Components in NPPs that have experienced irradiation degradation include [1, 4]: - BWRs; RPV internals, control rod drive mechanisms and emergency diesel generators, - PWRs; RPV supports, - LWRs; RPV, RPV internals, - HWR RPVs; calandria vessel, pressure tubes and calandria tubes. 3.7 Thermal ageing Thermal ageing refers to gradual and progressive changes in the micro-structure and properties of a material exposed to an elevated temperature for an extended period of time [6]. The dominating parameters responsible for thermal ageing processes are [4]: level of temperature, material state (micro-structure) and ex- posure time. In addition to these phenomena, the environment can have an effect to the ma- terial state as well. In general, thermal ageing is a damage process typically causing decrease in the material strength properties, hardness, ductility and toughness. Under certain conditions, however, an increase in ductility can be observed, too. One of the dominating factors is the chemical composition of the material and its thermo-mechanical pre-service treatment [4]. 3. Degradation mechanisms concerning power plant components 51 The mechanisms occurring in the micro-structure during thermal ageing are of the following types [4]: - precipitation of particles, - transformation of phases, - growth of precipitated phases and the grains of the matrix, - dissolution of precipitates. The rate of the above mentioned reactions is strongly dependent on the level of temperature. Such threshold temperatures can exist below which some of the processes are not being activated. The local chemical composition and the diffu- sion coefficient of the different phases and atoms play an important role [4]. Cast austenitic-ferritic stainless steels (duplex structures) and martensitic stainless steels are susceptible to thermal ageing in the normal operating temper- ature range of PWRs [34]. 3.8 Other degradation mechanisms Other mechanisms or loading phenomena degrading power plant components include erosion-cavitation, wear, loss of prestressing, water hammer, environ- ment effects, concrete degradation and differential settlement. Here environment effects concern processes or phenomena like wetting-drying and freeze-thaw cycling and chemical attack [6, 17]. Examples of NPP components that have experienced these degrading mecha- nisms or loading phenomena are presented in the following. Wear has occurred in the steam generator tubes due to contact with anti-vibration bars. Typical components subject to a loss of prestressing are the prestressed cables in the primary containment. Concrete components may be affected by several degrada- tion mechanisms, e.g. aggressive environment can increase porosity, and perme- ability as well as reduce concrete strength. Atmospheric carbon dioxide can cause concrete carbonation reducing the structural properties of the concrete. Flowing water over concrete surfaces can remove significant amount of con- crete. Severe differential settlement can cause concrete cracking and/or misa- lignment of equipment and can lead to high stresses within the structure [6]. 3.9 Interaction of degradation mechanisms It has been noticed that some degradation mechanisms can provide a joint effect so that it is more severe than the arithmetically added result of their separate 3. Degradation mechanisms concerning power plant components 52 effects. In these cases, the resulting degradation phenomenon is often quite complex. Due to this and also to the fact that the interaction of various degradation mech- anisms is still a quite recent research topic, the physical mechanisms of most interaction phenomena are not clear. However, two cases of interacting degrada- tion mechanisms are covered here, those being corrosion fatigue and creep fa- tigue, as good results concerning their assessment have already been achieved. Corrosion fatigue In case of fatigue, in corrosive environments the number of load cycles needed to crack initiation is considerably smaller than that needed in inert environment. In addition to crack initiation, the corrosive environment also influences the ensuing cyclic crack growth rate. The interaction of mechanical alternating stresses and corrosion attack usually leads to transgranular cracking with a low associated deformation. Since the corrosion mechanism is mainly time depend- ent and the fatigue mechanism is controlled by the number of cycles, a complex interaction between these mechanisms occurs. No threshold conditions exist for corrosion fatigue with respect to the corrosion system and stress amplitude [4]. In corrosion fatigue, the crack growth rate may be influenced by the following features and conditions [4]: - characteristics of the corrosive environment, such as pH value, temper- ature, electrochemical potential or oxyen content in water, - loading frequency and wave form, - mean load and load amplitude, - sulphur content of steel. Several corrosion fatigue mechanisms have been proposed to explain the en- hanced crack growth rates with varying degrees of success. The generalised cor- rosion fatigue cracking mechanism involves the single or mutual occurrence of hydrogen induced cracking and/or anodic dissolution at the crack tip. According to current understanding there appears to be at least four possible mechanisms of anodic dissolution [7]: - slip dissolution, - brittle film rupture, - corrosion tunnelling, - selective dissolution (dealloying). 3. Degradation mechanisms concerning power plant components 53 Hydrogen embrittlement appears to be divided into the following five types of mechanisms [7]: decohesion, pressure, adsorption, deformation and brittle hydride. The balance between various phenomena is difficult to sort out. A common approach is to assume that dissolution and hydrogen induced cracking processes are competitive, so that only one of them makes the major contribution to crack- ing and the others can be ignored. However, both processes (dissolution and hydrogen evolution) can occur simultaneously at the crack tips over ranges of potential that have been measured or that are suspected to occur [7]. Components in conventional power plants in which corrosion fatigue degrada- tion can occur are those mentioned earlier experiencing fatigue, but which are also exposed to corrosive environment [29]. Components in NPPs having experienced corrosion fatigue degradation include [1, 4]: - BWRs; RPV internals, - HWR RPVs; calandria vessel, pressure tubes and calandria tubes, - steam generator; tubes, - steam and water piping vessels valves, - main coolant pump; casing and shaft impeller, - turbine; casing, blades and shaft, - condensor tubes. Creep fatigue In power plant components which operate at relatively high temperatures, changes in prevailing conditions during operation cause transient temperature gradients. If these transients are repeated, the differential thermal expansion during each transient results in a thermally induced cyclic stress. The extent of the resulting fatigue damage depends on the nature and frequency of the transi- ent, on thermal gradient in the component and on material properties. In case of such components, the damage caused by both fatigue and creep has to be taken into account [11]. The principal method of studying creep fatigue interactions has been to con- duct strain controlled fatigue tests with and without a holding period (hold time) during some part/phase of the test. With higher frequencies and shorter hold times, fatigue dominates the degradation process, and the flaws initiate near the specimen surface and propagate transgranularly. With lower frequencies and longer hold times, creep begins to play a significant role in the degradation pro- 3. Degradation mechanisms concerning power plant components 54 cess, and the initiating flaws are of a mixed mode involving both fatigue crack- ing and creep cavitation. With prolonged hold times and only seldom occurring load cycles, creep dominates the degradation process [11]. There is ample evidence to show that rupture ductility has a major influence on creep fatigue interaction. In case of ferritic steels and austenitic stainless steels, the lower the ductility, the lower is the creep fatigue endurance. In addi- tion, long hold times, small strain ranges and low ductility favour creep domi- nated failures, whereas short hold times, intermediate stress ranges and high ductility favour creep fatigue interaction failures [11]. As an example, the effect of tensile hold time on fatigue endurance of type 316 stainless steel is shown in Figure 3.9-1 in the following. Figure 3.9-1. Effect of tensile hold time on fatigue endurance of type 316 stainless steel, from ref. [43]. Due to operational temperature range in NPPs being in general below the creep range of the metallic pressure boundary components, creep fatigue mainly relates to conventional power plants, where it can degrade those components mentioned earlier which have experienced fatigue, but which are also operated at tempera- tures that are near or exceed 0.4 times the melting point of the involved materials. 4. Deterministic analysis methods 55 4. Deterministic analysis methods 4.1 Introduction A brief overview of deterministic analysis methods is presented in the following. This is divided to structural mechanics based analysis methods, including frac- ture mechanics, and to some considerations concerning analysis methods based on other fields of science. A description concerning the structural mechanics based analysis methods is provided with emphasis and an example on power plant components and condi- tions. In this connection, no actual computation equations/procedures are pre- sented, but instead some relevant approaches and analysis tools are considered. Whereas in case of the fracture mechanics based analysis methods, which issue is also dealt with further in this thesis, some relevant computational parameters and procedures are briefly presented as well. Often, the connection between structural mechanics and fracture mechanics is such that the results obtained with the former procedures provide input data for computations carried out with those of the latter. Analysis methods based on other fields of science are mainly beyond the scope of this thesis, however some relevant connecting topics are pointed out. 4.2 Analysis methods based on structural mechanics Structural mechanics mainly deals with mathematical modelling of structures, together with associated loading and supports, to compute stresses/forces and deformations they experience. The basic function of any structure is to carry loads and transmit forces. In case of power plants, typical considered structures include straight pipe components, pipe bends, pipe T-joints and pressure vessels, as well as reinforced concrete structures, such as slabs and walls, which often support the aforementioned components. 4. Deterministic analysis methods 56 The structural models describe the geometry and material properties of the con- sidered structure. Depending on the strived accuracy, as well as shape and sup- ports of the considered structure, the model can have one, two or three dimension- al geometry. There are also several possible material models, for instance linear- elastic, elastic ideally-plastic, and elastic non-linear plastic. Also, the stress/strain hardening behaviour can be history dependent, with typical modelling options being then to use isotropically or kinematically evolving yield surface, or their combination. Further, more advanced material models exist, such as ORNL model [35] that covers both elastic-plastic material behaviour and thermal creep. Generally, loads on structures fall into several categories, with some notable examples of these being dead loads, live loads and accident loads. In power plant environments the dead loads typically cover self-weights of the load bearing concrete and steel structures. Also, the self-weights of process components, such as those mentioned above, should be considered as dead loads as they are not movable, even though at time intervals component replacements may have to be carried out. Live or imposed loads are movable or actually moving loads. In power plant environments, these mainly include temperature, pressure and flow rate, all caused by the process fluid, being either water or steam. At operational conditions these load parameters have most of the time constant values, however during anticipated/budgeted controlled load transients, such as start-up and shut- down, they vary as well. The structures carrying these loads form in power plants the pressure boundary. When including loads to a structural model, some idealisations are performed. As power plant components are mainly exposed to mechanical and thermal loads, only they are considered in the following. De- pending on the capability and scope of the structural model, mechanical loads can be included as pressures against surfaces, line loads distributed along a sur- face line, and as point specific forces. Also translations and rotations can be used as loads. Thermal loads are typically set to act on component surfaces. Depend- ing on the approach, loads can be given as time dependent or independent. There is a large variety of possibilities how a structure can be supported. The stiffness of support conditions can vary from rigid to very flexible. For instance, the support conditions of power plant piping systems are often relatively flexi- ble, so as to allow thermal deformations to occur during load transients without causing excessive restrained thermal stresses. Whereas the NPP RPVs often have a support skirt, a steel structure that is welded around them at some height in the upper half, with the aid of which the RPVs firmly rest on a reinforced concrete structure. When including supports to a structural model, some idealisa- 4. Deterministic analysis methods 57 tions are performed. Depending on the scope of the structural model, support conditions can be set on surfaces, lines, and points. Typical support modelling options include limiting at one or several locations translations and/or rotations in/around one, two or three directions to zero. Support conditions can be time dependent or independent. When carrying out a structural mechanics based analysis, a model for the con- sidered structure must be prepared first. Then convenient mathematical methods are sought for solving the unknown model parameters. In the simple case, the model can be a set of analytical equations. In case of power plant components, numerical structural models are often used. These are typically prepared using finite element method (FEM) based analysis codes. The basic principle concern- ing this methodology is to provide approximate solutions for the governing set of partial differential equations in a finite number of nodes using suitable inter- polation functions. In the modelling process, the considered geometry is divided to a finite number of sub-regions, called elements, within and on the edges of which are nodes, to which in turn the solution values are interpolated from the integration points. There are numerous advanced commonly used general pur- pose FEM codes. Such FEM codes also allow using elements of differing types in the same model, as well as provide tools for sub-modelling, with which local- ly more dense element meshes can be realised. Other numerical techniques exist as well, such as finite difference method. To achieve sufficient accuracy using numerical methods, dense enough computation grids are to be used, and in time dependent analyses small enough time increments. In addition, case specifically combined methods can be used, i.e. FEM solutions supplemented with those computed using analytical equations. In the following are structural mechanics modelling approaches used for ana- lysing power plant components/systems, and some application examples: - analytical equations; dimensioning of single components, computing stresses caused by inner pressure for straight pipe components, - truss models; to describe behaviour of simple auxiliary support struc- tures, - beam models; dimensioning and stress distribution analyses of piping systems, - shell models; pipe bends, some pressure vessels, - solid three dimensional models; RPV, piping T-joints. 4. Deterministic analysis methods 58 The types of analyses often performed for power plant components/systems include: - linear or non-linear time dependent or independent static stress/strain analyses, - heat transfer analyses, - combined sequential or coupled heat transfer and stress/strain analyses, - dynamic eigenvalue and/or eigenfrequency analyses. As for the obtained results, typically they are time dependent or independent stress/strain fields and temperature fields. From the stress/strain analysis results, local/global maximum/minimum values are often sought, and such can take place e.g. in inner corner with relatively small radius or where wall thickness changes abruptly. The maximum stress values are compared to corresponding material strength, e.g. to see whether the component in question meets the asso- ciated structural strength requirements. Whereas in case of thermal results, in addition to maximum and minimum values often also temporary temperature gradients through wall are sought, because they in turn cause thermal stresses. Of particular interest of the time dependent stress/strain results are the local stress/strain fluctuations, because they form the corresponding stress/strain cy- cles needed in fatigue analyses often performed for power plant components. As for the dynamic analyses, their results are needed e.g. to assess the structural response to assumed earthquake loads. Such results as nominal stresses, local stresses and stress fluctuations, from structural mechanics analyses, are often needed as input data in fracture mechan- ics based analyses. Most analyses of mechanics are deterministic. In this approach, single or fully pre-determined values are used for all input data parameters, such as yield strength or frequency of load cycle occurrence. The input data parameters often having markedly distributed characteristics in reality, such as yield strength, are also considered as distributed in the probabilistic analyses applying structural reliability methods. Such distributions are formed by fitting a suitable distribu- tion function to applicable experimental or simulated data. The results from de- terministic analyses provide no data concerning their uncertainty/reliability, whereas those from structural reliability analyses are probabilities or probability distributions. 4. Deterministic analysis methods 59 As an example concerning structural mechanics analyses, in the following is presented a numerical lifetime analysis for an outlet header in a conventional power plant. The analysis data are from the article [38]. The outer diameter of the outlet header is 711 mm, whereas its wall thickness and total length are 36 mm and 18 272 mm, respectively. The material of this component is ferritic- martensitic steel X20CrMoV121. As for loads, mainly load transient start-up and pressure variation during operation are considered. During the former of these load events both temperature and pressure rise in a controlled process within approximately three hours from room temperature and one bar to operational values of 535 °C and 2.6 MPa. The overall element mesh of an Abaqus FEM model of a section of the outlet header and the tubes connecting to it is shown in Figure 4.2-1. For more details concerning the Abaqus code, see refs. [36, 37]. The outlet header is supported by the tubes, which are cut on certain length in the model, and appropriate defor- mation boundary conditions are given to the cut surfaces, as well as to those of the header region. For both the heat transfer and static analyses of the outlet header region of the model, isoparametric four node linear tetrahedron elements are used, whereas those used for the connecting tubes are isoparametric eight node linear hexahedron “brick” elements. The former element type contains one integration point, whereas the latter contains four. An example of the stress re- sult distributions for the most stressed header/tube joint, namely that concerning Row5 tube, is shown in Figure 4.2-2 below for a selected time instant from the duration of load transient start-up [38]. 4. Deterministic analysis methods 60 Figure 4.2-1. The overall element mesh of the FEM model of the outlet header section and tubes connecting to it. The number of elements/nodes is approximately 150 000/46 000, from ref. [38]. Figure 4.2-2. Distribution of axial stresses [MPa] in the Row 5 joint middle section of the FEM model of the outlet header and the tubes in the middle of load case start-up, from ref. [38]. naming of tube rows, starting from left and proceeding clockwise: Row1, Row2, Row3, Row4, Row5 4. Deterministic analysis methods 61 4.3 Fracture mechanics based analysis methods Fracture mechanics concerns the design and analysis of structures which contain cracks or flaws. On some size scale, all materials contain flaws, which are either microscopic, due to cracked inclusions, debonded fibers etc., or macroscopic, due to corrosion, fatigue, welding flaws, etc. Thus fracture mechanics is in- volved in any detailed design or safety assessment of a load bearing structure. As cracks can grow during service due to e.g. fatigue, fracture mechanics as- sessments are required throughout the lifetime of a structure or component, not just in the beginning of the lifetime. Some theoretical background of fracture mechanics is presented in the follow- ing. At the top level, fracture mechanics is divided into linear-elastic fracture mechanics (LEFM) and elastic-plastic fracture mechanics (EPFM). Materials with relatively low fracture resistance that fall below their so called plastic col- lapse strength can be analysed on the basis of elastic concept through the use of LEFM. For other materials, the use of EPFM is often necessary. For certain cracked component geometries subjected to external forces, it is possible to derive closed form expressions for stresses, assuming linear-elastic material behaviour. Westergaard [39] and Irwin [40] were among the first to publish such solutions. There are three types of loading that a crack can experience, as Figure 4.3-1 illustrates. Mode I, where the principal load is applied normal to the crack plane, tends to open the crack. Mode II corresponds to in-plane shear and tends to slide the crack faces with respect to each other. Mode III refers to out-of-plane shear. A cracked component can be loaded with any of these modes, or with a combi- nation of two or three modes. 4. Deterministic analysis methods 62 Mode I Mode II Mode III Figure 4.3-1. The three modes of loading that can be applied to a crack, from ref. [30]. Of the fracture mechanics parameters, the stress intensity factor, K, completely characterises the crack tip conditions in a linear-elastic material. Usually K is provided with a subscript to denote the mode of loading, i.e. K I , K II or K III . The stress fields ahead of a crack tip in an isotropic linear-elastic material can be written with polar coordinates as: , , , , , , u t o I ij I I ij r f r K 2 lim 0 = ÷ (4.3-1a) , , , , , , u t o II ij II II ij r f r K 2 lim 0 = ÷ (4.3-1b) , , , , , , u t o III ij III III ij r f r K 2 lim 0 = ÷ (4.3-1c) where o ij is stress tensor, r and u are the radial and angular coordinates of the polar system, and f ij is a dimensionless function of u, respectively. In a mixed mode problem, i.e. when more than one loading mode is present, the individual contributions to a stress component are additive [30]: 4. Deterministic analysis methods 63 , , , , , , , , III ij II ij I ij total ij o o o o + + = (4.3-2) If one assumes that the material fails locally at some critical combination of stress and strain, then it follows that fracture must occur at critical stress intensity, K IC . Thus K IC is a measure of material fracture toughness. Failure occurs when K I = K IC . In general, the stress intensity factor (SIF) has the following form [30]: a BS K t = (4.3-3) where a [mm] is the crack depth (dimension of the crack in the principal direc- tion of the crack growth), S [N/mm 2 ] is the remote stress resulting from the ap- plied load, and B = B(a) [ - ] is a factor that accounts for the geometry. For any crack in any practical problem, only the factor B needs to be derived. For many geometries B can be found in handbooks. Typically, rapidly occurring brittle fracture is associated with fracture toughness parameter, K IC . When fracture is accompanied with considerable plastic deformation, EPFM is applied. An often used fracture parameter in EPFM is called J-integral. J is simply the strain energy rate, and it can be derived from a conservation of ener- gy criterion. In EPFM fracture occurs when [30]: R J J = (4.3-4) where J R [J/mm 2 ] is the fracture energy, which represents the material fracture resistance. Consider an arbitrary counter-clockwise path I around the tip of the crack, as is shown in Figure 4.3-3. The J-integral is defined as [30]: } | . | \ | c c ÷ = ds x u T wdy J i i (4.3-5) where w [N/mm 2 ] is the strain density, T i [N/mm 2 ] are the components of the traction vector, u i [mm] are the displacement vector components, ds is a length increment along the contour I, whereas x and y are coordinate variables as shown in Figure 4.3-2. The strain energy density w in equation (4.3-5) is defined as [30]: 4. Deterministic analysis methods 64 } = ij ij ij d w c c o 0 (4.3-6) where o ij [N/mm 2 ] and c ij [mm/mm] are the stress and strain tensors, respective- ly. Typically, stabily and more slowly occurring ductile fracture is associated with fracture toughness parameter J R . Another applicable parameter used in EPFM is crack tip opening displacement (CTOD). Unlike in the case of linear-elastic materials, the crack tip may experi- ence plastic deformation by blunting in strain hardening materials. The opening at the crack tip, i.e. CTOD, is a measure of fracture toughness. Crack tip blunt- ing and CTOD, denoted here by o, are illustrated in Figure 4.3-3. Figure 4.3-2. Arbitrary contour around the crack tip. 4. Deterministic analysis methods 65 Figure 4.3-3. Blunting of the crack tip and CTOD, from ref. [30]. A general form for CTOD can be expressed as [30]: E m K y I o o 2 CTOD = = (4.3-7) where o y [N/mm 2 ] is the yield strength, E [N/mm 2 ] is the Young’s modulus and m is a dimensionless constant that is approximately 1.0 for plane stress and 2.0 for plane strain. There exist a number of alternative geometric definitions for CTOD. Hutchinson [41] as well as Rice and Rosengren [42] independently showed that J-integral characterises crack tip conditions in a non-linear-elastic material. They each assumed a power law relationship between plastic strain and stress. The result of their work is called the HRR singularity, which is one of most commonly used EPFM models. According to this model, the actual stress and strain distributions are obtained by applying the appropriate boundary conditions as follows: , , u o oo o o , ~ 1 1 2 0 0 n r I EJ ij n n ij + | | . | \ | = (4.3-8) 4. Deterministic analysis methods 66 , , u c oo oo c , ~ 1 2 0 0 n r I EJ E ij n n n ij + | | . | \ | = (4.3-9) where o 0 is reference stress (most often yield strength), r is polar coordinate with origin at the crack tip, o is dimensionless constant, n is strain hardening expo- nent and I n is an integration constant that depends on n, whereas ij o ~ and ij c ~ are dimensionless functions of n and u. These parameters depend also on the stress state, i.e. plane stress or plane strain. The J-integral defines the amplitude of the HRR singularity, just as SIF characterises the amplitude of the linear elasticity. Thus, J-integral completely describes the conditions within the plastic zone. A structure in small scale yielding has two singularity dominated zones: one in the elastic region, where the stress varies as 1/\r, and one in the plastic zone, where stress varies as , , 1 1 + ÷ n r . The latter zone often persists long after the linear-elastic singularity zone has been taken over by crack tip plasticity. In the following are analysis approaches used for fracture mechanics assess- ments of power plant components/systems, and some application examples: - analytical equations; K, J-integral and CTOD values for static crack postulates in single material components with regular geometry and load distribution shape, - weight/influence function based analysis codes; K and J-integral values for static and growing crack postulates in single material components with regular geometry and arbitrary load distribution shape, - general purpose FEM codes; K, J-integral and CTOD values for static and growing crack postulates in single and multi-material components with arbitrary geometry and load distribution shape. Fracture mechanics based structural integrity assessment procedures are covered in more detail further in this thesis, including both deterministic and probabilistic approaches, see Chapters 7, 8 and 12. Most fracture mechanics analyses are deterministic, e.g. a single value of frac- ture toughness is used to estimate failure stress or critical crack size. Much of what happens in the real world, however, is not predictable. For instance, frac- ture toughness data in the ductile to brittle transition region are widely scattered. With structural reliability analyses, these distributed characteristics can be taken into account. The results from deterministic analyses provide no data concerning 4. Deterministic analysis methods 67 their uncertainty/reliability, whereas those from structural reliability analyses are probabilities or probability distributions. 4.4 Analysis methods based on other fields of science A field of science necessary for any kind of structural analyses concerning pow- er plant systems/components is that associated with material proper- ties/behaviour. Namely power plant environments not only include a relatively large number of different materials, for instance ferritic and austenitic steels of several types in the pressure boundary, but also provide a range of challenging mechanical loading, thermal and chemical conditions. Thus it is necessary to clarify how the material properties vary as a function of e.g. operational temper- ature range as well as how the inner surfaces of components respond to various kinds of chemical attacks. For instance, both the yield strength and ultimate strength decrease as a function of temperature, the extent of which is a material specific issue. Typically, relatively thin protective oxide layers form on the inner surface of power plant components. However, the layers can locally rupture as depending on changes in local chemical conditions, which in turn increases cor- rosion rate as fresh metal becomes exposed to corrosive conditions. The assess- ment of thermal ageing, which was mentioned earlier in Section 3.7, requires material science associated data as well. The clarification of these material be- haviour/property issues requires experimental research. Fluid mechanics is needed when simulating the flow patterns in more complex flow conditions, such as turbulent mixing of water flows of differing tempera- tures in piping T-joints. Results from these analyses include fluid temperature and pressure distributions as well as distributions for the heat transfer coefficient between the fluid and component inner surface. A more recent development is numerical fluid-structure interaction analyses. When an abrupt increase in flow rate occurs, e.g. caused by sudden closing of a valve, the pressure impulse can be of such magnitude that it may cause temporarily notable deformations to the component containing fluid. In such cases, these phenomena need to be taken into account in the structural integrity analyses. The results from the fluid me- chanics analyses produce input data to structural mechanics based computations, including fracture mechanics. Particle physics is needed when assessing the characteristics of the neutron ir- radiation that causes changes in microstructure and properties of the materials exposed to it. The extent of these changes depends on the material, neutron flux, 4. Deterministic analysis methods 68 flux spectrum, fluence and irradiation temperature. The effect of neutron irradia- tion on metals mainly increases the yield and ultimate strength and reduces the fracture toughness. Thus, it is obvious that these effects need to be taken into account in structural integrity analyses. In practise, irradiation effects concern only NPP RPVs, as was explained in Section 3.6. 5. Probabilistic analysis methods 69 5. Probabilistic analysis methods 5.1 Introduction There exists a number of methods to determine the probabilities of the varia- ble(s) of interest of a system. Monte Carlo simulation is a simple and most accu- rate procedure to analyse probability distributions. This procedure has one major drawback, though, which is the often remarkable amount of computation work required in order to achieve sufficient accuracy for the results. This limiting or even prohibiting feature is emphasised especially in the situations where a very low probability is to be determined. Refined Monte Carlo methods provide ra- ther sophisticated procedures, such as stratified sampling and importance sam- pling, which can help to overcome these difficulties [76]. As the efficiency, ca- pacity and affordability of available computers keeps improving swiftly, Monte Carlo simulations become easier and more economical to perform. Due to Monte Carlo calculations being often rather costly and laborious to perform, approximation methods for the calculation of variable probabilities are of great interest. Examples of these are first-order second-moment methods (FORMs) and second-order reliability methods (SORMs). Determining probabilities from scarce and incomplete input data is another problem. The first question that arises is how to handle the data basis in order to end up with input distributions reflecting all the information available. This problem is dealt with in the statistical estimation theory. Once an appropriate statistical model has been set up and the parameters of this model have been determined from the input data, the probability of interest can be computed in a straightforward manner [76]. There are two basic philosophical schools in modern probability theory, one based on a frequency interpretation, and one based on a Bayesian interpretation or degree of belief. These are also known as the objective and the subjective interpretation, respectively. According to frequency interpretation, a probability 5. Probabilistic analysis methods 70 is an objective property of some event. In terms of failure probability, it is the expected probability of failure that is reflected. In Bayesian philosophy, a proba- bility is considered as a subjective degree of belief, for example, in the chances of a particular event occurring. This probability, or degree of belief, depends on the amount of information available. Bayesian probabilities are also called sub- jective [45]. Before going on to describe in more detail the characteristics of various prob- abilistic methods, some basic definitions are presented first. 5.2 Failure probability and reliability Failure probability, P f , is often defined as the mean frequency with which the specified failure event would be expected to occur in a given period of opera- tion, typically one year. When assessing the probability of failure, it is important to consider the future deterioration rate from all potential mechanisms. The rate of degradation may increase in time due to interaction between mechanisms (e.g. corrosion and fatigue). Factors, such as overload, misuse, or accidental damage that cannot be easily predicted should be assumed to occur at a constant average rate [3]. Reliability, R, is closely related to the failure probability, and expresses the probability of a component not failing [46]. The relation between reliability R and probability of failure P f is expressed as [47]: f P R ÷ =1 (5.2-1) A rigorous structural reliability assessment involves taking into account all sources of uncertainty that may affect the failure of the component or system. This clearly involves taking into account all fundamental quantities entering the problem, and also the uncertainties that arise from lack of knowledge and ideal- ised modelling. These terms are referred to as basic variables. The common en- gineering quantities they represent include: component diameter, wall thickness, material and contents density, yield strength, maximum operating pressure, max- imum operating temperature, corrosion rate, etc. [45]. The structural reliability analysis procedure is outlined by the following steps [45]: 5. Probabilistic analysis methods 71 1) Identify all significant modes of failure of the structure or operation under consideration, and define failure events. 2) Formulate a failure criterion or failure function for each failure event. 3) Identify the sources of uncertainty influencing the failure of the events, model the basic variables and parameters in the failure functions and specify their probability distributions. 4) Calculate the probability of failure or reliability for each failure event, and combine these probabilities where necessary to evaluate the failure probability or reliability of the structural system. 5) Consider the sensitivity of the reliability results to the input, i.e. basic variables and parameters. 6) Assess whether the evaluated reliability is sufficient by comparison with a target. 5.3 Uncertainty and sensitivity The sources of uncertainty that are relevant to structural reliability analysis can be primarily classified into two categories: 1) aleatoric uncertainties; being asso- ciated with physical uncertainty or randomness and, 2) epistemic uncertainties; being associated with understanding or knowledge [45]. Aleatoric uncertainty refers to the natural randomness associated with an un- certain quantity, and is often termed Type I uncertainty in reliability analysis. It can be further divided into subcategories as shown in Figure 5.3-1. Aleatoric uncertainty is quantified through the collection and analysis of data. The ob- served data may be fitted by theoretical distributions, and the probabilistic mod- elling may be interpreted in the relative frequency sense. Epistemic uncertainty reflects the lack of knowledge or information about a quantity, and is often termed Type II uncertainty in reliability analysis. Epistem- ic uncertainty can further be divided into subcategories as shown in Figure 5.3-1. 5. Probabilistic analysis methods 72 Figure 5.3-1. Two primary types of uncertainty and associated subcategories, from ref. [45]. Model uncertainty arises because many of the engineering models used to de- scribe natural phenomena, to analyse stresses and to predict failure of compo- nents are imperfect. Models are often based on idealised assumptions, they may be based on empirical fits to test results or observed behaviour, and variables of lesser importance may be omitted for reasons of efficiency (or ignorance) [45]. There are difficulties in quantifying model uncertainty adequately. The errors in the model may be known in relation to more advanced models, but at any level of modelling there are errors relative to the unknown reality. Model uncer- tainty is often assessed on the basis of experimental results, but this too has a number of limitations. Tests themselves are idealisations or simplifications of reality, and are affected by scale, boundary effects, load rates, measurement errors, etc. Tests are often expensive, and the data need to be carefully screened to ensure that they are consistent. Ideally, the test data should cover uniformly the full range of the applicability of the model. Statistical uncertainty can be considered to arise from [45]: - Parameter uncertainty. This occurs when the parameters of a distribu- tion are determined from a limited set of data. The smaller the data set the larger the parameter uncertainty. 5. Probabilistic analysis methods 73 - Distribution type uncertainty. This uncertainty arises from the choice of a theoretical distribution fitted to empirical data. It is a particular problem when deriving extreme value distributions. It may not be possible to distinguish from each other the two types of statistical uncertainty in practice, since with limited data both the parameters and distribu- tion type may be uncertain. Development of mathematical models for structural analysis purposes consists of several logical steps, one of which should be the determination of parameters which are most influential on model output. A sensitivity analysis (SA) of the input parameters can serve as a guide to any further use of the model [48]. In general, SA is conducted by defining the model and its output parameters, to- gether with their probability density functions and output variable(s), and then assessing the influences or relative importance of each input parameter on the output variable [48]. There is a number of reasons for conducting a SA, including the needs to de- termine which input parameters contribute most to output variability, which parameters are insignificant and can be eliminated from the final model, and which (group of) parameters interact with each other [48]. 5.4 Response function and performance function A random function can often be described by a number of parameters. In the ideal situation, the parameters are determined from an infinitely large sample. In practise, however, available data are usually limited or non-existent, so a combi- nation of judgement and experience must often be used. Statistical tools can be used to measure the quality of the estimated parameters, i.e. to quantify the un- certainty in the estimates [52]. A common way to define a random variable in the literature is to denote it with a capital letter, X, and a certain value of it with a small letter, x. The distri- bution of a continuous random variable is controlled by probability density func- tion, f x (x). Whereas the cumulative distribution function, F x (x), corresponds to a definite integral of the probability density function over some selected region. The parameter used in the description of a physical problem is called a re- sponse function, Z [47]. This function describes a structural or mechanical re- sponse, such as stress, strain or crack growth in a structure. Response function can be a complete FEM model as well. The response function, Z, is expressed as a function of the random variable, X i , as follows: 5. Probabilistic analysis methods 74 , , n X X X X Z Z ,..., , , 3 2 1 = (5.4-1) The performance or limit state function, g, gives the failure boundary or critical conditions of a physical problem. It is defined by: , , 0 ,..., , , 3 2 1 0 0 = ÷ = ÷ = n X X X X Z Z Z Z g (5.4-2) where Z 0 is the limiting value of Z. The region g 0 corresponds to failure, whereas g > 0 is the safe domain [51]. The probability of failure can now be defined mathematically as: j ¦ , , } s = s = 0 0 g x f dx x f g P P (5.4-3) In some cases, equation (5.4-3) can be integrated analytically. In principle, the probability of failure or reliability can be evaluated using numerical integration, e.g. trapezoidal rule or Simpson’s rule, etc. In practice, this is not generally fea- sible in probabilistic analysis because of the number of dimensions of the prob- lem, being one dimension for each basic variable, and because the area of inter- est is usually in the tails of the distributions. Nevertheless, this equation is occa- sionally used in numerical form, and with dense enough computation grid it can potentially give accurate enough results [45]. There are procedures to overcome the difficulties encountered in calculating the failure probability, which issue is covered in more detail further in this chapter. 5.5 Commonly applied distribution functions The number of continuous reliability distributions in available handbooks which empirically describe the scatter of material test data is considerable. The most commonly applied distribution functions include [46]: exponential distribution, normal or Gaussian distribution, log-normal distribution, Weibull distribution, gamma distribution and Gumbel distribution. Some of these functions are sym- metric, such as normal distribution, whereas others are asymmetric, such as ex- ponential distribution. Using these functions, probabilistic distributions have been developed e.g. for fracture toughness, initial flaw size and frequency of occurrence of load cycles. 5. Probabilistic analysis methods 75 5.6 Review of probabilistic analysis procedures 5.6.1 First-order second moment method (FORM) To overcome the invariance problem with the failure function, it is necessary to transform the basic variables into independent standard normal variables. A space defined by independent standard normal variables is termed U-space, and basic variable space is termed X-space. The transformation of independent variables can be undertaken from the cu- mulative distribution function, F x (x), i.e. from the identity [45]: , , , , x F u x = u (5.6-1) , , j ¦ u F x x u = ÷1 (5.6-2) where , , u u is the standard normal distribution function. Various techniques and software tools for the transformation of the non-independent variables are available. FORM methods involve estimating the failure probability by linearising the failure surface at the closest point to the origin in standard normal space, or U- space, using e.g. Taylor series of the normalised random variables. Iteration is usually necessary to determine the closest point to the origin, and a number of iterative and optimisation techniques are available. The space outside the tangent hyperplane to the failure surface at the closest point to the origin corresponds approximately to the probability of failure. Even though the failure surface in standard normal space is rarely planar the curvature at the point closest to the origin is usually so small that the first-order linearisation is valid for most purposes. The basic variable transformation and the first-order reliability estimate are illustrated in Figure 5.6-1 with two basic variables. 5. Probabilistic analysis methods 76 Figure 5.6-1. Illustration of transformation from basic variable space (left) to U-space (right) and first-order reliability estimate, from ref. [45]. The closest point from the failure surface to the origin in U-space is the most probable point (MPP) or design point. The distance between these two points is equal to the first-order reliability index, |, denoted as beta in Figure 5.6-1 [45]. This is also referred to as the geometrical reliability index or Hasofer-Lind relia- bility index [53]. After the first-order transformation, the level curves of the joint density func- tions become origin centered circles. The limit state surface is a hyperplane in the space of the transferred random variables, meaning the U-space. An approx- imation of the probability of failure is then given by: , , | ÷ u ~ f P (5.6-3) 5.6.2 Second-order reliability method (SORM) Second-order methods improve the accuracy of the first-order probability esti- mates by including curvature information at the point |, and by approximating the failure surface with a quadratic surface. First is necessary to iteratively iden- tify the point |. The difference between the first- and second-order estimates 5. Probabilistic analysis methods 77 gives an indication of the curvature of the failure surface. If there is a significant difference it suggests that perhaps Monte Carlo methods should be used to con- firm the probability of failure estimate. For most practical reliability applica- tions, there is usually a small difference between FORM and SORM estimates [45], only. In many engineering structural problems, the SORM method has satisfactory accuracy, but in some cases, it may lead to erroneous results. A reason for this is that the quadratic approximations may appear to be insufficient when the failure surface is oscillatory in nature or too irregular in the vicinity of the point of max- imum likelihood. Moreover, a significant disadvantage of the FORM/SORM methods is that no estimation of the error made is available, and so it is very difficult to validate results [54]. Various methods have been developed to over- come these problems, see e.g. ref. [55]. A general but complicated expression for the failure probability of quadratic forms is derived in ref. [56]. A more convenient but approximate expression has been derived by Breitung as [57]: , , , , [ = ÷ + ÷ u = n i i f P 1 5 . 0 1 |k | (5.6-4) where k i denotes the principal curvatures of the limit state at the minimum dis- tance point. 5.6.3 Fast probability integration (FPI) In FORM and SORM, the basic random variables are transformed into inde- pendent standard normal variables by exact mapping. In some cases, this is not easy to apply and in addition, the original limit state function may become dis- torted by this transformation. The FPI approach has been developed to overcome these drawbacks. With the FPI, the original distributions are approximated by standard normal variables, instead of transforming them into standard normal. Thus the FPI methods used for linearisation and normalisation are numerical procedures for solving multidimensional integral equations concerning general failure probability or reliability analyses [52]. For instance, the FPI method published by Wu and Wirsching [58] is a SORM, employing a three-parameter normal tail approximation. In this method, the equivalent normals are constructed using a least squares approach. A second 5. Probabilistic analysis methods 78 order approximation of the failure surface is used and then transformed into lin- ear one in order to reduce the computational effort. It has been demonstrated, see ref. [59], that the FPI method in the ref. [58] is efficient and provides accurate failure probability estimates. 5.6.4 Mean value methods Mean value method (MV) When the response function is implicitly defined by a computer code, the gen- eral failure probability or reliability methods are difficult to apply. The MV method [60] is a first-order method that is suitable to be used under these cir- cumstances. This method is based on the assumption that a first-order Taylor expansion of the response function exists. By omitting the higher order terms, the response function is expressed at the mean values as: , , , , , , ¿ ¿ = = + = ÷ c c + = n i n i i i i i i X a a X X Z Z X Z 1 1 0 µ µ (5.6-5) where parameters a 0 and a i are coefficients, and the derivatives are evaluated at the mean value points µ i . The coefficients can in general be computed by numer- ical differentiation and the minimum number of performance function evalua- tions (computer runs) is (n+1), where n is the number of random variables [52]. Based on this linear approximation the cumulative distribution function for re- sponse function can be obtained directly, since the distributions for the random variables X i are fully defined and the response function is explicit [61]. Advanced mean value method (AMV) For non-linear performance functions, the MV solution is often not accurate enough. The AMV method [60, 62] takes the MV solution one step further and is capable of establishing full probability distributions even when the response function is not monotonic, which can result in truncated distributions. In the general failure probability or reliability methods, an iterative optimisation algo- rithm locates the MPP and then the response function is approximated by a Tay- lor expansion in the vicinity of this point. In the AMV method, the mean value approximation is used to locate the MPP and then, the corresponding failure 5. Probabilistic analysis methods 79 probability is calculated. Then, the analysis is performed again at this point in order to obtain the correct response value. Advanced mean value method with iteration In the AMV with iteration, the accuracy of the MPP location, as obtained with the mean value approximation, is improved with an iteration procedure. A new Taylor series expansion is then performed at the MPP obtained from the mean value solution. Another MPP may then be obtained by using the first-order fail- ure probability or reliability method, after which the analysis is performed again at this point. The analysis process is continued in this iterative manner until suf- ficient accuracy is achieved [52]. 5.6.5 Monte Carlo simulation (MCS) Often when a mathematical formulation for probabilistic structural/fracture me- chanics analysis can be defined, it is not possible to solve analytically. In these cases, a probabilistic solution may be obtained through Monte Carlo simulation (MCS). This simulation method is simply a repeated process of generating de- terministic solutions to a given problem. Each solution corresponds to a set of deterministic values of the underlying random variables. The main element of a MCS procedure is the generation of random numbers from a specified distribu- tion. There exist systematic and efficient methods for generating such random numbers from several common probability distributions [63]. Because a MCS solution generally requires a large number of repetitions, in particular for problems involving very rare events, its application to complex problems can be computationally costly. Hence, the MCS should be used with some caution. Often, MCS solutions may be the only means for checking or validating an approximate probability computation method [63]. Plain MCS offers a direct method for estimating failure probability. In es- sence, the technique involves sampling a set of values of the basic variables at random from the probability density function, and evaluating the failure function for the values to see if failure occurs. By generating a large number of samples, or trials, the probability density function is simulated, and the ratio of the num- ber of trials leading to failure to the total number of trials tends to the exact probability of failure [45]. In order to evaluate the failure probability corresponding to a known perfor- mance function, g(X i ), the MCS would consist of the following steps [64]: 5. Probabilistic analysis methods 80 1. Given the predefined probabilistic density functions (PDFs) of the ran- dom variables in the performance function, generate a single value of each variable. 2. Assess the performance function: if g(X i ) < 0, a system or component failure takes place. 3. Repeat steps 1 and 2 for N times. 4. Estimate the probability of failure by P f = N f /N, where N f is the num- ber of failures. In order to evaluate the failure probability corresponding to an unknown perfor- mance function, the MCS method would consist of the following steps [64]: 1. Given the predefined PDFs of the random variables involved in the de- terministic structural analysis (e.g., FEM), generate a single value of each random variable. 2. Perform the deterministic analysis, and record if failure is predicted. 3. Repeat steps 1 and 2 for N times. 4. Estimate the probability of failure by P f = N f /N, where N f is the num- ber of failures. MCS offers a number of advantages [65]: - The distributions of the model variables do not have to be approximated. - Correlations and other inter-dependencies can be modelled. - The level of mathematics required to perform a MCS is quite basic. - Commercial software is available to automate the tasks involved in the simulation. - Greater levels of accuracy can be achieved by simply increasing the number of calculated iterations. - MCS is widely recognised as a valid technique so its results are likely to be accepted. - The behaviour of the model can be investigated with great ease. The main drawback with the plain MCS is the computational effort involved. To produce a reasonably accurate estimate of the failure probability, at least 100/P f 5. Probabilistic analysis methods 81 trials are required. For instance, for failure probabilities around 10E-04, this requires that at least 10E+06 simulations are performed [45]. MCS methods rely on the use of random numbers, which are most convenient- ly generated numerically with conventional computers. A number of random number generator types are available. However, it is important to realise that all generator types produce pseudo-random numbers that form a long sequence of numbers which, although may be expected to pass all standard tests for random- ness, will eventually repeat. For most applications, standard random number generators, often available as functions in software libraries, are acceptable. However, there may be problems if a poor generator is used to generate many millions of samples in a problem involving a large number of basic (random) variables [45]. A variety of techniques have been developed to reduce the number of needed simulations. Generally speaking, these are called variance reduction techniques, and in favourable circumstances, they can be very efficient. These techniques include: - Importance sampling; modifying the sampling density function to ‘im- portant regions’ of the failure space, - Stratified sampling; the sampling space is divided into subspaces from which the sampling is performed, - Directional sampling; involves sampling along random vectors, - Adaptive sampling; successive updating of the sampling density function, - Axis orthogonal simulation; a semi-analytic technique. These techniques can also be combined together. Knowledge of the failure re- gion can be utilised to significantly improve the efficiency of MCS by tailoring the sampling scheme to the particular conditions. 5.6.6 Response surface methods When the performance function g of a system is not explicitly known, FORM or SORM are not directly applicable. Therefore, MCS seems to be a suitable ap- proach. The advantages of MCS are obvious, but an innate disadvantage of it is the formidable computational effort for problems involving low probability of failure or the problems that require a considerable amount of computation in each sampling cycle. To overcome this disadvantage, numerous variance reduc- 5. Probabilistic analysis methods 82 tion techniques have been proposed, such as those mentioned in the previous section. For certain problems, FORM or SORM can be effectively combined with MCS [66]. However, there are many practical problems that cannot be dealt with the above mentioned methods. Recourse must, therefore, be made to other method- ologies which may not be as accurate as the above methods but are nevertheless feasible for a wider spectrum of problems. The response surface method (RSM) is such a method [66]. The basic idea of the RSM is to approximate the performance function g, which may be implicit and/or very time consuming to evaluate, with a response surface function (RSF) that is easier to deal with. The RSF typically takes the form of a set of polynomials. Regression is usually performed to determine the RSF by the least squares method. After the response surface has been fitted at a sequence of sampling points, the reliability analysis can then be carried out. The crux of the RSM is to fit the RSF to g at the sampling points, in particular in the neighbourhood of the design point [66]. 5.6.7 Importance sampling (IS) The main idea of IS is to restrict the sampling domain to the tail parts of the joint distribution of the basic random variables. For instance, the sampling domain can be restricted so that they remain inside a sphere the radius of which is |, which is the distance from the origin to the MPP. No failures occur inside this sphere, as MPP is the point in the failure space that is closest to the origin. This method has to be combined with an optimisation algorithm or analytical reliabil- ity method to find the MPP. The purpose of the IS is often to verify or to im- prove the accuracy of a solution that is obtained by employing an analytical method [52]. The IS is a particularly useful tool when the failure surface has a disadvanta- geous shape, i.e. a convex shape that produces a considerably higher failure probability than the corresponding hyperplane. Also, when there exist singulari- ties at the failure surface or when there are several MPPs, e.g. if the failure sur- face is a circle the number of MPPs is infinite, IS is a very suitable analysis ap- proach [52]. The achieved advantage with the IS as compared to plain MC is the consider- ably reduced amount of computational work required. Various IS methods in- 5. Probabilistic analysis methods 83 clude sphere-based importance sampling, adaptive importance sampling, Latin hypercube simulation (LHS) and stratified sampling. Sphere-based importance sampling The principal in this method is to formulate a spherical surface, inside which no sampling is performed. The radius of this spherical surface is defined in relation to the MPP. The smaller the searched probability, the more efficient is the solu- tion, when compared to the MCS. The efficiency correspondingly decreases when the number of random variables increases, as the radius of the spherical surface becomes very small and hence the difference to the MCS remains very small. An example of the application of the sphere-based importance sampling is presented in Figure 5.6-2 [67]. Figure 5.6-2. Sampling techniques employed in the MCS (left) compared to the sampling technique employed in the sphere-based importance sampling (right), from ref. [67]. Adaptive importance sampling (AIS) AIS method is in principle similar to sphere-based importance sampling. The difference between these two methods is that in the AIS the region that limits sampling is modified in such a way that it stays in the so called safe side during the computational analysis. This is done because the first analysis result of the MPP is always approximate and thus one cannot entirely trust to the correctness of the limiting region that is based on it [67]. 5. Probabilistic analysis methods 84 5.6.8 Latin hypercube simulation (LHS) Models that contain a large number of parameters can be difficult to analyse using Monte Carlo methods. The difficulty arises in trying to obtain a repre- sentative sample of parameter values from their distributions. Having a large number of parameters requires a large number of Monte Carlo simulations to produce defensible results. One approach to obtaining a representative set of model output data is to use LHS to obtain a representative set of input samples to evaluate using the model. LHS is based on stratified sampling of probability distributions. The approach is to divide each distribution into n intervals of equal probability. In the simula- tions, one point from each interval is sampled, so that the interval specific loca- tion is chosen randomly. Thus if each distribution is divided into two parts and there are p parameters, there will be 2 to the power of p sampling intervals. When a random approach is used for selection, then the order of intervals from which the points are sampled is also selected randomly [68]. An alternative is to use the median of each interval in the analysis [69]. LHS generally gives better results in calculating the tails of the distribution of risk and requires fewer simulations relative to ordinary Monte Carlo analysis. This may be of considerable importance when the number of parameters is large [69]. 6. Risk analysis methods 85 6. Risk analysis methods 6.1 Introduction Risk analysis is a technique for identifying, characterising, quantifying and eval- uating hazards. It is widely used to support regulatory and resource allocation decisions. The main goals of risk management are to minimise the occurrence of accidents by reducing the likelihood of their occurrence, reduce the impacts of uncontrollable accidents and transfer risk (e.g. via insurance coverage). The estimation of probability or frequency of hazard occurrence depends greatly on the reliability of the system components, the system as a whole and human- system interactions [49]. There are two major parts in risk analysis [49]: - Determination of the probability of an undesirable event. - Evaluation of the consequence of this hazardous event. In general, there is a wide range of risk analysis methods and related theories. Here, the scope is to present a brief overview of this extensive topic. 6.2 Determination of risk values In most risk assessments, the likelihood of an event is expressed in terms of probability, P i , of that event. Alternatively, a frequency per year or per event (in units of time) may be used. Consequence, C i , is a measure of the impacts of an event. This can be in the form of mission loss, payload damage, damage to prop- erty, number of injuries, number of fatalities, etc. [49]. The results of risk estimation are then used to interpret the various contribu- tors to risk, which are compared, ranked and placed in perspective. The engi- neering definition of risk is now accepted as being the product of the likelihood and the consequence of the event [2]. Thus the risk assessment for an individual 6. Risk analysis methods 86 risk, R i , and the total risk, R, can be carried out by applying the following two equations [49]: i i i C P R = , for a possible event i in the examined system (6.2-1) ¿ = = n i i i C P R 1 , for all possible events in the examined system (6.2-2) 6.3 Qualitative, quantitative and semi-quantitative risk analysis approaches The risk analysis approaches can be divided to the following three categories [3]: - qualitative, - quantitative, - semi-quantitative. Qualitative risk analysis is based primarily on engineering judgements. The like- lihood and consequences of failure are expressed descriptively and in relative terms. In a qualitative approach risks are usually presented in a risk matrix as combinations of the likelihood and consequences. Semi-quantitative risk analy- sis determines single numerical values for the probability of failure and the con- sequences from every cause and effect. The analysis of accident scenarios should generally be more numerically based and detailed than the qualitative approach, but may still contain a large element of engineering judgement. In a fully quantitative risk analysis, the likelihood and consequences of equipment failure are determined for each accident scenario from the underlying distribu- tions of the variables using reliability analysis methods [3]. Both qualitative and quantitative risk assessment techniques are valuable for analysing the safety of nuclear and non-nuclear facilities. Their aim is to help to identify the important safety aspects of the design, operation, and maintenance of a facility by estimating the sources and magnitude of risk. Then, priorities for resource allocation can be established on the basis of risk [70]. Qualitative risk analysis approach Qualitative risk approaches are based on assigning subjective scores to the dif- ferent factors that are thought to influence the probabilities and consequences of 6. Risk analysis methods 87 failure. The scores are then combined using simple formulae to give an index representing the level of risk. The resulting indices for different components (e.g. pipe regions, failure modes, or hazards) can then be ranked to determine components with the highest risk [45]. Examples of the relative terms used in qualitative risk analysis are very unlikely, possible, reasonably probable and probable for likelihood of failure, and high, medium and low for consequences of failure. For qualitative analysis to be used consistently, criteria for the de- scriptive categories of likelihood and consequence of failure should be defined [3]. Quantitative risk analysis approach Quantitative risk analysis (QRA) systems are based on estimating the level of risk by direct assessment of the probability and consequences of failure. Most of the quantitative risk systems are based on Bayesian decision theory [45]. 6.4 Review of risk analysis procedures 6.4.1 Failure mode and effect analysis (FMEA) With the FMEA procedure, it is attempted to predict possible sequences of events that lead to a system failure, determine their consequences and device methods to minimise their occurrence or reoccurrence. FMEA is inductive in nature and, in practise, is used in all aspects of system failure analysis through- out the design process [49]. In general, FMEA procedure consists of a sequence of steps starting with the analysis at one level or a combination of levels of abstraction, such as system functions, subsystems or components. In FMEA analysis it is assumed that a failure mode is present and that it causes a failure. The effect of failure is then determined as well as the causative agent for the failure, i.e. the failure mecha- nism. The effect of the failure can be determined at various levels of abstraction starting at the component level. A criticality rating can also be determined for each failure mode and its resulting effect. The rating is normally based on the probability of the failure mode occurrence, the severity of its effect(s) and its detectability. Failures that score high in this rating can potentially be the source of system unreliability, and the corresponding failure modes should be removed [49]. One drawback of FMEA is that it provides very limited insight into probabil- istic representation of system reliability. Another limitation is that FMEA is performed for only one failure at a time. This might not be adequate for systems 6. Risk analysis methods 88 in which multiple failure modes can occur, with reasonable likelihood, at the same time. However, FMEA provides a lot of valuable qualitative information about the system design and operation [49]. The product of FMEA is a table of information that summarises the analysis of all relevant failure modes. 6.4.2 Fault tree Analysis Fault trees represent relationships between a fault in a system and the associated events. There are two ways in which fault trees are used in the life cycle of a safety or mission critical system. In the first case, fault trees describe the faults that can happen in a system and relate them to their causes. In the second case, fault trees are taken as specifications of a part of the system requirements. Fault trees are applied in diverse industries, such as nuclear energy, process control, aerospace and aviation [73]. The fault tree analysis is a deductive process whereby an undesirable event, called the top event, is postulated, and the possible means for this event to occur are systematically deduced. For example, a typical top event looks like “failure of control circuit A to send a signal when it should”. The deduction process is performed so that the fault tree embodies all component failures that contribute to the occurrence of the top event. It is also possible to include individual failure modes of each component as well as human errors during the system operation. The fault tree itself is a graphical representation of various parallel and sequen- tial combinations of faults that lead to the occurrence of the top event [49]. In general, there are three types of symbols in fault trees [49]: events, gates and transfers. The listing and meaning of these symbols can be found from text- books and handbooks [49, 71, 72], and are not presented here. When postulating events in the fault tree, it is important to include not only the undesired compo- nent states (e.g. applicable failure modes), but also the time they occur. A cut set consists of those basic events the occurrence of which leads to the failure of the system [32]. Determining the fault tree cut sets involves some straightforward Boolean manipulation of events. The quantitative evaluation of fault trees in- volves the determination of the probability of the occurrence of the top event. Accordingly, unreliability or reliability associated with the top event can also be determined. An example of a fault tree is shown in Figure 6.4-1. It shows an automotive brake system that consists of master cylinder, brake lining, wheel cylinder and brake fluid. The top event of this system is labeled as brake system failure. The 6. Risk analysis methods 89 system will fail in case of the occurrence of any one of the following events [72]: master cylinder fails to produce required pressure, insufficient amount of brake fluid to transmit the pressure to the wheel cylinder, wheel cylinders fail to provide adequate braking, and brake lining fails to provide adequate braking. Figure 6.4-1. An example of a fault tree for an automotive break system, from ref. [72]. 6.4.3 Event tree analysis The event tree analysis is a convenient tool for analysing systems the successful operation of which depends on an approximately chronological but discrete op- eration of its components or units. For a simple system, the use of this method may not be necessary, but for complex systems, such as NPPs, event trees are particularly useful [49]. The range of applicability of event tree analyses is as wide as that of fault tree analyses. The event trees are horizontally built structures that start from the left end, where the initiating event is modelled. This event describes a situation where a demand for the operation of a system occurs. Development of the tree proceeds chronologically, with the demand on each component or subsystem being postu- lated. The first component demanded appears first. At a branch point, the upper branch of an event shows the success of the event heading and the lower branch shows its failure. Event headings in event trees represent discrete states of the systems. These states can be represented by fault trees. This way the event tree sequences and the logical combinations of events can be considered. Evaluation 6. Risk analysis methods 90 of event trees is straightforward. Each branch in an event tree is evaluated to determine its probability. The probability of each overall outcome is given by multiplying together the individual probabilities of the branches leading to that outcome [49]. When the initiating events in a system have been identified, event trees and required fault trees are constructed for each group of initiating events. An event tree connects an initiating event with damage states. Systems and components used to model the accident sequence progression are placed along the top of the event tree. The assumed success or failure of these systems leads to multiple event tree branches that result in specified damage states. The event tree branch- es occur at branch points that specify whether or not the system or component along the top of the event tree is functioning [32]. A simple event tree example of a failure of the corrosion protection (CP) system in a pipeline possibly leading to pipeline rupture is illustrated in Figure 6.4-2 [45]. Figure 6.4-2. An example of a simple event tree, from ref. [45]. 6.4.4 Expert opinion The role of experts is important because their judgements can provide valuable information, particularly in view of the limited availability of technical data re- garding many important uncertainties in risk analyses. Because uncertainties are often represented in terms of probability distributions, the information provided by expert opinion is often treated as probabilistic distributions. The motivation 6. Risk analysis methods 91 for the use of multiple experts is simply the desire to obtain as much information as possible. The combination of probability distributions based on expert opinion summarises the accumulated information for risk analysts and decision makers. Procedures for combining probability distributions are often divided as mathe- matical aggregation methods or behavioural approaches [74]. These probabilities may be difficult to estimate even though reasonable engineering judgement is applied. This occurs because expert opinion under incomplete knowledge and limited data is inherently imprecise. In this case, the concept of uncertainty about a probability value is both intuitively appealing and potentially useful [75]. Application areas of expert opinion, also called expert judgement, are diverse, including nuclear engineering, aerospace, military intelligence, seismic risk and environmental risk from toxic chemicals [74]. Current PRA/PSA methodology uses expert opinion in the assessment of rare event probabilities [75]. The Delphi method [77, 78] is by far the most known method for eliciting and synthesizing expert opinions. The purpose and steps of the Delphi method de- pend on the nature of use. Primarily, the targets of application can be categorized into (1) technological forecasting, and (2) policy analysis. The technological forecasting relies on a group of experts on a subject matter of interest. The ex- perts should be those having most knowledge on the issues or questions of con- cern. The issues and/or questions need to be stated by the study facilitators or analysts or monitoring team and high degree of consensus is sought from the experts [76]. A more recent useful guideline concerning the use of expert judgement/panel for NPP applications is “ENIQ Recommended Practice 11: Guidance on Expert Panels in RI-ISI” [80]. According to this guideline the role of an expert panel is to synthesise the views of various experts and identify and characterise the un- certainties in their analyses. A structured approach is needed in order to find a balance between the (often contrasting) arguments of experts representing dif- ferent disciplines. Furthermore, an expert panel is important as it compels the different experts to openly discuss the technical bases of their arguments both among each other and with the decision maker. The guideline describes a com- plete approach for using an expert panel. This includes defining the role and responsibilities of the expert panel in RI-ISI as well as its composition. The de- scribed process covers planning, preparation and realisation of the expert panel, as well as the necessary associated documentation to be prepared. Concerning applications for Finnish NPPs, the expert panel approach has been used to combine the deterministic information on degradation mechanisms and 6. Risk analysis methods 92 probabilistic information on pipe break consequences. Then, the expert panel served both as a critical review of the preliminary results and as a decision sup- port for the final definition of risk categories of piping [79]. 6.4.5 Hazard and operability studies (HAZOP) HAZOP is a structured brainstorming exercise by a group discussion designed to identify potential variations and deviations from the technical design or operat- ing intent and their consequences. A list of guidewords is sometimes used to stimulate the discussions. Typically, these focus initially on credible variations of process parameters (flow, level, temperature) before branching out to consider human factors and less likely scenarios [3]. HAZOP is a procedural tool designed to highlight and identify hazards and operability problems in industrial plants that could reduce the ability of the plant to achieve productivity in a safe manner. Studies tend to be wide-ranging and threats to the integrity of pressure systems may only be considered briefly. This procedure is a tool for hazard analysis and it aims to ensure that weaknesses in the design intent are detected and acted upon [3]. 6.5 Risk based regulations, guidelines and standards There is a general trend towards the use of risk based regulations in areas where complex technological systems are in use. The use of quantitative risk analysis as a foundation for rational decision making is increasing in a number of engi- neering areas, e.g. aviation industry, space industry, nuclear industry, civil and marine engineering [81]. Some risk based regulations, guidelines and standards for risk analyses are listed below. The main emphasis is on nuclear industry, otherwise the selection is based on applicability of the documents in a wide range of areas. Regulations and guidelines: - U.S. Environmental Protection Agency: Policy for use of Probabilistic Analysis in Risk Assessment and Guiding Principles for Monte Carlo Analysis, EPA/630/R-97/001, EPA 1997. - CPR, Committee for the Prevention of Disasters, Methods for the cal- culation of damage, “Green book”. Voorburg: Ministry of social affairs and employment, 1990. 6. Risk analysis methods 93 - CPR, Committee for the Prevention of Disasters, Methods for deter- mining and processing probabilities, “Red book”. The Hague: SDU, 1997a. - CPR, Committee for the Prevention of Disasters, Methods for the cal- culation of physical effects, “Yellow book”. The Hague: SDU, 1997b. - USNRC: An Approach for Using Probabilistic Risk Assessment in Risk-Informed Decisions on Plant-Specific Changes to the Licensing Basis, Regulatory Guide 1.174, U.S. Nuclear Regulatory Commission, 1998. - USNRC: Use of Probabilistic Risk Assessment Methods in Nuclear Activities: Final Policy Statement, Federal Register, Vol. 60, p. 42622 (60 FR 42622), August 16, 1995. - Probabilistic Safety Analyses. YVL Guide 2.8. Radiation and Nuclear Safety Authority (STUK), 2003. - Management of Health and Safety at Work Regulations 1999. Ap- proved Code of Practice and Guidance L21 (Second edition). Health and Safety Executive (HSE), 2000. - Guide to Risk Assessment Requirements. Health and Safety Executive (HSE), 1996. - CEOC – Risk Assessment: A Qualitative and Quantitative Approach. Confédération Européenne d'Organismes de Contrôle. R 35/CEOC/CR1 87 Def. - Risk Based Inspection (RBI): A Risk Based Approach to Planned Plant Inspection. Health and Safety Executive (HSE) - Hazardous Installa- tions Division, 1999. Standards: - International Standard ISO 2394: General Principles on Reliability for Structures, Second Edition, 1998-06-01. - DS-Information DS/INF 85: Risk Analyses, requirements and termi- nology, Danish Standards Association, 1993. 6. Risk analysis methods 94 - European Standard EN 1050:1996: Safety of machinery – Principles for risk assessment, European Committee for Standardisation, 1996. - IEC International standard nr 60300-3-9: Dependability management- Part 3: Application guide- Section 9: Risk analysis of technological systems, International Electrotechnical Commission, 1995. - British Standard BS 8444: Risk management, Part 3 Guide to analysis of technological systems – application guide, British Standards Institu- tion, 1996. - Australian/New Zealand Standard AS/NZS 3931: Risk analysis of technological systems – application guide, Standards Australia, Stand- ards New Zealand, 1998. - Canadian Standard CAN/CSA-Q850-97: Risk management: Guideline for Decision-Makers, Canadian Standards Association, 1997. 6.6 Concerns with risk assessment In the following are briefly described some of the concerns that have been raised in the literature with risk assessment. It is important to be aware of the limita- tions of the used methodologies and the uncertainty or lack of confidence in the results. Inclusion of model uncertainty Model uncertainty is caused by the use of simplified or idealised mathematical models that are needed as operational tools in the risk analysis. By its very na- ture, model uncertainty is very difficult to assess and model, and it is often omit- ted. However, in many cases, when model uncertainty is allowed for it is an important influence on the evaluated probability [45]. If model uncertainty is for some reason omitted from a risk analysis, the re- sults should be interpreted with some caution. The tails of failure probability distributions The sensitivity problem of the tails of the failure probability distributions is a common concern within structural reliability analyses. In a risk analysis, it is the probability content defined by the shape of the distribution tails that most influ- 6. Risk analysis methods 95 ences the evaluated failure probability. By definition, data points located in the tail of the distribution are very unlikely to occur in a population of data. Thus, even in the fortunate situation where a large data sample is available to define the distribution for a basic variable, very few data points at the tail of the distribution influence the modelling of the variable. It is often pointed out that statistics based on data at the centre of a distribution carry no information about the extremes, and so extrapolation is inherently untrustworthy. This is clearly an important concern when physical mechanisms governing the shape of the extreme tails are different from those governing the central part of the distribution [45]. In a well constituted problem with well defined basic variable models, tail sensitivity is rarely a concern. However, in a particularly sensitive situation, or where the modelling for the most sensitive variables is limited, the sensitivity of the reliability can be examined by using different, valid probability distributions. Clearly, a valid distribution should fit the data well, and should comply with any physical constraints or limitations. Small failure probabilities and lack of data It has been pointed out that typical probabilities of failure often evaluated from structural reliability analyses have no conceivable physical meaning if interpret- ed in a frequency manner. By their very nature, failures of structural components and components of pressure systems, and corresponding failure data are rare. Often, to obtain sufficient data to assess a failure rate for a specific application, it is necessary to consider very broad categories [45]. The problem with too little data can be managed by deriving confidence limits that can be determined from statistical analyses. By carrying out this type of analyses on the various data sources, an estimation of the variability of such data can be established [3]. The lack of statistically sufficient amount of empirical data necessary to estimate new parameters is one important reason to apply expert opinion [49]. Validation The small failure probabilities for some components mean that evaluated failure probabilities cannot be properly and completely validated. To do so, it would either be necessary to observe a small number of similar components for a very long period, or to observe a very large population of structures for a shorter more practical period. Unfortunately, whilst this is to some extent possible for manu- factured items, it is not possible for most pressure systems which are typically 6. Risk analysis methods 96 more unique items, thus there is no population of nominally identical structures under nominally identical conditions that might be observed. However, for most component types there is a large enough population of components of similar type for at least allowing to obtain some crude comparative values. It may also be possible to calibrate the failure models from test data [45]. Completeness There can never be a guarantee that all accident situations, causes, and effects have been considered. Indeed, a number of failures have occurred because the scenario was not and could not have realistically been envisaged. Therefore it is important to undertake rigorous hazard identification studies. 7. Deterministic degradation modelling methods for power plant components 97 7. Deterministic degradation modelling methods for power plant components 7.1 Introduction Modelling methods of the major degradation mechanisms concerning power plant components are considered in the following. Of these mechanisms, the three most often encountered ones in NPP environments are fatigue, SCC and irradiation embrittlement [32]. However, of the NPP types and components, only the PWR RPVs and LWR internals are susceptible to the last one of these. In conventional power plants, irradiation embrittelement is obviously not an issue. However, due to relatively high operational temperatures, in conventional power plants also creep has to be often taken into account in addition to fatigue and corrosion. A brief overview is also presented of the various commonly applied compo- nent structural integrity assessment methods. These include methods that are defined in fitness-for-service procedures, codes, standards and research arti- cles/reports, and which have been developed by both research institutes and pri- vate companies. 7.2 Fatigue Fatigue is the progressive, locally confined, and permanent structural change that occurs in a material subjected to repeated or fluctuating strains at nominal stresses that have maximum values less than the static yield strength of the mate- rial. Fatigue may lead to emergence of cracks and cause fracture after a suffi- cient number of fluctuations. Fatigue damage is caused by the simultaneous action of cyclic stress, tensile stress, and plastic strain. If any one of these three is not present, a fatigue crack will not initiate and propagate. The plastic strain 7. Deterministic degradation modelling methods for power plant components 98 caused by cyclic stresses initiates the crack, and tensile stresses promote crack growth. In the process of fatigue failure in an originally intact metal, micro- cracks arise, coalesce or grow to macro-cracks that propagate until the fracture toughness of the material is exceeded and final fracture will occur [7]. One of the most successful applications of the theory of fracture mechanics is in the characterisation of fatigue crack propagation. For an overview of the frac- ture mechanics based analysis methods, see Section 4.3. A fracture mechanics based analysis of fatigue flaw growth inevitably requires a thorough understand- ing of the assumptions, significance and limitations underlying the development of various crack tip parameters. In mechanical components, like boiler tubes, the principal sites of fatigue crack nucleation include voids, inclusions, dents, scratches, forging laps and folds, macroscopic stress concentrations, as well as regions of micro-structural and chemical non-uniformity [97]. All in all, there are several computational approaches for the cumulative fa- tigue damage analysis. Since the introduction of damage accumulation concept by Palmgren [98] and linear damage rule by Miner [99], both published several decades ago, a multitude of cumulative fatigue damage models have been devel- oped. These models can be divided into six categories [100]: - linear damage rules, - non-linear damage curves and two-stage linearisation approaches, - life curve modification methods, - approaches based on crack growth concepts, - continuum damage mechanics models, and - energy based theories. Approaches based on crack growth concepts and fracture mechanics are covered in more detail here. That is because most of the other cumulative fatigue models consider fatigue crack initiation or it reaching a macroscopic but still relatively small size as the limiting criteria, rather than the crack propagating to some lim- iting depth, through wall or to a size leading to plastic collapse. In case of cumu- lative continuum fatigue models it is also important to note that the order of loading events in load cycle sequences formed and assembled with commonly used procedures deviate from the actual chronological load history. A typical example of this approach is the often used rainflow load cycle counting proce- dure in its different modifications. As for fracture mechanics based crack growth computations, they strictly require the realistic chronological order of the load 7. Deterministic degradation modelling methods for power plant components 99 history to be preserved, as the magnitude of each crack growth increment is de- pendent both on load amplitude and the current crack size. The procedures for analysing constant amplitude fatigue under small scale yielding conditions are fairly well established, although a number of uncertain- ties remain. Variable amplitude loading, multi-axial loading, large scale plastici- ty, crack closure, overloads, weld residual stresses and short cracks introduce additional complications that are not yet fully understood. Of the numerous developed cumulative fatigue damage models only some most notable and/or commonly used ones are considered here. The method relating the stress level and number of load cycles, S-N curves, is used to predict the number of cycles to failure at a single stress level. The S-N curves conveniently display basic fatigue data on a plot of cyclic stress level versus the number of cycles to failure. Analytical representation of S-N curves is given in the form [32]: k NS b = (7.2-1) where b and k are material parameters estimated from test data obtained using identical specimens. Linear damage rules The Palmgren-Miner rule (or Miner’s rule) is a linear damage accumulation rule used to predict the cycles to failure under variable amplitude loading. The Palmgren-Miner rule asserts that the damage fraction A i at any stress level S i is linearly proportional to the ratio of n i , the number of cycles of operation under this stress amplitude to N i , the total number of cycles that would produce a fail- ure at that stress level. The damage fraction is computed as follows [99]: i i i N n = A (7.2-2) where i i N n s . If the stress amplitude is changed, a new partial damage is cal- culated for this new amplitude level, where the appropriate N i is found form the S-N curve. The total accumulated damage D is then given by [99]: 7. Deterministic degradation modelling methods for power plant components 100 ¿ ¿ = A = i i i i i N n D (7.2-3) and failure is assumed to occur when 1 > D . The main deficiencies with linear damage rule used are its load level independence, load sequence independence and lack of load interaction accountability. Non-linear damage curves and two-stage linearisation approaches Attempting to overcome the inaccuracies of linear damage rules, non-linear damage rule procedures have been developed. A representative and early exam- ple of non-linear damage rules is as follows [101]: ¿ ¿ | | . | \ | = A = i i x i i x i i i N n D (7.2-4) where x i is an exponent parameter related to the ith loading level. As for two-stage linearisation approaches, they provide improvement to the shortcomings of the linear damage rule, while still retaining its simplicity. In the two-stage linearisation approaches, the fatigue damage is divided to two phases, those being crack initiation and crack propagation. A representative example of a double linear damage rule is as follows [102]: ¿ | | . | \ | = i i I I I N n D crack initiation phase, (7.2-5a) ¿ | | . | \ | = i i II II II N n D crack propagation phase, (7.2-5b) where N I = N – N II , , , o N N N B N r II · · = , N r is a reference level for N, B = 0.65, and o = 0.25. Here, the two constants B and o are determined from regression analysis of the experimental data. The presented values for them mainly corre- spond to high strength steels. 7. Deterministic degradation modelling methods for power plant components 101 Life curve modification methods In the life curve modification methods, the conventional S-N curve is modified to better follow the fatigue test data. An example of this approach accounting for load interaction effect is [103, 104]: , , ¿ = E ´ 1 i i N N | (7.2-6) where N E is the total accumulated life, whereas | i and N i ´ are the frequency of load cycles (n i /N E ) and the modified life associated with loading level o i , respec- tively. The experimental verification results show that this model can provide accurate predictions of fatigue lives under repeated block loading. This predic- tive theory is also expanded to stochastic loading histories. Continuum damage mechanics models Continuum damage mechanics deals with the mechanical behaviour of a deterio- rating medium at the continuum scale. The success of continuum damage me- chanics application in modelling the creep damage process has led to extend this approach to ductile plastic damage, creep-fatigue interaction, brittle fracture and fatigue damage. For the one-dimensional case, the cyclic fatigue damage evolu- tion can be generalized by a function of the load condition and damage state. The following non-linear damage evolution equation has been developed as based on measured changes in tensile load carrying capacity and using the effec- tive stress concept [110, 111]: , , , , j ¦ , , , , j ¦ , , | o | o + ÷ + ÷ ÷ ÷ = ÷ ÷ = 1 1 1 1 1 1 1 1 1 1 1 1 r N n D f (7.2-7) where n is number of load cycles, N f is number of load cycles to failure, | is a material constant and o is a function dependent on the stress state. This damage model is highly non-linear in damage evolution and is able to account for the mean stress effect. This model allows for the growth of damage below the initial fatigue limit, when the material is subjected to prior cycling above the fatigue limit. The model is able to take into account the influence of initial hardening effect by introducing a new internal variable which keeps track of the largest plastic strain range in the prior loading history. In addition, the mean stress ef- fect is directly incorporated in the model. However, since a scalar damage varia- 7. Deterministic degradation modelling methods for power plant components 102 ble is employed and the model is written in its uniaxial form involving the max- imum and mean stresses, difficulties will inevitably occur when the model is extended to multiaxial loading conditions. It has been noted that there is a connection between stress-strain cycling loop hysteresis energy and fatigue, which in turn led to the development of energy based cumulative damage theories, mainly considering strain energy [105]. It has been realised that an energy based damage parameter can unify the damage caused by different types of loading such as thermal cycling, creep, and fatigue. Energy based damage models also allow to include mean stress and multiaxial loads since multiaxial fatigue parameters based on strain energy have been de- veloped [106, 107]. Energy based damage theories By examining constant strain amplitude test data, it was found according to refs. [108, 109] that while the values of the cyclic strain hardening coefficient changed during the cycling process, the corresponding change for cyclic strain hardening exponent was negligible. This led to the development of the following cyclic stress-strain relation [109]: | c o r K n p * 2 2 | | . | \ | A = A - (7.2-8) where Ao is stress range, Ac p is plastic strain, K * and n * are cyclic strain harden- ing coefficient and exponent determined near failure (r = n/N f = 1 ), and | is the cyclic hardening rate. The expression for | is given as: b p a | | . | \ | A A = 2 2 c o | (7.2-9) where a and b are constants, the values for which are obtained as based on ex- perimental data. The exponent b is dimensionless, whereas the dimension of the parameter a is such that its product with Ao [MPa] and Ac p [%] is dimensionless. The incremental rate of plastic strain energy, dW/dN, is then derived as: 7. Deterministic degradation modelling methods for power plant components 103 * 1 2 1 1 4 n p K r n n dN dW + - - - | | . | \ | A | | . | \ | + ÷ = c | (7.2-10) where again r = n/N f and the energy accumulation is defined by introducing a parameter called the fraction of plastic strain energy, u = W/W r = r 1+| . Finally, the fatigue damage function is expressed as: , , , , j ¦ , , o | o + + · + = u = ' 1 1 ' 1 n n r D (7.2-11) where , , a b p · A A = 2 4 c o o and n´ is the cyclic strain hardening exponent. This model represented by equation (7.2-11) is a non-linear, load dependent damage accumulation model. It accounts for the load interaction effect and the change in strain hardening through the stress response. This damage model is particularly suitable for materials which exhibit cyclic hardening. Fracture mechanics based crack growth computation procedures Fracture mechanics can be used to model fatigue of components. An overview concerning fracture mechanics is presented in Section 4.3. Materials with rela- tively low fracture resistance that fall below their so called plastic collapse strength can be analysed on the basis of elastic concept through the use of LEFM. For other materials, the use of EPFM is often necessary. Fracture will occur when the stresses at the crack tip become too high for the material to bear. Under linear-elastic circumstances, the stress intensity factor K determines the entire crack tip stress field, whereas in case of ductile material accompanied with considerable plastic deformation J-integral applies. The cor- responding fracture toughness/resistance, J R , tends to increase during the fracture process, so that fracture will be slow and stable initially, until at some point in- stability occurs causing the fracture to become fast and uncontrollable. Thus, stable fracture may slow down and stop unless the stress level is further in- creased [32]. The growth of the crack that eventually leads to fracture occurs by mecha- nisms entirely different from the fracture itself. During most part of the cracking process, the crack is much smaller than the one that would cause fracture at the prevailing stress. Therefore, the crack tip stress field is less severe, and the size of the plastic zone smaller than at the time of fracture. Due to this small plastic zone, LEFM can often be applied. If the crack size is close to critical (fracture), 7. Deterministic degradation modelling methods for power plant components 104 this may no longer be true, but because by far the longest time is spent in the growth of much smaller cracks, it is justifiable to use LEFM to obtain the time for crack growth with good accuracy [7]. For fatigue, the crack propagation equations are models that relate the crack growth rate to the level of cyclic stress. These fracture mechanics based compu- tation procedures can be further divided to those intended for constant and vari- able amplitude loading, respectively. Apparently the models of the latter type can also be applied with sufficient accuracy to those of the former. However, this should be case specifically checked. One of the earliest and also most often used fatigue crack propagation models is the Paris-Erdogan equation. It is an empirical formula that relates the cyclic crack growth rate to stress intensity factor range, as follows [112]: , , m K C dN da A = (7.2-12) where a [mm] is crack depth, N [ - ] is the number of load cycles, AK [MPa\m] is the stress intensity factor range, i.e. K max - K min , whereas C and m are material and environment specific constants. The exponent m is dimensionless, whereas the dimension of the parameter C is such that its product with (AK) m is mm. The Paris-Erdogan equation assumes that the crack growth depends only on the stress intensity factor range. It also assumes that the stress amplitude is constant and that it is small enough so that LEFM is applicable and that the crack growth rate is independent of the previous load history. Failure occurs when the stress inten- sity factor exceeds the fracture toughness. In addition, the Paris-Erdogan equa- tion describes crack growth only at intermediate values of fatigue crack growth curve, see Region II in Figure 7.2-1 below. Region II represents the intermediate crack propagation zone where the length of the plastic zone ahead of the crack tip is long compared with the mean grain size, but much smaller than the crack length, whereas Region I contains the stress intensity factor range threshold be- low which fatigue cracks do not propagate and Region III is characterised by rapid and unstable crack growth just prior to final failure [114]. A fatigue damage propagation model proposed by Forman [113], which co- vers both the intermediate and high AK regions, is: 7. Deterministic degradation modelling methods for power plant components 105 , , , , , , , ,, , max 1 1 K K R K C K K R K C dN da C m F C m F y y ÷ ÷ A = A ÷ ÷ A = (7.2-13) where C F and m y are material and environment specific constants, K C [MPa\m] is fracture toughness and R = K min /K max is stress ratio. Equation (7.2-13) indicates that as K max approaches K C , crack growth rate da/dN in turn tends to infinity. With equation (7.2-13), it is possible to predict both stable intermediate crack growth and accelerated crack growth rates for various stress ratios. Figure 7.2-1. Typical fatigue crack growth behaviour in metals. McEvily [115] developed another equation that can be fit to the entire fatigue crack growth curve, as follows: , , | | . | \ | ÷ A + A ÷ A = max 2 1 K K K K K C dN da C th (7.2-14) where K th [MPa\m] is crack propagation threshold. Equation (7.2-14) is based on a simple physical model rather than a purely empirical fit. The expression m II III I Fracture Threshold | . | \ | dN da log K C log(AK) AK th 7. Deterministic degradation modelling methods for power plant components 106 distinguishes between two crack advancement modes; ductile striation and static mode. It is noted that for tests in non-aggressive environments the number of adjustable parameters becomes zero when AK values are large with respect to AK th [114]. In general, it is considered that fatigue cracks do not grow when being closed, which can be the case when the surrounding stress field has turned to compres- sion. As there is no unique definition of crack closure or crack opening, due to the gradual nature of the crack behaviour between the points of complete open- ing and closure, usually an average closure/opening stress intensity factor, K, is employed. As inconsistencies in assessing the K values for the first contact be- tween the crack flanks could occur due to the fracture surface asperities, the use of an average value is recommended. The difference between the K values con- cerning opening, K op , and closure, K cl , is quite insignificant, thus using a single value K cl has proven to be an adequate approximation. According to Elber [116], a crack only propagates while its flanks are separated, and is then driven by the effective stress intensity factor range, AK eff , as follows: K U K K K cl eff A = ÷ = A max (7.2-15a) where U is a factor depending primarily on the stress ratio R. By replacing the stress intensity factor range with the effective stress intensity factor range of equation (7.2-15a) in the Paris-Erdogan equation (7.2-12) yields [116]: , , , , m m eff K U C K C dN da A = A = (7.2-15b) Concerning the factor U above, Shih and Wei [117] have reported a definite dependence on K max , whereas Hudak and Davidson [118] attributed the confu- sion and controversy over the effect of loading variables on closure to experi- mental factors. McClung [119] has reported of three distinct K dependent re- gimes of crack closure and concluded that no single equation could describe the closure in all three regimes. Other attempts have included linking U to the spec- imen geometry, stress state and environment. Wheeler [120] developed a fatigue damage propagation model based on yield zone concept for tensile overloads. To account for crack growth retardation, a retardation parameter, C p , is introduced. This retardation parameter is dependent 7. Deterministic degradation modelling methods for power plant components 107 on the current plastic zone, overload plastic zone and a shaping exponent. The Wheeler model equation together with definitions for its parameters can be ex- pressed as [120]: , , m p K C C dN da A = (7.2-16) where: C p is retardation parameter, 1 m p y p a a R C | | . | \ | ÷ = for (a + R y ) < a p , and C p = 1 for (a + R y ) > a p , R y is extent of current yield zone, a p is the sum of a and largest prior yield zone, (a p ÷ a) is distance from a crack tip to the boundary of the yield zone caused by the last tensile overload in a sequence, m1 is shaping exponent, C, m are defined in connection of Paris-Erdogan model, see equation (7.2-12). This model allows to use other crack propagation formulations as well, thus the right side of equation (7.2-16) can be written as: C p f(AK), where f(AK) is a func- tion describing the crack growth. When using the model, among the needed in- put data is initial crack length, a 0 , and the computation process follows the load cycles, i.e. crack grows increment by increment. The limitation of this model [120] is that the values for the shaping exponent must generally be obtained experimentally. The model considers mainly variable amplitude loading condi- tions. Further, this model does not predict acceleration caused by compressive overloads (underloads). Johnson [121] has developed a systematic technique for modeling fatigue crack growth under variable amplitude loading to account for interaction effects. The procedure includes a multi-parameter yield zone model, which accounts for crack growth retardation, acceleration and underload effects by decreasing or increasing the stress ratio. Modified Forman type crack growth equation was used in developing the model, as follows: 7. Deterministic degradation modelling methods for power plant components 108 , , , , K K R K C dN da C eff m A ÷ ÷ A = 1 (7.2-17) where: m = 1 at R 0, and n = 2 at R < 0, threshold K range for variable amplitude loading; , , th eff th K R K A ÷ = A - 1 , eff eff red red eff K K K K K K R max, min, max min = ÷ ÷ = , maximum allowable stress ratio; 6 . 0 2 . 0 max + | . | \ | = t Z R OL , Z OL = plastic zone diameter for the applied K max , t = material wall thickness. This model [121] includes the assumption that the load interactions are a result of the residual stress intensity due to plastic deformation at the crack tip. Further, constant amplitude loading data was used for computation of threshold AK under variable amplitude loading. Wang et al. [122] proposed a simple model for the fatigue crack growth rate which considers the plastic component of the J-integral as a damage factor. The proposed damage accumulation theory assumes that: the total plastic strain ener- gy density absorbed by the material is prior to reaching its ultimate state constant to be determined experimentally, the elastic strain energy density stored by the material does not cause damage and is released upon unloading. This model is expressed as: , , , , 2 2 2 2 2 2 4 max 1 1 1 1 1 1 1 1 ( ( ¸ ( ¸ ÷ ÷ ÷ ( ¸ ( ¸ ÷ ÷ ÷ | | . | \ | = ç ç ç ç o o R R K dN da y (7.2-18) where: , , eff K K 2 2 max 2 = ç , a K y eff t o = , 7. Deterministic degradation modelling methods for power plant components 109 o = constant determined according to the material and loading conditions, y o = local yield strength, o y = average local yield strength. According to this fatigue damage model [122] the accumulated plastic strain energy will cause the material damaging and lead to failure. This plastic damage only occurs after the maximum elastic component has been reached. Based on the research to obtain the procedure equation (7.2-18), it was concluded that the crack growth rate is not only a function of AK, but also of average local yield strength, fracture toughness and amplitude of the applied effective stress intensi- ty factor in Regions II and III. The mode1 was verified using the test results published in ASTM STP 789 [123]. The test data and equation predictions were then found to be in reasonably good agreement. A more advanced development is the fatigue crack growth rate model by Forman and Mettu [124], which follows a cycle-by-cycle integration method as using the sigmoidal crack growth rate relationship, and which model is now also included in the quite recently published European Fitness For Service Network (FITNET) structural integrity assessment procedure/guideline [125]. The model equation is given as: , , , , q C p th n K K K K K R f C dN da max 1 1 1 1 ÷ A A ÷ ( ¸ ( ¸ A | . | \ | ÷ ÷ = (7.2-19) where C, n, p and q are parameters for which material specific values are given in the procedure documentation. The values for parameters f, AK th , K max and K C are obtained from associated functions which are dependent e.g. on stress ratio R, constraint conditions and thickness. The FITNET procedure covers a wide range of materials and takes into account the crack closure effect on the crack growth behaviour in more detail. On the other hand, the procedure contains many parameters the values for which need to be obtained experimentally in order to assure reliable computations. 7. Deterministic degradation modelling methods for power plant components 110 Discussion on damage caused by multiaxial fatigue loading Power plant components can be subjected to multiaxial fatigue loading, in which the cyclic loads in various directions may be applied at different frequencies and/or with a phase difference. In these circumstances of non-proportional mul- tiaxial loading, the corresponding principal directions and/or principal stress ratios vary during a loading cycle or loading block. During recent decades, fa- tigue damage evaluation of components/structures under non-proportional multi- axial loading has become a growing research interest as conventional multiaxial fatigue criteria are based on proportional fatigue data, and hence, they are not applicable for non-proportional loading due to changing directions and/or ratios of the principal stresses [126]. The multiaxial fatigue degradation models proposed in the literature may be categorised into three groups: stress based, strain based and energy based meth- ods [127, 128, 129]. In their comprehensive more recent review on the subject, Marquis and Socie [130] consider methods based on cyclic J-integral as a sepa- rate group from the energy based methods. For long life fatigue problems, most of the multiaxial fatigue criteria are stress based. In order to handle non- proportional loading effects on fatigue resistance, many new methodologies have been developed. They are based on various concepts, such as the critical plane approach, integral approach, mesoscopic scale approach, etc. [126]. An example of multiaxial fatigue degradation models based on equivalent stress intensity is presented in the following. For mixed mode loading, the fa- tigue crack growth rate may also be expressed by a simple Paris-Erdogan type model, see equation (7.2-12) earlier, where the stress intensity factor range is replaced by equivalent stress intensity factor range, AK eq , so that the model equation becomes: , , m eq K C dN da A = (7.2-20) The basis for the model by Tanaka [131] is that displacements behind the crack tip reach a critical value. This leads to the following formulation for the equiva- lent stress intensity factor range: , , j ¦ 25 . 0 4 4 4 1 8 8 v ÷ A + A + A = A III II I eq K K K K (7.2-21) 7. Deterministic degradation modelling methods for power plant components 111 where AK II and AK III are the ranges for mode II and III stress intensity factors, see Section 4.3, and v is Poisson’s coefficient. The material constants in equa- tion (7.2-20) are taken from standard mode I crack growth experiments. Crack growth under other loading conditions can be obtained with the use of the ap- propriate equivalent stress intensity factor range given by equation (7.2-21). Other forms of the equivalent stress intensity factor range have also been used. As for strain energy density fatigue degradation models, Sih [132] proposed a criterion for mixed mode loading known as the strain energy density factor, S. It is based on the strain energy density around the crack tip and is expressed as: 2 33 2 22 12 2 11 2 III II II I I K a K a K K a K a S + + + = (7.2-22) where the coefficients a 11 , a 12 , a 22 and a 33 under plane strain are dependent on elastic modulus, Poisson's coefficient and the inclination angle of the crack, u. Crack extension occurs when the strain energy density factor reaches a critical value in a direction defined by u 0 . This will be the direction of minimum strain energy density. For cyclic loading a cyclic strain energy density factor range is defined as: , , , ,, , j + A + A + A = A II I I II I I K K K K a K K a S mean mean 0 12 mean 0 11 2 u u (7.2-23) , , , , ¦ III III II II K K a K K a A + A + mean 0 33 mean 0 22 u u This equation includes both stress range and mean stress. Before computing the strain energy density factor, the direction of crack growth u 0 must be determined from the necessary and sufficient conditions for crack growth, as follows: 0 = c c u S at u = u 0 (7.2-24) 0 2 2 > c c u S at u = u 0 Then the crack growth rate can be computed as: , , 2 m S S C dN da A = (7.2-25) 7. Deterministic degradation modelling methods for power plant components 112 where the values of the constants can be determined from the standard crack growth constants in equation (7.2-20) as follows: , ,, , 2 1 2 1 2 m S E C C ( ¸ ( ¸ + ÷ = v v t (7.2-26) where v is Poisson’s coefficient. Concerning equivalent strain intensity models, any of the equivalent stress intensity models could be used with the appropriate substitution of EAc or GA¸ for the stress, where Ac is strain range, G is shear modulus and A¸ is shear strain range. The strain intensity factor range, AK I (c) is often used for elastic-plastic loading conditions. Further, an effective strain in- tensity factor range, AK eq (c), can be written in terms of the strain ranges and crack geometry factors for modes I and II, the latter two being denoted as F I and F II . Equation (7.2-27) gives the effective strain intensity factor range based on the strain energy release rate as [130]: , , , , , , a E F E F K I II eq t c ¸ v c 5 . 0 2 2 1 2 ( ( ¸ ( ¸ A + | | . | \ | A + = A (7.2-27) In cyclic J-integral approach, crack growth is analysed by assuming that the cyclic J-integral range, AJ, controls the crack growth as [130]: , , 2 m J J C dN da A = (7.2-28) For elastic loading, AK and AJ are related and the constant C J can be evaluated as: 2 m J CE C = (7.2-29) Alternatively, crack growth data could be fitted in terms of AJ to obtain the con- stants directly. 7. Deterministic degradation modelling methods for power plant components 113 7.3 Corrosion Of the various identified forms of corrosion, see Section 3.4, modelling of those which mainly concern power plant components are considered here. These cor- rosion mechanisms are general corrosion, pitting and stress corrosion cracking (SCC). Of the numerous models that have been developed for these corrosion mechanisms, only some notable ones are presented here. 7.3.1 General and pitting corrosion General corrosion is the result of a chemical or electro-chemical reaction be- tween a material and its environment. It is typically characterised by uniform attack resulting in material dissolution and sometimes corrosion product build- up. At ordinary temperatures and in neutral or near neutral media, both oxygen and moisture must be present to corrode steel. Generally, when humidity ex- ceeds 70 %, a moisture film forms on the steel surface providing an electrolyte [133]. The ratio between humidity and temperature determines the thickness of the film. The thinner the film, the easier the diffusion of oxygen through the film that drives the corrosion reaction. Corrosion rates of carbon and low-alloy steels increase with rising surface temperatures until the electrolyte is evaporated [134]. For instance, in LWRs the steam generator tubes are susceptible to gen- eral corrosion [4]. Pitting corrosion is a localised corrosion attack in aqueous environments con- taining dissolved oxygen and chlorides. This corrosion degradation is more common in austenitic stainless steels than in carbon steels. When passivity breaks down at a spot on a surface, an electrolytic cell is formed with the anode at the minute area of active metal, and the cathode at the larger area of passive metal. The large electric potential difference between the two areas accounts for considerable flow of current with rapid corrosion at the anode. The anode does not spread because it is surrounded by passive metal, and as the mechanism con- tinues it propagates deeper into the metal forming a pit [134]. For instance, in LWRs piping components of stainless steel are susceptible to pitting corrosion [4] A number of computation procedures for general corrosion rates using electro- chemical and thermo-dynamic models exist both for carbon steels and austenitic stainless steels, see e.g. refs. [135, 137]. However, such models are not suitable for mechanical analyses of e.g. wall thinning due to corrosion, as the computed results mainly concern corrosion potential parameters, such as current density at 7. Deterministic degradation modelling methods for power plant components 114 the corrosion site. A straightforward equation for assessment of general corro- sion propagation depth, d(t), is [138]: , , n Ct t d = (7.3-1) where t is time since the start of corrosion process, C is effective corrosion rate in absence of coating, and n is power law exponent for non-linear behaviour (where applicable), with C and n being empirical material and environment spe- cific parameters. A number of electro-chemical and thermo-dynamic models exist for pitting corrosion. The model for assessment of pitting propagation rate, V, by Engel- hardt and Macdonald [139] is: , , , , n t t V t V 0 0 1+ = (7.3-2) where V 0 is initial pitting propagation rate, t is time since the start of corrosion process, t 0 is constant, and n is empirical material and environment specific con- stant. It is important to note that in many cases the period of time over which the approximation; V(t) V 0 = constant, is valid can be comparable with the obser- vation time (or even with the service life of the component/system). 7.3.2 Stress corrosion cracking (SCC) Crack propagation is the favoured action once the stress intensity factor exceeds a critical value, which is mainly determined by the surface energy on the plane of crack propagation, and by the energy dissipated in the plastic zone ahead of the crack. In SCC, the high chemical corrosion rate at the crack tip is driven by the fast diffusion rate of embrittling impurities towards the crack tip. The surface energy is thereby reduced and the critical stress intensity factor decreases. This is an unstable process, which is possibly limited by the corrosion and/or the dif- fusion rate of impurities [32]. The computation models describing SCC and fatigue induced crack growth for intermediate region are often similar. The data resulting from SCC tests have been presented in both S-N curves and fracture mechanics formats [32]. Howev- er, often the scatter of SCC data plotted as crack propagation rate against K I is remarkable, see e.g. ref. [140], thus leaving more accurate assessment of the dependence between these two parameters unclarified. 7. Deterministic degradation modelling methods for power plant components 115 In simple fracture mechanics based treatments, SCC and fatigue crack growth computation procedures it is assumed that a stress field created at the crack tip causes the crack propagation. Note, that for fatigue, the time variable of interest is measured in number of load cycles, whereas for SCC it is measured in time units. The stress intensity factor K for SCC computations is based on a single tensile stress, whereas the stress intensity range AK is based on an alternating stress for fatigue crack propagation [32]. An illustration of SCC propagation rate as a function of K for metallic materials is shown in Figure 7.3-1. The mathematical expression for intermediate (Stage 2) SCC rate equation is [136]: n I CK dt da = (7.3-3) where a is crack depth, t is time and K I is the crack tip value of type I stress inten- sity factor, whereas C and n are material and environment specific constants. The popularity of this SCC equation is based on its simplicity and availability of values for constants C and n, which in turn are assessed based on experimental data. Figure 7.3-1. An illustration of SCC propagation rate as a function of K for metallic materials, showing the stages 1, 2 and 3 as well as the plateau velocity. 2 1 3 subcritical crack propagation total failure K C K th K plateau dt da dt da 7. Deterministic degradation modelling methods for power plant components 116 When considering smaller scale structure of metallic materials, two SCC modes are identified, namely intergranular SCC (IGSCC) and transgranular SCC (TGSCC), as mentioned earlier in Section 3.4. Values for material and environment specific constants C and n are presently available only for IGSCC. However, this SCC mode is encountered much more often in power plant environments than TGSCC. A comprehensive explanation for Stage 1 behaviour has this far not been de- veloped. According to one approach the crack tip strain rate increases rapidly with K. According to another approach the crack tip corrosion rate and transport of species within the crack increase rapidly with increasing crack volume and hence with K [33]. A further developed and in some applications useful modification of the Stage 2 SCC rate equation that also takes into account K threshold is [141, 142]: , , | o th ref g K K T T R Q dt da ÷ ( ( ¸ ( ¸ | | . | \ | ÷ ÷ = 1 1 exp (7.3-4) where: g Q is thermal activation energy for crack growth = 130 kJ/mol, R is universal gas constant = 8.314E-03 kJ/mol·K, T is absolute operating temperature at crack location [K], T ref is absolute reference temperature used to normalise data = 598.15 K (325 °C), o is crack growth rate coefficient = 2.67E-12 at 325 °C for da/dt in units of m/s and K in units of MPam, K is crack tip stress intensity factor [MPam], K th is crack tip stress intensity factor threshold = 9 MPam, | is exponent = 1.16. Equation (7.3-4) is mainly applicable to alloy Inconel 600, which is a commonly used material for NPP pressure boundary components [141, 142]. A similar approach as equation (7.3-4) has been developed for Alloys 182 and 82 which are also commonly used NPP component materials. This equation is [143, 144]: 7. Deterministic degradation modelling methods for power plant components 117 , , | o K f f T T R Q dt da ref g n orientatio alloy 1 1 exp ( ( ¸ ( ¸ | | . | \ | ÷ ÷ = (7.3-5) where: f alloy = 1.0 for Alloy 182 and 0.385 for Alloy 82, f orientation = 1.0, except for crack propagation that is clearly perpendicular to the dendrite solidification direction, where f orientation = 0.5, and otherwise the parameter explanations/values are the same as for equation (7.3-4). Such more advanced SCC propagation models have been developed that take also into account electro-chemical characteristics and consequent behaviour of protective oxide layer. An example of these models is the SCC propagation equation by Shoji et al. [145]: , , × | . | \ | + ( ¸ ( ¸ | . | \ | ÷ | | . | \ | | | . | \ | ÷ = r dt da K dt dK n n E t m F z Mi dt da m y m f 2 1 1 0 0 o | c µ (7.3-6) , , 1 2 2 ln ÷ ( ( ¸ ( ¸ | | . | \ | × n m y r K o ì where: M is atomic mass [kg/mol], i 0 is initial value of decay curve of electric current on newly exposed surface [A/mm 2 ], z is charge change at electrolysis (dimensionless), µ is density [kg/m 3 ], F is Faraday constant [Coulomb/mol], m is decay constant of current at newly exposed material surface (di- mensionless), c f is strain caused by break of protective film at crack tip (dimensionless), t 0 is starting time of current decay at newly exposed material surface [s], |, ì are constants (dimensionless), o y is yield stength [MPa], 7. Deterministic degradation modelling methods for power plant components 118 E is Young’s modulus [MPa], n is work hardening exponent in Ramberg-Osgood equation (dimen- sionless), r is radial distance from crack tip [mm]. The SCC model expressed by equation (7.3-6) contains the assumption that the SCC propagation is regarded as a cyclic corrosive process assisted by strain relaxation. When protective oxide film on the surface of steel is broken by oper- ational load, newly exposed surface corrodes following the Faraday’s law for electrolysis to form a new film. This process is depicted in Figure 7.3-2 below. However, equation (7.3-6) contains the uncertain variables r and m. As for m, it is possible to approximately estimate an adequate value by considering the solution conductivity, corrosion potential and the degree of chromium depletion on the grain boundaries. A similar derivation of crack growth rate can be made based upon the slip-oxidation mechanism [146]. break of protective film break of next film Figure 7.3-2. Current decay after break of protective film followed with formation and break of new film, from ref. [145]. A quite recent predictive methodology for SCC propagation using a mechano- chemical model based on a slip formation/dissolution was presented by Saito and Kuniya [147]. This model consists of combined kinetics of the plastic de- 7. Deterministic degradation modelling methods for power plant components 119 formation process as a mechanical factor and the slip dissolution-repassivation process as an environmental factor at the crack tip. The model equations are: n m ct C A dt da | | . | \ | = c 0 (7.3-7a) , , n t i zF M A n m ÷ = 1 0 0 0 µ (7.3-7b) , , 2 slip cos 2 b n N C d d m u µ = (7.3-7c) where: ct c is crack tip strain rate, n is numerical constant correlated with the degree of sensitisation, water conductivity and corrosion potential, M is atomic weight of the metal, z is number of electrons involved in the reaction rate, F is Faraday’s constant, µ m is density of the metal, i 0 is initial dissolution current density at the bare metal surface, t 0 is short time constant, µ d is dislocation density, u is angle between the direction of the slip plane and tensile stress, N slip is the number of active slip bands, n d is the number of dislocations contributing to formation of a slip band, and b is Burgers vector. Crack tip strain rate ct c is calculated from a separate equation, which is a func- tion of for example K and K ISCC , which is the threshold stress intensity factor for SCC. According to the ref. [147] the SCC propagation model enables to predict quantitatively the crack growth for a wide combination of materials, environ- ments and stress conditions. 7. Deterministic degradation modelling methods for power plant components 120 For instance, for austenitic stainless steel of type 304, which is a commonly used NPP piping component material, in water with temperature of 288 °C equa- tion (7.3-7) is written after incorporation of associated parameter data as [147]: , , ( ( ¸ ( ¸ | | . | \ | × ÷ × × ÷ × ÷ × × × × = ÷ ÷ ÷ ÷ 21 3 1 20 19 10 07 10 74 . 7 2 10 5 . 1 10 3 exp 10 5 . 2 10 1 . 1 I K dt da (7.3-8) The results presented in ref. [147] are in good agreement with the experimental SCC measurements performed for the same conditions. Some remarks concerning the application of SCC propagation computation models are presented in the following. Firstly, all these models contain from a few to several material and environment dependent parameters, the values for which have to be assessed based on experimental data. Often, such parameter data are at least partly missing in the available technical literature. On the other hand, due to SCC tests being relatively expensive to carry out, most/all experi- mental SCC data are proprietary and thus not available. As for the published parameter data, the available values have mostly/invariably been conservatively developed as upper bound solutions based on the experimental data. As the scat- ter of SCC data sets is typically wide, the upper bound approach can in this con- nection lead to excessively conservative SCC equation parameter values. Most- ly/invariably SCC model equations are limited to concern only sensitised mate- rials, whereas in the operating NPPs, there are numerous components of SCC susceptible materials that have not experienced sensitising treatments/conditions. Such components are at most weakly susceptible to SCC. However, due to weld- ing process the weld regions and adjacent heat affected zones (HAZs) are often sensitised. Consequently, in structural integrity assessments concerning SCC the crack postulates are often located to these regions. On the other hand, in weld and HAZ regions locally confined weld residual stresses have to be taken most- ly/fully into account as well. In commonly used fitness-for-service handbooks, these stresses have also been developed as upper bound solutions based on the underlying experimental data, in this case conservatively on the tensile side. Furthermore, of the K values over the crack front, typically only that from the crack tip is used in the incremental SCC propagation computations, as in most cases it is also the maximum value. However, the variation of the K values along the crack front is typically relatively large, and consequently the crack tip value, being a conservative choice, describes quite poorly the overall/average growth 7. Deterministic degradation modelling methods for power plant components 121 potential of the crack front. Thus the SCC propagation computations are often burdened with conservative assumptions coming from several sources at the same time. When performing computational SCC analyses, it is quite difficult to avoid ending up with excessively conservative results, and thus it appears that further procedure and test/input data treatment development would be necessary. Obviously, this calls for contributions from several fields/sources of science. 7.4 Creep Creep degradation occurs in power plant components that operate at high tem- peratures relative to the material melting point. See Section 3.5 for a more de- tailed description of the issue. From a continuum point of view, the most rele- vant parameters with regard to creep degradation are the strain rate, c , and time to rupture, t r . To obtain these, constitutive equations relating these parameters to stress and temperature have been developed. One of the earliest and still commonly used constitutive equations for model- ling creep is that by Norton [148], which expresses the relation between strain rate, c , and prevailing stress, o, as follows: n Ao c = (7.4-1) where A is the rate coefficient (considered temperature dependent) and n the creep exponent, both being material specific independent constants for which values are obtained experimentally. This equation describes creep behaviour mainly in the secondary regime. Later on, more advanced and also more com- plex constitutive equations for modelling creep have been developed, e.g. those by Graham and Walles [149], Garofalo [150], Dyson and McClean [151] as well as Merckling [152]. Larson and Miller developed a time temperature parameter approach [154] that relates the stress state to the time of rupture. The approach combines the Arrhenius law for the thermally activated process rate with the constitutive equa- tion by Norton (7.4-1), resulting with the following equation: , , , , j ¦ , , o o log log N M C t T P r LM ÷ = + = (7.4-2) where P LM is Larson-Miller parameter, T is temperature, t r is time of rupture, M and N are material dependent parameters and , , nA C log = . Other time tempera- 7. Deterministic degradation modelling methods for power plant components 122 ture parameter equations have been developed as well, such as that by Manson and Haferd [153]. Equation (7.4-2) indicates that the Larson-Miller parameter is linear against logarithmic time. However, most often all time temperature pa- rameters are fitted as polynomial stress functions. Cumulative damage equations for creep exist as well. The most common ap- proach to calculation of cumulative creep damage is to compute the amount of life expended by using time or strain fractions as measures of damage. When the fractional damages add up to unity, then failure is postulated to occur. One commonly used cumulative creep damage equation, called life fraction rule, is expressed as [155]: 1 = ¿ ri i t t (7.4-3) where t i is time spent at condition i and t ri is life to rupture at the same condi- tions, as obtained from associated standard/normative documents. As according to commonly used notation, the number of all considered conditions, n, is not shown in the equation. A similar creep damage model as that expressed by equa- tion (7.4-3) but based on strains, called strain fracture rule, is expressed as [156]: 1 = ¿ ri i c c (7.4-4) where c i is strain cumulated at condition i and c ri is rupture strain at the same conditions. Then, a creep damage approach combining time and strain, called mixed rule, is expressed as [157]: 1 2 1 2 1 = | | . | \ | | | . | \ | ¿ ri i ri i t t c c (7.4-5) Finally, a further developed combining model as compared to that expressed by equation (7.4-5), is [158]: , , 1 1 = | | . | \ | ÷ + | | . | \ | ¿ ¿ ri i ri i k t t k c c (7.4-6) 7. Deterministic degradation modelling methods for power plant components 123 where k is constant. For the equations (7.4-3) to (7.4-6), creep failure is postulat- ed to occur when the fractional damages add up to unity. Numerous constitutive equations for modelling creep exist, and for application they all require a sufficient amount of experimental creep data for the material, time scale and temperature range in question. Typically at least partly/mostly such data are available on the technical literature. Here, however, local creep degradation modelling procedures are more on focus than those considered this far, being based on continuum approach corresponding to more globally occur- ring creep degradation. Fracture mechanics based methods can also be applied for modelling creep degradation. Components that operate at high temperatures relative to the melt- ing point of the material may fail by slow and stable extension of a macroscopic crack. Traditional approaches to design in the creep regime apply only when creep and material damage are uniformly distributed. Time dependent fracture mechanics approaches are required when creep failure is controlled by a domi- nant crack in the structure. Deformation at high temperatures can be divided into four regimes: instanta- neous (elastic) strain, and the earlier mentioned creep regimes primary, second- ary and tertiary creep. The elastic strain occurs immediately upon application of the load. Whereas microscopic failure mechanisms, such as grain boundary cavi- tation, nucleate during tertiary creep. During the growth of a macroscopic crack at high temperatures, all four types of creep response can occur simultaneously in the most general case, see Figure 7.4-1. The material at the tip of a growing crack is in the tertiary stage of creep, since the material is obviously failing lo- cally. The material may be elastic remote from the crack tip, and in the primary and secondary stages of creep at moderate distances from the tip. 7. Deterministic degradation modelling methods for power plant components 124 Figure 7.4-1. Creep zones at the vicinity of the crack tip, from ref. [30]. A formal fracture mechanics approach to creep crack growth was developed soon after the J-integral was established as an elastic-plastic fracture parameter. Landes and Begley [159], Ohji et al. [160] and Nikbin et al. [161] independently proposed what became known as the C * -integral to characterize crack growth in a material undergoing steady-state creep. The C * integral is defined by replacing strains with strain rates, and displace- ments with displacement rates in the J-integral, see equation (4.3-5) in Section 4.3, as follows: } I | . | \ | c c ÷ = ds x u n dy w C i ij ij o * (7.4-7) where i u are displacement rate components and w is the stress work rate (power) density, defined as: } = kl ij ij d w c c o 0 (7.4-8) The analogy by Hoff [162] implies that the C * integral is path independent, be- cause J-integral is path independent. When strain rate in secondary creep follows a power law: 7. Deterministic degradation modelling methods for power plant components 125 n ij ij Ao c = (7.4-9) where A and n are material constants from the Norton equation (7.4-1), then it is possible to define a HRR type singularity for stresses and strain rates near the crack tip as: , , , , u o o , ~ 1 1 n r AI C ij n n ij + - | | . | \ | = (7.4-10a) and: , , , , u c c , ~ 1 1 n r AI C ij n n ij + - | | . | \ | = (7.4-10b) where the constants I n , ij o ~ , and ij c ~ are identical to the corresponding parameters in the HRR model, see equations (4.3-8) and (4.3-9) in Section 4.3. Note that in the present case, n is a creep exponent rather than a strain hardening exponent. Just as the J-integral characterizes the crack tip fields in an elastic or elastic- plastic material, the C * -integral defines crack tip conditions in a viscous materi- al. Thus, the time dependent crack growth rate in a viscous material should de- pend only on the value of C * . Experimental studies [159–162] have shown that creep crack growth rates correlate well with C * , provided steady state creep, i.e. secondary regime creep, is the dominant deformation mechanism in the specimen. The C * parameter applies only to crack growth in the presence of global steady state creep. Explained in another way, C * applies to long time behavior. When the crack grows with time, the behavior of the structure depends on the crack growth rate relative to the creep rate. In brittle materials, the crack growth rate is as fast that it overtakes the creep zone, and then crack growth can be characterized by K I because the creep zone at the tip of the growing crack re- mains small. At the other extreme, if the crack growth is sufficiently slow for the creep zone to spread throughout the structure, C * is the appropriate characteriz- ing parameter. Riedel and Rice [163] defined a characteristic time for the transi- tion from short time to long time behaviour, t 1 , as: , , , , - + ÷ = EC n K t I 1 1 2 2 1 v , or; (7.4-11a) 7. Deterministic degradation modelling methods for power plant components 126 , , - + = C n J t 1 1 (7.4-11b) When significant crack growth takes place over time scales much less than t 1 , the behavior can be characterized by K I , while C * is the appropriate parameter when significant crack growth requires times >> t 1 . Based on a finite element analysis, Ehlers and Riedel [164] suggested the following simple equation to interpolate between small scale creep and extensive creep (short and long time behaviour, respectively): , , , , 1 1 + ~ - t t C t C (7.4-12) where t is time. Unlike K I and C * , a direct experimental measurement of C(t) under transient conditions is usually not possible. Saxena [165] defined an alter- nate parameter C t which deviates to some extent from C(t). The advantage of C t is that it can be measured relatively easily. Saxena proposed the following inter- polation between small scale creep and extensive creep: , , , , - + A A ÷ = C C C t SSC t t 1 (7.4-13) where (C t ) SSC is separately computed small scale creep limit for C t , A is total displacement rate and t A is creep displacement. In the limit of long time behav- ior, C * /C t = 1.0, but this ratio is less than unity for small scale creep and transient behaviour. The C(t) parameter characterizes the stresses ahead of a stationary crack, while C t is related to the rate of expansion of the creep zone. The latter quantity appears to be better suited to materials that experience relatively rapid creep crack growth. Both parameters approach C * in the limit of steady state creep. The primary creep may have an appreciable effect on the crack growth behav- iour as well if the size of the primary zone is significant. Riedel [166] introduced a new parameter C h * , which is the primary creep analog to C * . The characteristic time that defines the transition from primary to secondary creep, t 2 , is defined by: , , , , p p h C p C t 1 2 1 + - - ( ¸ ( ¸ + = (7.4-14) 7. Deterministic degradation modelling methods for power plant components 127 where p is material specific constant for primary creep. When primary creep strains are present also the interpolation scheme for C(t) is modified [166]: , , , , - + ( ( ¸ ( ¸ + | . | \ | + ~ C t t t t t C p p 1 1 2 1 (7.4-15) The creep damage models presented here contain from a few to several material and environment specific parameters, the values for which have to be assessed based on experimental data. At least partly, this parameter data may be missing in the available technical literature. As for crack growth rates during creep, Nikbin, Webster et al. [167, 168, 169] have presented a model which is based on fracture mechanics to determine the time taken for the creep damage to accumulate at the crack tip starting from first initial elastic loading. Essentially, the steady state creep crack growth rate, da/dt, can be correlated with sufficient accuracy to the C * as follows: | - = C D dt da 0 (7.4-16) where D 0 and | are material constants which can be measured experimentally using the NSW model, see ref. [167]. For most engineering metals equation (7.4-16) can be approximated as: - · - = f s C dt da c 85 . 0 3 (7.4-17) where da s /dt is the steady state crack growth rate and - f c is taken as the uniaxial failure strain. For plane stress conditions this failure strain is f f c c = - , whereas for plain strain conditions it is 30 f f c c = - . This range describes the effects of constraint on crack growth due to both material properties and size/geometric factors. Then, the initial crack growth rate, da 0 /dt, can be approximated in rela- tion to that in the steady state as: 1 0 + = n dt da dt da s (7.4-18) 7. Deterministic degradation modelling methods for power plant components 128 For most engineering materials the values of n usually fall within the range of 5 to 10, which suggests that da 0 /dt is approximately 10 % of the corresponding steady state value. In design and for predicting remaining life in operation of components in ele- vated temperatures, extensive use is made of test data obtained with specimens subjected to uniaxial loading [11]. However, components in service generally experience multi-axial loading conditions. It is therefore necessary to establish effective stress criteria governing creep and rupture under multi-axial stress con- ditions and to be able to interpret them in terms of uniaxial test data. In addition, the redistributions of stresses occurring with creep must be taken into account. For a brief review of the former of these two aspects, see the paper by Roberts et al. [170]. 7.5 Irradiation embrittlement The material changes caused by neutron irradiation that may affect the use of the NPP reactor pressure vessels (RPVs) are connected to the fracture resistance of materials. As a rule, irradiation reduces the fracture toughness of the materials and therefore it is important to know the rate of reduction as a function of irradi- ation dose. Here, the description concerning computation of irradiation embrit- tlement of NPP RPVs corresponds mainly to that in the U.S. ASME code Sec- tion XI [173, 174] together with the U.S. Regulatory Guide 1.99, Revision 2 [171]. Most European approaches for evaluating irradiation effects in reactor pressure vessels are based on the ASME code [173, 174] procedure. In some countries, like U.K., no accepted standard approach exists, but a case by case best estimate analysis is carried out [177]. Many countries apply the ASME code directly, but in some countries like France [175, 176] the approaches have, how- ever, experienced a national variation. The reduction in the fracture toughness is evaluated with the aid of material specific reference nil-ductility temperature, RT NDT . This temperature is deter- mined from the nil-ductility temperature (NDT) and the Charpy-V impact test. The NDT temperature is determined in accordance with the ASTM Standard E 208 [172]. If the minimum impact energy and lateral expansion in the Charpy-V test (3 specimens) are at least 68 J and 0.9 mm, respectively, at a temperature equal to NDT + 33 °C, the NDT temperature is taken to represent RT NDT . If the Charpy-V properties do not meet the above criterion, RT NDT is taken as the tem- 7. Deterministic degradation modelling methods for power plant components 129 perature at which the requirements are reached, minus 33 °C. Normally, for modern steels RT NDT is equal to NDT. The ASME code is based upon linear-elastic fracture mechanics. The French approach originates from the ASME methodology, but it is more flexible, e.g. to account for possible plasticity effects. Both approaches assume that the tempera- ture dependence of fracture toughness is not affected by irradiation, enabling the fracture toughness temperature dependence to be described by a single curve. The shift is either determined based upon Charpy-V impact tests or from the chemical composition of the material, applying a chemistry factor. The ASME code Section XI [173] includes both a static fracture initiation ref- erence curve and a crack arrest reference curve. It is additionally assumed that the difference between static initiation and crack arrest is constant. Both curves are given as a function of effective temperature (T-RT NDT ), where RT NDT is as explained above. For the fracture mechanical assessment of normal operation conditions, the crack arrest curve also describes crack initiation. Reference [173] provides also equations for both of these curves. The crack arrest reference curve is intended to describe normal operation conditions, and the static fracture initia- tion reference curve emergency and faulted conditions. In addition to the differ- ent reference curves, also safety factors are applied. The French approach ap- plies essentially the ASME fracture toughness reference curves, but the safety factors are different and also the definition of upper limiting plateau, i.e. “upper shelf”, differs from that in ASME. In the ASME code approach, the irradiation induced shift of the RT NDT is de- fined as the shift in the 41 J impact energy transition temperature, J 41 TK . It is assumed that the true static fracture toughness shift and crack arrest shift is equal to or less than ATK 41J (ART NDT ). Reference [171], requires the use of an addition- al margin in the determination of RT NDT . For welds, the margin is 31 °C and for base metal, 19 °C. In both cases, the margin is not, however, more than ART NDT /2. This margin is put on top of the experimental ART NDT . Based upon ART NDT the fluence dependence can be calculated as [171]: , , , , , , f NDT f CF RT log 10 . 0 028 · ÷ = (7.5-1) where CF is the radiation embrittlement coefficient and f is the neutron fluence (×10E-19 n/cm 2 ) with E > 1 MeV. When two or more credible surveillance data 7. Deterministic degradation modelling methods for power plant components 130 sets are available, the radiation embrittlement coefficient is determined from the experimental data by least square fitting. Definitions of critical fracture toughness for the lower bound crack initiation value K IC and the lower bound crack arrest value K Ia are presented in Appendix A of ASME Section XI [173]. The effects of irradiation on the crack initiation and arrest fracture toughness can be estimated by applying the shift in the RT NDT to shift the ASME lower bound curves by moving the curves by the same shift amount, but leaving the shapes unaltered. These relationships are expressed as [32, 178]: , , j ¦ ¹ ´ ¦ + ÷ ÷ · × + = 200 100 02 . 0 exp 806 . 2 2 . 33 min 0 NDT NDT IC RT RT T K (7.5-2) , , j ¦ ¹ ´ ¦ + ÷ ÷ × × + = 200 160 0145 . 0 exp 233 . 1 78 . 26 min 0 NDT NDT Ia RT RT T K (7.5-3) where 0 NDT RT is the initial NDT temperature for unirradiated material, while K IC and K Ia are expressed in units of ksiin, and T, 0 NDT RT and ART NDT in units of °F. The French approach differs from the ASME methodology. The irradiation induced shift of the RT NDT is defined as the shift in the 56 J impact energy transi- tion temperature, 6J 5 TK , or the shift in the 0.9 mm lateral expansion transition temperature, mm 9 . 0 TK , whichever is greater. The lateral expansion correspond- ing to a certain energy level is affected by the material yield strength. Increasing the yield strength makes plastic deformation of the specimen more difficult. Therefore, ART NDT is generally controlled by mm 9 . 0 TK A . The French approach does not apply additional safety margins. The French codes [175, 176] allow the fluence dependence of ART NDT to be determined experimentally, but do not give any recommendations for the type of expression to be used. As for the chemistry factor, the European approaches also apply that concept in an effort to determine the irradiation shift directly from the steel chemistry. For the ASME methodology type steels, ref. [171] gives different chemistry factors for welds and base metal in a tabulated form. Even when ART NDT is cal- culated with the chemistry factor, one is required to use the additional safety margin prescribed in ref. [171]. The French codes contain chemistry factor equa- tions which are dependent on phosporus, copper and nickel. 7. Deterministic degradation modelling methods for power plant components 131 7.6 Modelling of interaction of degradation mechanisms Modelling of interaction of degradation mechanisms is considered in the follow- ing. Theoretically and in several cases in practise, the number of possible com- binations for interacting degradation mechanisms is relatively high. More than two degradation mechanisms can participate in the interaction. However, the scope is here limited to cover the major degradation mechanism interaction phe- nomena encountered/experienced in the power plant environments. These cases of interacting degradation mechanisms are corrosion fatigue and creep fatigue. In the first case, joint damage by crack growth is considered, whereas in the second case, more general cumulative damage is covered as well. Concerning corrosion fatigue, environmental fatigue correction factor approach for incorpo- rating effects of environment into fatigue evaluations is briefly described as well. Corrosion fatigue Corrosion fatigue is fatigue enhanced by corrosion reactions. Since fatigue fail- ures usually occur at stress levels below the yield strength after a number of cycles, the presence of a corrosive environment reduces the number of cycles to failure as well as the stress level at which failure occurs. The nominal fatigue limit, if any, is eliminated. Thus, corrosion fatigue is the reduction of the fatigue resistance of a material due to the presence of a corrosive environment. Alt- hough corrosion fatigue is characterized by cracking like SCC, corrosion fatigue is not environment specific, i.e., all environments will reduce the fatigue life of a component. In almost all corrosion fatigue cases, cracking is transgranular [138]. Many models which combine fatigue and stress corrosion effects have been suggested for use in predicting the corrosion fatigue crack growth rate as a function of fatigue loading frequency. These models can be divided into three categories: - superposition models [179], - competition models, and - models for environmentally modified material deformation and fatigue properties [180–183]. The superposition models and the competition models are developed based on the following assumptions: (1) an environmental fracture process is independent from the mechanical fatigue fracture process, and (2) the mechanical fatigue fracture pro- cess in a corrosive environment is identical with the process in an inert environment. 7. Deterministic degradation modelling methods for power plant components 132 Corrosion fatigue can be load cycle dependent, time dependent, or dependent on both of these. Of these the load cycle dependent corrosion fatigue corre- sponds to a simple acceleration of the fatigue crack growth that is insensitive to the loading frequency. The crack growth rate can be represented by an accelera- tion factor, u, multiplied by the inert growth rate [30]: inert aggressive | . | \ | u = | . | \ | dN da dN da (7.6-1) This expression can be applied for AK values above the fatigue threshold AK th for the inert environment. The acceleration factor u may be a constant or it may vary with AK. Load cycle dependent corrosion fatigue normally occurs in envi- ronments that do not result in significant environmentally assisted cracking (EAC) under static loading and where mass transport and electrochemical reac- tions that contribute to fatigue acceleration are very rapid. Time dependent cor- rosion fatigue can be modelled by a simple superposition of the inert fatigue crack growth rate with the environmental cracking rate, as follows [30]: EAC inert aggressive 1 | . | \ | + | . | \ | = | . | \ | dt a d f dN da dN da (7.6-2) where , , EAC dt a d is the average environmental crack growth rate over a loading cycle, and f is the loading frequency. Most material/environment combinations display both cycle dependent and time dependent behaviour. Combining equa- tion (7.6-1) and equation (7.6-2) gives the following more general expression for corrosion fatigue [30]: EAC inert aggressive 1 | . | \ | + | . | \ | u = | . | \ | dt a d f dN da dN da (7.6-3) Concerning the equations (7.6-1) to (7.6-3), the inert terms are computed with a suitable fatigue crack growth procedure in Section 7.2 and the EAC terms with a suitable procedure in Section 7.3.2, respectively. By definition, the growth rate is independent of frequency for cycle dependent corrosion fatigue. The time dependent corrosion fatigue is sensitive to the frequency, as equation (7.6-2) indicates. At high frequencies, the crack growth rate approaches the inert rate 7. Deterministic degradation modelling methods for power plant components 133 because the EAC growth per cycle is negligible. At low frequencies, the envi- ronmental crack growth per cycle dominates over fatigue, and the rate is propor- tional to 1/f. When there is a combination of time dependent and cycle depend- ent acceleration of crack growth rate, the former dominates at low frequencies and the latter dominates at high frequencies. Another variable that can affect the corrosion fatigue crack growth rate is the wave form of the cyclic loading. If, for example, the cyclic loading followed a square wave form instead of a sine wave, one might expect the average EAC growth rate to be greater because the maximum load is held for a sustained peri- od with a square wave form. A saw-tooth waveform might be expected to pro- duce less environmental cracking per cycle, all else being equal. A mitigating factor is that environmental cracking is often faster during periods when K is increasing. In those instances, a square wave form might actually result in less environmental cracking per cycle because a sustained load is less damaging than a continually rising load. Environmental fatigue correction factor approach The report NUREG/CR-6909 [184] by USNRC provides an environmental fa- tigue correction factor (F en ) methodology that is considered acceptable for incor- porating the effects of reactor coolant environments on fatigue usage factor evaluations of metal components for new reactor construction. The methodology reflects the earlier development on the subject carried out in Japan, presently it is included in the JSME codes for nuclear power generation facilities, see refs. [185, 186]. In the ref. [184] the environmental fatigue correction factor method- ology for performing fatigue evaluations is described for the four major catego- ries of structural materials, those being carbon steel, low-alloy steels, wrought and cast austenitic stainless steels, and Ni-Cr-Fe alloys, respectively. The effects of reactor coolant environments on the fatigue life of structural materials are expressed in terms of a nominal environmental fatigue correction factor, F en,nom , which is defined as the ratio of fatigue life in air at room tempera- ture, N air,RT , to that in water at the service temperature, N water , as follows [184]: water RT air, nom en, N N F = (7.6-4) For the above mentioned four major categories of structural steels the nominal environmental fatigue correction factor has the following form: 7. Deterministic degradation modelling methods for power plant components 134 , , - - - - × × × × ÷ = O T S 2 C 1 C exp F nom en, (7.6-5) where S * , T * , O * and - are transformed sulphur content, temperature, level of dissolved oxygen and strain rate, whereas C1 and C2 are constants, respectively. For these four transformed input data parameters, the definitions are given steel category specifically. The values of the two constants C1 and C2 vary case spe- cifically between 0 and 1. For carbon and low-alloy steels, all transformed input data parameters are defined as a function of their nominal values, whereas for wrought and cast austenitic stainless steels and Ni-Cr-Fe alloys, the same applies otherwise but S * = 1. For the two latter steel categories also O * is constant, though with steel category and in some cases water chemistry specifically differ- ing values. The minimum value of F en,nom is one, corresponding to conditions below the strain amplitude threshold. The necessary input data for this fatigue evaluation procedure includes partial fatigue usage factors U1, U2, U3, …, Un, as determined in ASME Section III [174] for Class 1 fatigue evaluations. To incorporate environmental effects into the Section III fatigue evaluation, the partial fatigue usage factors for a specific stress cycle or load set pair is multiplied by the environmental fatigue correction factor: en,1 1 en,1 F U U = (7.6-6) where the fatigue usage factor values are determined according to the current code fatigue design curves, as given e.g. in Appendix I of ASME Section III [174]. The cumulative fatigue usage factor, U en , considering the effects of reac- tor coolant environments is then calculated as: n en, n i en, i en,3 3 en,2 2 en,1 1 en F U ... F U ... F U F U F U U + + + + + + = (7.6-7) where F en,i is the nominal environmental fatigue correction factor for the “i”th stress cycle, see Paragraph NB-3200 [174], or load set pair, see Paragraph NB- 3600 [174]. Because environmental effects on fatigue life occur primarily during the tensile loading cycle (i.e. rising ramp with increasing strain or stress), this calculation is performed only for the tensile stress producing portion of the stress cycle constituting a load pair. The values for parameters such as strain rate, tem- perature, dissolved oxygen in water, and for carbon and low–alloy steels sulphur content, are also needed to calculate F en for each stress cycle or load set pair. 7. Deterministic degradation modelling methods for power plant components 135 Nearly all of the existing fatigue c-N data concerning ref. [184] are obtained under uniaxial loading histories with constant strain rate, temperature, and strain amplitude. However, the actual loading histories encountered during service of NPPs are far more complex, and the stress/strain responses are in general three dimensional. The modified rate approach has been proposed to predict fatigue life under changing test conditions. It allows calculating F en under conditions where temperature and strain rate are changing. The correction factor, , , T , F en , is assumed to increase linearly from 1 with increments of strain from a minimum value c min [%] to a maximum value c max [%]. Then, F en for the total strain transi- ent is computed as: , , | | . | \ | ÷ A = ¿ = min max k n 1 k k k k en, en T , F F (7.6-8) where n is the total number of strain increments, and k is the subscript for the “k”th incremental segment. An illustration of the application of the modified rate approach is presented in Figure 7.6-1 below. Figure 7.6-1. Application of the modified rate approach to determine the environmental fatigue correction factor F en during a transient, from ref. [184]. 7. Deterministic degradation modelling methods for power plant components 136 However, the description of the modified rate approach appears insufficient for computing the strain values and consequent F en values for actual NPP piping components concerning typical load transients with varying load parameter val- ues. Namely, the partial usage factors U i are computed according to the Para- graphs NB-3200 and NB-3600 of ref. [174] for load cycles as defined therein, and the correspondence of these cycle definitions and the rising ramps with in- creasing strain or stress considered by the modified rate approach is unclear for the various possible kinds of load cycles encountered in the actual NPP envi- ronments. Moreover, the choice of sign for the cycle specific stress intensities computed according to the Paragraph NB-3200 of ref. [174] is not unambiguous, and neither is the choice of the associated strain component. Other gaps concern- ing the conduct of application of the F en approach in practise exist as well. The main shortcoming of the NUREG/CR-6909 [184] F en approach appears to be its uniaxial nature, which remarkably limits its feasibility to actual plant conditions where the stresses/strains experienced by components are in general three di- mensional. More recently, ASME has provided Code Case N-792, Fatigue Evaluations Including Environmental Effects [187], to supplement fatigue assessment proce- dures in ref. [174]. This document [187] contains Fen based methodology for incorporating the effects of reactor coolant environments on fatigue usage factor evaluations of metal components, which is mostly the same as that presented in the report NUREG/CR-6909 [184]. The new developments include general guidance on combining plant load transients to load cycles for fatigue analysis purposes, also presented are slightly updated versions of the ANL S-N fatigue design curves in air given in ref. [184] for carbon steel, low-alloy steels, wrought and cast austenitic stainless steels and Ni-Cr-Fe alloys. It is also mentioned in ref. [187], that for the four mentioned material types it can be used for evalua- tion of thermal and mechanical fatigue, as caused by the LWR environment dur- ing service. On the other hand, the provided approach for assessment of strain rate is more limited than e.g. the modified rate approach described in the NUREG/CR-6909 [184], see equation (7.6-8) above. Namely, it is simply given as the stress intensity divided by Young’s modulus multiplied by the time from the start of load transient for the stress difference corresponding to the stress intensity to reach the maximum value, i.e. a strictly linear approach. However, it is also mentioned in ref. [187], that the presented strain rate computation ap- proach has interim status, and that the rules for determination of strain rate for NPP piping components are in the course of preparation. 7. Deterministic degradation modelling methods for power plant components 137 Creep fatigue interaction Changes in loading conditions during operation cause transient temperature gra- dients to power plant components operating mostly at relatively high tempera- tures. Due to restrained thermal expansion these transients cause thermally in- duced stress cycles. The extent of ensuing fatigue damage depends on the severi- ty and frequency of occurrence of these transients as well as of the material properties. Components which are subject to thermally induced stresses general- ly operate within creep range so that damage caused by both fatigue and creep has to be taken into account [11]. The most common approach to take into account the simultaneous effect of creep and fatigue is based on linear superposition. This procedure, combining the damage summation of Robinson for creep [155] and that of Miner for fatigue [99], is expressed as [188]: ´ D t t N N f f = + ¿ ¿ (7.6-9) where N/N f is the cyclic portion of the life fraction, in which N is the number of cycles at a given strain range and N f is the pure fatigue life at that strain range. The time dependent creep life fraction is t/t r , where t is the time at a given stress and t r is the time to rupture at that stress. The stress relaxation period is divided into time blocks during which an average, constant value of stress prevails, and for each time block t/t r is computed and summed up. D´ is the cumulative dam- age index. When D´ = 1, failure is assumed to occur. A strain based damage rate approach which takes into account the rate of damage accumulation has been proposed by Majumdar and Maiya [189-192]. They view the total damage as consisting of fatigue induced crack growth, da FA /dt, and creep induced cavity growth, da CA /dt. If the two damage mecha- nisms are additive, the damage rate is given by the sum of the following two equations: ¦ ¹ ¦ ´ ¦ = | | . | \ | 2 2 1 k p s p k p s p FA FA C T dt da a c c c c for tension, (7.6-10) for compression. 7. Deterministic degradation modelling methods for power plant components 138 and: ¦ ¹ ¦ ´ ¦ ÷ = | | . | \ | 3 3 1 k p s p k p s p CA CA G G dt da a c c c c for tension, (7.6-11) for compression. where T, C, G, k2, k3 and s are material parameters dependent on temperature, environment and the metallurgical state of the material, c p and p c are the current and absolute values of plastic strain and plastic strain rate, respectively, whereas a FA and a CA are the crack and cavity size at time t. The parameters T and C are included to account for differences in growth rates in tension and compression. Equation (7.6-10) describes the crack growth induced damage due to fatigue, whereas equation (7.6-11) describes the crack growth induced damage due to creep, respectively. The parameters T and C are included to account for differ- ences in growth rates occurring in tension and compression. The parameter G is given the appropriate sign for the tensile or the compressive stress regime. Final failure is computed as the reciprocal of the sum of the crack and cavity damage. For cases where the fatigue and creep damages are interactive (not additive), Majumdar and Maiya have proposed the following slightly modified equation [193]: , , 2 0 , ln 1 1 k p s p CA CA FA FA a a C T dt da a c c o ( ( ¸ ( ¸ | | . | \ | + v = | | . | \ | (7.6-12) Here it is assumed that cavities of size larger than a CA,0 interact with fatigue in- duced crack and increase the crack growth rate. As for parameter o, when it equals zero the fatigue crack growth is unaffected by the cavities and reverts to continuous cycling as according to equation (7.6-10). The equation for the creep induced cavity growth does not alter, i.e. equation (7.6-11) is applied as such. The damage rate approach allows taking into account the effects of various wave shapes on fatigue life. This model has been successfully applied e.g. to 1Cr-Mo- V steels [194] and austenitic stainless steels [189-192]. The crack growth rate computation equations proposed Majumdar and Maiya contain several parameters the values for which have to be determined experi- mentally. Nikbin, Webster et al. [169, 195] concluded that a more straightfor- ward model containing less parameters to be assessed experimentally can be 7. Deterministic degradation modelling methods for power plant components 139 derived for computation of crack growth rate due to joint effect of fatigue and creep. This model equation for computing the total/combined crack growth rate, (da/dN) TOTAL , is: , , f dt da K C dN da m + A = | . | \ | TOTAL (7.6-13) where the first term on the right side is the Paris-Erdogan equation for fatigue induced crack growth, see equation (7.2-12), f is frequency and da/dt is comput- ed according to equation (7.4-16), respectively. The creep fatigue interaction damage models presented here contain from a few to several material and environment specific parameters, the values for which are obtained as based on experimental data. At least partly this parameter data may be missing in the available technical literature. 7.7 Commonly applied component degradation assess- ment procedures One convenient way to categorise various structural integrity/degradation as- sessment procedures is to divide them into low and high temperature procedures [196]. Low temperature procedures can be further divided into assessment pro- cedures based on failure assessment diagram (FAD) calculation procedures and fitness-for-service guides, assessment procedures based on crack driving force (CDF) techniques and other procedures. More recently, also procedures that apply for both high and low temperatures have been developed. Low temperature procedures Low temperature structural integrity/degradation assessment procedures can be further divided to the following subgroups [196]: - assessment procedures based on FAD approach; calculation procedures, fitness-for-service guides, - assessment procedures based on CDF techniques, - others. The FAD methodology is based on the use of an integrated graphical representa- tion where fracture failure and plastic collapse are simultaneously evaluated by 7. Deterministic degradation modelling methods for power plant components 140 means of two non-dimensional variables (L r and K r ). These two variables repre- sent the ratio between the applied value of either stress or stress intensity factor and the resistance parameter of the corresponding magnitude (yield strength or fracture toughness). Once the plotting plane is determined by the variables L r and K r , each procedure defines a critical failure line which establishes the safe or acceptable zone. This zone is the area bounded by the mentioned line and the axes. The procedures which apply FADs have been divided into two different groups according to differences in structural applications and objectives [196]. The following documents contain a FAD calculation procedure: - R6 Method, Revision 4 [200], - BSI PD6493 [202], - SSM; A Combined Deterministic and Probabilistic Procedure for Safe- ty Assessment of Components with Cracks [203]. The following documents, in addition to containing a FAD procedure, contain a fitness-for-service Guide: - EXXON, Fitness-for-service Guide [204], - MPC, Fitness-for-Service Evaluation Procedures for Operating pres- sure vessels, tanks and piping in refinery and chemical service [205], - API 579, Recommended Practice for Fitness-in-Service [206]. The assessment methodology of the procedures based on the use of CDF dia- grams is different from the philosophy of FAD based documents. The evaluation of fracture failure and plastic collapse are not computed simultaneously. There- fore, two independent steps are needed. On the one hand, a direct comparison between applied stress or load and flow stress or limit load, and on the other hand, a diagram should be plotted where the applied stress intensity factor, J- integral or crack tip opening displacement (CTOD) is compared with the corre- sponding toughness values. The CDF diagrams are easier to interpret in a physi- cal sense than the FADs. Moreover, if the CDF technique is used together with the J-R curve, the full crack propagation history is described [196]. The following documents present a CDF technique: - GE-EPRI Approach [207, 208], - ETM, The Engineering Treatment Model for Assessing the signifi- cance of crack-like defects in engineering structures [209] and The En- 7. Deterministic degradation modelling methods for power plant components 141 gineering Treatment Model for assessing the significance of crack-like defects in joints with mechanical heterogeneity (strength mismatch) [210]. Documents that present other procedures than FAD or CDF: - ASME Section XI [173] code, which contains figures and tables to which the real situations can be compared to, - RSE-M Code [211], which is very similar to ASME code, - KTA Code [212], which is very similar to ASME code. High temperature procedures The early approaches to high temperature life assessment show methodologies which are based on defect free assessment codes. The fracture mechanics based procedures basically followed the low temperature assessment procedures that have been developed within Europe and elsewhere [196]. The high temperature procedures include: - ASME Code Case N-47 [213], - RCC-MR Code [214], - R5 Method [215], - BS PD 6539 [216]. The early approaches to high temperature life assessment methodologies include ASME Code Case N-47 [213] and the French RCC-MR [214]. These procedures have many similarities, and both of them are based on lifetime assessment of uncracked structures. ASME Code Case N-47 [213] is the first design code to formally include linear damage summation as a method of predicting material failure at high temperature, consisting of a fatigue (Miner cycle summation) component and a creep (Robinson time summation) component. The French Code RCC-MR [214] incorporates many of the concepts behind ASME Code Case N-47 [213] but with modified stress analysis procedures. One advantage of the more recent R5 Method [215] approach is that a defect may be postulated and the subsequent behaviour may be predicted for likely service cycles. Where- as a step-by-step high temperature creep crack growth computation procedure is given in the British Standard document PD 6539 [216]. 7. Deterministic degradation modelling methods for power plant components 142 Combined high and low temperature procedures During recent years, four European structural integrity/degradation assessment procedures that apply for both high and low temperatures have been published. These are SINTAP (Structural Integrity Assessment Procedures for European Industry) [197], FITNET (European Fitness For Service Network) [125], HIDA procedure [198] and FKM guideline [199]. These structural integrity/degradation assessment procedures cover all types of power plants as well as most of other types of industry plants. They also include compendia of analytical structural and fracture mechanics parameter solutions for most common industry plant component types, such as plates, straight pipes, pipe bends, T-joints and pressure vessels. 8. Probabilistic degradation modelling methods for power plant components 143 8. Probabilistic degradation modelling methods for power plant components Time dependent probabilistic models can be used to identify possible trends and to predict the future behaviour of components. Different probabilistic approaches to model ageing phenomena are used depending on the degradation mechanism in question. In straightforward reliability calculations, the components are often assumed to have a constant failure rate, i.e. the component failures occur according to a homogeneous Poisson process. In the Poisson process, it is assumed that the occurrences of successive events are exponentially distributed with the same parameter. The component failure rates generated from large amount of data in reliability databases are usually based on the assumption of constant failure rate [217]. The ageing of components which have a mean time to failure that is signifi- cantly shorter than the assumed plant lifetime is called short-term ageing. Typi- cally, there is a lot of failure data available concerning such components and thus statistical analyses can be performed. This mainly concerns active compo- nents, such as pumps and valves. In the case of long-term ageing, the amount of failure data is very limited. This mainly concerns passive components, such as pipes. Then, information of the degree of component degradation is obtained by surveillance and condition monitoring. The evaluation of the expected remaining lifetime is based on monitored parameters. The behaviour of degradation mech- anisms is often stochastic and thus probabilistic techniques should be applied in the modelling. A brief description concerning some probabilistic approaches to model ageing of power plant component is presented in the following. 8. Probabilistic degradation modelling methods for power plant components 144 8.1 Probabilistic structural mechanics based modelling methods Here, the scope of described probabilistic modelling methods mainly concerns initiation and growth of cracks in passive power plant components. Because the critical crack size depends on the magnitude of the applied load, a fracture crite- rion that relates the stress amplitude and the crack length to failure must be ap- plied if the stresses are varying. Thus, to estimate the fatigue reliability of a component, four factors must be explicitly considered. These are load history, initial crack distribution, crack growth history as a function of the load history and a failure criterion. Because there is much scatter in empirical crack propaga- tion data, the crack growth rate should be modelled probabilistically [32]. There are three basic classes of probabilistic models for crack propagation. The most commonly used type involves randomisation of an appropriate crack growth equation, e.g. the Paris-Erdogan model, see equation (7.2-12). Other types of models include evolutionary probabilistic models (Markov or diffusion models) and cumulative jump models [32]. The randomisation is carried out by introducing appropriate quantities charac- terising random effects into the empirical crack growth law. For instance, one approach to modelling random fatigue is to treat the parameters of the Paris- Erdogan equation, C and m, as random variables. By doing so, the variability in the stress range, material properties and environmental factors can be characterised [32]. 8.2 Probabilistic fracture mechanics based methods 8.2.1 Introduction Of the many possible degradation mechanisms, failure due to crack growth is the dominant degradation mechanism for most power plant components that are made of metal(s). Crack initiation and growth in these components can be caused by various phenomena, such as fatigue and corrosion, or their combina- tion. Many of the shortcomings of traditional deterministic analysis methods can be alleviated or overcome by applying probabilistic analysis methods. The main disadvantages of all deterministic analysis methods are that [45]: a) Properties and partial safety factors (where used) are often not given as best estimates or most likely values, with the result that it is not possi- ble to estimate the most likely strength of the structure. 8. Probabilistic degradation modelling methods for power plant components 145 b) The risk of failure or collapse, or the overload necessary to cause fail- ure or collapse, may vary widely for different structural members and components, and different types of structure. c) The assumption that most design parameters are known constants ra- ther than statistical variables is in most cases a gross simplification. d) The safety factor approach is not so easy to apply in assessment of ex- isting structures and for making maintenance decisions. Most fracture mechanics analyses are deterministic, e.g. a single value of frac- ture toughness is used to estimate failure stress or critical crack size. Much of what happens in the real world, however, is not predictable. For instance, since fracture toughness data in the ductile to brittle transition region are widely scat- tered, it is not appropriate to view fracture toughness as a material constant hav- ing a single value. Other factors also introduce uncertainty into fracture analyses. A structure may contain a number of flaws of various sizes, orientations and locations. Extraordinary events, such as earthquakes and accidents can result in stresses significantly higher than the intended design level. Because of these complexities, crack growth should be viewed probabilistically rather than deter- ministically [30]. Probabilistic fracture mechanics (PFM) provides an approach for probabilistic modelling of initiation and growth of cracks in components. The advantages of PFM include also the possibility of modelling clearly the uncertainties related to the material degradation process, and thus being able to perform sensitivity analyses of the factors affecting this process. For ageing management purposes it is of interest, for example, to evaluate how changes in operating conditions can affect the failure probability of the structure [87]. The earliest PFM developments include aircraft applications. However, the emphasis in the first efforts was more on crack initiation than fracture mechanics aspects. Recently, the most common object of application has been pressurised components, primarily RPVs and piping in NPPs. Weld joints have received a particular interest, as fabrication defects and cracks tend to concentrate in those regions [90]. PFM is based on deterministic fracture mechanics (DFM) procedures, but considers one or more of the input variables to be random instead of having a single deterministic value. For example, initial crack size is typically one of the rarely well known variables. Rather than assuming a fixed given initial size, this parameter can be projected over a range of sizes with probabilities of occurrence 8. Probabilistic degradation modelling methods for power plant components 146 and detection estimated for each size. In the simplest case, a DFM analysis would then be performed after inspection for each size to provide the crack size distribution as a function of time (or stress load cycles) in service. The failure probability at any time is then equal to the probability of having a crack larger than the critical size (which can also be a random variable) [90]. Often one or more of the fracture mechanics variables is not known with suffi- cient certainty, and many of the input variables should be considered random. Frac- ture mechanics input parameters typically considered to be random include [90]: - Initial crack size; depth, length, location, - Crack detection probability; detection probability as a function of size, uncertainty in crack size for a given level of indication, - Material properties; subcritical crack growth characteristics, fracture toughness, tensile properties, - Service conditions; stress levels, load cycle rate, temperature, environment. Not all of these parameters are considered random in every application. Only those variables with uncertainty or scatter producing the greatest effects on life or reliability calculations need to be recognised as such. The failure probability at any given time may be determined by combining conventional fracture me- chanics calculations with an appropriate statistical approach. The results can then be used to ascertain the suitability of the component for service. 8.2.2 PFM analysis procedure A typical PFM analysis is described in the following. First, one or more parame- ters or variables are randomised, depending on the characteristics of the em- 8. Probabilistic degradation modelling methods for power plant components 147 ployed analysis model. Parameters or variables that can be considered as random were presented in Section 8.2.1. Then crack growth simulations are performed. Fracture mechanics models employed are based on LEFM, e.g. Newman-Raju solutions, weight and/or influence functions, or on EPFM or on a combination of LEFM and EPFM. During the crack growth simulation, pre- and in-service in- spections are considered and failure judgements of failure states, e.g. leak or rupture, are performed. Cumulative failure probabilities are calculated as a func- tion of time in operation [44]. A flow chart example of a PFM analysis is shown in Figure 8.2-1 below. Figure 8.2-1. A flow chart of a PFM analysis, from ref. [44]. Most of the applications of PFM for power plant components lead to very low failure probabilities which have to be determined by sophisticated numerical procedures from rather scarce and incomplete input data. 8.2.3 Engineering models and PFM Many components, which could be otherwise rejected, can be saved by remov- ing such arbitrary and unrealistic worst case assumptions as simultaneous occur- 8. Probabilistic degradation modelling methods for power plant components 148 rence of improbable events, and allowing the model to account probabilistically for actual variation in critical parameters which are based in test results and/or experience. PFM is applied to remove such unrealistic conservatism in compo- nent reliability analyses, though in their pure form they have limitations similar to, but not as pronounced as those concerning deterministic analyses. Two methods are available to alleviate these shortcomings in pure PFM. The first is to use conservative bounds for input variable probability distributions where accurate distributions are unavailable. The resulting conservative pure PFM analysis represents a notable improvement compared to deterministic mod- el on which it is based. The primary reason for this consequence is that PFM in any form terminates the assumption that all conceivable deleterious events will occur simultaneously. The second method consists of calibrating the PFM model against all available relevant experience. Here calibrated PFM essentially repre- sents a combination of the database and pure PFM approaches [90]. When considered individually, probabilistic engineering models contain a number of inherent limitations with respect to reliability prediction. These same limitations are present when models are constructed as a component of a pure PFM analysis [90]. PFM models are based on conventional DFM principles. Hence uncalibrated tools cannot treat problems for which fracture mechanics tools are not available. Any of the components of a DFM analysis, e.g. crack size and stress history, can be treated as a random variable, in which case the model becomes a probabilistic one. The basic assumptions inherent for DFM are naturally included in the respec- tive PFM models. Additional assumptions are usually made when a PFM model is constructed for a particular application. It may also be assumed, for example, that only one crack will exist and that the crack will be situated in a weld [90]. Currently, there are many methods and applications for PFM in various manu- facturing industries, nearly all of which are based on LEFM models. In contrast, probabilistic analyses based on EPFM models are not widespread and are only currently gaining notice, particularly for applications in pressure boundary com- ponents. It is well established that EPFM models provide more realistic measures of fracture behaviour of cracked structures with high toughness and low strength materials compared to LEFM models. Cracked components com- posed of these materials, when used in power plants, pose a serious threat to structural integrity. In addition, the operational temperature region of power plants is above the brittle-to-ductile transition temperature of the component materials. Then, the fracture response is essentially ductile, and the material is 8. Probabilistic degradation modelling methods for power plant components 149 capable of considerable inelastic deformation. As such, EPFM should be applied in fracture analyses of these components [91]. The probabilistic aspects of crack repair are drawing attention nowadays as well. It is often assumed that all detected cracks are repaired and that no new cracks or adverse conditions are introduced during the repair process, even though this assumption may not be realistic. Still, assumptions have to be made in the development of a probabilistic treatment. Typical assumptions might in- clude consideration of crack depth, aspect ratio, fracture toughness and subcriti- cal crack growth characteristics to be independent [90]. 8.2.4 Numerical PFM methods Numerical techniques for generation of failure probabilities from PFM models are generally required for more complex problems. Application of these tech- niques does not necessarily lead to a large amount of computations or complicat- ing factors, as workable techniques are readily available. Numerical results from the construction of PFM models are produced in a variety of ways. The numeri- cal techniques include: stress/strength overlap, variance/covariance, convolution and MCS. The stress/strength overlap can be calculated once the statistical distributions of both parameters are known or estimated. However, estimating these distribu- tions can be difficult. In the case of fatigue crack growth, the time dependent strength distribution would at least depend on the following input parameters and their interactions: initial crack size, fracture toughness, material properties and environment. In all but the simplest cases, a separate numerical generation technique would have to be employed to calculate the individual stress and strength distributions. Variance/covariance analysis by perturbation techniques is based on the as- sumption that some variables will deviate by relatively small amounts from their respective means. This technique is also known as the method of system mo- ments [92]. It provides estimates of the moments (mean, variance, skewness, etc.) of the dependent variable from the moments of the independent variables and the partial derivatives of the functional relationship between the independent and dependent variables. However, this technique is only approximate. It in- creases in accuracy as the variances of the independent variables decrease. Only the moments of the distribution of the dependent variables are provided [90]. 8. Probabilistic degradation modelling methods for power plant components 150 The convolution method, which is sometimes referred to as “direct synthesis”, derives the probability density function of the dependent output random variable from the functional relationship between the dependent and independent varia- bles and the probability density functions of the independent variables. In prin- ciple, this method is capable of providing analytical results. However, the deri- vation of the multidimensional integrals is often a complicated task. Difficult numerical integrations are usually required and care must be taken to obtain accurate results. The incorporation of relatively simple statistical and physical dependencies among the input variables is also very difficult [90]. MCS, being a general numerical technique for determining the distribution of the dependent variables, is broadly applicable to the generation of numerical results from PFM models [93, 94]. The inclusion of dependencies between input variables is a straightforward procedure with MCS. As described earlier in Section 5.6.5, MCS consists of selecting a value for each input variable at random from its assumed statistical distribution for substi- tution into the underlying deterministic model. In the context of a typical PFM problem, initial crack size, a 0 , crack growth rate and nominal stress values are randomly sampled, and the effective crack size, a, is computed after a number of load cycles, N. Similarly, a value for K IC is sampled and used along with the sampled values of stress to provide the critical crack size value, a c . The propor- tion of times that a exceeds a c is equal to the probability of failure with N cycles. A single simulation computation is therefore precisely equivalent to a single application of the corresponding DFM analysis, with the exception that some of the input values have been selected randomly [90]. In many instances, the probability of failure will be small. Hence, many simu- lations would be required for accurate enough results. This can be alleviated by selective sampling from the distributions of the input variables, thus drawing from the tails of the distributions controlling the failure probabilities. Selectivity is then compensated for during the analysis of the numerically generated results. Sampling procedures are discussed in Section 5.6. In the following is described an example concerning importance sampling. If it is known that large initial crack sizes associated with small probability are required for the failure to occur, it is not necessary to randomly sample from the initial crack size distribution. This is because the vast majority of sampled cracks would be from the much more likely portion of small cracks. Then, importance sampling would involve sampling only from the large crack end of the initial 8. Probabilistic degradation modelling methods for power plant components 151 crack size distribution. The results would then be compensated by the probabil- ity of having an initial crack in the portion of the sampled distribution. Stratified sampling is another sampling method that can be used to reduce the large amount of computational work related to assessment of small probabilities. Figure 8.2-2 illustrates a typical stratification of an initial crack sampling space composed of normalised crack depth, a/t, and aspect ratio, a/c, of a semi- elliptical surface crack, where t is wall thickness and c is half of the crack length. The sampling space is divided into a number of mutually exclusive small sub- spaces, called cells. A predetermined number of samples are then taken from each of those cells. Within each cell, an individual sampling is carried out ac- cording to a postulated initial crack distribution [44]. The cumulative failure probability up to time t is computed as: , , , , ¿ = s m j j j f f P N t N t T P (8.2-1) where m is the total number of cells, N j is the number of samples taken from the j:th cell, N f (t) is the number of failed samples in the j:th cell up to time t, and P j is the probability that an initial crack exists in the j:th cell. Figure 8.2-2. The stratification of sampling space, from ref. [44]. 8. Probabilistic degradation modelling methods for power plant components 152 As shown in Figure 8.2-2, the crack samples located in the upper part of the sampling space seem more likely to fail than those in the lower parts. A region of uncertainty may exist between “Failure” and “No failure” regions. Since the samples taken from the “No failure” region would never lead to failure, consid- erable reduction in the amount of computational work can be expected by ignor- ing these cells in a sampling plan. Since the assessment of the region of uncer- tainty cannot be known before calculation, many cells and samples would be selected from the whole sampling space to ensure computational accuracy. Computation time would increase as a function of the number of cells and sam- ples employed. To overcome this problem, a parallel processing algorithm com- bined with the stratified MC method is very useful [95, 96]. 8.3 Statistical modelling methods 8.3.1 Short-term ageing models The models discussed in the following are applicable to estimate component ageing from failure data. Modelling of ageing with probability distribution functions The time dependence in failure occurrence can be expressed with the failure rate, ì(t), defined as the probability of failure given the component has been in opera- tion for a certain time [32, 217]: , , , , , , t R t f t = ì (8.3-1) where t is time, f(t) is the probability density function of component failure and R(t) is reliability function. The failure rate is also called the hazard rate. The reliability of the component at time t can be defined as [32, 217]: , , , , ( ( ¸ ( ¸ ÷ = } t d t R 0 exp t t ì (8.3-2) If the probability of a failure within a short time interval (t, t + At) is independent of the age of the component, t, the failure times are exponentially distributed and 8. Probabilistic degradation modelling methods for power plant components 153 the failure occurrence is independent of time. In this case, the failure rate is con- stant. A distribution is an increasing failure rate distribution if [218]: , , , , ( ( ¸ ( ¸ ÷ = } +t x x d x t R t t ì exp (8.3-3) is increasing when 0 > x for each 0 > t . When the failure rate is increasing, it is said that component is ageing. Several distributions commonly used in reliability engineering are suitable to describe events with an increasing failure rate. These include Weibull, gamma and log-normal distributions. In reliability engineering, Bayesian methods are of special interest since the amount of failure data of power plant components is often very sparse. These methods are used to include all relevant information to the estimation of model parameters. In the Bayesian approach, the probabilities are always conditioned to the background information. Thus, the Bayesian approach requires the defini- tion of the prior distributions. The choice of the form of the prior distribution is generally based on computational reasons. The parameter values of prior distri- butions may be based on the existing data and/or expert opinions. The basic principles of Bayesian approach are described for example in ref. [219]. Modelling approaches to short-term ageing In general, the failure rate of components changes over time as is shown in Fig- ure 8.3-1, which depicts the so called bathtub curve. The first interval, from time t 0 to t 1 , represents early failures due to material and/or manufacturing defects, and is usually referred to as the burn-in or infant mortality period. The second interval, from time t 1 to t 2 , is the period of random failures and is usually approx- imated by a constant hazard rate. It is often referred to as the useful life of the component or system. The part of the curve after t 2 represents the so called wear- out failures, where the failure rate increases as the component deteriorates. With respect to ageing, this is the period of the component lifetime that is of concern [32]. The probabilistic short-term ageing models of power plant components pre- sented here are divided to three groups [217]: - component replacement models, - ageing models of repairable components, - ageing models of tested stand-by components. 8. Probabilistic degradation modelling methods for power plant components 154 The repair or replacement time of components is assumed to be very short as compared to the failure time and is thus neglected. The uncertainty of the times between failures is described with a distribution which is called the intensity of the process [217]. Figure 8.3-1. Component failure rate as a function of time. If the failed components are replaced with new ones, which can be considered as identical to the failed components in the beginning of their lifetime, the failure histories can be described with renewal processes [220]. A renewal process, F, is a sequence of independent, identically distributed non-negative random varia- bles, of which all do not disappear. If the time for the replacement of a compo- nent is negligible and the time to failure of each component is distributed identi- cally, the successive intervals between failures can be treated as random varia- bles of a renewal process. The expected number of renewals (i.e. failures) during the time interval (0, t) is called the renewal function, M(t). It can be expressed in terms of the underlying distribution as [220]: , , , , , , ¿ · = = 1 k k t F t M (8.3-4) where , , , , t F k is the k-fold convolution of F. t 0 t 1 t 2 Time ì(t) F a i l u r e r a t e Burn-in period Useful life Ageing The linear approximation to the Ageing portion 8. Probabilistic degradation modelling methods for power plant components 155 As the successive failure times are distributed according to the same distribu- tion in a renewal process, the mean times between component failures remain also constant. The statistical estimation of renewal processes reduces to the es- timation of the parameters of the underlying life length distribution, if the subse- quent life lenghts have been observed. The renewal models cannot be applied in the case of repaired or maintained damaged components. This is because the times between events cannot be straightforwardly considered as independent and identically distributed random variables. If the development of the hazard function is assumed to be influenced by component repairs, it becomes dependent on the history. In ageing analyses, it is assumed that a baseline failure rate is increasing due to the environmental conditions during the component service [217]. Some hypothetical examples of the behaviour of the failure function are shown in Figure 8.3-2. Case (a) of Figure 8.3-2 illustrates a situation where the failure rate is increasing in time and is not affected by the failures, i.e. after re- pair the condition of the components is assumed to be the same as just before the failure. A more realistic model is illustrated in Case (b), where a basic failure rate is assumed to be increasing but after the failure a decrease in the failure rate describes the effect of maintenance. In Case (c) the maintenance is assumed to have a fractional effect on the failure rate [217]. It may also be assumed that the increase in failure rate begins in the “Ageing” region of the bathtube curve. The failure rate mode presented at Case (a) of Figure 8.3-2 can be modelled as a non-homogeneous Poisson process. The probability p n (t) that failures have occurred during time interval (0, t) is [217]: , , , , , , j ¦ t n t t p n n ÷ = exp ! (8.3-5) where , , t is the integral of the intensity, called the cumulative failure function, which in turn is defined as: , , , , } = t d t t 0 t ì (8.3-6) 8. Probabilistic degradation modelling methods for power plant components 156 When ì(t) is constant the model reduces to the homogeneous Poisson process. For ageing components, ì(t) is increasing with time. The most commonly used non-homogeneous Poisson process is the Weibull process. In a linear ageing model, the effects of the ageing process on the component failure probability are expressed as a simple linear equation that relates the com- ponent failure rate to exposure time of the component to the ageing mecha- nism(s). The failure function is defined in this case as [221]: , , t t o ì ì + = 0 (8.3-7) where ì 0 is the constant random failure rate and o is called the ageing accelera- tion rate. The second term of equation (8.3-7) describes the “Ageing” region of the total failure rate, see Figure 8.3-1. Other frequently used ageing models are the exponential failure rate model and the Weibull failure rate model. They are defined, in this order, by [222]: , , , , t t o ì ì exp 0 = (8.3-8) , , o ì ì | | . | \ | = 0 0 t t t (8.3-9) where t 0 is time for start of operational lifetime, as shown in Figure 8.3-1. 8. Probabilistic degradation modelling methods for power plant components 157 Figure 8.3-2. Realisations of failure rates of some ageing models with imperfect repair, from ref. [217]. f a i l u r e r a t e f a i l u r e r a t e f a i l u r e r a t e 8. Probabilistic degradation modelling methods for power plant components 158 Each of the three models presented in equations the (8.3-7) to (8.3-9) has its historical roots in machine and material lifetime analyses, the Weibull failure rate being the most commonly used of the three. The linear failure rate, which is the simplest of these models, has been used in determining age dependent risk [221, 223]. Wolford et al. [224] observed that the linear failure rate performed poorly in more instances than the other two models and due to its non- exponential form was most difficult to work with analytically. If there is a sample of recorded component lifetimes that is large enough and the parametric constraints do not greatly contradict the true underlying lifetime distribution, the data will most likely fit these three models adequately. Howev- er, as mentioned earlier, failure data pertaining to component ageing is scarce and justification for parametric models is sometimes doubtful. It is not certain how much data are exactly needed for accurate definition of model parameters. In ref. [225], it is suggested that the size of a data sample should be at least 100 in the case of Weibull rate model. In ref. [32], it is suggested that good approximate tests and confidence intervals can be obtained with somewhat smaller samples. When it is assumed that maintenances cause a decrease in the otherwise in- creasing failure rate, as is in Case (b) of the Figure 8.11, the failure rate is de- fined as a function of time and number of failures/maintenances by [217]: , , , , ¿ = ÷ = k j i t a t t 1 ì ì , j , 1 , + e k k T T t (8.3-10) where ì t (t) is increasing failure rate, T k are maintenance times and a i is the maintenance specifically constant effect of i:th maintenance. The maintenance effect a i must be defined so that ì(t) remains positive. It may also be assumed that the maintenance affects the failure rate in a multiplicative way, as is illus- trated in Case (c) of the Figure 8.11. Now the failure rate can be defined as [217]: , , , , , , t i a t t ì ì = , j , 1 , + e k k T T t (8.3-11) The decrease may be expressed for instance with the fractional learning model, where the failure rate is a function of events, as [217]: , , i Ac t = ì , j , 1 , + e i i T T t (8.3-12) 8. Probabilistic degradation modelling methods for power plant components 159 where c is the reduction factor and A is the event rate parameter before the first event. More generally, the repair may be assumed to have an effect on some parameter u of the failure rate as [217]: , , , , i t t t u ì ì , = (8.3-13) where u i is the maintenance specifically constant value of u at i:th maintenance. The models above present only some specific cases of repairable component ageing models, several other corresponding models may be developed. Besides the models of repairable components, the ageing of tested stand-by components may be modelled in terms of failure probability in the test situation. In such models, the time may be discretised to describe for instance i:th test in- terval which may facilitate the computations. The probability of failure is as- sumed to be constant during the test interval. The increase of the number of fail- ures in a test may be due to the fact that components wear between the tests, and degraded but not yet failed items are not identified in the test. Furthermore, the repaired component may not correspond to a new one after the repair. An exam- ple of an ageing model for tested stand-by components is the modified binomial model. In this model, the successful cases in tests are binomially distributed but the distribution parameter is allowed to evolve in time. This kind of model can be used to detect a trend in the results [217]. Unless the ageing model suits perfectly the underlying failure rate for the ex- amined components, the model results may not be directly applicable. To over- come this drawback, it is suggested [32] to use a non-parametric inference based procedure. This includes a test for increased failure rate, a test for increased fail- ure rate average, and test for “new better than used”. 8.3.2 Long-term ageing models Information about long-term ageing of components is obtained via condition monitoring. This data can be used e.g. to assess the remaining lifetime of the components. Examples of long-term ageing mechanisms include crack growth in metals, which issue is discussed in Chapter 7, and the combined thermal and radiation ageing of polymeric insulating materials in cables. The evaluation of long-term ageing is often based on the expected physical phenomena causing the component degradation. 8. Probabilistic degradation modelling methods for power plant components 160 The use of probabilistic models instead of deterministic ones is needed, when the experiments show a significant scatter in the results. Examples of such un- certainties are variations in material properties and changes in environmental conditions. Modelling approaches to long-term ageing The probabilistic long-term ageing models of NPP components presented here are divided into three groups [217]: - randomised deterministic models, - stress-strength-time models, - cumulative damage models that apply Markov chains. In the randomisation of deterministic models, parameters describing uncertain phenomena are treated as random variables, or a random noise term is added in the model to account for variations. The scatter in the results of experiments is used to estimate the distribution parameters. One approach to randomisation of deterministic models is PFM, see Section 8.2. The advantage of PFM is the pos- sibility of modelling clearly the uncertainties related to the ageing mechanism, and thus being able to perform sensitivity analyses for the factors affecting it. The time evolution of structural reliability can be described with a time depend- ent model, where the applied stress or load and the strength of the structure are stochastic processes. Stresses and strengths may vary during long operation in- tervals as a function of time or the number and/or severity of stress applications. Ageing denotes the decrease in strength as a function of time, cyclic damage denotes the decrease in strength as a function of the number of stress cycles and cumulative damage denotes the case in which the decrease in strength is by both the number and severity of stress applications. Component failure occurs when strength has decreased below a stress dependent level. Failure can also occur when the cyclic or cumulative damage has exceeded a certain limit correspond- ing to the strength. The reliability of the component can be defined as [32]: , , , , , , , , t t X Y P t R > = , j ¦ t , 0 e t (8.3-14) where Y(t) denotes the strength of the component and X(t) denotes the applied stress. 8. Probabilistic degradation modelling methods for power plant components 161 The stress-strength-time models may be classified according to the uncertainty about stress and strength [226]. Both stress and strength can basically be classi- fied as deterministic, random-fixed or random-independent. Illustrations of some stress-strength-time modelling cases are presented in Figure 8.3-3. Case (a) of Figure 8.3-3 describes conditions where strength is assumed to be constant and stress is an increasing random function. If the degree of degradation corresponding to the accumulated stress can be described with a Wiener process [227], the failure time distribution has the form: , , , , ( ¸ ( ¸ ÷ ÷ = 2 2 2 exp 1 2 o µ t o t y t t y t f (8.3-15) where y is the constant value for strength, µ denotes mean, o denotes standard deviation and t is time. Case (b) of Figure 8.3-3 illustrates a cumulative damage model, where load peaks occurring at random weaken the strength a discrete amount. If the times between the occurrences of loading events are exponentially distributed with the same parameter and the strength decrease is constant, this case can be modelled as a Poisson process. Additional uncertainty arises if the strength of the compo- nent is not necessarily reduced by the shock [217]. Case (c) of the Figure 8.3-3 illustrates a case where the occurrences and dura- tions of loads are random, but the level of stresses is constant. It is assumed that the decrease in strength during each loading depends on the duration of the load- ing. If the occurrences of the loads are Poisson distributed and the duration of each load is exponentially distributed, the reliability of the structure can be com- puted [217]. Let us assume that the stress rate and the duration are exponentially distribut- ed with parameters ì and µ. The failure occurs when the strength has decreased below a critical level, which is called d. The reliability of the structure is the probability that during the time interval (0, t) the accumulated time in the stress state (T(t)) is less than the critical value d [228]: , , , , , , , , , , j ¦ ¦ ) ¦ ` ¹ ¦ ¹ ¦ ´ ¦ ÷ ÷ + = s } ÷ ÷ · ÷ d y d t dy y d t I y e d t e d t T P 0 1 2 1 ìµ ìµ µ ì (8.3-16) 8. Probabilistic degradation modelling methods for power plant components 162 where I 1 (x) is the modified Bessel function of order 1 [229]: , , | | . | \ | = ÷ i i xe J e x I t t 2 1 1 2 1 1 (8.3-17) where J 1 is the Bessel function of the first kind of order 1. Case (d) of the Figure 8.3-3 illustrates circumstances in which there are two sources for stress, one being constantly in effect (e.g. temperature or radiation) and the other consisting of a variety of discrete shock loads. The accumulated stress is then [217]: , , , , , , } + = t c t aN dt t X t X 0 (8.3-18) where X c is the continuous load and N(t) is a point process describing the occur- rence of shocks. The failure occurs when the accumulated stress X(t) exceeds the strength. The strength may be assumed as random but independent of time. 8. Probabilistic degradation modelling methods for power plant components 163 Figure 8.3-3. Some simplified stress-strength-time modelling situations, from ref. [217]. The simplified cases above are applicable only to very limited situations. In practise, the formulation of the functions Y(t) and X(t) causes problems. There 8. Probabilistic degradation modelling methods for power plant components 164 are almost invariably difficulties in obtaining stresses and especially in obtaining strength. Discrete time Markov models are applicable to describe situations where the damage is caused by cyclic loadings. If, in addition, the amount of damage is discretised to specific damage states, the cumulative damage can be modelled with Markov chains [230]. An example of an ageing mechanism, which can be described with a station- ary, discrete time and state Markov process, is fatigue crack growth. The crack growth can be modelled with a transition probability matrix, where the states represent the depth of the crack and the discrete time describes the fatigue cy- cles. The final state, here defined as n, corresponds to the situation where the crack has propagated through the component wall, e.g. that of a pipe. The initial size of the crack, t 0 , is described with the following state vector: j ¦ n i p p p p · · · · · · = 1 0 0 t (8.3-19) where p i are the initial probabilities of various damage states. After k cycles, the crack size distribution t k is: k k P · = 0 t t (8.3-20) where P is the transition probability matrix, which contains the transition proba- bilities from each damage state to all considered more severe damage states. If a detected crack is repaired and no new cracks initiate during the inspections, the inspection procedure can be described with matrix D, which contains the detec- tion probabilities of cracks. The crack depth distribution after m cycles with one inspection between the cycles k and k+1, when k < m, can be computed as: k m k m ÷ · · · = P D P 0 t t (8.3-21) This approach concerns the cyclic loading and the Markov property is assumed only between the crack tip at the end of the cycle. In this case, the crack propa- gation is not continuous in time. In this example, it was assumed that at the time instant t the number of loadings needed to determine the distribution of the crack depth is known. The model becomes continuous if the cycles are assumed to occur randomly according e.g. to a Poisson process. 9. Risk analysis applications for power plant systems 165 9. Risk analysis applications for power plant systems 9.1 Introduction Risk analysis applications presented here are used in several industries. Due to their complexity, wide scope and importance in power plant applications, they are presented here in this separate chapter. The considered risk analysis applica- tions are PRA/PSA and RI-ISI. A detailed description of these approaches would be beyond the scope of this thesis, thus instead an overview is presented here. 9.2 Probabilistic risk/safety assessment Introduction PRA/PSA is a technique used to identify combinations of events that compose accident sequences and estimate their frequency of occurrence together with consequences. An accident sequence typically includes an initiating event fol- lowed by any number of failures or successes of control and mitigating func- tions, enabling conditions that retard or allow the accident sequence, and the responses of systems or equipment to developing conditions. The scope of ap- plicability of PRA/PSA is wide, it can be used for analysing of almost any com- plex technical system. In addition to power plants the technique is also applica- ble e.g. for car manufacturing, aerospace, chemical and process industries. In most countries, the method is referred to as PSA. In the United States the meth- od is referred to as PRA. Recently, The Finnish Radiation and Nuclear Safety Authority (STUK) has adviced/announced that also in Finland this method should be referred to as PRA. Despite having two names, the technique is practi- cally the same. 9. Risk analysis applications for power plant systems 166 Accident sequences are usually developed using event trees. Different paths through each event tree represent different accident sequences beginning with the same initiating event but having a different outcome because of system fail- ures or conditions that occur during the course of the accident. Branches of an event tree that represent failures of complex systems or processes are often de- veloped using fault trees. Fault trees are used to help the analyst systematically deduce the possible causes of a failure or fault. To estimate the probability of an accident sequence, each event in the se- quence is assigned a probability or frequency from operational data, calculations and/or expert judgements. The frequency for an accident sequence is estimated using probability mathematics to combine the probabilities of the events which in turn combine to create the sequence. Uncertainty in the probability or fre- quency estimates is included in the frequency estimates of the accident sequenc- es using simulation and other mathematical techniques. Accident sequences for complex systems or processes can then be ranked according to the severity of their consequences and the estimated frequency of their occurrence [82]. PRA/PSA usually answers three basic questions [1, 49, 71]: 1. What can go wrong with the studied technological entity, or what are the initiators or initiating events that lead to adverse consequences? 2. What and how severe are the potential detriments, or the adverse con- sequences that the technological entity may be eventually subjected to as a result of the occurrence of the initiator? 3. How likely to occur are these undesirable consequences, or what are their probabilities or frequencies? Motivations to perform PRA/PSA In many modern technological applications (e.g., nuclear power, chemical pro- cessing industry, etc.), PRA/PSA has proven to be a systematic, logical, and comprehensive tool to assess risk (likelihood of unwanted consequences) for the purpose of [84]: - increasing safety in design, operation and upgrade, - saving money in design, manufacturing or assembly and operation. In general, a PRA/PSA is needed when decisions need to be made that involve high stakes in a complex situation. Appropriate resource allocation depends on a 9. Risk analysis applications for power plant systems 167 well formed risk model. Developing a comprehensive scenario set is a special challenge, and systematic methods are essential [85]. Scope of PRA/PSA As for power plant environments, presently PRA/PSA is applied mainly for NPPs. Concerning NPPs, three PRA/PSA levels are usually defined. These are described as follows [83, 86]: - Level 1 PRA/PSA: Conditional core damage (CDF) is usually estimat- ed in this level. In general, in this level the events and event sequences that can lead to the exceeding of plant safety limits are identified and their probabilities of occurrence are quantified. - Level 2 PRA/PSA: Containment failure and radionuclide release fre- quencies are usually estimated in this level, given that a core damage state occurs. Level 2 PRA/PSA builds on the results of level 1 PRA/PSA, identifying the ways in which releases of fission products can occur from the plant. In addition to quantifying the likelihoods of the releases, also the actual amounts of fission products released are es- timated in this level. - Level 3 PRA/PSA: The offsite consequences from a release, e.g. early and latent cancer fatalities, given a radionuclide release occurs, are es- timated in this level. Level 3 PRA/PSA builds on the results of a level 2 PRA/PSA, estimating the public health risks and environmental con- sequences. This level provides insight into the accident mitigation and emergency response provisions. Steps in conducting a PRA/PSA The steps in conducting a PRA/PSA are listed in the following [49]: - methodology definition; development of an inventory of possible tech- niques for the desired analysis, - familiarisation and information assembly; collection of general data, the physical layout of the system or process (e.g. facility, plant, design), administrative controls, maintenance and test procedures as well as protective systems, 9. Risk analysis applications for power plant systems 168 - identification of initiating events; identification of events (abnormal events) that could result in a hazard, - sequence or scenario development; definition of a complete set of sce- narios that include all the potential propagation paths that can lead to the loss of confinement of the hazard following the occurrence of an initiating event, - system analysis; application of event tree and fault tree techniques to considered systems, - internal events external to the process; correspond to events that origi- nate within a system, - external events; correspond to events that originate outside of the system, - dependent failure considerations; identification of dependencies be- tween event paths, such as those due to coupling between their failure mechanisms, and their inclusion to both event trees and fault trees, - failure data analysis; predicting future availability of equipment based on past experiences and test data, quantification of initiating events, component failures and human errors, - quantification; fault tree and event tree sequences are quantified to de- termine the frequencies of scenarios and associated uncertainties in the calculation. Uncertainty and sensitivity in PRA/PSA Randomness (variability) in the physical processes modelled in the PRA/PSA imposes the use of probabilistic models. The development of scenarios introduces model assumptions and model parameters which are based on what is currently known about the physics of the relevant processes and the behaviour of systems under given conditions. It is important that both natural variability of physical pro- cesses and the uncertainties in knowledge of them are properly accounted for [85]. The most widely used method for determining the uncertainty in the output risk assessment is to use a sampling process, in which values for each basic event probability are derived by sampling randomly from the probability distri- bution of each event. These probabilities are then combined through the risk expression to determine the value of the risk assessment for that sample. This 9. Risk analysis applications for power plant systems 169 sampling process is repeated many times to obtain a distribution on the risk as- sessment [85]. Sensitivity analyses are also frequently performed in a PRA/PSA to indicate analysis inputs or elements whose value changes cause the greatest changes in partial or final risk results. They are also performed to identify components in the analysis to whose quality of data the analysis results are or are not sensitive [85]. Summary of PRA/PSA approach PRA/PSA is generally used for low probability and high consequence events for which insufficient statistical data exist. If enough statistical data exist to quantify system or subsystem failure probabilities, use of some of the PRA/PSA tools may not be necessary [84]. An illustration of the main steps of a PRA/PSA anal- ysis is shown in Figure 9.2-1. All in all, PRA/PSA analyses are carried out to support decisions. Figure 9.2-1. A diagram of the main steps of PRA/PSA analysis approach, from ref. [84]. Living PRA/PSA Nuclear facilities, because of their complex nature, are subject to change with time. These changes can be physical (resulting from plant modifications, com- ponent ageing, etc.), operational (resulting from enhanced procedures, etc.) and organisational. In addition, there are also changes in our understanding of the plant, due to the analysis of operational experience, implementation of data collec- tion systems, development of improved models, etc. Therefore, if the PRA/PSA is 9. Risk analysis applications for power plant systems 170 to be of continuing use in the enhancement and understanding of plant safety, the PRA/PSA must be updated or modified when necessary to reflect the above changes. This has led to the concept of a “living PRA/PSA”, which can be de- fined as a PRA/PSA which is updated as necessary to reflect the current design and operational features with providing detailed documentation of the updating process [88]. Acceptable risk levels The acceptability of risk by individuals depends on the degree of control over the risk producing activity that they perceive they have. Typically, people de- mand much lower risks from activities over which they have no control, e.g. commercial airliners [85]. Establishing the boundaries and guidelines for risk acceptability is a government responsibility. These levels may change as tech- nology improves or emphasis changes with regard to environmental issues [50]. The Nuclear Regulatory Commission (NRC) of U.S.A. has established safety goals for NPPs as follows [236]: - The individual early fatality risk in the region between the site bounda- ry and one mile beyond this boundary will be less than 5E-07 per year (one thousandth of the risk due to all other causes). - The individual latent cancer fatality risk in the region between the site boundary and 10 miles beyond this boundary will be less than 2E-06 per year (one thousandth of the risk due to all other causes). These goals were established using as a criterion the requirement that risks from NPPs should be smaller than other risks by a factor of 1000. Thus, the individual latent cancer fatality risk due to all causes was taken to be 2E-03 per year. Be- cause of the large uncertainties in the estimates of fatalities, the NRC is using subsidiary goals in its daily implementation of risk informed regulation. These are the reactor core damage frequency and the large early release frequency. The latter refers to the release of radioactivity to the environment. The Finnish Radiation and Nuclear Safety Authority (STUK) has also defined a set of numerical design objectives for Finnish NPPs, which are given in Guide YVL 2.8, Probabilistic Safety Analyses [89]. According to STUK the following numerical design objectives cover the whole NPP: - The mean value of the probability of core damage is less than 1E- 05/year. 9. Risk analysis applications for power plant systems 171 - The mean value of the probability of a release exceeding the target value defined in section 12 of the Council of State Decision (359/91) must be smaller than 5E-07 per year. However the containment has to be designed in such a way that its integrity is maintained with a high likelihood in case of both low and high pressure core damage. According to STUK the risks associated with various accident sequences of the PRA/PSA are to be compared with each other to ensure that no dominant risk factors deviating from the common risk level remain at a plant [89]. Some benefits and drawbacks of PRA/PSA After completing the compulsory PRA/PSA efforts, performing organisations have usually discovered benefits beyond mere compliance with regulation. The- se have included new insights into and an in-depth understanding of [1]: - Design flaws and cost effective ways to eliminate them in design prior to construction and operation. - Normal and abnormal operation of complex systems and facilities even for the most experienced design and operating personnel. - Design flaws and hardware related, operator related and institutional reasons impacting safety and optimal performance at operating facili- ties and cost effective ways to implement upgrades. - Approaches to reduce operation and maintenance costs while meeting or exceeding safety requirements. - Technical bases to request and receive exemptions from unnecessarily conservative regulatory requirements. The current PRA/PSA models usually include the following drawbacks [32]: - As a general practise, event tree and fault tree analyses of current PRAs/PSAs do not include passive SSCs, such as RPV and reactor coolant piping. The argument for doing so is that the failure rates for the passive SSCs are negligibly small. Although this argument may be reasonable when SSCs are new, a critical evaluation of this assumption is needed when the subjects of the investigation are older. - The traditional PRA/PSA approach assumes a constant rate for each component failure mode. 9. Risk analysis applications for power plant systems 172 - Except for failure rates of identical components that have been covered by common cause failure analyses and by complete state-of-knowledge correlation, the current PRA/PSA methodology does not model other kind of dependencies. - Failure probabilities of events other than hardware failure, such as hu- man error rates and component unavailability because of testing and maintenance, could be different for older plants than for new plants. The current PRA/PSA methodology does not make such a distinction. 9.3 Risk informed in-service inspection (RI-ISI) 9.3.1 Introduction Presently, the risk-informed in-service inspection (RI-ISI) analysis approaches are applied to NPP piping systems, whereas risk-based inspection (RBI) ap- proach is applied for conventional power plants and other industry plants involv- ing structural systems bearing relatively severe loads and involving periodic inspections. RI-ISI reflects recent developments in PRA/PSA methodology, the understanding of degradation mechanisms and the experience gained from the operating experience of NPPs. RI-ISI aims at rational plant safety management by taking into account the results of plant specific risk analyses. The fundamen- tal idea is to identify high-risk locations where the inspection efforts should be concentrated. The objective is to provide an ongoing improvement in the overall plant safety, measured by risk, together with reduced irradiation doses for the inspection teams [237]. In-service inspection (ISI) is an essential element of the defence in depth con- cept. ISI consists of non-destructive examination (NDE) as well as pressure and leakage testing. ISI helps to confirm that basic nuclear safety functions are pre- served and that the probability of radioactive materials breaching containment is reduced. The development of a RI-ISI programme requires expertise from a number of different disciplines including inspection, maintenance, design, materials, chem- istry, stress analysis, systems, PSA, operations and safety. It also requires a long- term co-ordinated management commitment through inspection qualification, inspection result analysis and final feedback to the risk analysis, in order to maintain a living ISI programme. Before resorting to a risk-informed pro- 9. Risk analysis applications for power plant systems 173 gramme, it is essential, therefore, to obtain the backing and commitment of the utility and plant management [237]. An impending failure and its consequences are not prevented or changed by RI-ISI unless additional mitigating actions are taken. Inspection is an initiator for actions such as the repair or replacement of deteriorating equipment, or a change to the operating conditions. By identifying potential problems, RI-ISI increases the chances that mitigating actions will be taken, and thereby reduces the frequency of failure [3]. RBI requires a wide range of information in order to assess the probability and consequences of equipment failure and develop an inspection plan. A plant data- base containing an inventory of the equipment and associated information is a useful way of managing the relevant data [3]. In the top level, the RI-ISI analysis is divided into the following three parts: - assessment of consequences, - assessment of degradation potential, - assessment of risks and the consequent forming of a risk informed inspection program. The plant specific PRA/PSA tool is used for the quantification of piping failure consequences. The PSA allows the calculation of several risk importance measures, such as risk increase factor, RIF (known also as risk achievement worth, RAW), and risk reduction factor, RDF (known also as risk reduction worth, RRW). In connection to RI-ISI, conditional core damage probability, CCDP, and conditional large early release probability, CLERP, have become commonly used consequence measures. There are basically the following three options/approaches for quantitative as- sessment of leak and rupture probabilities/frequencies of piping components: - use of PFM and/or other structural reliability approaches, - statistical estimation from experience data, e.g. large databases, - use of formal expert judgement, e.g. based on deterministic structural models. A best result is achieved with a combination of all these approaches. However, how much each of these approaches weighs as compared to others under various conditions depends on several aspects, the foremost of which typically being the amount and quality of available and applicable degradation data. When NPP 9. Risk analysis applications for power plant systems 174 piping degradation data are available only scarcely, the role of structural reliabil- ity approaches is strongly emphasised, and to a considerable extent that of expert judgement as well. 9.3.2 On principles of RI-ISI RI-ISI aims at a rational plant safety management strategy by taking into ac- count the results of plant specific risk analyses. The fundamental principles of any risk informed inspection programme are [237]: 1) The ability to define the consequence, probability and risk associated with structural failures so that the ISI programme can focus on an inte- grated defence in depth strategy. 2) The identification of an inspection programme that will decrease the risk from selected sites as far as it is practical and with due considera- tion of the costs and accumulated radiation dose to plant workers. The advantage of such an approach is the optimisation of the inspection efforts. The term optimise in this context is seen as a process that maintains defence in depth, whilst [237]: a) improving or at least maintaining the overall plant safety, b) minimising the radiation dose to personnel involved in the inspection activities, c) providing improved plant reliability. 9.3.3 Process of risk-informed inspection planning The main elements constituting the process of risk informed inspection planning are as follows [237]: 1) Assurance of the long term commitment of senior management to the risk informed methodology. 2) Formation of the RI-ISI assessment team. 3) Definition of the scope of the equipment/structures to be considered in the application. 4) Collection and analysis of the information required to carry out the risk assessment. 9. Risk analysis applications for power plant systems 175 5) Definition of the level of the evaluation. 6) Assessment of the probability of failure for all the components in- cluded in the scope of the application. 7) Assessment of the consequences of failure for all the components in- cluded in the scope of the application. 8) Ranking of the risks associated with all the components. 9) Performing sensitivity studies to determine the impact of changes in key assumptions or data. 10) Choice of the components to be inspected according to chosen criteria. 11) Assessment of the implication on inspection qualification. 12) Feed back of the obtained information (after completing the inspection). 9.3.4 Qualitative, quantitative and semi-quantitative methods for RI-ISI In principle, three different approaches to the use of risk concepts in developing ISI programs can be considered: the purely quantitative, the semi-quantitative and the purely qualitative approach. In the quantitative approach, both failure probabilities and consequences are assessed with physical models and expressed in physical units. In the qualitative approach, the failure probabilities are as- sessed qualitatively, taking into account only the involved degradation mecha- nism(s) and some factors affecting their severity, with simply dividing these mechanisms to classes having such descriptive names as high, medium, low and none, corresponding to the assumed levels of severity/threat. Both in quantitative and qualitative approaches, the consequences are considered and expressed with the same quantitative measures. Of the commonly applied RI-ISI methods, a notable example of a qualitative approach is the EPRI RI-ISI procedure [17, 240, 241], whereas a representative example of a quantitative one is the PWROG Methodology [242, 238, 239]. The semi-quantitative approach combines features from both quantitative and qualitative approaches. In the nuclear industry, the risks are assessed with PRA/PSA, which is a quan- titative approach. The use of PSA is the foundation of risk informed approaches in the nuclear industry, and thus a purely qualitative approach to RI-ISI that does not make use of PRA/PSA insights would be difficult to justify. 9. Risk analysis applications for power plant systems 176 Ideally, a purely quantitative approach should be chosen whenever feasible, as only in such framework, it is possible to quantify the risk change achieved when implementing the RI-ISI program. However, it is recognised that the current state of knowledge and understanding of some of the degradation mechanisms and the availability of the required plant data can be insufficient to accurately quantify the probability of failure. Likewise, many current PRA/PSA analyses may not be detailed enough for the level required in an ISI application, and thus the consequence analysis must be complemented by some degree of qualitative assessment. Thus, the most feasible approach to the development of any RI-ISI program today would arguably be the semi-quantitative one, based on a plant specific PSA and with due recognition of the current mechanistic understanding of any degradation mechanisms. 9.3.5 Brief overview and comparison of two main RI-ISI methodol- ogies The two main RI-ISI methodologies that are also widely applied in Europe are: PWROG methodology [242, 238, 239] and EPRI methodology [17, 240, 241]. Recently has been published a notable IAEA report [243] on current status, is- sues and development of RI-ISI methodologies. Some European countries have adapted these methodologies to their particular context, but have not added de- velopments that differ from the original methods. Flow charts presenting the methodologies of these two RI-ISI methodologies are presented in Figures 9.3-1 and 9.3-2 in the following. 9. Risk analysis applications for power plant systems 177 Figure 9.3-1. Flow chart of PWROG RI-ISI methodology, from refs. [242, 238, 239]. 9. Risk analysis applications for power plant systems 178 Figure 9.3-2. Flow chart of EPRI RI-ISI methodology, from refs. [17, 240, 241]. 9. Risk analysis applications for power plant systems 179 The main steps or structure in both the EPRI and PWROG methodologies are rather similar: scope definition, segmentation, probability of failure analysis and failure consequence analyses, risk ranking and finally selection of inspection sites. The main differences arise from the way of performing the failure probability analyses. In the EPRI methodology, the approach is basically qualitative, and quantification is basically done by using bounding values. In the PWROG meth- odology, a specific structural reliability analysis tool is used to quantify the fail- ure probabilities. The use of risk importance measures is somewhat different. In the PWROG methodology, the risk ranking is based on RRW, while EPRI methodology uses CCDP (and CLERP). There are also other minor differences related e.g. to the segmentation, and the final selection of inspection sites follows different rules. The impact of these differences has been studied in an OECD-JRC co-ordinated RI-ISI benchmark- ing project RISMET [251]. 9.3.6 Probability of detection of cracks The effectiveness of inspection is an important input parameter in RI-ISI anal- yses. A quantitative measure of inspection effectiveness is needed in order to calculate the reduction in risk associated with inspection. Quantitative estimates of the inspection capability enable a better optimisation of ISI. The reliability of NDT techniques is usually quantified using flaw detection reliability in terms of the probability of detection (POD). The detection probabil- ity is typically presented in the form of POD functions, which describe the de- tection probability as a function of the flaw size, e.g. flaw depth or length. How- ever, the construction of a POD function requires a considerable amount of data before statistical confidence is achieved. In case of NDT methods, it is often expensive and time consuming to produce such a large amount of data. For a recent notable document on POD issues, see ref. [254]. It contains a literature review of some important papers and reports, a review of the statistical models that have been proposed for quantification of inspection reliability as well as descriptions and recommendations on statistical best practices for producing POD curves. The quantification of the POD is not a straightforward task. The output from the European Network for Inspection Qualification (ENIQ) qualification process [252] is generally a statement concluding whether or not there is high confidence 9. Risk analysis applications for power plant systems 180 that the required inspection capability will be achieved in practice, for the speci- fied inspection system, component and defect range. However, the ENIQ meth- odology is not designed to provide a quantitative measure of inspection capabil- ity of the type which can be used by quantitative RI-ISI. It also means it is diffi- cult to “benchmark” the confidence associated with any given inspection qualifi- cation. The POD curves used in U.S. applications, e.g. in the SRRA code of the PWROG RI-ISI methodology [239], have been subject to criticism due to some- times unrealistically high detection capability assumptions. The development of full POD curves seems both impossible, and for RI-ISI applications also impractical. Simplified estimates would be of sufficient accu- racy in RI-ISI applications. Studies on the sensitivity of the risk reduction to POD have been documented e.g. in ref. [252]. The simplified POD step func- tions can then be implemented in the analysis process and used for calculating the effect of inspections, as is shown e.g. in recent VTT reports [244, 253]. If the failure probability is not evaluated using a PFM approach, but an esti- mate is given based on expert judgement or a statistical approach from experi- ence data, a simplified approach can be used for the evaluation of the impact of inspections. Then only a single POD value is used, without accounting for the detection limit. 9.3.7 RI-ISI approach developments by VTT VTT has aimed at developing a RI-ISI approach which uses quantitative classifi- cation for both failure potential and consequence assessment, but recognising the above mentioned limitations in failure probability quantification [244]. In RI-ISI approaches, NPP piping systems are for risk analysis purposes typi- cally divided into regions/parts called segments. In the VTT approach, the seg- mentation follows the basic principle, that within a segment, the degradation mechanism(s) and the consequence(s) of a piping failure remain unaltered. This is similar to the segmentation in the EPRI methodology. The VTT approach also includes an expert panel to review the initial risk ranking. This is in line with the ENIQ Framework Document [237] and the Finnish regulations. As a result, a more accurate and refined modification of the qualitative EPRI RI- ISI risk matrix procedure was suggested at VTT. The modifications include [245]: - changing the degradation category in the risk matrix from qualitative to quantitative, 9. Risk analysis applications for power plant systems 181 - assessing the piping segment failure probabilities with a PFM based analysis tool developed at VTT, instead of evaluating them roughly based just on operational/process conditions as is done in the EPRI procedure, and - assessing and optimising the piping segment risks with a Markov sys- tem based analysis tool developed at VTT. The overall method of the applied discrete time Markov procedure for piping risk analyses can be summarised in six steps as follows [245, 246]: 1. crack growth simulations based on PFM, 2. construction of degradation matrix transition probabilities from PFM simulations and database analysis of crack initiation frequencies, 3. model for inspection quality, which is used to construct inspection ma- trix transition probabilities, 4. Markov model to calculate pipe rupture probabilities for chosen in- spection schemes, 5. obtaining pipe rupture consequences from plant specific PSA, and 6. comparison of results for different inspection strategies, where measures of interest include yearly rupture probability, yearly core damage probability and average values for both of these over plant lifetime. The PFM part of the VTT RI-ISI approach has also been benchmarked against some of other PFM analysis tools in a recent international project called PODRIS, see ref. [247]. These other PFM/RI-ISI analysis tools were NURBIT by Det Norske Veritas (DNV) [248, 249, 250] and JRC approach (which is based on a probabilistic implementation of the R6 Method, Rev. 4 [200], and a discrete time Markov process analysis). In PODRIS, the main interest was to investigate whether it is justifiable to use a simplified POD curve. The PFM analysis tools were applied to a small but representative set of NPP pipe compo- nents, from both BWR and PWR units, whereas the considered degradation mechanism was SCC. A good correspondence between the PFM results comput- ed with these three tools was achieved. As for other significant results, they indi- cate that the use of a simplified POD could be justifiable in RI-ISI applications. 9. Risk analysis applications for power plant systems 182 9.3.8 RISMET project – benchmarking of RI-ISI methodologies The RISMET project was launched in 2005 by the Joint Research Centre of the European Commission (JRC) together with the Nuclear Energy Agency of the OECD (NEA), with the aim of benchmarking several RI-ISI methodologies. It featured more than twenty participating organisations from Europe, U.S.A., Canada and Japan, representing utilities, regulators and research organisations. Also VTT participated in RISMET [251]. In RISMET, various RI-ISI methodologies were applied to the same case, consisting of four selected piping systems at the Swedish Ringhals 4 PWR unit [251]. The applied RI-ISI methodologies were: SKIFS [255], PWROG [242, 238, 239], PWROG-SE (an adaptation of the original PWROG methodology to the Swedish regulatory environment), EPRI [17] and ASME Code Case N-716 [256]. The RI-ISI applications were compared among each other and to the de- terministic ASME XI ISI selection procedure. The scope of the benchmark was limited to four systems, but the variety re- garding safety class, potential degradation mechanisms and pipe break conse- quences ensured a good coverage of issues for a comparative study. The risk- informed methodologies showed some significant differences and resulted in slightly different risk ranking and selection of inspection sites. However, the results of the benchmark indicated that the risk impact of these differences is small, and the RI-ISI approaches identify safety important piping segments that are ignored by approaches not using the PSA. The results of the benchmark ex- ercise RISMET improve the knowledge on differences in approaches and their impact on plant safety, and promote the use of RI-ISI [257]. 10. Power plant component ageing analyses 183 10. Power plant component ageing analyses Ageing of various SSCs affects the safety of power plants. The objective of component ageing analyses is to identify the degradation mechanisms of com- ponents and the increase in failure occurrences, to assess the remaining lifetime of components and to find suitable means to prevent or mitigate the effects of ageing. This chapter deals mainly with ageing analyses concerning NPP compo- nents. The ageing analyses should be focused on those components that are the most significant from the safety point of view. Other significant selection criteria, especially for components which are not safety critical, are availability and costs. For most PWRs, the most safety significant components are [231]: RPV, vessel internals, pressuriser, steam generators, main primary pipes, main primary pump, containment, cables and concrete structures. For most BWRs, the most safety significant components are [231]: RPV, vessel internals, pressuriser, steam generators, main primary pipes, recirculation pump, containment, cables and concrete structures. After the components are selected, the analysis methods for ageing detection and prediction are chosen. The suitability of a procedure depends on the selected component and on the available data. When the ageing is detected, suitable tech- niques are applied to prevent or mitigate the effects of ageing. Ageing research programmes have been conducted in several countries, e.g. in the U.S., Japan and France [32, 232, 233, 234]. 10.1 Selection of components for analyses The evaluation of safety significance and ageing sensitivity of components can be based on the analysis of operating experience, expert opinions and probabilis- tic techniques for prioritising and determining risk significance of ageing. The 10. Power plant component ageing analyses 184 results from PRA/PSA can (and usually are) be used in association with the probabilistic techniques [4]. If the collected degradation and failure data is properly analysed and reported, the operating experience data can be effectively used to identify systems and components which are susceptible to ageing. A NPP has a large variety of components, many of which are essential for overall plant safety. Some of the safety related components contribute more than others towards ensuring plant safety and the extent to which these components are susceptible to ageing also varies considerably [235]. It is neither practical nor necessary to evaluate and quantify the extent of the many thousands of individu- al power plant components. Therefore, a more rational and cost effective ap- proach is required. The approach presented here is based on a systematic selec- tion and prioritisation of safety important NPP components for more detailed ageing analyses. IAEA has proposed a component selection process for NPPs [235], which is based on a systematic examination of all systems and structures from a safety perspective. This examination is based on the following precepts: - Required safety margins for NPP components have to be maintained at all times. - Safety important components in a NPP have already been classified. - The failure of safety important components can result in different de- grees of loss of system safety functions, with or without severe conse- quences. - Current maintenance, testing and inspection programmes in NPPs in- clude different strategies of varying effectiveness to deal with ageing effects. - Safety important components may experience varying degrees of deg- radation due to ageing under different loading and environmental con- ditions. - Methods used for monitoring ageing effects in specific components and structures may vary significantly. - Existing databases pertaining to ageing effects in NPP components are not sufficiently comprehensive. 10. Power plant component ageing analyses 185 10.2 Identification of ageing mechanisms The process used for identifying deterioration mechanisms for pressure systems and other systems containing hazardous materials should be conducted in a wide ranging and systematic manner. It is more effective if it involves experienced staff from different disciplines rather than being the work of a single person. Acceptable processes could include combinations of the following [3]: - review of specific plant history and information from previous inspections, - review of experience across similar industries or plants, - expert elicitation of knowledge of structural integrity and materials, - use of check-lists and mechanism descriptions, - computer based expert systems. 10.3 Mitigation of ageing effects In the last phase of ageing analyses, decisions on management of ageing should be made on the basis of identified degradation mechanisms and their severity. The primary goal in this phase is to find suitable methods to prevent, mitigate or restore the effects of ageing [6, 217]. Thus the basic management methods are controlling and slowing down ageing as well as replacement of components. The application of these methods is component specific, depending on e.g. the expected ageing rate and mechanisms, the possibilities of replacement and early failure detection possibilities. In the case of repairable or replaceable compo- nents which will age relatively fast compared to the plant lifetime, the ageing management means following the efficiency of current maintenance procedures and reviewing the test programmes. In the case of components which are diffi- cult to replace, the efforts may be focused on the reduction of environmental stressors, re-evaluation of surveillance and condition monitoring methods. The effects of ageing on SSCs are detected through different mechanisms in plant operation, such as preventive/corrective maintenance, inspections, moni- toring and performance monitoring of SSCs. The mechanism that has given rise to the degradation of the behaviour or characteristics of component material can be determined through the subsequent analysis of these effects. The selection of an efficient method for mitigating ageing in component material depends on an accurate determination and evaluation of the degradation mechanism that caused the ageing [6]. 10. Power plant component ageing analyses 186 One principal concept in ageing and maintenance management is the concen- tration of effort on reduction of the failure probability of the most important components [87]. Reliability centered maintenance (RCM) is a method for estab- lishing a scheduled preventive maintenance programme resulting in improved component reliability and minimised costs [258]. The RCM approach intends to optimise the use of maintenance resources by identifying the most critical com- ponents with respect to safety, availability or maintenance costs, and selecting for these components the most appropriate maintenance procedures with the aid of decision logic. Commonly applied ageing management methods for SSCs in NPPs can be grouped into three main categories [6]: - change of SSC design, - change/recovery of material characteristics, - change of operating parameters. 11. Probabilistic ageing and risk analysis tools 187 11. Probabilistic ageing and risk analysis tools There are many software tools that can be used for system and component risk and ageing analyses. An overview of some of the noteworthy of these tools is presented in the following. The emphasis is on component analysis tools which are presented first, following with a more brief presentation of the system analy- sis tools. The emphasis of this presentation is on the component analysis tools. The overview ends with a summary of the presented analysis tools. In the general analysis applications, any failure mode, fracture or other, can be addressed using the appropriate limit states. Validation of such software is usual- ly carried out by benchmark exercises between different programs since often it is not feasible to compare results with real failure statistics, due to their scarcity. The probabilistic analysis approaches in the presented software vary from FORM and SORM to MCS. Some of the presented analysis tools are commer- cially available and some are so called in-house programs, the availability of which may be limited or non-existent. 11.1 Component ageing and risk analysis software PRAISE PRAISE is a PFM computer code for estimating probabilities of leaks and breaks in NPP cooling piping. The code was originally developed at Lawrence Livermore National Laboratory (LLNL) for the assessment of seismic events on the failure probability of PWR piping. PRAISE is an acronym for Piping Reliabil- ity Analysis Including Seismic Events [259]. Latest version of the code is named WinPRAISE, which is a Windows version of the original PRAISE code [260]. PRAISE considers the initiation and/or growth of crack-like defects in piping welds. This meaning that both fabrication cracks and those induced by service 11. Probabilistic ageing and risk analysis tools 188 are covered. The initiation analyses are based on the results of laboratory studies and field observations in austenitic piping material operating under BWR condi- tions. The considerable scatter in such results is quantified and incorporated into a probabilistic model. The crack growth analysis is based on deterministic frac- ture mechanics principles, in which some of the input parameters, such as initial crack size, are considered to be random variables. MCS, with stratified sampling on initial crack size, is used for probabilistic computations [261]. Presently PRAISE contains models for the analysis of both SCC and fatigue crack growth. In the case of SCC, the growth model is divided into two parts: the early growth of initiated crack and fracture mechanistic crack growth. The frac- ture mechanistic SCC growth rate model parameters are based on data on growth rates in fracture mechanics specimens. The code has features for fatigue crack growth analysis in both ferritic and austenitic steels. The probability of detection (POD) of cracks is based on a model where the POD is a function of the crack area [260]. PRO-LOCA The PFM based PRO-LOCA analysis code has been developed for predicting break probabilities for loss-of-coolant-accidents (LOCAs) in NPP piping sys- tems. In brief, the background of PRO-LOCA is that in 2003 the USNRC began its development [262] with the intension to adopt and apply advances in fracture mechanics models, and thereby to provide the successor to the PRAISE code. Like PRAISE, the PRO-LOCA code addresses the failure mechanisms associat- ed with both pre-existing cracks and service induced cracks. Both fatigue and intergranular SCC (IGSCC) are addressed. In addition, PRO-LOCA can also predict failure probabilities for primary water SCC (PWSCC). Other improved capabilities are in the areas of leak rate predictions and the prediction of criti- cal/unstable crack sizes. PRO-LOCA has also been incorporated an improved basis for simulating weld residual stress (WRS) distributions. In the recent MERIT project, PRO-LOCA has been developed further, and in summary the code has now the following features [263]: - crack initiation models for fatigue or stress corrosion cracking for pre- viously unflawed material, - sub-critical crack growth models for fatigue and SCC for both initiated and pre-existing circumferential defects, 11. Probabilistic ageing and risk analysis tools 189 - models for flaw detection by inspections and leak detection, and - crack stability. The PRO-LOCA code can thus predict the leak or break frequency for the whole sequence of initiation, subcritical crack growth until wall penetration and leak- age, and instability of the through-wall crack (pipe rupture) [263]. The results obtained with the PRO-LOCA code are a sequence of failure frequencies which represents the probability of a surface crack developing, a through-wall crack developing, and six different sizes of crack opening areas corresponding to dif- ferent leak flow rates or LOCA categories. Note that the level of quality assur- ance of the PRO-LOCA code is such that the code in its current state of devel- opment is considered to be more of a research code than a regulatory tool. NURBIT NURBIT developed by Brickstad and Zang [250] is a RI-ISI/PFM software. In order to perform the probabilistic evaluations, a computer code named PIFRAP (PIpe FRActure Probabilities) was developed. PIFRAP contains the analysis programs LBBPIPE and SQUIRT for the evaluation of crack growth and leak rates. PIFRAP calls upon these programs for different values of initial crack length and required quantities for the integration are obtained to give the leak and rupture frequency. Later, NURBIT was developed [250], which includes the features in PIFRAP for the failure frequency evaluation. In addition, NURBIT also performs a complete RI-ISI selection based on a risk ranking procedure including the selection of the most appropriate inspection interval. Here risk means the core damage frequency, commonly used in PSA studies. NURBIT version 2.0 also includes a cost-benefit analysis where the balance between in- spection costs and failure costs are made. The PIFRAP code is based on the model described by Nilsson et al. [266]. A more thorough description of the PIFRAP model with sensitivity analyses can be found from the SKI report describing the application to establish risk informed ISI priorities for Oskarshamn 1 piping [265]. The PIFRAP code is intended for calculation of the failure probability of a specific pipe cross section with a certain stress state and possibly containing a circumferential crack growing due to IGSCC. A number of assumptions were made for the probabilistic analysis [267]: 11. Probabilistic ageing and risk analysis tools 190 - stresses are deterministic, - crack growth is deterministic, - initial crack depth is fixed to 1.0 mm, - initial length of the crack is random (based on Swedish distribution data), - the probability of not detecting a crack at an ISI depends on crack length and crack depth (the actual function used was not dependent on the crack length, but a possible dependence on length is retained in the general equations), - there is a detection limit for leakage flow, and above this limit there is a probability of not detecting a leakage flow. Due to the assumptions made, the growth of the crack will be deterministic and will only depend on the initial crack length for a given geometry and given stresses. The probabilistic data to the PIFRAP code are [267]: - the probability that a crack with an assumed depth is initiated during a certain time interval, - initial length of the crack (initial crack length/depth is calculated from sta- tistical data backwards, to a calculated length when the depth is 1.0 mm), - the probability of not detecting a crack at an ISI, - the probability of not detecting a leak rate for a given leak rate detec- tion limit. In addition, the code requires deterministic data on pipe geometry, loading con- ditions and material properties [267]. ProSACC The analysis software ProSACC [268] developed by Det Norske Veritas (DNV) contains both deterministic and probabilistic analysis capabilities. The determin- istic part of the software ProSACC may be used both for assessment of detected cracks or crack like defects and for defect tolerance analysis. The procedure, which is based on the R6 Method [201], could be used to calculate possible crack growth due to fatigue or stress corrosion and to calculate the reserve mar- gin for failure due to fracture (if wanted, including a limited amount of stable crack growth) and plastic collapse. ProSACC also has an option, which enables 11. Probabilistic ageing and risk analysis tools 191 assessment of cracks according to the 1995 edition of the ASME Boiler and Pressure Vessel Code, Section XI [173], Appendices A, C and H for assessment of cracks in ferritic pressure vessels, austenitic piping and ferritic piping, respec- tively. ProSACC mainly considers surface crack postulates in plates, hollow cylinders and spheres. The probabilistic part of the software ProSACC is based on a PFM procedure that calculates two different failure probabilities: - probability of failure, defect size given by NDT/NDE, - probability of failure, defect not detected by NDT/NDE. Probabilistic VTTBESIT The analysis software VTTBESIT developed by both VTT [244, 246, 247, 269] and Fraunhofer-Institut für Werkstoffmechanik (IWM, Germany) [270, 271, 272], contains both deterministic and probabilistic analysis capabilities. Mainly, the probabilistic features are considered here. With VTTBESIT it is possible to quickly compute mode I stress intensity fac- tor values along the crack front as well as crack growth. The covered component geometries are hollow cylinders and straight plates. The code was originally intended for deterministic fracture mechanics based crack growth analyses, but its scope has been extended by adding probabilistic capabilities to the code. Both cracks caused by fabrication and those induced by service are considered. The probabilistically treated crack growth analysis input data parameters are as follows: - depth of initial cracks, - length of initial cracks, - frequency of load cycle occurrence. For the time being, other crack growth analysis input data parameters are con- sidered to have single deterministic values. In VTTBESIT analyses, the crack growth increment in each time step is com- puted from the respective crack growth equation. The covered crack growth mechanisms are fatigue and SCC. The simulation ends either when the crack depth reaches the outer pipe surface, or the time cycles reach the end of plant lifetime. For each analysed case, thousands of separate simulations are calculat- ed, and for each of these, values of the above mentioned distributed input data parameters/variables are sampled at random from the respective probabilistic 11. Probabilistic ageing and risk analysis tools 192 distributions. Typically each simulation run spans several decades of assumed plant operation with the crack depth calculated yearly, conditional on the exist- ence of an initial flaw. The annual crack depth information for each simulation is transferred to the second phase of the analyses. In the second phase, the analyses are based on Markov process. This method is a stochastic process in which the probability distribution of the current state is conditionally independent of the path of past states, a characteristic called the Markov property, see Section 8.3.2 for a more detailed description concerning Markov models. In the VTT application, the states of the Markov process corre- spond to crack penetration depths through component wall, and the transition probabilities those from a lower state to higher states (deeper cracks). On the other hand, the effects of inspections are included in the model as transitions from a cracked state to the flawless state, as each time a crack is detected it is assumed that the component in question is repaired or replaced with a new i.e. intact one. The effect of inspections is taken into account with POD functions. For the whole duration of the assumed operational plant lifetime the results obtained from a VTTBESIT analysis cover yearly piping component failure probabilities for different inspection strategies, including also the case of no inspections, for the considered degradation mechanism, being either fatigue or SCC. These results can then be used further e.g. in RI-ISI analyses. STRUREL STRUREL (STRUctural RELiability) is a general purpose reliability software series that has been developed by Reliability Consulting Programs GmbH [273] for performing different computational tasks. The program comprises several independent but interrelated programs: - STATREL; statistical analysis of data, simulation, distribution fitting and analysis of time series, - COMREL; time-invariant and time-variant analysis of component reliability, - SYSREL; reliability analysis of systems, - NASCOM; finite element code for structural analysis, - NASREL; module combining COMREL with NASCOM. STATREL enables appropriate distributions to be derived for datasets input from e.g. spreadsheets. Goodness of fit tests are also included to demonstrate the 11. Probabilistic ageing and risk analysis tools 193 best fitting method to be used. COMREL comprises 44 models and limit state equations can be input for failure modes not addressed. The program includes MC simulation, FORM and SORM methods, and for the case of the time-variant version it includes methods for incorporating random and point-in-time events. SYSREL enables multiple failure criteria for parallel and series systems to be evaluated, including conditional events. It links directly with COMREL, making it straightforward to check individual failure criteria before combining them in a system analysis. STRUREL has applications in many fields, including nuclear industry, and there exists many examples of its use for fracture, fatigue, collapse, corrosion and general strength problems [274]. ProSINTAP ProSINTAP (PRObabilistic Structural INTegrity Assessment Procedure) auto- mates MCS and FORM analysis of the failure assessment diagram and is only applicable to fracture and collapse failure modes [275, 276]. It consists of five input data decks: Geometry, Loading, Material, NDE and Analysis. The Geometry section comprises stress intensity factor solutions for a range of plate and hollow cylinder geometries with surface and through-thickness cracks. The load module enables through thickness distributions of applied and welding residual stress to be incorporated. In the material module, yield strength, rupture stress and fracture toughness and their associated statistical distributions are input. This requires the mean and standard deviation of each parameter as de- fined by the normal, log-normal or Weibull distribution. The NDE module ena- bles defect sizing as input data, but also allowing for treatment as an exponential distribution. The Analysis module enables the user to select MCS or FORM methods and to apply partial safety factors if required in order to achieve a spec- ified target reliability method. CALREL CALREL (CALibration of RELiability) is a general purpose structural reliability analysis program developed by the University of Berkeley, California [277]. The code is designed to work on its own or to operate as a shell program in conjunc- tion with other structural analysis programs. Structural failure criteria are de- fined in terms of one or more limit state functions. The specification is by the user through user defined subroutines. 11. Probabilistic ageing and risk analysis tools 194 CALREL is capable of computing the reliability of structural components as well as systems. Specific macro commands are available for the following types of analyses: - first-order component reliability analysis, - second-order component reliability analysis by both curvature fitting and point fitting methods, - first-order reliability bounds for series systems, - first-order reliability sensitivity analysis for independent and dependent variables with respect to distribution and limit state function parameters, - directional simulation for components and general systems, employing first or second-order fittings of the limit state surfaces, - MCS for components and general systems. CALREL has a large library of probability distributions for independent as well as dependent variables. Additional distributions may be included through a user defined subroutine. CALREL is available for purchase from UC Berkeley as both object and source code. STAR 6 STAR 6 [278] is a reliability software developed by British Energy for automat- ing reliability analyses of the fracture and collapse analysis procedures in R6 Method [200]. It is similar in approach to ProSINTAP and includes FORM and MCS, but a more extensive library of stress intensity factor solutions for differ- ent component geometries is included [200]. NESSUS NESSUS is a general purpose tool for computing the probabilistic response or reliability of structures, developed by Southwest Research Institute (SwRI) [279]. The framework of NESSUS allows the user to link traditional and advanced probabilistic algorithms with analytical equations, external computer programs including commercial finite element codes and general combinations of the two. The finite element codes NESSUS has interface to include ABAQUS, NASTRAN and PRONTO. Also, NESSUS provides a built-in finite element structural modelling capability [280]. 11. Probabilistic ageing and risk analysis tools 195 Eleven probabilistic algorithms are available in NESSUS. These include MCS, FORM, SORM, FPI, MV, AMV and LHS. In addition, NESSUS allows linking of different analysis packages or analytical functions to allow a general relationship of physical processes to predict the uncertainty in the performance of the system [280]. NESSUS can compute the cumulative distribution function of system perfor- mance (e.g. stress or strain) or include the strength of the system to compute the reliability. Traditional reliability analysis involves computing the probability of the stress, S, exceeding the strength, R, or P[R < S] or P[g < 0], where g is the limit state function. R and S may be complex models involving other random variables such that R = R(X i ) and S = S(Y i ). In NESSUS function g is formulated such that g = g(X i , Y i ), and thus it accounts for possible correlation between the stress and strength parts of the performance measure. NESSUS provides also a general formulation of function g that allows linking of different analysis codes and analytical functions. For example, a stress or strain from a finite element analysis can be used with a fatigue life equation or S-N curve to define the per- formance of a structure. The results for a component analysis in NESSUS are reliability or cumulative distribution function and importance factors. The im- portance factors indicate which variables contribute most to the unreliability and may be used for design changes or manufacturing process modifications [280]. NESSUS licences are available for purchase. NASGRO Deterministic analysis software NASGRO, developed by SwRI, is a suite of computer programs comprised of analysis modules linked together by graphical user interfaces (GUIs) [281]. The objective in developing NASGRO was to im- prove an earlier computer code to accommodate more recent advancements in fracture mechanics and crack propagation theory and to meet the needs for dam- age tolerance and durability analyses [282]. The capabilities of NASGRO include: - fatigue crack growth and fracture analysis, - structural life assessment, - stress computation, - fatigue crack growth properties processing and storing. 11. Probabilistic ageing and risk analysis tools 196 When using NASGRO, various types of analyses can be performed in an interac- tive mode using the GUIs for each module. Material properties for crack growth can be chosen from a large database by selection from a menu. The material property module can also be used to enter, edit and curve fit crack growth data. Analysis in complex two dimensional geometries with or without cracks to ob- tain stress intensity factors and stresses can be performed using the boundary element module of NASGRO. The software is commercially available. BRT-CICERO Electricité de France (EDF) has developed software BRT-CICERO [283] for both deterministic and probabilistic evaluation of pipe wall thinning due to flow accel- erated corrosion (FAC). It has been in use in every French NPP unit since 1994. In the modelling of FAC, the main parameters of influence that are taken into account are: effect of the water chemistry: pH, oxygen, steel composition (Cr, Cu, Mo), influence of mass transfer in single phase flow, influence of tempera- ture and geometry. BRT-CICERO provides e.g. the following results: - wear and wear rate of the pipe wall, - evolution of residual thickness determined from the nominal thickness and the tolerances of the nominal thickness. The probabilistic computations allow to: - follow the evolution of the fractiles of the residual thickness distribu- tion as a function of time, - follow the evolution of the probability that the residual component thickness is less than the critical thickness as a function of time, - evaluate the residual lifetime of selected component. The probabilistic module calculates the evolution of the 1E-03 fractile of the residual thickness distribution until it reaches the critical thickness. The proba- bility for thickness being less than critical is evaluated after this. When the 1E-03 fractile estimate appears to be higher than the critical value, plane MCS is ap- plied. If not, MC variance reduction technique is applied. Due to its proprietary nature, a detailed documentation of BRT-CICERO or the code itself are not available for purchase [1]. 11. Probabilistic ageing and risk analysis tools 197 PROPSE DNV has developed software PROPSE (PRObabilistic Program for Safety Eval- uation) [284] for computation of probability of failure when the NDT/NDE have found/missed a defect in NPP piping system inspections. PROPSE contains two different algorithms to calculate the probability of failure: simple MCS with an error estimate on probability of failure, and application of FORM with sensitivity factors using the most probable point of failure in a standard normal space. In PROPSE, the following assumptions are made: - crack depth is assumed as log-normally distributed, - constant crack aspect ratio is assumed, where a conservative assump- tion is to use a high value, e.g. 30, - fracture toughness is assumed as normally distributed, - material yield strength is assumed as normally distributed, and covari- ance for tensile strength is assumed to be the same as for the yield stress. According to PROPSE, the resulting failure probabilities are usually very small for shallow defects in moderately stressed components. This illustrates the gen- eral observation that there is a distinct difference, in terms of failure probabilities and risk for core damage, between those components which have an active dam- age mechanism and those which have not. In PROPSE, failure is predicted when the surface crack reaches a critical condition. Through-wall cracks are not con- sidered in the present program version. SRRA SRRA (The Structural Reliability & Risk-Assessment) [285, 242] is an analysis software for assessing pipe component failure probabilities. It is developed by Westinghouse to support piping RI-ISI analyses. Besides pipe components, SRRA is applicable for probabilistic integrity assessment of RPVs and RPV internals. The SRRA applies PFM approach and MCS to simulate the effect of time de- pendent degradation mechanisms and infrequent loading events and to derive the probability of piping failure for a variety of failure modes. In SRRA, the user is required to provide best estimate (median) values for various input parameters related to pipe material, geometry, operating conditions, NDE, service stresses and cycles. Each parameter is associated with a statistical distribution and stand- ard deviation which are based on industry data and detailed analyses are not 11. Probabilistic ageing and risk analysis tools 198 required to be entered by the user. Similarly, assumptions for initial flaw distri- butions are built-in to the program based on industry correlations with respect to weld geometry and pre-service inspection. The SRRA code has the capability to simulate the effect of a variety of time dependant material degradation mechanisms for components of carbon steel and stainless steels. These mechanisms include: - low-cycle fatigue crack growth of an existing (fabrication) flaw, - crack growth of an existing flaw due to SCC, - wall thinning due to material wastage.(e.g. by FAC), - high-cycle fatigue induced stresses exceeding the fatigue crack threshold. In each MC simulation trial, the code simulates the effect of the modelled degra- dation over time and at each time step the pipe stability against rupture is checked with respect to design limiting loading conditions. Design limiting stress input data may also be attributed with a frequency corresponding to the frequency of occurrence of the corresponding loading event, such as earthquake, waterhammer or a pipe break. SRRA computes the lifetime failure probability for three different failure mode types concerning piping pressure boundary: - small leak (i.e. through-wall flaw with minor leakage), - large leak (i.e. a through-wall flaw that leads to leakage beyond a user- defined value), - full break (i.e. complete severance of the pipe cross-section). Based on user defined input data for ISI interval and POD model, for the appli- cable NDE technique SRRA also calculates the (reduced) failure probability, assuming that flaws detected by ISI will be mitigated before failure can occur. By entering best estimate leak detection capability, credit may also be given to the reactor coolant leak detection system in the calculation of large leak and break probabilities. Again, it is assumed that mitigating actions would be taken on a detected through-wall flaw, before such a flaw would develop into a large leak or break. 11. Probabilistic ageing and risk analysis tools 199 VISA-II With VISA-II analysis software [286] developed by Oak Ridge National Labora- tory (ORNL), it is possible to assess the failure probability of a PWR RPV under pressurised thermal shock (PTS) conditions. The VISA II code is divided into two parts to define stress and strength mod- els needed for the following MC analyses. The deterministic part defines the stress model and the probabilistic part defines the strength model. In the first part, a deterministic fracture mechanics analysis is performed for a temperature and pressure transient entered by the software user. This analysis estimates val- ues of crack tip temperatures and applied stress intensity factors for several crack depths. These values are used by the probabilistic analysis performed in the second part. The second part treats flaw depth and fracture toughness of ves- sel materials as random variables. The sampled values of fracture toughness are compared with the stress intensity factors (estimated in the first part) at a sam- pled flaw depth to determine crack initiation and growth. The proportion of through wall cracks in a large number of passes through the simulation loop is an estimate of the conditional probability of through-wall cracking (vessel failure). The VISA II code uses closed form solutions for heat transfer and stress calcu- lations. It uses influence coefficients for calculating applied stress intensity fac- tors. The necessary input data for the deterministic analysis include: (a) pressure and temperature of the reactor coolant as a function of time for a given PTS tran- sient, (b) surface heat transfer coefficient, (c) material properties, and (d) wall thickness and radius of the vessel. Effects of cladding are included in the heat transfer and stress analysis. The software provides a selection of initial crack distributions. For each pass through the simulation loop, simulated values of initial RT NDT , fluence at the inner wall, flaw (crack) size, flaw location, and copper and nickel contents are selected from their respective distributions. These sampled values are eventually used to determine the fracture toughness. With these values fixed for a given pass, the software performs time history analysis for a given transi- ent. At the end of each time step, the simulated value of the fracture toughness is compared with the applied stress intensity factor at the crack tip. If the fracture toughness is less than the applied stress intensity factor, crack initiation occurs, otherwise, the simulation moves to the next time step. If crack initiation occurs, the crack is extended 0.25 in. and the crack arrest toughness K Ia is simulated. If crack arrest occurs, the simulation moves to the next time step, otherwise the 11. Probabilistic ageing and risk analysis tools 200 crack is extended another 0.25 in., and a new value of K Ia is simulated. This process continues until either the crack grows through wall or the transient is completed. The conditional probability of through wall cracks is obtained by dividing the number of simulations (passes) that resulted in through wall crack- ing by the total number of simulations that were made. For each time step, the sampled value of fluence at the crack tip is calculated. Then, the value of the shift in RT NDT is calculated using sampled values of cop- per and nickel and the attenuated fluence. The calculated value of the shift is then added to the sampled initial RT NDT to obtain the adjusted RT NDT , which is used to estimate the simulated fracture toughness. This fracture toughness is compared with the applied stress intensity factor determined in the first part of the analysis. RR-PRODIGAL The structural reliability software RR-PRODIGAL [299] by Rolls-Royce consists of the following three program parts: DANCER, ISIBREAK and SNOWBREAK. DANCER and ISIBREAK evaluate the probability of failure of a welded joint, whilst SNOWBREAK evaluates the probability of failure for an area, such as an area of stress concentration, which includes a period of defect initiation prior to crack growth to failure. The objective of the program DANCER is to simulate the number and size of defects generated during the welding process. Its output is a histogram or fre- quency plot of the defects that may formate during the normal build of a weld. The program is designed for multi-pass welds only and can handle ‘Single’ and ‘Double Vee’ weld constructions. In the software methodology, both the tech- nical expertise of welding process engineers and mathematical modelling are used to build a model that will simulate the weld manufacture and the errors that lead to different types of defects. The defects that may initiate in weld beads include center cracks, lack of fusion, slag, pores with tails and cracks in heat affected zones. Various welding processes are addressed including submerged metal arc welding. The model simulates the effects of both radiographic and dye penetrant surface inspections. With this methodology, the model attempts to build up, rather than measure, a defect distribution and density for a given type or family of welds [287, 288]. DANCER does not use MCS to obtain the failure probability. Instead, since the failure criteria are deterministic, a conditional probability of failure follows directly from the distributions of Paris-Erdogan 11. Probabilistic ageing and risk analysis tools 201 crack growth rate equation parameters. Thus, a set of conditional failures can be determined and hence the distribution of the conditional failure probabilities over the range of initial defects is determined [299]. The first function of ISIBREAK is to utilise the stress input data to estimate the conditional probability of failure for defects of varying size and position within a weld. The conditional probability of failure of a defect is the probability that it will cause failure given that it exists. ISIBREAK then obtains the calcu- lated weld defect distribution from a DANCER file and combines it, using a simple mathematical technique, with the conditional probability of failure to calculate an overall probability of failure for the weld. Of degradation mecha- nisms, only fatigue induced crack growth is considered, whereas the applied failure criteria are based on procedures in the R6 Method, Rev. 4 [200]. The effect on the probability of failure of one or several in-service inspections (ISI) during the life of a component can also be included to the analyses. It is the task of the software user to enter the PODs as ISIBREAK does not contain POD data. The SNOWBREAK routine in RR-PRODIGAL is an attempt to estimate the probability of failure from non-weld areas. Its primary use is for areas of stress concentration where defects could initiate at the surface by fatigue and then grow on through the wall of the component. When a defect has initiated, the crack growth to failure follows basically the same procedure as that used in ISIBREAK except that only one conditional situation, that of a 2 mm surface breaking defect, requires analysis. However, one important assumption is that there is a positive correlation between the crack initiation and the crack growth. PROST PROST (PRObabilistic STructure Analysis) [300] is a PFM analysis software to evaluate leak and break probabilities of piping systems in nuclear power plants. A graphical user interface supports the necessary data input. With PROST, leak and break probabilities from pre-existing semi elliptical shaped inner surface cracks driven by cyclic loading conditions (fatigue) can be estimated. The calculation of the subcritical crack growth and the final instabil- ity are based on deterministic fracture mechanics principles. The probabilistic nature is determined by the uncertainties of the input data entering the determin- istic routines. The deterministic fatigue crack growth rate is calculated by slight- ly modified Paris-Erdogan equation. 11. Probabilistic ageing and risk analysis tools 202 Uncertainties in the data on geometry, material properties and crack size of a given fatigue problem can be considered via different distribution functions. For each parameter the selection possibilities includes the normal, log-normal, weibull and exponential distribution function. The functions can be truncated on request by input of a minimum and maximum value. Additionally, it is possible to consider the parameter to be deterministic with a fixed value. Two different types of loading conditions are distinguished in PROST. The first are cyclic load sets recurring with user defined frequencies, which are dom- inant for fatigue problems. The second are stochastic load sets, like seismic events or transients, which occur with user defined probabilities. User defined weld residual stresses can be included as well. The user can choose if he wants to consider ISI in the computation, which is realised by entering a POD curve. The user can determine the POD curve by specifying ten crack depth values and their corresponding detection probabilities in percent together with the time inspection interval. Internally, this data are used to identify if during an inspection a crack is detected or not. Two probabilistic computation procedures are available in PROST. The first is the plain MCS. For this method, a combination of the distributed parameters will be selected randomly and the crack propagation for a given time period is com- puted. The second option is a stratification method based on a variance reduction technique. For this method, the user can select two out of all distributed parame- ters (usually the crack depth and the crack length to depth aspect ratio) to be integrated. The regular output file contains the information about all given input parame- ters including the run time variables. Then for all combinations that fail are listed the actual parameters, failure time, failure mode, crack depth, crack length, stress intensity factors for the maximum and minimum load at deepest and sur- face point and the combination probability. In a second output file, the accumu- lated failure probability is given as a function of time. If the stratification meth- od was selected, processed 5 %, 50 % and 95 % failure probability curves as a function of time are given too. The 50 % curve represents the expected failure probability that is equivalent with the results of a MCS run, depending strongly on the mean values of the distribution functions of the input parameters. 11. Probabilistic ageing and risk analysis tools 203 11.2 System ageing and risk analysis software CAFTA CAFTA is an integrated tool to perform PRA, incorporating linking event tree/fault tree methodology [292]. Using CAFTA one can build, quantify and analyse event tree/fault tree models of considerable size or complexity. CAFTA is developed by Data Systems & Solutions, a joint venture between Rolls-Royce and Science Applications International Corporation (SAIC). Since 1992, Data Systems & Solutions has worked with the Electric Power Research Institute (EPRI) to develop a suite of Windows based PRA/PSA software tools and appli- cations. One result of this effort is CAFTA, which has over 1500 users world- wide [292]. CAFTA includes e.g. the following integrated modules: The Fault Tree Editor for building, analysing and maintaining fault trees, The three level Reliability Database Editor for controlling model data, The Cutset Generator for generating minimal cut sets from the Fault Tree Editor, and The Cutset Editor for viewing probability and structural changes. SAPHIRE SAPHIRE (Systems Analysis Programs for Hands-on Integrated Reliability Evaluations) is a software application developed for performing a complete PRA/PSA using a PC. SAPHIRE is developed by the Idaho National Engineer- ing and Environmental Laboratory (INEEL), the funding for which is provided by USNRC [289]. The functions and modules in SAPHIRE include: functions for creating event trees and fault trees, Models And Results Database (MAR-D), System Analysis and Risk Assessment (SARA), Fault Tree, Event Tree and Piping Instrumenta- tion Diagram (FEP) and Graphical Evaluation Module (GEM). Presently, SAPHIRE has been distributed to thousands of users in the U.S.A. and through- out the world. These users include USNRC staff, national laboratories, industry contractors, vendors, utilities, architectural engineering firms, consultants, uni- versities, other government agencies and government contractors [72]. 11. Probabilistic ageing and risk analysis tools 204 WinNUPRA WinNUPRA is a self-contained, integrated, PSA/PRA software package for Level 1 PRA/PSA analyses [290]. The WinNUPRA software package is devel- oped by Scientech Inc. and the NUPRA User’s Group. WinNUPRA consists of six major analysis modules designed to generate and analyse minimal cut set solutions of various fault trees and cut set equations for accident sequences. Op- erations are provided for direct solution of fault trees, for their minimum cut sets, and for Boolean manipulation (merging) of cut set equations. WinNUPRA is used in several NPPs and chemical process facilities in the U.S.A. and other countries. RiskSpectrum PSA RiskSpectrum PSA [291] developed by Relcon Scandpower Inc. is a fault tree and event tree software tool licensed for use in many NPPs worldwide. The software provides a user interface for modelling from the basic fault tree with AND and OR gates to advanced fault tree and event tree integration of sequenc- es in linked event trees with boundary conditions and common cause failure (CCF) events. The integrated analysis tool is specially designed for solving large PSA models and covers e.g. the following analysis options: minimal cut set, sensitivity, importance and time dependent analysis. The latest program version was released in 2010. FaultTree+ FaultTree+ is an interactive graphics and analysis software for performing a PRA/PSA using integrated fault tree, event tree and Markov analyses. The soft- ware allows analysing large and complex fault and event trees producing the minimal cut representation for fault tree TOP events and event tree consequenc- es. The code is developed by Isograph Inc. [293]. FaultTree+ has been used to perform systems reliability analysis by a wide range of different industries for nearly three decades. FaultTree+ provides CCF analysis, importance analysis, uncertainty and sensi- tivity analysis facilities. The program allows constructing a single project data- base containing generic data and event tables, fault trees with multiple TOP events, event trees originating from different initiating events, CCF tables and consequence tables. Fault tree TOP events may be used to represent specific 11. Probabilistic ageing and risk analysis tools 205 columns in the event tree. Multiple branches are also handled to allow for partial failures. Users may feed the end branches of event trees into secondary event trees eliminating the need for the user to reproduce identical event tree structures leading to identical consequences. AvSim+ AvSim+, also developed by Isograph Inc., provides a MC simulation package for analysing systems availability and reliability problems using fault trees or reliability block diagrams [294]. AvSim+ provides means to construct fault tree or network diagrams (reliabil- ity block diagrams) by using drag and drop facilities. When using fault trees, AvSim+ automatically organises the diagram after receiving logical connection data. When using networks, AvSim+ automatically deduces the failure logic of the system after receiving block structure and logical connection data. Once a logical fault tree or network structure is defined one can define failure and maintenance models to represent the performance of components within the system. These models could be simple failure and repair models or they could represent complex dependencies including ageing, spares requirements, labour availability, operational phases, standby arrangements, etc. Historical data (times to failure and times to repair) are automatically analysed using the Weibull Analysis facility and connected directly through to component failure models. This allows the updating of their historical data records observ- ing the effects on predicted system performance. By allocating safety, environ- mental and operational consequences to selected system failures one is able to determine the frequency and duration of each type of consequence. Severity values may be assigned to a given consequence allowing safety, environmental and operational criticality values to be determined over the lifetime of the system. LOGAN Fault and Event Tree Analysis The LOGAN Fault and Event Tree module is a program developed by Reliass Inc. for the construction, evaluation and printing of fault and event trees and is widely used for quantitative risk assessment [295]. Fault trees can be linked to event trees by specifying a fault tree which calculates a branch probability. If the fault trees contain common events, the effects of the resulting non-independence are accounted for. The system lays out the diagrams automatically. 11. Probabilistic ageing and risk analysis tools 206 The fault and event tree capabilities of LOGAN Fault and Event Tree module include: fault/event tree integration, integrated reliability database, print-out in conventional top-down format or LOGAN horizontal format, minimal cut set analysis, various forms of sensitivity analysis, and optimisation of proof test intervals to minimise testing while achieving safety/reliability targets. Weibull++ Weibull++ 6 software [296] developed by ReliaSoft Corporation allows per- forming life data analyses utilising multiple lifetime distributions, including all forms of the Weibull distribution and generalised gamma distribution. Accord- ing to ReliaSoft [296], their software products and services are currently used by hundreds of companies, including e.g. ABB, Siemens and TÜV, with an active reliability engineering program. The capabilities of the code include analysis of data from various censoring schemes, degradation analysis, warranty analysis, non-parametric analysis and analysis of variance. The Degradation Analysis utility allows extrapolating the failure times of a component based on its performance over a period of time. One can analyse the extrapolated failure times as times-to-failure data. In addi- tion, one can use the competing failure modes analysis option to assign a sepa- rate distribution to each failure mode identified in the data set. CARA-FaultTree CARA-FaultTree [297], developed presently by ExproSoft, is a tool for fault tree analysis and construction. Performance measures calculated by the code include: unavailability, average availability, survival probability, mean time to failure, frequency of TOP event and failure frequency distribution. Computed compo- nent importance measures include: Birnbaum's reliability, Birnbaum's structural, Vesely-Fussell, criticality importance, improvement potential and order of smallest cut-set. Other notable computed results include: importance measures at cut-set level, unavailability at cut-set level and uncertainty analysis. ITEM ToolKit ITEM ToolKit software [298] is a suite of prediction and analytical modules developed by Item Software Inc. It uses globally recognised standards and methodologies to analyse components, systems and projects. ITEM ToolKit 11. Probabilistic ageing and risk analysis tools 207 allows taking a total system approach in analysing individual systems and com- ponents. This allows the optimisation of design targets with respect to compo- nent selection, increased safety and reduced liability. One can analyse reliability and availability at the component or system level and view the entire project. The analysis capabilities of the software include: FMECA, computation of unre- liability and unavailability, analysis of uncertainty and sensitivity, analysis of CCFs and creation of minimal cut sets. Each of the several modules of the soft- ware is designed to analyse and calculate the failure rates of components and systems in accordance with the appropriate standard. According to software developer [298], ITEM ToolKit is widely used throughout the world by compa- nies in defence, aerospace, electronics and other industries. FinPSA STUK started to develop living PSA computer code SPSA in 1988. Level 1 part of this code was taken into use in 1991. Since then, the algorithm has been con- tinuously developed, and its efficiency and robustness have been demonstrated in many benchmarks and in practical PSA use. FinPSA is a new version of SPSA for Microsoft Windows [301]. FinPSA is developed for teamwork. PSA models can be shared or private models. Shared models reside in a server, and can be accessed and edited simul- taneously by several users. The PSA team can solve minimal cut sets together with several computers utilizing parallel computation. FinPSA searches for cut sets for each event tree sequence. After the search, the cut sets of individual sequences are automatically combined and grouped according to userdefined hierarchy of consequences. On the top level, FinPSA produces cut set files classified by consequence only. Any number of intermedi- ate levels can be defined. FinPSA works together with common Windows programs for importing and exporting data base tables, fault trees, events trees, and almost every item in the model. Results are available in several formats to Windows clipboard, printer and files. Preliminary version of FinPSA level 1 is available. The development of FinPSA will continue with new features like: - task-oriented model for control systems (from RELVEC code), - asymmetric common cause failures, additional CCF models, - level 2 (dynamical containment event trees already implemented in SPSA). 11. Probabilistic ageing and risk analysis tools 208 11.3 Summary of ageing and risk analysis software There exists a large number of software for ageing and risk analysis of systems and components. It would have been way beyond the scope of this thesis to go through all or even most of them. Thus, only some of the most noteworthy ex- amples of these analysis tools are covered in this thesis. It appears that most of the available ageing degradation analysis tools consider only one or two degradation mechanisms. This is a drawback, as the number of failure mechanisms encountered in NPPs is much greater. Thus, it appears that for the time being several analysis codes are needed if one wishes to cover the degradation of all safety significant NPP systems and components. Many of the system analysis tools presented in this thesis, e.g. those with fault and event tree analysis capabilities, can be applied within many industries, e.g. nuclear, aerospace, chemical, process and transportation. There exists, however, a number of system analysis tools that are specifically developed for power plant industry purposes. These include e.g. SAPHIRE and PSA Professional. Probabilistic models that predict rare events cannot be validated by comparing the results to observed failure and degradation data. One way to verify these models to some extent is to benchmark the independently developed models describing the same phenomena against each other. Thus, the main sources of difference can be identified, based on which the models can be developed fur- ther. Comparisons of this kind have been made e.g. for the following models [247, 264, 302, 303]: - SRRA against PRAISE, - PROPSE against STAR6, - ProSINTAP against STAR 6, - NURBIT against WinPRAISE, - probabilistic VTTBESIT against NURBIT, - RR-PRODIGAL, ProSACC, PROST and WinPRAISE against each other. In several cases these comparisons have resulted in identification of errors and model improvements. 12. Probabilistic lifetime analysis application for piping components 209 12. Probabilistic lifetime analysis application for piping components 12.1 Introduction Two analysis codes are used in the lifetime analysis application for piping com- ponents. The first and obvious one of them is the probabilistic VTTBESIT, de- veloped at VTT, and the second one is PIFRAP, both of which are described in more detail in Chapter 11. With these analysis tools, it is possible to calculate e.g. yearly failure probabilities for the considered piping cross-sections. To al- low performing the present computational analyses, the probabilistic VTTBESIT was developed further to some extent. As for PIFRAP, the analysis code version 2.0 rev. 0 published in 1997 is used. The analyses consist of a set of Monte Carlo simulations with probabilistic VTTBESIT, and of some PIFRAP computations for comparison purposes. The considered degradation mechanism is SCC. All examined pipe cross-sections are similar butt welds. As in the NPP piping components SCC typically occurs as a circumferentially oriented inner surface crack in the heat affected zone (HAZ) adjacent to weld, such postulates are also considered in the present anal- yses. The examined characteristics in the computations cover e.g. three different initial crack distribution assumptions, three pipe sizes and three considered time spans. The last issue means the service time in years from the start of operation. 12.2 Examined piping components and associated input data The computational probabilistic analyses are performed for three pipe sizes. These sizes are assumed to correspond to representative small, medium and large pipes in Finnish BWR NPP reactor circuit piping systems, respectively, 12. Probabilistic lifetime analysis application for piping components 210 and they were provided by expert personnel from TVO [307], which is gratefully acknowledged. 12.2.1 Geometry The geometry input data concerning all computations are presented in the fol- lowing. As the considered pipe cross-sections are assumed to be located in straight pipe components, the needed geometry input data are quite limited. The cross-section dimensions of the three pipe component sizes considered in the probabilistic lifetime analyses are presented in Table 12.2-1. Table 12.2-1. Representative Small, Medium and Large BWR reactor circuit pipe sizes selected for probabilistic lifetime analyses. Pipe Outer diameter [mm] Wall thickness [mm] Outer radius/wall thickness [ - ] Small 60 4.0 7.50 Medium 170 11.0 7.73 Large 510 26.0 9.81 12.2.2 Material properties The base material of the considered pipe components is assumed to be austenitic stainless steel TP 304, which is a commonly used material type for NPP piping system components. As the material property data of welds are often more diffi- cult to obtain, the commonly used option to assume the material properties of the adjacent base material for the weld material is also applied here. In most cases, this is a conservative assumption, as typically the strength properties of the weld materials are to some extent better than those of the base materials. Material property values for various metallic NPP component materials are taken from Section II of the ASME code [305]. The obtained data are in the form of tables, and presented here for a representative range of temperatures. On the other hand, for the operational temperature of 286 °C of Finnish BWRs the material property values were interpolated, as for that particular temperature the material property values were not given in the ref. [305]. The presented yield strength values cor- respond to 0.2 % of strain in uniaxial tension test. 12. Probabilistic lifetime analysis application for piping components 211 Table 12.2-2. Strength properties of type 304 austenitic stainless steel as a function of temperature, see Tables 2a, U and Y1 in Section II of ASME code [305]. Temperature [°C] Yield strength [MPa] Design stress [MPa] Tensile strength [MPa] 20 172 115 483 100 146 115 452 150 132 115 421 200 121 110 406 250 114 103 398 275 111 286 109 102 397 300 108 98 393 Table 12.2-3. Some mechanical properties of type 304 steel as a function of temperature, see Tables TCD, TE-1 and TM-1 of Part D of Section II of ASME code [305]. Tempera- ture [°C] Elastic modulus [GPa] Thermal conductivity [W/m°C] Specific heat [J/kg°C] Coefficient of ther- mal expansion [(1/°C)*10 -06 ] 21 195 14.9 484 15.2 38 15.1 486 15.4 66 15.6 496 15.6 93 190 16.1 506 15.8 121 16.6 516 16 149 186 17 520 16.2 177 17.5 529 16.4 204 183 18 535 16.5 232 18.3 539 16.7 260 178 18.9 544 16.9 286 176 19.2 548 17.0 316 174 19.6 551 17.2 12. Probabilistic lifetime analysis application for piping components 212 The assumed welding type is shielded metal arc welding (SMAW), which is a commonly used method for fabricating NPP piping welds. According to ref. [203], when the base material of the pipe is austenitic stainless steel and temper- ature is 288 °C, the fracture toughness data for this welding type are: K IC = 182 MPa·\m, J IC = 168 kJ/m 2 , and for the base material: K IC > 350 MPa·\m, J IC > 620 kJ/m 2 . Table 12.2-4 presents the values of parameters C and n used in the SCC rate equation for the considered austenitic stainless steel (SS) TP 304. Table 12.2-4. The values of parameters C and n used in the SCC equation for the con- sidered material. The dimensions used in the crack growth equation are: [da/dt] = mm/s, [K I ] = MPa·\m. The dimension of C is [(mm/s)/((MPa·\m) n )] whereas n is dimensionless. Material C n Environment Refs. SS TP 304 4.5E-12 3.00 water [308] 12.2.3 Loads and failure mechanisms The loads considered in the analyses are mechanical and thermal. The crack postulates are assumed to be located in the HAZ, as that region is more suscepti- ble to SCC than the base or weld material. Due to the choice of examined loca- tion, the residual stress distribution caused by the welding process must be con- sidered as well. Thus, the considered mechanical loads are operational process pressure, as caused by the water inside pipe components, and residual stress distribution in and near the weld. Thermal loading is also caused by the water inside the pipe components. The outer surfaces of NPP piping components are typically thermally insulated, and for most of the time the temperature outside the insulation is of the scale of room temperature. Both applied analysis tools, VTTBESIT and PIFRAP, allow considering two degradation mechanisms, which are intergranular SCC (IGSCC) and fatigue (in case of PIFRAP limiting to high cycle vibration). Here, the only considered deg- radation mechanism is IGSCC. The considered crack growth type is mode I, with corresponding stress intensity factor K I [MPam] as the governing load parameter. As all considered crack postulates are circumferentially oriented and located in the inner surface, only axial stresses are taken into account, i.e. stresses acting perpendicular to crack faces. Under operational conditions, the values of inner pressure caused by and tem- perature of the fluid are 7.0 MPa and 286 °C, respectively. The axial membrane 12. Probabilistic lifetime analysis application for piping components 213 stresses, o AXIAL [N/mm 2 ], caused by the inner pressure are computed with the following analytical equation [306]: , , t t D p 2 2 AXIAL ÷ = o (12.2-1) where p [N/mm 2 ] is pressure, D [mm] is outer diameter and t [mm] is wall thickness. The residual stress distributions present in a structure are the result of the manufacturing history and the elastic-plastic properties of the structure. The former referring to the mechanical and thermal processes executed during the whole production sequence and the latter to the elastic-plastic behaviour of the structure. Typically, in NPP piping component welds, the local maximum values of WRSs are of the scale of material yield stress in tension, and local minimum values of the same scale in compression, respectively. Thus, the locally confined WRSs often provide the dominant component to total stress distribution within and near the weld region. The distributions of the axial WRSs in the weld centre-line and HAZ are as- sumed according to two commonly applied procedures, namely the ASME rec- ommendations [309, 310] and the SINTAP procedure [311, 312]. These two procedures provide conservatively defined WRS distributions e.g. to NPP piping component welds. When actual WRS measurement data are not available, as is in the present case, handbook WRS distribution assumptions are often used. The WRS distributions could also be simulated more accurately by FEM, but again due to lack of data concerning the actual welding process, this option was not resorted to, as such simulations require input data for a relatively large number of parameters. For Medium and Large pipe sizes, the WRS distributions were assumed according to the SINTAP procedure [311, 312], but for Small pipe size according to the ASME recommendations [309, 310], as the former procedure is limited to cover wall thicknesses from 9 to 84 mm. Mainly, these limits concern ferritic steels, but to ensure that appropriate WRS distributions are applied to the Small pipe size of austenitic stainless steel and with wall thickness of 4.0 mm, the WRS equations in the ASME recommendations [309, 310] were used. The axial stress distributions applied in the probabilistic lifetime analyses of the Small, Medium and Large pipe components are presented in Figures 12.2-1 to 12.2-3. The horizontal axis in all these figures corresponds to radial coordinate through wall with the origin in the inner surface. The dominant role of the WRSs in the total axial stress distributions can clearly be seen in these figures. 12. Probabilistic lifetime analysis application for piping components 214 In some piping components, thermal expansion can cause thermal stresses. This can happen, when the piping component supports resist thermal expansion. Totally rigid support conditions can cause very high thermal stresses to the pip- ing component. However, mostly the NPP piping components are supported quite flexibly. One potential location for thermal stresses is pipe bends. Due to lack of specific data, no thermal stresses are included in the computations. -300 -200 -100 0 100 200 300 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t wall [mm] a x i a l o [ M P a ] pressure ASME-Au p+WRSs-Au Figure 12.2-1. The axial stress distribution through wall used in the probabilistic lifetime analyses of the Small pipe component. The colours of the curves correspond to: green; stress caused by pressure, light red; WRSs acc. to ASME recommendations, blue; total stress. 12. Probabilistic lifetime analysis application for piping components 215 -200 -100 0 100 200 300 400 0.0 2.0 4.0 6.0 8.0 10.0 t wall [mm] a x i a l o [ M P a ] pressure SINTAP-Au p+WRSs-Au Figure 12.2-2. The axial stress distribution through wall used in the probabilistic lifetime analyses of the Medium pipe component. The colours of the curves correspond to: green; stress caused by pressure, light red; WRSs acc. to SINTAP procedure, blue; total stress. -200 -100 0 100 200 300 400 0.0 5.0 10.0 15.0 20.0 25.0 t wall [mm] a x i a l o [ M P a ] pressure SINTAP-Au p+WRSs-Au Figure 12.2-3. The axial stress distribution through wall used in the probabilistic lifetime analyses of the Large pipe component. The colours of the curves correspond to: green; stress caused by pressure, light red; WRSs acc. to SINTAP procedure, blue; total stress. 12. Probabilistic lifetime analysis application for piping components 216 12.2.4 Initial crack size distributions As mentioned earlier, three different initial crack distribution assumptions are used in the probabilistic analyses. These distributions are briefly described in the following. Existing fabrication cracks The assessment of probabilistic distributions for depth and length of the existing fabrication/manufacturing cracks in stainless steel piping used here are from ref. [313] by Khaleel and Simonen. In developing these to some extent conservative distributions, it was assumed that the welds for all pipe sizes used the manual metal arc process and that the welds (except for the 0.25 inch wall thickness) were inspected using radiographic inspection. The initial cracks were circumfer- ential, and were conservatively placed at the inner pipe surface. The semi- elliptical surface cracks had depths between zero and (with a low probability) the full wall thickness. Surface lengths were between a semi-circular shape and (with a low probability) the full pipe circumference. Log-normal distributions were used to characterize the crack depths. The parameters of these log-normal distributions are listed in Table 12.2-5 in the following. The log-normal distribution equation used in defining depth probabilities, f x (x), for existing manufacturing cracks in stainless steel piping is [313]: , , , , ( ( ¸ ( ¸ ÷ ÷ = 2 2 2 ln exp 2 1 y y y x x x x f o µ o t , and (12.2-2a) X y ~ ln = µ (12.2-2b) where x is the variable for which the probability is calculated, o y is the shape parameter, µ y is the scale parameter and X ~ is the median of x. 12. Probabilistic lifetime analysis application for piping components 217 Table 12.2-5. Parameters for log-normal crack depth probability distribution for existing manufacturing cracks in stainless steel piping, from ref. [313]. For wall thicknesses falling between the presented ones the corresponding parameter values are to be interpolated. Pipe wall thickness Parameters Median flaw depth Shape pa- rameter Flaw density per inch of weld Flaws per weld [inch] [inch] [ - ] [1/inch] [ - ] 0.250 0.1063 0.1784 0.0047 0.0440 0.562 0.0991 0.2669 0.0028 0.0480 1.000 0.0892 0.3672 0.0035 0.0960 2.500 0.0555 0.4993 0.0256 2.0508 The probability distribution for lengths of existing manufacturing cracks in stainless steel piping is defined using crack aspect ratio, which is given here as | = b/a, where b is the total length of the surface crack and a is the crack depth. The probability distribution for aspect ratio | is given by [313]: , , , , 1 , ln 2 1 exp 2 1 , 0 2 2 2 1 0 > ¦ ¹ ¦ ´ ¦ ( ( ¸ ( ¸ | | . | \ | ÷ < = | | | ì t ì| | | | m C f (12.2-2c) where ì = 0.5382, C | = 1.419, and | m = 1.136. This aspect ratio distribution was assumed to be independent of the flaw depth. SCC induced initial cracks in VTTBESIT The assessment of probabilistic distributions for depth and length of SCC in- duced initial cracks in stainless steel piping used here are from ref. [245] by Cronvall et al. The development of these distributions was based on the flaw data from nine Swedish BWR units, see ref. [304]. This data consists of 98 de- tected SCC cases, all of which are circumferentially oriented cracks at the inner pipe surfaces. In the assessment of probabilistic distributions for initial crack sizes, not all of the above mentioned 98 crack cases were considered, which may affect the applicability of these distributions for the present analyses to some extent. 12. Probabilistic lifetime analysis application for piping components 218 The fitted exponential probabilistic density functions for estimated depth and length distributions of SCC induced initial cracks are presented as probability density against relative crack dimension in Figure 12.2-4. 0.00 0.02 0.04 0.06 0.08 0.10 5 15 25 35 45 55 65 75 85 95 initial crack depth/wall thickness [%] p r o b a b i l i t y d e n s i t y [ 1 / % ] ' f(x) = 0.070*exp(-0.062*x) Figure 12.2-4a. The fitted exponential probabilistic density function for estimated depth distribution of SCC induced initial cracks, from ref. [245]. In the legend “f” stands for prob- ability density and “x” for relative initial crack depth. 12. Probabilistic lifetime analysis application for piping components 219 0.00 0.02 0.04 0.06 0.08 0.10 5 15 25 35 45 55 65 75 85 95 initial crack length/inner circumference [%] p r o b a b i l i t y d e n s i t y [ 1 / % ] ' f(x)=0.070*exp(-0.062*x) Figure 12.2-4b. The fitted exponential probabilistic density function for estimated length distribution of SCC induced initial cracks, from ref. [245]. In the legend “f” stands for prob- ability density and “x” for relative initial crack length. SCC induced initial cracks in PIFRAP The assessment of probabilistic distributions for depth and length of SCC in- duced initial cracks in stainless steel piping used in PIFRAP is based on the same flaw data from the nine Swedish BWR units [304], as the above described distributions developed by Cronvall et al. [245]. In PIFRAP, the depth for SCC induced initial cracks in stainless steel piping is assumed to be fixed to 1.0 mm. For developing a distribution for initial crack lengths, the original data of detected cracks were first modified so as to better correspond to assumed initial sizes. This treatment was such that the measured length of each detected crack was reduced two times the distraction of the meas- ured crack depth and 1.0 mm. A truncated exponential function was fitted to thus obtained modified set of crack lengths. 12. Probabilistic lifetime analysis application for piping components 220 12.3 Analysis characteristics The scope of the probabilistic analyses is described below. This is followed by some relevant details concerning the analysis flow of the two applied analysis codes, probabilistic VTTBESIT and PIFRAP. 12.3.1 Scope of probabilistic analyses The scope of the probabilistic analyses with VTTBESIT: - considered pipe sizes; the earlier described Small, Medium and Large pipe components, - considered initial crack size distributions; fabrication cracks from ref. [313] and SCC induced cracks from ref. [245], i.e. altogether two dif- ferent distributions for both initial crack depth and length, - considered crack depths in relation to wall thickness; 25 %, 50 %, 75 % and 100 %, the last one corresponding to pipe failure, - considered time spans from the start of operation; 20 years, 40 years and 60 years, - thus the scope of analyses is all in all 72 computed probabilities. The scope of the probabilistic analyses with PIFRAP: - considered pipe sizes; the earlier described Small, Medium and Large pipe components, - considered initial crack size distributions; SCC induced cracks from ref. [304], - considered crack depths in relation to wall thickness; 100 %, as corre- sponding to pipe failure, note that with PIFRAP version 2.0 rev. 0 it is not possible to compute probabilities for cracks with depth less than 100 %, - considered time spans from the start of operation; 20 years, 40 years and 60 years, - thus the scope of analyses is all in all 9 computed probabilities. 12. Probabilistic lifetime analysis application for piping components 221 The effect of periodic pipe component inspections is not considered here. How- ever, both applied analysis codes do allow the inclusion of inspections, which are typically performed in NPPs yearly. The probability of detecting a crack in inspections is dependent of its size, so that the larger it is the more probable is its detection. Typically, crack detection probabilities are assessed with POD func- tions, as mentioned earlier. In all analysed cases the operational lifetime was assumed to be 60 years. 12.3.2 Details concerning the analysis flow of the two applied analysis codes Details concerning the analysis flow of probabilistic VTTBESIT The probabilistic VTTBESIT calculates the probability of failure for a loaded pipe with circumferential or axial cracks that may propagate due to SCC or low/high-cycle fatigue. Failure is taken to mean the event when the crack postu- late has grown to just reach the opposite surface of the pipe component, most often the outer surface. Account is taken to the probability that the crack remains undetected during successive inspections. As mentioned above, inspections are not taken into account here. The theory and procedure behind the probabilistic VTTBESIT are given in refs. [246, 247]. The following assumptions are made for the probabilistic analyses [246, 247]: - for SCC analyses the stresses are assumed to be deterministic, for fa- tigue induced crack growth analyses their cyclic frequency of occur- rence is assumed as Poisson distributed, - the crack growth equation and its parameters are assumed to be deter- ministic, - both depth and length of initial crack postulates are assumed to be ran- dom with exponential probability density functions. Due to these assumptions, the growth of the crack due to SCC will be determin- istic and will only depend on the initial crack depth and length for a given geom- etry and given stresses. As mentioned earlier, to allow performing the computational analyses the prob- abilistic VTTBESIT was developed further to some extent. More precisely, the possibility to compute the probabilities for growing crack postulate to reach the depths of 25 %, 50 % and 75 % of wall thickness were is to the analysis code. 12. Probabilistic lifetime analysis application for piping components 222 The basic failure probability equation of VTTBESIT can be expressed as: , , , , 0 , , , i i FS tj i FS f N a a N t p > = (12.3-1) where , , t p tj i FS , , [ - ] is failure probability corresponding to selected degrada- tion/failure state, t [years] is time, , , i FS a a N , > [ - ] is the number of times the Monte Carlo simulations have reached or exceeded the selected degrada- tion/failure state in terms of crack depth a [mm], N [ - ] is the total number of Monte Carlo simulations, and f i0 [ - ] is the rate of crack initiation due to SCC. The rate of crack initiation due to SCC, f i0 , was here based on the mentioned Swedish IGSCC data with the value of this parameter being 4.08E-04, see ref. [304]. In order to be on the safe side, the fracture toughness properties used in the analyses were those of the weld material. Other used material properties were those of the base material. Details concerning the analysis flow of probabilistic PIFRAP PIFRAP calculates the probability of failure for a loaded pipe with circumferen- tial cracks that may propagate due to IGSCC or IGSCC in combination with high cycle fatigue, water hammer loads, seismic loads etc. Failure is taken to mean a complete break due to J –integral controlled fracture or plastic collapse. The value of the J-integral is calculated according to the R6 procedure. Account is taken to the probability that the crack remains undetected during successive inspections. As mentioned above, inspections are not taken into account. The theory and procedure behind PIFRAP are given in reference [315]. The following assumptions are made here for the probabilistic analyses [304]: - the stresses are assumed to be deterministic, - the crack growth law and its parameters are assumed to be deterministic, - the initial crack depth is assumed to be fixed to 1.0 mm, - the probability that a crack with the assumed depth is initiated during the time interval (t i , t i + dt) is given by the function f i (t i )dt. Due to these assumptions, the growth of the crack will be deterministic and will only depend on the initial crack length for a given geometry and given stresses. The growth of the above mentioned crack postulate is calculated with the proce- dure presented in ref. [248]. 12. Probabilistic lifetime analysis application for piping components 223 The failure probability equation of PIFRAP can be expressed as [304]: , , , , , , } = i c R l al f f dl l f l T t p T t p t 2 0 0 0 0 , , , (12.3-2) where t [years] is time, T [years] is service life, p f0 [ - ] is conditional probability of rupture given the initial crack length l 0 . The lower length limit l c [mm] is the solu- tion of the equation t F (l c ) = T, where t F [years] is time of rupture. Finally, p f is di- vided by the time (T - t), i.e. p f represents the probability for a rupture, measured per reactor year as a mean value for the remaining operating time of the plant. In order to be on the safe side, the fracture toughness properties used in the analyses were those of the weld material, and the rest of the material properties used were those of the base material. The authors of the program use the same approach in ref. [304]. 12.3.3 Sub-critical crack growth In the sub-critical crack growth analyses, the propagation of an initial surface crack through the pipe wall is assessed. In both the probabilistic and deterministic modules of VTTBESIT, the fracture mechanics based crack growth rate equation used in the simulations depicting the intermediate (stage 2) SCC, as obtained from refs. [136, 314], is: n I CK dt da = (12.3-3) which was presented earlier in Section 7.3.2. Values for these constants are pre- sented in Section 12.2.2. In PIFRAP the crack growth calculations are performed by the PC program LBBPIPE, which is a program for leak-before-break analysis of circumferential cracks in pipes subjected to IGSCC or fatigue. LBBPIPE estimates the growth of an initial surface crack through the pipe wall to leakage, and from that point the growth of a through thickness crack to final failure of the pipe. Crack opening areas and mass leak rates are also calculated for the leaking crack. The crack growth rate concerning IGSCC can be defined in two ways. The first way is by providing the crack growth rate equation coefficients. The crack growth rate equation used in PIFRAP is [304]: 12. Probabilistic lifetime analysis application for piping components 224 , , n I C K C dt da 1 0 ÷ = (12.3-4) where the input data parameters are otherwise the same as in equation (12.3-3), with the value of additional parameter C 1 being here zero, i.e. no threshold for propagation of SCC is considered. Thus here, equation (12.3-4) is upon applica- tion the same as equation (12.3-3). The rate of crack initiation due to SCC, f i0 , is the earlier mentioned 4.08E-04 [304]. 12.4 Results and discussion The numerical results from the probabilistic VTTBESIT and PIFRAP analyses for to the considered three pipe sizes are presented in the following, see Tables 12.4-1 and 12.4-2 as well as Figures 12.4-1 and 12.4-2. As mentioned earlier, the assumed operational lifetime is 60 years. Table 12.4-1. The results from the probabilistic SCC analyses with VTTBESIT for pipe sizes Small, Medium and Large with initial crack size distribution as fabrication cracks. Time from start of opera- tion [years] Reached crack depth through wall [%] Probability; Small pipe [1/year/weld] Probability; Medium pipe [1/year/weld] Probability; Large pipe [1/year/weld] 20 25 4.08E-04 3.36E-04 1.22E-06 20 50 4.05E-04 2.53E-05 0.00E+00 20 75 1.15E-04 1.02E-06 0.00E+00 20 100 0.00E+00 0.00E+00 0.00E+00 40 25 4.08E-04 3.86E-04 9.34E-06 40 50 4.07E-04 8.41E-05 0.00E+00 40 75 1.22E-04 5.67E-06 0.00E+00 40 100 0.00E+00 0.00E+00 0.00E+00 60 25 4.08E-04 4.01E-04 2.63E-05 60 50 4.07E-04 1.34E-04 0.00E+00 60 75 1.27E-04 1.18E-05 0.00E+00 60 100 0.00E+00 0.00E+00 0.00E+00 12. Probabilistic lifetime analysis application for piping components 225 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 SVTB MVTB LVTB p r o b a b i l i t y o f f a i l u r e [ 1 / y e a r / w e l d ] 20y, 25% 20y, 50% 20y, 75% 20y, 100% 40y, 25% 40y, 50% 40y, 75% 40y, 100% 60y, 25% 60y, 50% 60y, 75% 60y, 100% Figure 12.4-1. The results from the probabilistic SCC analyses with VTTBESIT for pipe sizes Small, Medium and Large with initial crack size distribution as fabrication cracks. Here SVTB, MVTB and LVTB correspond to Small, Medium and Large pipe sizes. 12. Probabilistic lifetime analysis application for piping components 226 Table 12.4-2. The results from the probabilistic SCC analyses with VTTBESIT for pipe sizes Small, Medium and Large with initial crack size distribution as SCC induced initial cracks. Time from start of opera- tion [years] Reached crack depth through wall [%] Probability; Small pipe [1/year/weld] Probability; Medium pipe [1/year/weld] Probability; Large pipe [1/year/weld] 20 25 3.65E-04 3.61E-04 3.23E-04 20 50 3.41E-04 3.37E-04 1.95E-05 20 75 3.02E-04 3.02E-04 3.43E-06 20 100 2.19E-04 2.28E-04 6.53E-07 40 25 3.91E-04 3.90E-04 3.87E-04 40 50 3.80E-04 3.79E-04 2.56E-05 40 75 3.56E-04 3.58E-04 3.67E-06 40 100 2.82E-04 2.94E-04 1.18E-06 60 25 3.98E-04 3.97E-04 3.97E-04 60 50 3.89E-04 3.89E-04 1.75E-04 60 75 3.69E-04 3.71E-04 4.04E-06 60 100 2.82E-04 2.94E-04 1.22E-06 12. Probabilistic lifetime analysis application for piping components 227 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 SVTB MVTB LVTB p r o b a b i l i t y o f f a i l u r e [ 1 / y e a r / w e l d ] 20y, 25% 20y, 50% 20y, 75% 20y, 100% 40y, 25% 40y, 50% 40y, 75% 40y, 100% 60y, 25% 60y, 50% 60y, 75% 60y, 100% Figure 12.4-2. The results from the probabilistic SCC analyses with VTTBESIT for pipe sizes Small, Medium and Large with initial crack size distribution as SCC induced initial cracks. Here SVTB, MVTB and LVTB correspond to Small, Medium and Large pipe sizes. The results from the probabilistic SCC analyses with PIFRAP for pipe sizes Small, Medium and Large with initial crack size distribution as SCC induced initial cracks [304] are presented in Table 12.4-3, as well as in Figure 12.4-3. As mentioned earlier, the assumed operational lifetime is 60 years. Table 12.4-3. The results from the probabilistic SCC analyses with PIFRAP for pipe sizes Small, Medium and Large with initial crack size distribution as SCC induced initial cracks [304]. Note that with PIFRAP version 2.0 rev. 0 it is not possible to compute probabilities for cracks with depth less than 100 % of wall thickness. Time from start of opera- tion [years] Reached crack depth through wall [%] Probability; Small pipe [1/year/weld] Probability; Medium pipe [1/year/weld] Probability; Large pipe [1/year/weld] 20 100 1.84E-04 2.50E-04 3.01E-04 40 100 2.26E-05 3.19E-05 3.33E-05 60 100 0.00E+00 0.00E+00 0.00E+00 12. Probabilistic lifetime analysis application for piping components 228 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 SPFP MPFP LPFP p r o b a b i l i t y o f f a i l u r e [ 1 / y e a r / w e l d ] 20y, 100% 40y, 100% 60y, 100% Figure 12.4-3. The results from the probabilistic SCC analyses with PIFRAP for pipe sizes Small, Medium and Large with initial crack size distribution as SCC induced initial cracks [304]. Here SPFP, MPFP and LPFP correspond to Small, Medium and Large pipe sizes. As for the analysis result diagrams, the scaling of them has been selected so that most of the results could be included visibly enough together with the associated scatter. As the scaling of the vertical axis in the diagrams is logarithmic, the cases with result value of zero are not included in them, however these and all other result cases can be found in the corresponding result tables. As for SCC analysis results obtained with probabilistic VTTBESIT, in some cases, there was notable variation in the results, whereas in others they were quite conformable. Within the considered time spans of 20, 40 and 60 years, the probability to reach 25 % of wall thickness was greater than zero in all cases, varying approximately between 10E-06 to 10E-04. Concerning fabrication cracks, in case of Small and Medium pipe sizes the probabilities for crack postu- lates to reach 50 and 75 % of wall thickness varied approximately between 10E- 06 to 10E-04, whereas for Large pipe size these probabilities were zero. Con- cerning SCC induced initial cracks, for all pipe sizes the probabilities for crack postulates to reach 50 and 75 % of wall thickness was greater than zero, varying 12. Probabilistic lifetime analysis application for piping components 229 again approximately between 10E-06 to 10E-04, however with the mean value being to some extent smaller than that corresponding to probability to reach 25 % of wall thickness. Concerning fabrication cracks, for all pipe sizes the probability for crack postulates to reach 100 % of wall thickness was zero, whereas concerning SCC induced initial cracks this probability varied for all pipe sizes approximately between 10E-07 to 10E-04. Concerning both fabrica- tion cracks and SCC induced initial cracks, the maximum probability for crack postulates to reach 100 % of wall thickness within the considered time spans of 20, 40 and 60 years was 2.94E-04, whereas the corresponding minimum proba- bility value was zero. As for SCC analysis results obtained with PIFRAP, they were in all cases quite conformable as compared to VTTBESIT results. As mentioned earlier, with PIFRAP version 2.0 rev. 0 it is not possible to compute probabilities for cracks with depth less than 100 % of wall thickness. Within the time spans of 20 and 40 years, in all cases the probability for SCC induced initial cracks to reach 100 % of wall thickness was greater than zero, varying approximately between 10E-05 to 10E-04. However, for the time span of 60 years this probability was in all cases zero. The maximum probability for crack postulates to reach 100 % of wall thickness within the considered time spans of 20, 40 and 60 years was 3.01E-04, whereas the corresponding minimum probability value was zero. When concerning the effect of pipe size to VTTBESIT results, the outer diam- eter plays no role in them, as only the wall thickness matters here. From the re- sults it can be seen, that the higher the total tensile stress, the higher the failure probabilities. As in all cases here the tensile stresses are at maximum in the inner surface, and consequently the initial crack postulates that all open to that surface are prone to propagation. The most severe total stress distribution is associated with the Small pipe size, which consequently leads to the highest crack growth and failure probability results. Here the cases with fabrication cracks and SCC induced initial cracks lead to quite matching probability results, those remaining within the range from 10E-04 to 10E-03. However, for Medium and Large pipe sizes, there are notable differences between the probability result sets concerning fabrication cracks and SCC induced initial cracks. With fabrication cracks the scatter of probability results is of several decades for these pipe sizes, whereas with SCC induced initial cracks they are considerably more conformable. Fur- ther, with fabrication cracks the probability results are for these pipe sizes in general lower than with SCC induced initial cracks. This difference is explained by initial depth of fabrication cracks being limited on average to approximately 12. Probabilistic lifetime analysis application for piping components 230 2.5 mm, whereas the SCC induced initial cracks can be deeper than that (with a small probability), and with larger wall thicknesses this limitation leads to com- parably lower crack growth and failure probabilities. Similar effects as those described for VTTBESIT results above can be recog- nized also in the corresponding PIFRAP results. Namely, again in case of the Small pipe size, being associated with the most severe total stress distribution, the resulting failure probabilities are highest. The Medium pipe size cases result with the second highest failure probabilities, and the Large pipe size cases with the lowest failure probabilities, respectively. Not only the maximum total tensile stresses through wall affect the crack growth and failure probabilities, but the shapes of through wall stress distribu- tions need to be considered as well. Namely, for the Small and Medium pipe sizes the shape of the total stress distribution is linear, whereas that for the Large pipe size is non-linear. In both stress distribution shape cases, there are regions both in tension and in compression. This is obviously realistic, as the dominant stress component, being WRSs, has to be self-balancing in the axial direction. The nearer to the inner surface the total stress distribution turns from tension to compression, the slower is the ensuing crack growth. For the Small and Medium pipe sizes, this turning point is approximately in the middle of the wall, whereas for the Large pipe size it is approximately 25 % from the inner surface. This partly explains why in case of this latter pipe size the resulting failure probabili- ties are also the lowest. The analysis results obtained with probabilistic VTTBESIT are more scattered than those obtained with PIFRAP. This can be mainly attributed to the differ- ences in the used initial crack assumptions, all being in the form of probabilistic density functions. Namely, concerning probability against size, the distribution of SCC induced initial cracks in PIFRAP falls mostly somewhere between the distributions of fabrication cracks and SCC induced initial cracks in VTTBESIT. This is reflected in the results so that the failure probabilities obtained with PIFRAP reside mainly between those obtained by VTTBESIT. It could be as- sumed that with exactly same initial crack distributions these two codes will provide quite accurately matching results for the same set of analysis cases. All in all, the thicker the pipe wall, the generally less severe the WRSs are and consequently also the total stress distribution. In the light of the obtained proba- bilistic SCC analysis results, this leads to lower failure probabilities for pipe components with thicker walls. 12. Probabilistic lifetime analysis application for piping components 231 The presented probabilistic results are supposed to be conservative, due to the assumptions made with input data. As the actual WRS distributions were not available, the to some extent conservative distributions given in refs. [309, 310, 311, 312] were used. As the material property data of the weld material were not available, those of the base material were used, and typically the strength properties of the materials of the latter type are lower than those of the former. Moreover, the material strength data of the base material were obtained from a normative ref. [305], where the given values are invariably lower than those obtained from the actual material test result data, which in this case were not available either. As or more significant than looking at single pipe failure probability values, is to compare these values against each other. This viewpoint also serves the needs of RI-ISI analyses for NPP piping systems. The formation of component groups according to risk level in RI-ISI is preceded by formation of component groups according to failure probability and failure consequences. Some characteristics of both probabilistic VTTBESIT and PIFRAP as model- ling tools are discussed in the following. It is a drawback that quite few input data parameters are allowed to be random, those being for VTTBESIT the depth and length of initial cracks as well as frequency of occurrence of cyclic loads, and for PIFRAP only the length of the initial cracks. These are features which cannot be changed or modified by the program user. However, the values of many material properties are distributed and therefore a single value cannot cor- rectly describe them. For instance, fracture toughness is a distributed parameter. The inclusion of realistic load histories consisting of typical NPP load transients of various types is possible with VTTBESIT but is not with PIFRAP. Neither VTTBESIT nor PIFRAP do allow considering a change in water chemistry dur- ing a single analysis run, as both codes accept only constant values for SCC equation parameters. As a single analysis typically covers several decades of plant operation, this limitation unrealistically does not allow taking into account the improvements of water chemistry carried out in many NPPs, e.g. when SCC has been detected after some number of years in operation. The advantages of both VTTBESIT and PIFRAP include relatively short analysis run times, due to which it is easy to perform sensitivity analyses. Further, the inclusion of physical models is another advantage of both VTTBESIT and PIFRAP. Compared to purely statistical modelling tools, which completely omit the modelling of phys- ical degradation phenomena, the PFM approach of VTTBESIT and PIFRAP describes much more realistically component degradation. 12. Probabilistic lifetime analysis application for piping components 232 In the light of the discussion above concerning the scope and characteristics of VTTBESIT and PIFRAP, the probabilistic analysis results presented in this sec- tion are at least to some extent suggestive. Arguably, the best use of the single result values can be achieved by comparing them against each other, which would give the quantified order of proneness to degradation/failure. More pre- cisely, this would provide for considered single failure probabilities such inter- pretation as how many times more/less a particular pipe weld is prone to fail than the others. All in all, in order to achieve more accurate/realistic analysis results, the shortcomings concerning input data and analysis scope of these two analysis codes should be improved. 13. Summary and suggestions for future research 233 13. Summary and suggestions for future research Lifetime analyses of structural systems and components are discussed in this the- sis. These analyses necessitate a thorough knowledge of structural properties, loads, the relevant ageing mechanisms and prevailing environmental conditions. In general, the nature of ageing models of structural systems and components can be deterministic, probabilistic or a combination of these two types. Ageing models of all these kinds are presented. When the resulting failure probabilities from proba- bilistic analyses are combined with knowledge of system or component failure consequences, it is also possible to perform a risk analysis. In practise, the ageing analyses are usually performed with a suitable analysis tool, i.e. with analysis software. A selection of probabilistic system and component ageing and risk anal- ysis tools together with an application example are also presented in this thesis. The power plant components have generally substantial safety margins when properly designed and constructed. However, the available margins for degraded structures are not well known. In addition, age related degradation may affect the dynamic properties, structural response, structural resistance/capacity, failure mode and location of failure initiation. A better understanding of the effect of ageing degradation on structures and components, especially passive compo- nents, is needed to ensure that the current licensing basis is maintained under all loading conditions [10]. Ageing degradation can be observed in a variety of changes in physical properties of metals, concrete and other materials in a power plant. These materials may undergo changes in their dimensions, ductility, fa- tigue capacity, mechanical or dielectric strength. Ageing degradation presents itself as a variety of ageing mechanisms, physical or chemical processes such as fatigue, cracking, embrittlement, wear, erosion, corrosion and oxidation. These ageing mechanisms act on components due to a challenging environment with high heat and pressure, radiation, reactive chemicals and synergistic effects [5]. 13. Summary and suggestions for future research 234 Both physical and probabilistic ageing models for power plant components are presented in the thesis. Component ageing analyses are discussed as well. In the top level they consist of selecting components for analyses, identification of ageing mechanisms and mitigation of ageing effects. Concerning structural as well as risk analysis methods for power plant com- ponents, those presented and discussed in this thesis include: - deterministic analysis methods, - probabilistic analysis methods, - risk analysis methods, - deterministic degradation modelling methods, - probabilistic degradation modelling methods, - ageing and risk analysis applications, - conduct and process of ageing degradation analyses. As for structural and risk analysis applications, the emphasis here is on the com- ponent analysis tools. Some of the presented tools are capable for analyses of both individual components and systems comprising several components. In the general applications, any failure mode, fracture or other, can be addressed using the appropriate limit states. Validation of such software is usually carried out by benchmark exercises between different programs since it is not feasible to com- pare results with real failure statistics due to their scarcity. The probabilistic analysis approaches of the presented software vary from FORM and SORM to MCS. Some of the presented analysis tools are commercially available and oth- ers are proprietary. Presently, RI-ISI is a very topical subject in Finland, because the Finnish Ra- diation and Nuclear Safety Authority (STUK) now requires RI-ISI analyses for all piping systems of both existing and planned/future NPPs [316]. Keeping this in mind, it has been attempted to cover RI-ISI here from a number of viewpoints to get a wider description of this subject. The RI-ISI associated issues considered in this thesis include: - introduction and basics of RI-ISI, - principles of RI-ISI, - process of risk-informed inspection planning, - qualitative, quantitative and semi-quantitative methods for RI-ISI, - RI-ISI approach developments by VTT, - brief overview and comparison of three main RI-ISI methodologies, and - benchmarking of RI-ISI methodologies in the recent RISMET project. 13. Summary and suggestions for future research 235 The computational application part of this thesis concerns probabilistic failure and lifetime analyses to a relatively small but representative set of NPP piping components. In addition to data concerning structural properties, primary and secondary loads, supports and environment, also data concerning relevant degra- dation mechanisms and failure modes are needed. Essential sources for piping degradation and failure data are international and plant specific databases. In the probabilistic failure analysis computations, two analysis codes were used, those being probabilistic VTTBESIT developed by VTT and IWM, and PIFRAP de- veloped by DNV. The matching of the obtained analysis results is at least good. The main source causing in some cases deviations in the results is the differ- ences and limitations in the treatment of input data parameters, whereas the ap- plied PFM analysis procedures are quite similar. All in all, lifetime and risk analyses of power plant components and systems are disciplines that require combined knowledge from several fields of science. These include structural mechanics, fracture mechanics, probability mathemat- ics, and material science. As lifetime and risk analysis procedures are in a devel- oping stage, there are many aspects and features that call for further research. Some drawbacks and concerns of risk assessment that would need further re- search/development are presented in the following: - model uncertainty, - the sensitivity of the tails of the failure probability distributions, - the knowledge that is represented by data based on too few statistics, - the small failure probabilities for some components mean that evaluat- ed failure probabilities cannot be properly and completely validated, - in risk analyses, quantities are omitted either intentionally, e.g. human errors, or unintentionally because of a lack of data, or full understand- ing of the system. Incorporation of unavoidable ageing effects of SSCs into PRAs/PSAs is a sub- ject that would benefit from further research/development. In a fairly recent and thorough study [1], it was attempted to apply linear ageing failure rate model to incorporate ageing effects into PRA/ PSA. The results showed that this task is cumbersome, and not all questions were answered. Living PRA/PSA is one way with which problems related to time dependent ageing can be approached. How- ever, as a living modification PRA/PSA is simply just updated as necessary to 13. Summary and suggestions for future research 236 reflect the current design and operational features, it does not have the capability to predict how time dependent SSC ageing affects failure probabilities and risk. The advantage of PFM is the possibility of modelling clearly the uncertainties related to the structural degradation process, and thus being able to perform sen- sitivity analyses for the factors affecting this process. However, there are still methodological problems and there is no standard format for how to perform PFM analyses. This is a problem that can be overcome through the development of computational procedures. Concerning this issue, ENIQ has quite recently published a notable report [317] on ensuring the applicability of PFM and other structural reliability approaches. This report summarises the verification and validation requirements that structural reliability models and associated analysis codes should satisfy in order to be suitable for such purposes. As for PRA/PSA approach, it has been used extensively to analyse entire complex systems in nu- clear industry and elsewhere. PFM has had a more limited area of application and has been used for more specialised purposes concerning certain single com- ponents e.g. in primary systems of NPPs. Since both techniques have the same goal, a better utilisation of them could potentially be beneficial. A difference between PFM and PRA/PSA that causes difficulties is on which level the input data are based on actual observations. It is highly desirable to obtain standard- ised methods for PFM analyses like the one that has been developed for PRA/PSA. While PFM procedures lack standardisation, any comparison be- tween components that have been analysed with different methods will be at least to some extent questionable. Even though most degradation phenomena are quite well known, there are still degradation phenomena the physical mechanisms of which are not clear. An example of such degradation phenomena is SCC. These degradation phenomena need further research. Through improved knowledge of various degradation phenomena, more accu- rate models of them can be formulated. In general, physical degradation models need development in many areas, which include scope, range of validity, accuracy and realistic consideration of the underlying physical phenomenon or phenomena. For instance, the considerably high level of conservatism in the current SCC propagation computation approaches could be relieved. This could include fur- ther developing the computation procedures as well as providing more realistic estimates for certain model input parameters. Namely, the strictness in the upper bound approach used in the treatment of the underlying data concerning experi- mentally defined model parameters characterising environment and material as 13. Summary and suggestions for future research 237 well as WRS distributions could be to some extent relaxed. The aim would be to obtain a reasonably conservative SCC propagation computation approach. Another topical example is the environmental fatigue correction factor ap- proach, i.e. the Fen procedure [184]. The approaches to compute strain rates, combine load transients to cycles as well as to select and treat strain and stress components should be clarified. Presently, the scope and accuracy of this ap- proach is not sufficient for application concerning NPP environments. Often, modelling problems are caused by scarce and unreliable input data. One problem that would also benefit from further research is the modelling of several degradation phenomena acting simultaneously. Statistical ageing models of SSCs are not hampered by the above mentioned problems. On the other hand, since statistical ageing methods omit the modelling of the underlying physical phenomenon or phenomena, their applicability is limited. Improved degradation models would also be beneficial from the viewpoint of RI-ISI. With more accurate and realistic degradation models, more reliable fail- ure probabilities could be calculated. These results could in turn be used to per- form a more accurate quantitative risk analysis. Based on the results from such risk analysis, a more realistic proposal of a new inspection program could be prepared. The number of locations included in the inspection program is often considerably smaller in new RI-ISI programs than in older and more conserva- tive inspection programs developed with deterministic approach. Thus, remarka- ble benefits could be achieved in the forms of financial savings and shorter times which NDT personnel have to work under radiation exposure. It appears that most of the available NPP SSC lifetime analysis codes consider only one or two degradation mechanisms, which is a drawback, as the number of failure mechanisms encountered in NPP environments is much greater. Thus, it appears that for the time being several codes are needed, if one wishes to cover the degradation of all safety significant NPP systems and components. This causes problems, as the failure probability results given by various analysis codes may not be comparable due to differences in analysis models and required input data. Thus, it would be necessary to develop either one analysis program that considers all relevant degradation phenomena or a family of analysis pro- grams, each of which would cover one or two degradation phenomenon or phe- nomena and the failure probability results of all of which would be comparable. The analysis programs should also be able to consider the distributed nature of at least all those parameters the actual nature of which is markedly distributed. Another capability that is needed when considering power plant environments is 13. Summary and suggestions for future research 238 that the applied degradation modelling tools should be able to consider plant specific features and characteristics. Other capabilities that degradation model- ling tools of components should be able to consider are small failure probabili- ties and quantification of the effectiveness of NDT. The effect of NDT tech- niques is usually quantified using flaw detection reliability in terms of POD curves, which describe the detection probability as a function of the flaw size, e.g. flaw depth or length. However, the construction of a POD curve requires a considerable amount of data before statistical confidence is achieved. In the case of NDT methods, it is often expensive and time consuming to produce such a large amount of data. One possibility for further research would be to collect all available good quality NDT data and develop simple POD functions environ- ment, degradation mechanism and material type specifically for a representative set of power plant pipe sizes. Databases are one possible solution to the problems related to the input data needed in the risk and ageing analyses. The amount of NPP piping component degradation data in the international databases has increased and quality im- proved, respectively, which consequently allows improving accuracy in the sta- tistical piping degradation estimates. An example of such a database is OPDE (OECD Piping Failure Data Exchange), which is an advanced good quality pip- ing failure database containing data from 12 countries using nuclear energy, see refs. [318, 319]. On the other hand, the inclusion of plant specific features and characteristics in the degradation analyses necessitates the availability of plant specific databases. The development and better availability of component degra- dation/failure databases will hopefully provide means for how the problems related to small failure probabilities, i.e. to very rare events, could be alleviated or even overcome. 239 References 1. Smith, C. L., Shah, V. N., Kao, T. Apostolakis, G. 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Evaluation of Flaws in Austenitic Steel Piping. Journal of Pressure Vessel Technology, Vol. 108, 1986, pp. 352–366. 310. Shack, W. J. et al. Environmentally assisted cracking in light water reactors: An- nual report, October 1981 – September 1982. Report NUREG/CR-3292, Washing- ton D.C., U.S. Nuclear Regulatory Commission (USNRC), June 1983. 311. SINTAP. Structural integrity assessment procedures for European industry. Final Procedure: November 1999. Project funded by the European Union (EU) under the Brite-Euram Programme: Project No. BE95-1426, Contract No. BRPR-CT95-0024. 312. Barthelemy, J. Y., Janosch, J. J. Structural integrity assessment procedures for European industry. SINTAP; Task 4; Compendium of Residual Stress Profiles; Fi- nal Report: 18.5.1999. Project funded by the European Union (EU) under the Brite-Euram Programme: Project No. BE95-1426, Contract No. BRPR-CT95-0024. 40 p. + app. 18 p. 313. Khaleel, M. A., Simonen, F. A. Effects of alternative inspection strategies on piping reliability. Nuclear Engineering and Design 197, 2000, pp. 115–140. 264 314. Hazelton, W. S., Koo, W. H. Technical report on material selection and processing guidelines for BWR coolant pressure boundary piping: Final report. Report NUREG-0313, Rev. 2-F, Final Technical Report. U.S. Nuclear Regulatory Com- mission (USNRC), January 1988. 47 p. 315. Bergman, M. et al. A procedure for estimation of pipe break probabilities due to IGSCC. SAQ/FoU-Report 97/06, SAQ Kontroll AB, Stockholm, Sweden, 1997. 316. YVL Guide 3.8. Nuclear power plant pressure equipment. In-service inspection with non-destructive testing methods. Säteilyturvakeskus (STUK, Radiation and Nuclear Safety Authority), Helsinki, Finland, 22 September 2003. 30 p. 317. ENIQ Report No. 30, ENIQ recommended practice 9: Verification and validation of structural reliability models and associated software to be used in risk-informed in- service inspection programmes. EU Joint Research Centre (JRC), Institute for Energy, Petten, the Netherlands. 12 p. 318. OECD-NEA piping failure data exchange project (OPDE). Workshop on database applications. OPDE/SEC(2004)4, 2005. Nuclear Energy Agency, Issy-les-Moulineaux, France. 319. OPDE 2006:1. Coding guideline (OPDECG) & OPDE User instruction. PR01 Ver- sion 02, Restricted Distribution. Nuclear Energy Agency, Issy-les-Moulineaux, France, 2006. Series title, number and report code of publication VTT Publications 775 VTT-PUBS-775 Author(s) Otso Cronvall Title Structural lifetime, reliability and risk analysis approaches for power plant components and systems Abstract Lifetime, reliability and risk analysis methods and applications for structural systems and components of power plants are discussed in this thesis. These analyses involve many fields of science, such as struc- tural mechanics, fracture mechanics, probability mathematics, material science and fluid mechanics. An overview of power plant environments and a description of the various degradation mechanisms damaging the power plant systems and components are presented first. This is followed with a descrip- tion of deterministic structural analysis methods, covering e.g. structural mechanics and fracture mechan- ics based analysis methods as well as the disadvantages of the deterministic analysis approach. Often, physical probabilistic methods are based on deterministic analysis methods with the modification that one or more of the model parameters are considered as probabilistically distributed. Several probabilistic analysis procedures are presented, e.g. Monte Carlo Simulation (MCS) and importance sampling. De- scription of probabilistic analysis methods covers both physical and statistical approaches. When the system/component failure probabilities are combined with knowledge of failure consequences, it is possi- ble to assess system/component risks. Several risk analysis methods are presented as well as some limitations and shortcomings concerning to them. Modelling methods for various degradation (or ageing) mechanisms are presented. These methods are needed in the lifetime analyses of structural systems and components of power plants. In general, the lifetime analyses in question necessitate a thorough knowledge of structural properties, loads, the rele- vant degradation mechanisms and prevailing environmental conditions. The nature of degradation models of structural systems/components can be deterministic, probabilistic or a combination of these two types. Degradation models of all these kinds are presented here. Some important risk analysis applications are described. These include probabilistic risk/safety assessment (PRA/PSA) and risk informed in-service inspections (RI-ISI). In practise, lifetime and risk analyses are usually performed with a suitable analysis tool, i.e. with anal- ysis software. A selection of probabilistic system/component degradation and risk analysis software tools is presented in the latter part of this thesis. Computational application of probabilistic failure and lifetime analyses to a representative set of NPP piping components with probabilistic codes VTTBESIT and PIFRAP are presented after that. The thesis ends with a summary and suggestions for future research. ISBN 978-951-38-7760-6 (soft back ed.) 978-951-38-7761-3 (URL: http://www.vtt.fi/publications/index.jsp) Series title and ISSN Project number VTT Publications 1235-0621 (soft back ed.) 1455-0849 (URL: http://www.vtt.fi/publications/index.jsp) Date Language Pages December 2011 English, Finnish abstr. 264 p. Keywords Publisher Structural mechanics, fracture mechanics, risk analysis, reliability, PFM, RI-ISI VTT Technical Research Centre of Finland P.O. Box 1000, FI-02044 VTT, Finland Phone internat. +358 20 722 4520 Fax +358 20 722 4374 Julkaisun sarja, numero ja raporttikoodi VTT Publications 775 VTT-PUBS-775 Tekijä(t) Otso Cronvall Rakenteellisen eliniän, varmuuden ja riskien analysoimisen lähestymistapoja voimalaitosten komponenteille ja järjestelmille Tiivistelmä Tämän lisensiaattityön aiheina ovat voimalaitosten rakennejärjestelmien ja -komponenttien käyttöiän, luotettavuuden j a riskitarkastelujen analyysimenetelmät ja sovellukset. Näihin analyysimenetelmiin liittyy usea tieteen ala, kuten lujuusoppi, murtumismekaniikka, todennäköisyysmatematiikka, materiaalitiede ja virtausmekaniikka. Ensin esitetään yleiskatsaus voimalaitosympäristöistä sekä kuvaukset erilaisista voimalaitosten raken- nejärjestelmiä ja -komponentteja koskevista vaurioitumismekanismeista. Sitten käsitellään deterministiset rakenneanalyysimenetelmät, kuten lujuusopin ja murtumismekaniikan menetelmät, sekä erittelyä deter- ministisen lähestymistavan puutteista. Usein fysikaaliset probabilistiset menetelmät perustuvat deter- ministisiin vastaaviin sillä muunnelmalla, että yksi tai useampi malliparametri on asetettu probabilistisesti jakaantuneeksi. Työssä esitetään useita probabilistisia analyysimenetelmiä, kuten Monte Carlo -simulaatio ja tärkeysperusteinen otanta. Todennäköisyysperusteisia analyysimenetelmiä koskeva kuvaus kattaa sekä fysikaaliset että tilastolliset lähestymistavat. Kun rakennejärjestelmien/-komponenttien todennäköi- syydet yhdistetään tietämykseen seurausvaikutuksista, voidaan arvioida vastaavat riskit. Työssä esitetään useita riskianalyysimenetelmiä, sekä eräitä niitä koskevia rajoituksia ja puutteellisuuksia. Työssä esitetään valikoima erilaisia vaurioitumismekanismeja koskevia mallinnusmenetelmiä. Näitä menetelmiä tarvitaan rakennejärjestelmien ja -komponenttien käyttöikäanalyyseissa. Yleisesti ottaen kyseiset käyttöikäanalyysit edellyttävät perusteellisia tietoja rakenteellisista ominaisuuksista, kuormista, merkittävistä vaurioitumismekanismeista sekä vallitsevista olosuhteista. Rakennejärjestelmien/-komponenttien vaurioitumista kuvaavat mallit voivat olla tyypiltään deterministisiä, probabilistisia tai näiden yhdistelmä. Työssä esitetään kaikkiin näihin tyyppeihin lukeutuvia vaurioitumismalleja. Esitetään muutamia merkittäviä riskianalyysin sovelluksia. Näihin lukeutuu todennäköisyyspohjaiset turvallisuus- ja riskianalyysit (PSA j a PRA) ja riskitietoiset tarkastusohjelmat (RI-ISI). Käytännössä käyttöikä- ja riskianalyysit tehdään yleensä jollakin sopivalla analyysityökalulla, eli sovellus- ohjelmalla. Työn jälkipuoliskolla esitetään valikoima rakennejärjestelmien/-komponenttien vaurioitumisen ja riskien analyysisovelluksia. Sen jälkeen edustavalle valikoimalle ydinvoimalan putkistokomponentteja esitetään analyysiohjelmilla VTTBESIT ja PIFRAP tehdyt todennäköisyysperusteiset vikaantumis- ja käyttöikäanalyysit. Työn lopuksi esitetään yhteenveto sekä jatkotutkimusaiheita koskevia ehdotuksia. ISBN 978-951-38-7760-6 (nid.) 978-951-38-7761-3 (URL: http://www.vtt.fi/publications/index.jsp) Avainnimeke ja ISSN Projektinumero VTT Publications 1235-0621 (nid.) 1455-0849 (URL: http://www.vtt.fi/publications/index.jsp) Julkaisuaika Kieli Sivuja Joulukuu 2011 Suomi, engl. tiiv. 264 s. Avainsanat Julkaisija Structural mechanics, fracture mechanics, risk analysis, reliability, PFM, RI-ISI VTT PL 1000, 02044 VTT Puh. 020 722 4520 Faksi 020 722 4374 VTT CREATES BUSINESS FROM TECHNOLOGY Technology and market foresight • Strategic research • Product and service development • IPR and licensing • Assessments, testing, inspection, certifcation • Technology and innovation management • Technology partnership • • • V T T P U B L I C A T I O N S 7 7 5 S T R U C T U R A L L I F E T I M E , R E L I A B I L I T Y A N D R I S K A N A L Y S I S A P P R O A C H E S F O R P O W E R P L A N T C O M P O N E N T S . . . . ISBN 978-951-38-7760-6 (soft back ed.) ISBN 978-951-38-7761-3 (URL: http://www.vtt.f/publications/index.jsp) ISSN 1235-0621 (soft back ed.) ISSN 1455-0849 (URL: http://www.vtt.f/publications/index.jsp) Lifetime, reliability and risk analysis methods and applications for structural sys- tems and components of power plants are considered in this thesis. There are vari- ous degradation mechanisms to which power plant systems and components are susceptible to. Probabilistic structural analysis procedures can be used to assess their effects. For power plant components, probabilistic fracture mechanics is an of- ten used approach. When the system/component failure probabilities are combined with knowledge of failure consequences, it is possible to assess system/component risks. Examples of important risk analysis applications for power plants include probabilistic risk/safety assessment (PRA/PSA) and risk informed in-service inspec- tions (RI-ISI). In general, the lifetime analyses of power plant systems/components necessitate a thorough knowledge of structural properties, loads, the relevant deg- radation mechanisms and prevailing environmental conditions. Several objectives were set for this thesis. One objective is to collect both the methods and background theories for evaluation of operational lifetime, reliability and risk of structural systems and components of power plants. This was pursued by making an extensive literature survey. Another objective is to collect informa- tion concerning the relevant existing probabilistic degradation and risk analysis tools, with which one can assess the remaining lifetime and structural integrity of systems/components of power plants. The scope and characteristics of such tools are described. The last objective is to provide an application example. This was car- ried out by making a failure probability analysis to a representative set of nuclear power plant piping components. VTT PUBLICATIONS 775 Structural lifetime, reliability and risk analysis approaches for power plant components and systems Otso Cronvall ISBN 978-951-38-7760-6 (soft back ed.) ISSN 1235-0621 (soft back ed.) ISBN 978-951-38-7761-3 (URL: http://www.vtt.fi/publications/index.jsp) ISSN 1455-0849 (URL: http://www.vtt.fi/publications/index.jsp) Copyright © VTT 2011 JULKAISIJA – UTGIVARE – PUBLISHER VTT, Vuorimiehentie 5, PL 1000, 02044 VTT puh. vaihde 020 722 111, faksi 020 722 4374 VTT, Bergsmansvägen 5, PB 1000, 02044 VTT tel. växel 020 722 111, fax 020 722 4374 VTT Technical Research Centre of Finland, Vuorimiehentie 5, P.O. Box 1000, FI-02044 VTT, Finland phone internat. +358 20 722 111, fax + 358 20 722 4374 Technical editing Marika Leppilahti Kopijyvä Oy, Kuopio 2011 2 Otso Cronvall. Structural lifetime, reliability and risk analysis approaches for power plant components and systems [Rakenteellisen eliniän, varmuuden ja riskien analysoimisen lähestymistapoja voimalaitosten komponenteille ja järjestelmille]. Espoo 2011. VTT Publications 775. 264 p. Keywords structural mechanics, fracture mechanics, risk analysis, reliability, PFM, RI-ISI Abstract Lifetime, reliability and risk analysis methods and applications for structural systems and components of power plants are discussed in this thesis. These analyses involve many fields of science, such as structural mechanics, fracture mechanics, probability mathematics, material science and fluid mechanics. An overview of power plant environments and a description of the various degradation mechanisms damaging the power plant systems and components are presented first. This is followed with a description of deterministic structural analysis methods, covering e.g. structural mechanics and fracture mechanics based analysis methods as well as the disadvantages of the deterministic analysis approach. Often, physical probabilistic methods are based on deterministic analysis methods with the modification that one or more of the model parameters are considered as probabilistically distributed. Several probabilistic analysis procedures are presented, e.g. Monte Carlo Simulation (MCS) and importance sampling. Description of probabilistic analysis methods covers both physical and statistical approaches. When the system/component failure probabilities are combined with knowledge of failure consequences, it is possible to assess system/component risks. Several risk analysis methods are presented as well as some limitations and shortcomings concerning to them. Modelling methods for various degradation (or ageing) mechanisms are presented. These methods are needed in the lifetime analyses of structural systems and components of power plants. In general, the lifetime analyses in question necessitate a thorough knowledge of structural properties, loads, the relevant degradation mechanisms and prevailing environmental conditions. The nature of degradation models of structural systems/components can be deterministic, probabilistic or a combination of these two types. Degradation models of all these kinds are presented here. Some important risk analysis applications are described. These include probabilistic risk/safety assessment (PRA/PSA) and risk informed in-service inspections (RI-ISI). 3 Computational application of probabilistic failure and lifetime analyses to a representative set of NPP piping components with probabilistic codes VTTBESIT and PIFRAP are presented after that.e. with analysis software. The thesis ends with a summary and suggestions for future research.In practise. 4 . A selection of probabilistic system/component degradation and risk analysis software tools is presented in the latter part of this thesis. i. lifetime and risk analyses are usually performed with a suitable analysis tool. varmuuden ja riskien analysoimisen lähestymistapoja voimalaitosten komponenteille ja järjestelmille]. 5 . RI-ISI Tiivistelmä Tämän lisensiaattityön aiheina ovat voimalaitosten rakennejärjestelmien ja -komponenttien käyttöiän. VTT Publications 775. murtumismekaniikka. kuten lujuusoppi. Yleisesti ottaen kyseiset käyttöikäanalyysit edellyttävät perusteellisia tietoja rakenteellisista ominaisuuksista. Todennäköisyysperusteisia analyysimenetelmiä koskeva kuvaus kattaa sekä fysikaaliset että tilastolliset lähestymistavat.Otso Cronvall. että yksi tai useampi malliparametri on asetettu probabilistisesti jakaantuneeksi. Näihin analyysimenetelmiin liittyy usea tieteen ala. Structural lifetime. Työssä esitetään useita probabilistisia analyysimenetelmiä. Kun rakennejärjestelmien/-komponenttien todennäköisyydet yhdistetään tietämykseen seurausvaikutuksista. kuten lujuusopin ja murtumismekaniikan menetelmät. Avainsanat structural mechanics. 264 s. PFM.ja riskianalyysit (PSA ja PRA) ja riskitietoiset tarkastusohjelmat (RI-ISI). materiaalitiede ja virtausmekaniikka. reliability. Lisäksi esitetään muutamia merkittäviä riskianalyysin sovelluksia. merkittävistä vaurioitumismekanismeista sekä vallitsevista olosuhteista. probabilistisia tai näiden yhdistelmä. Ensin esitetään yleiskatsaus voimalaitosympäristöistä sekä kuvaukset erilaisista voimalaitosten rakennejärjestelmiä ja -komponentteja koskevista vaurioitumismekanismeista. sekä eritellään deterministisen lähestymistavan puutteita. Näihin lukeutuvat todennäköisyyspohjaiset turvallisuus. kuten Monte Carlo -simulaatio ja tärkeysperusteinen otanta. voidaan arvioida vastaavat riskit. Työssä esitetään valikoima erilaisia vaurioitumismekanismeja koskevia mallinnusmenetelmiä. Espoo 2011. reliability and risk analysis approaches for power plant components and systems [Rakenteellisen eliniän. Näitä menetelmiä tarvitaan rakennejärjestelmien ja -komponenttien käyttöikäanalyyseissa. Rakennejärjestelmien/ -komponenttien vaurioitumista kuvaavat mallit voivat olla tyypiltään deterministisiä. Työssä esitetään kaikkiin näihin tyyppeihin lukeutuvia vaurioitumismalleja. fracture mechanics. Sitten käsitellään deterministiset rakenneanalyysimenetelmät. kuormista. luotettavuuden ja riskitarkastelujen analyysimenetelmät ja sovellukset. Usein fysikaaliset probabilistiset menetelmät perustuvat deterministisiin vastaaviin sillä muunnelmalla. todennäköisyysmatematiikka. Työssä esitetään useita riskianalyysimenetelmiä sekä eräitä niitä koskevia rajoituksia ja puutteellisuuksia. risk analysis. 6 . Työn jälkipuoliskolla esitetään valikoima rakennejärjestelmien/-komponenttien vaurioitumisen ja riskien analyysisovelluksia. Sen jälkeen edustavalle valikoimalle ydinvoimalan putkistokomponentteja esitetään analyysiohjelmilla VTTBESIT ja PIFRAP tehdyt todennäköisyysperusteiset vikaantumis. Lopuksi esitetään yhteenveto sekä jatkotutkimusaiheita koskevia ehdotuksia.ja riskianalyysit tehdään yleensä jollakin sopivalla analyysityökalulla eli sovellusohjelmalla.ja käyttöikäanalyysit.Käytännössä käyttöikä. to MSc Jouni Alhainen for computational support and music related discussions. Espoo. The author also expresses his gratitude to DrTech Heli Talja for encouragement and preparing the way for this thesis in its earlier phase. December 2011 Author 7 . Gratitude is also addressed to many fellow researchers from VTT and certain experts from the domestic energy industry for providing useful technical background information and advice during the preparation of this thesis. in particular during the completion phase of the thesis. and to Niina for general support and caring understanding. The author expresses his gratitude to Professor Juha Paavola for supervising the thesis and to DrTech Kaisa Simola for being the inspector for this thesis as well as for valuable advice and comments.Foreword The author expresses his gratitude to VTT for providing both working quarters and altogether nearly three person work months worth of resource support for the preparation of this licentiate thesis. ................................ 71 Response function and performance function ............................. 79 Response surface methods ........................... 5 Foreword................2 4....................................................................................................... 74 Review of probabilistic analysis procedures ................................................................................6...... 82 Latin hypercube simulation (LHS) .............. 73 Commonly applied distribution functions ............. 28 3.........................7 3............ 75 Second-order reliability method (SORM) ............................... 55 Analysis methods based on structural mechanics ..........................................................................3 5...................................................................................4 5. 69 5.......................... 31 Fatigue ...........................................................................................................................................6........................................................3 5..............................................................................6................................................. 55 Fracture mechanics based analysis methods ........................................................... 11 List of abbreviations ...........................1 5...........................................................................6............................................. 24 Degradation mechanisms concerning power plant components .................................... 36 Corrosion .............. 3........................................ 49 Thermal ageing ........................... 51 Interaction of degradation mechanisms .........................8 3....................................................Contents Abstract ....5 5... 51 4....1 5.......................................................................................................................9 Introduction .................6... 55 4.. 7 List of symbols .....................................................................................1 3............................. 3 Tiivistelmä .............5 5....2 5.......................................... 77 Mean value methods ....................... Deterministic analysis methods ................................4 Introduction ....... 50 Other degradation mechanisms ....................................................................................................................... 42 Creep ..................4 5.....3 4............................................ 23 Introduction............................................................ 2......................................................................... 47 Irradiation embrittlement ..........6 Introduction ................................ 75 5...........................2 5............. 84 8 ............................ 76 Fast probability integration (FPI) .........................................................................6 3.2 3......................................................................... Objective of the thesis........................................................................................ 81 Importance sampling (IS) .............................................................................................................................................................3 3.................................................................................................................................................................................................1 4......................... 69 Failure probability and reliability ............................... 67 5.... 61 Analysis methods based on other fields of science ............................................... Probabilistic analysis methods ..........................................................6.......................6 5...................................4 3............. 28 Overview of conventional and nuclear power plant environments .............................................7 5.......... 78 Monte Carlo simulation (MCS) ................8 First-order second moment method (FORM) ..............6.........................................6.............. 20 1.....................................................5 3..................... 70 Uncertainty and sensitivity ....................................... ................................4 9......................... 144 PFM analysis procedure ................6..2.................................. 172 On principles of RI-ISI .............................................................................................................. 87 6.................2 6..............................................................................5 6......................2 7..3 6................................ 143 8..................1 9........................3........... 144 8............... 131 Commonly applied component degradation assessment procedures .............................................. 147 Numerical PFM methods ............................................................................................. 149 Short-term ageing models .............1 8............................................................................................ 121 Irradiation embrittlement .. 85 Determination of risk values ..2........................................................................4.....................3....... 87 Fault tree Analysis......................4........................................................................................................2 7.......... 139 8....... 182 9 .3 8.. 144 Probabilistic fracture mechanics based methods ..............1 8..... 94 7..................................... 97 Corrosion ................................................ quantitative and semi-quantitative methods for RI-ISI ..............................3 Introduction .....1 7............ quantitative and semi-quantitative risk analysis approaches ....................... 128 Modelling of interaction of degradation mechanisms........... 97 7.... 85 6.................................. 175 Brief overview and comparison of two main RI-ISI methodologies ........................................1 8.. 92 Concerns with risk assessment ......4 8.................................. 165 9.2 9........................................................... 88 Event tree analysis ......................................................4............................3.. Risk analysis methods ...........................4..........................................................................3............................5 7.......................... 97 Fatigue ......................... 172 9........5 9.............................7 9.......3 8.................... Deterministic degradation modelling methods for power plant components.......................6 9.............................................................. 113 Stress corrosion cracking (SCC) ...........................2 Introduction ........ 176 Probability of detection of cracks .................... 92 Risk based regulations.................. 90 Hazard and operability studies (HAZOP) .......3..2................................. 159 Statistical modelling methods ........ 179 RI-ISI approach developments by VTT ...... 174 Qualitative.4 7................................................... 85 Qualitative...................................3........................................................5 6...........................................3...........4 6................................4................................. 146 Engineering models and PFM ........1 6............................................................................................................................... Risk analysis applications for power plant systems.................... Probabilistic degradation modelling methods for power plant components ....................................................................4 Introduction .......... 113 7...................................................... 152 Long-term ageing models ..........................................................................................8 Introduction ............2 9......................6 Failure mode and effect analysis (FMEA) ................... 165 Risk informed in-service inspection (RI-ISI) ...3.... guidelines and standards ............3.................. 89 Expert opinion .... 86 Review of risk analysis procedures ..........................................3 6......... 180 RISMET project – benchmarking of RI-ISI methodologies ......3 9...........................2 6........................1 9..... 174 Process of risk-informed inspection planning ................ 114 Creep .....................2 8..................................6 7...................................2 Probabilistic structural mechanics based modelling methods .......................2..............3....................3.................................................................. 152 9..................................3..............................7 General and pitting corrosion ...3 Introduction ...........................1 7.............................................................................................................................................................. 165 Probabilistic risk/safety assessment .........................................................................1 6................................................ ............. Summary and suggestions for future research .......................................................... 216 Scope of probabilistic analyses ....................................................................... 209 12.................................................................................................................3 Geometry ..................................1 Selection of components for analyses.............. 203 11................10............2.........................................................3 Mitigation of ageing effects ....................................4 12...........................................................................2 12................................. 239 10 ......................................3...............................2..................................................... 212 Initial crack size distributions ............. 185 10....................................................1 12....... 209 12........................... 220 Details concerning the analysis flow of the two applied analysis codes ........................4 Results and discussion ............. 209 12...................... 185 11....2 Identification of ageing mechanisms ................ 224 13.......................1 Component ageing and risk analysis software .................................. 233 References ............................................................................. Power plant component ageing analyses .............. 183 10.................................................................3......................................................... 220 12..............2 Examined piping components and associated input data .......................... Probabilistic ageing and risk analysis tools ............. Probabilistic lifetime analysis application for piping components ...............1 Introduction ...........1 12........................... 223 12........... 183 10... 187 11..............................................3............ 208 12...........................................2 System ageing and risk analysis software .................2......................3 Analysis characteristics ..3 12...2 12...................................................................................... 210 Material properties ..................................... 221 Sub-critical crack growth ....................... 187 11......................2........3 Summary of ageing and risk analysis software.............................. 210 Loads and failure mechanisms ...................... k c c C C(t) Ct (Ct)SSC C.List of symbols Latin symbols A A a a0 ac aCA aCA. a33 B b b. C2 event rate parameter before the first event rate coefficient in constitutive equation for creep crack depth initial crack depth. a22. C1 C1. CS. CF. CJ C C0.0 aFA ai an a11. a12. initial crack length critical value of crack depth depth of creep induced cavity/crack threshold depth of creep induced cavity/crack for onset of interaction with fatigue induced crack growth depth of fatigue induced crack effect of i:th maintenance final crack length coefficients in strain energy density equation geometry factor Burgers vector material parameters in S-N curves method half of the crack length reduction factor effective corrosion rate in absence of protective coating interpolation parameter between small scale creep and extensive creep interpolation parameter between small scale creep and extensive creep is separately computed small scale creep limit for interpolation parameter between small scale creep and extensive creep coefficient characterising material and environment in fatigue crack growth rate equation coefficient characterising material and environment in stress corrosion crack growth rate equation stress corrosion crack growth rate equation coefficients constants in equation for nominal environmental fatigue correction factor 11 . a sequence of independent.nom fi0 fij FI. n. which contains the detection probabilities of cracks total accumulated damage for crack initiation phase total accumulated damage for crack propagation phase material constants in creep crack growth rate equation general corrosion propagation depth elastic modulus.i Fen. of which not all are zero depth probability for existing manufacturing crack material type dependent coefficient in stress corrosion cracking rate equation environmental fatigue correction factor nominal environmental fatigue correction factor for the stress cycle i nominal environmental fatigue correction factor occurrence rate of stress corrosion cracking dimensionless function of angular coordinate of the polar system having origin at the crack tip crack geometry factors for fracture modes I and II crack propagation direction dependent coefficient in stress corrosion cracking rate equation fx(x) falloy Fen Fen. i.e. d(t) E f F F f f(t) F(t) coefficient in probabilistic equation for aspect ratio of fabrication cracks radiation embrittlement coefficient consequence retardation parameter C* integral parameters in fatigue crack growth rate equation critical strength level total accumulated damage outer pipe diameter matrix. Young’s modulus loading frequency Faraday’s constant renewal process neutron fluence time dependent probability density function of component failure time dependent renewal process. q d D D D DI DII D0. p.C CF Ci Cp C* C. identically distributed non-negative random variables. FII forientation 12 . Kr l0 m m m 13 .fx (x) Fx (x) fxy (x. the strain energy rate Bessel function of first kind and of order 1 mode I fracture toughness fracture energy. PDF cumulative distribution function of random variable X.0 for plane stress and 2.0 for plane strain KIa KIC KI. limit state function shear modulus pipe wall thickness integration constant initial dissolution current density at bare metal surface initial value of decay curve of electric current on newly exposed surface modified Bessel function of order 1 J-integral. CDF joint probability density function of random variables X and Y performance function. fracture toughness constant General expression for stress intensity factor mode I stress intensity factor mode II stress intensity factor mode III stress intensity factor mode I stress intensity factor corresponding to mean stress mode II stress intensity factor corresponding to mean stress mode III stress intensity factor corresponding to mean stress lower bound crack arrest value for stress intensity factor mode I fracture toughness Maximum value of mode I stress intensity factor Minimum value of mode I stress intensity factor crack propagation threshold value for stress intensity factor strain hardening coefficient determined near failure non-dimensional variables in failure assessment diagram approach initial crack length total number of cells in stratified sampling decay constant of current at newly exposed material surface dimensionless constant that is approximately 1.max KI. y) g G h In i0 i0 I1(x) J J1 JIC JR k K KI KII KIII K Imean mean K II mean K III probability density function of random variable X.min Kth K* Lr. number of load cycles strain hardening exponent exponent characterising material and environment in stress corrosion crack growth rate equation power law exponent for non-linear behaviour in general corrosion dimensionless work hardening exponent in Ramberg-Osgood equation numerical constant correlated with the degree of sensitisation. corrosion potential exponent in constitutive equation for creep total number of strain increments the number of times Monte Carlo simulations have reached or exceeded the selected degradation/failure state in terms of crack depth a time dependent point process describing the occurrence of shocks number of load cycles for crack initiation phase number of load cycles for crack propagation phase fatigue life in air at room temperature number of dislocations contributing to formation of slip band number of failures. water conductivity.i N(t) NI NII Nair. N N n n n n n n n N a a FS . number of load cycles to failure number of cycles of operation under certain stress amplitude total number of cycles that would produce a failure at certain stress level number of samples taken from the j:th cell in stratified sampling modified life at some considered loading level total accumulated life number of active slip bands fatigue life in water at service temperature cyclic strain hardening exponent determined near failure transformed level of dissolved oxygen transition probability matrix internal pressure 14 .m. my M M(t) m1 M.RT nd Nf ni Ni Nj Ni ´ N Nslip Nwater n* O* P p exponent characterising material and environment in fatigue crack growth rate equation atomic weight of the metal time dependent renewal function shaping exponent material dependent factors sample size. plate thickness time in service since start of operation. time since start of corrosion process temperature expected total time in service time in general short time constant p FS .tj t pi Pj PLM Qg R R R R R r R(t) Reff Ri Ri Rmax RTNDT RTNDT0 Ry S S S S* t t T T t t0 15 .i . remote stress resulting from the applied load stress strain energy density factor transformed sulphur content wall thickness.P. Pi Pf pf pf0 probability failure probability rupture failure probability for pipe section conditional fracture probability for a given crack length failure probability corresponding to selected degradation/failure state initial probabilities of various damage states probability that an initial crack exists in the j:th cell in stratified sampling Larson-Miller parameter thermal activation energy for crack growth total risk reliability stress ratio universal gas constant strength polar coordinate with origin at the crack tip time dependent component reliability effective stress ratio individual risk inner radius maximum allowable stress ratio reference nil-ductility temperature of the material initial NDT temperature for unirradiated material extent of current yield zone stress level. 9 mm lateral expansion transition temperature material parameters which are function of temperature. s U Uen ui Ui ui V V0 w w x x ~ X X(t) X. C. k2. k3. G. Y Xc xi y Y(t) time constant in pitting corrosion model starting time of current decay at newly exposed material surface transition time from short time to long time behaviour transition time from primary to secondary creep time of rupture time spent at condition i components of traction vector maintenance times time to rupture time to rupture at condition i absolute reference temperature used to normalise data transformed temperature 41 J impact energy transition temperature 56 J impact energy transition temperature 0. relative initial crack length Median value time dependent applied stress random variable continuous load exponent parameter related to considered loading level constant value for strength time dependent strength of a component 16 .9 mm T. environment and the metallurgical state of the material factor depending primarily on the stress ratio cumulative fatigue usage factor components of displacement vector partial fatigue usage factor displacement rate components pitting corrosion propagation rate initial pitting corrosion propagation rate strain density stress work rate (power) density certain value of the random variable relative initial crack depth.t0 t0 t1 t2 tF ti Ti Tk tr tri Tref T* TK41J TK 56J TK 0. Z z z Z0 ZOL response function number of electrons involved in reaction rate charge change at electrolysis limiting value of response function plastic zone diameter Greek symbols ageing acceleration rate dimensionless constant crack growth rate coefficient Parameter describing the degree of interaction between fatigue induced crack growth and creep induced cavity growth first-order reliability index. geometrical reliability index. HasoferLind reliability index cyclic hardening rate frequency of load cycles at some considered loading level exponent in stress corrosion cracking rate equation . aspect ratio of fabrication crack dimensionless constants in stress corrosion cracking rate equation crack tip opening displacement total displacement rate time dependent creep displacement strain range p p i t Parameter related to plastic strain current value of plastic strain absolute value of plastic strain rate shear strain range damage fraction cyclic J-integral range effective stress intensity factor range equivalent stress intensity factor range fatigue threshold in terms of stress intensity factor range strain intensity factor range shift in reference nil-ductility temperature of the material p i J Keff Keq Kth KI( ) RTNDT 17 . cumulative failure function mean value point general mean value scale parameter Poisson’s coefficient initial crack degradation/size state crack degradation/size state at time k location parameter. angular coordinate around crack tip angle between the direction of the slip plane and tensile stress TK 0.S TK41J TK56J cyclic strain energy density factor range stress range shift in 41 J impact energy transition temperature shift in 56 J impact energy transition temperature shift in 0.9 mm ct f f i max min ri ~ ij (u) i . (t) 0 m (t) i y 0 k 18 .9 mm lateral expansion transition temperature strain rate crack tip strain rate strain caused by break of protective film at crack tip failure strain as a fraction strain accrued at condition i maximum strain minimum strain rupture strain at condition i transformed strain rate dimensionless function of work hardening exponent and angular coordinate around crack tip acceleration factor for fatigue induced crack growth rate equation standard normal distribution function contour around crack tip principal curvatures of the limit state at the minimum distance point coefficients in probabilistic equation for aspect ratio of fabrication cracks time dependent component failure rate constant random failure rate time dependent integral of the intensity. inclination angle of the crack 0 d m inclination angle of the crack at the direction of minimum strain energy density material density dislocation density density of the metal standard deviation reference stress (most often yield strength) axial membrane stress components of stress tensor/matrix dimensionless function of work hardening exponent and angular coordinate around crack tip yield strength. average local yield strength shape parameter y 0 AXIAL ~ y y ij ij local yield strength 19 . List of abbreviations AGR AIS AMV ASME BWR CANDU CCDP CCF CDF CDF CLERP CTOD DFM DNV EAC EDF ENIQ EPFM EPRI ETA FAC FAD FBR FEM FITNET FMEA FMECA FORM FPI FTA GCR GUI HAZ advanced gas-cooled reactor adaptive importance sampling advanced mean value method American Society of Mechanical Engineers boiling water reactor Canadian deuterium uranium reactor conditional core damage probability common cause failure conditional core damage crack driving force conditional large early release probability crack tip opening displacement deterministic fracture mechanics Det Norske Veritas environmentally assisted cracking Electricité de France European Network for Inspection and Qualification elastic-plastic fracture mechanics Electric Power Research Institute event tree analysis flow accelerated corrosion failure assessment diagram fast breeder reactor finite element method European Fitness For Service Network failure mode and effect analysis failure mode effect and criticality analysis first-order second-moment method fast probability integration fault tree analysis gas cooled reactor graphical user interface heat affected zone 20 . HAZOP HIDA HSE HWR IAEA IASCC IGSCC INEEL IS ISI IWM JRC JSME LEFM LHS LLNL LOCA LWR MCS MIC MPP MV NDE NDT NDT NPP OECD OPDE ORNL PDF PFM POD PRA PSA PTS hazard and operability study HIgh temperature Defect Assessment procedure Health and Safety Executive heavy water reactor International Atomic Energy Agency irradiation assisted stress corrosion cracking intergranular stress corrosion cracking Idaho National Engineering and Environmental Laboratory importance sampling in-service inspection Fraunhofer-Institut für Werkstoffmechanik Joint Research Centre Japan Society of Mechanical Engineers linear-elastic fracture mechanics Latin hypercube simulation Lawrence Livermore National Laboratory loss of coolant accident light water reactor Monte Carlo simulation microbiologically influenced corrosion most probable point mean value method non-destructive examination non-destructive testing nil-ductility temperature nuclear power plant Organisation for Economic Co-operation and Development OECD Piping Failure Data Exchange Oak Ridge National Laboratory probability density function probabilistic fracture mechanics probability of detection Probabilistic Risk Assessment Probabilistic Safety Assessment pressurised thermal shock 21 . PWR PWROG PWSCC QRA RAW RBI RCM RDF RIF RI-ISI RPV RRW RSM SA SAIC SCC SINTAP SKI SMAW SORM SSC SSM STUK SwRI TGSCC TVO USNRC VTT WRS WWER pressurised water reactor Pressurized-Water Reactor Owners Group primary water stress corrosion cracking quantitative risk analysis risk achievement worth risk based inspection reliability centered maintenance risk reduction factor risk increase factor risk informed in-service inspection reactor pressure vessel risk reduction worth response surface method sensitivity analysis Science Applications International Corporation stress corrosion cracking Structural Integrity Assessment Procedures for European Industry Swedish Nuclear Inspectorate shielded metal arc welding second-order reliability method systems. Nuclear Regulatory Commission Technical Research Centre of Finland weld residual stress water-cooled water-moderated energy reactor 22 . structures and components Swedish Radiation Safety Authority The Finnish Radiation and Nuclear Safety Authority Southwest Research Institute transgranular stress corrosion cracking Teollisuuden Voima Oyj U.S. Objective of the thesis Several objectives were set for this thesis. with which one can assess the remaining lifetime and structural integrity of systems/components of power plants. This was pursued by making an extensive literature survey. reliability and risk of structural systems and components of power plants. This was carried out by making a failure probability analysis to a representative set of nuclear power plant piping components. Another objective is to collect information concerning the relevant existing probabilistic degradation and risk analysis tools. The last objective is to provide an application example. One objective is to collect both the methods and background theories for evaluation of operational lifetime. 23 . The scope and characteristics of these tools are described.1. Objective of the thesis 1. Other associated fields of science are material science. is presented in Chapter 4. and stress corrosion cracking (SCC) of components may cause cracking. the relevant ageing mechanisms and prevailing environmental conditions. high-cycle thermal fatigue. Introduction 2.2. supports. where the emphasis is on structural mechanics and fracture mechanics based analysis methods. thermal ageing of cast stainless steel components and vibration fatigue of small diameter piping may cause rupture. loads. An overview of conventional and nuclear power plant environments and a description of the various degradation mechanisms damaging the power plant systems and components are presented first. see Chapter 3. Due to distributed nature of many of these properties or phenomena. The disadvantages of the deterministic analysis methods are discussed. Introduction Structural lifetime. and (2) those that may cause cracking. reliability and risk analysis methods and applications for structural systems and components of conventional and nuclear power plants are considered in this thesis. Radiation embrittlement. see Chapter 5. The emphasis here is on the latter power plant type. Low-cycle fatigue. chemistry and radiation physics. The probabilistic analysis methods are often based on deterministic analysis methods 24 . Lifetime analyses of structural systems and components necessitate a thorough knowledge of their structural properties. A description of deterministic structural analysis methods. The mechanisms that have potential to cause rupture are likely to have significantly more risk impact [1]. probabilistic modelling methods are deemed suitable for the lifetime analyses. Failure probabilities combined with knowledge of system or component failure consequences allow performing risk analyses. Probabilistic analysis methods are dealt with after that. Ageing degradation mechanisms may be divided into two groups based on the resulting failure modes: (1) those that may cause rupture. The trend towards a risk based approach in power plants. At the same time. The engineering definition of risk is now accepted as being the product of the likelihood and the consequence of an event [2]. Inspection trials have produced a greater appreciation of the limits of NDT performance and reliability. with analysis codes. This includes e. see Chapters 7 and 8.g. Deterministic and probabilistic computational modelling methods for various ageing degradation mechanisms damaging power plant components are presented after this. In general. Introduction with the modification that one or more of the model parameters are considered as probabilistically distributed. Deterministic fracture mechanics is an example of deterministic ageing modelling approaches. There exist nowadays a number of risk analysis procedures. mean value methods. is a modelling approach which combines both deterministic and probabilistic features. supports. Monte Carlo simulation (MCS) and importance sampling. which is discussed in more detail. probabilistic risk/safety assessment (PRA/PSA) 25 . loads. In practise. i. Probabilistic fracture mechanics. the lifetime analyses necessitate a thorough knowledge of structural properties. Degradation models of all these kinds are presented. A brief description of risk analysis methods is presented then. Statistic failure models. A brief description of some important risk analysis applications is presented in Chapter 9. the relevant degradation mechanisms and prevailing environmental conditions.e. the degradation analyses are usually performed with suitable analysis tools.2. Risk based regulations and standards as well as concerns related to risk assessment are considered also. second-order reliability methods (SORM). see Chapter 6. which are based on observed failure statistics. The nature of degradation models of structural systems/components can be deterministic. probabilistic or a combination of these two types. and especially in nuclear power plants (NPPs). The presented probabilistic analysis procedures include first-order second moment (FORM) methods. are an example of probabilistic ageing modelling approach. The considered risk analysis procedures include failure mode and effect analysis (FMEA). These methods are needed in the lifetime analyses of structural systems and components of power plants. fault tree analysis (FTA). event tree analysis (ETA) and expert opinion. developments in non-destructive testing (NDT) technology have increased the scope and efficiency of examinations that can be undertaken. is being supported by extensive plant operating experience. improved understanding of material degradation mechanisms and the availability of fitness-for-service assessment procedures [3]. and up to removal from service. Computational application of probabilistic failure and lifetime analyses to a relatively small but representative set of NPP piping components is presented after that. PRA/PSA has emerged as an increasingly popular analysis tool in several other industries as well. through design. The objective of component ageing analyses is to identify the degradation mechanisms of components and the increase in failure occurrences. Despite having two names. Ageing of various systems. also data concerning relevant degradation mechanisms and failure modes are needed in the probabilistic failure and lifetime analyses of NPP piping components. 26 . construction and operation. Another difference in approaches of these programs is that one part of them contains models of one or more specific degradation mechanisms. A selection of various probabilistic system/component degradation and risk analysis tools is presented then. while the other part consists of general structural reliability analysis codes. These are divided into component and system analysis applications. to assess the remaining lifetime of components and to find suitable means to prevent or mitigate the effects of ageing. primary and secondary loads. see Chapter 12. The inspection plan can then target the high risk equipment and be designed to detect potential degradation before fitness-forservice could be threatened [3]. In the probabilistic failure analysis computations two analysis codes were used. supports and environment. the method is referred to as PRA. the method is referred to as PSA. In addition to the energy industry. see Chapter 11. Introduction and risk informed in-service inspection (RI-ISI). Essential sources for piping degradation and failure data are international and plant specific databases. see Chapter 10. Component ageing analyses are discussed after this. Recently The Finnish Radiation and Nuclear Safety Authority (STUK) has adviced/announced that also in Finland this method should be referred to as PRA. PRA/PSA is a systematic and comprehensive methodology to evaluate risks associated with every life-cycle aspect of a complex engineered technological entity or component from concept definition. structures and components (SSCs) affect the safety of power plants. The purpose of the risk analysis is to identify the potential degradation mechanisms and threats to the integrity of the equipment and to assess the consequences and risks of failure. RI-ISI involves the planning of an inspection on the basis of the information obtained from a risk analysis of the equipment. In most countries. especially during the last two decades [1]. the technique is practically the same. In the United States.2. In addition to data concerning structural properties. developed by VTT and Fraunhofer-Institut für Werkstoffmechanik (IWM. respectively.2. and PIFRAP developed by Det Norske Veritas (DNV). Germany). Introduction those being probabilistic VTTBESIT. The thesis ends with a summary and suggestions for further research. 27 . wear. In this thesis degradation is used as an equivalent expression to material ageing. due to the 28 . From the safety perspective. Ageing degradation can be observed in a variety of changes in physical properties of metals. corrosion and oxidation. changing power output) and equipment testing. Ageing degradation is caused by a variety of ageing mechanisms. can also create stresses for plant SSCs [5]. and fairly similar operating conditions. Some degradation mechanisms can also act simultaneously. physical or chemical processes such as fatigue. Degradation mechanisms concerning power plant components 3. However. ageing degradation has the potential to reduce the safety of operating power plants. which include normal operation and transient conditions. Degradation mechanisms concerning power plant components 3. ductility. such as power plant cycling (i. this implies that ageing degradation of SSCs important to safety remains within acceptable limits. These ageing mechanisms act on SSCs due to a challenging environment with relatively high heat and pressure. erosion..1 Introduction Ageing degradation in power plants should be managed by ensuring that the design functions remain available throughout the service life of the plant. The technical definition of ageing given by the International Atomic Energy Agency (IAEA) is the following [4]: Ageing is the continuous time dependent degradation of materials due to normal service conditions. reactive chemicals and synergistic effects. radiation. cracking. concrete and other materials in a power plant. There is a fairly limited set of degradation mechanisms.3.e. embrittlement. like for example fatigue and creep. Some operating practices. a large commonality in used materials. These materials may undergo changes in their dimensions. fatigue capacity and mechanical or dielectric strength. and that procedures and personnel training remain adapted. Unchecked. safety requirements may increase with time. The plant life will then be the result of the consideration of ageing in a changing regulatory. In the context of stable safety requirements. although similar. A reduction of the applied loading or an increase of component performance would result in an increase of the safety margin. and costs of other energy sources may decrease. Figure 3. ageing in power plants can be taken to mean evolution of personnel and procedure adequacy and evolution of material or equipment properties. after a certain time may not be compatible with the required safety margins. Even near twin units at the same site can have substantial differences in the remaining life of major SSCs. Repair or replacement of components. as well as a better knowledge of component performance can result in a modification of the demonstrated safety margin. political. are unique for each plant. as well as change in service conditions for a better compatibility with component reduced capacities are possible [6]. operating conditions and histories as well as maintenance practices. construction and used materials. In general. 29 . shows that a better evaluation of the applied loading. or with an economic functioning of the plant. based on subtle design or material differences and operating histories [5]. which is given for illustrative purpose only and does not take into account the possibility of component replacement. following the evolution of the public acceptance. the specific effects of ageing. limiting as a result the economic interest of continued operation.1-2 illustrates the combined influence of the above considerations with a possible evolution of safety requirements. the evolution of the safety margins for a given component can be illustrated as shown in Figure 3. This figure. In parallel. which. Degradation mechanisms concerning power plant components diversity in plant design. technical and economic context. and consequently of the potential component life.3.1-1. 1-2. [9].1-1. [8]. but also through a better use and a better evaluation of the real evolution of component performance. Illustration of possible component margin evolution during service. Illustration of combined evolution of component margins and safety requirements. from ref.3. Figure 3. a better evaluation of existing 30 . for example a better prediction of material properties evolution.1-1 and 3. from ref. As shown in Figures 3. Degradation mechanisms concerning power plant components Figure 3.1-2. lifetime extension is not only obtainable through repair or replacement of components. swelling. structural response. However. thinning. The processes of degradation can result in material and geometry changes of which the most important are [4]: reduced toughness (embrittlement). Depending of the metal or alloy in question and the intensity of the external loading as well as environmental impact. or a better knowledge of operating conditions. In addition. Degradation mechanisms concerning power plant components defects and mechanical behaviour of real defects. The most common turbine type is steam turbine. The working fluid can be wind. denting and pitting. cracking. which employs steam produced from burning fossil fuels in a boiler or from the heat produced by atomic reactions inside a nuclear reactor [11]. especially passive components. structural resistance/capacity.2 Overview of conventional and nuclear power plant environments The principal route for producing electricity in power plants is the conversion of mechanical energy of rotation into electrical energy using a generator. is needed to ensure that the current licensing basis is maintained under all loading conditions [10].3. the available margins for degraded structures are not well known. The large generators used by electric utilities employ a shaft comprising the magnetic field (rotor) which rotates inside a stationary electric field containing conducting wires (stator). age related degradation may affect the dynamic properties. Rotation of the shaft is achieved by coupling it to a turbine in which the kinetic energy of a moving fluid is converted into mechanical energy of rotation. A better understanding of the effect of ageing degradation on structures and components. steam or combustion gases. 3. failure mode and location of failure initiation. there may be a high potential for the degradation. water. Power plant structures generally have substantial safety margins when properly designed and constructed. the power plant environments are characterised by high system temperatures and 31 . Due to the nature and requirements of the energy production methods. This thesis considers mainly the ageing degradation of metallic SSCs. 2-1. The pressure. and during start-up they are correspondingly raised to the operational level. These accidents are.g. usually to atmospheric conditions. take place.g. e. the power plants have to be shutdown when certain accidents. after which the operation is started again. Besides the planned and usually yearly outages. The chemistry of the working fluid can affect the structural integrity of the power plant components as well. Many power plant components are also exposed to large temperature and pressure variations. with several auxiliary systems. however. e. the power plants are shutdown at specific intervals for maintenance and inspection outages. which can in certain cases occur quite quickly. Degradation mechanisms concerning power plant components pressures. temperature and specific volume of the steam at various stages in coal fired power plants are illustrated in Figure 3. Besides the operational conditions. malfunctions or structural failures. and the power plants are also designed to sustain their safety in the case of their occurrence. During shutdown the system temperature and pressure are lowered. relatively rare. In general the temperatures in conventional power plants vary from atmospheric conditions during shutdown to nearly 1000 C in the combustion bed of some boiler types under operational conditions.3. 32 . 3. 33 . from ref. temperature and specific volume in the various components of a large steam power plant. circulating fluidised bed combustion. [13]. Two major factors that influence the temperature and pressure levels in conventional power plants are the boiler type and used fuel. The most common boiler types of conventional power plants are [12]: grate-fired furnace. 1000 ksi corresponds to approximately 7 MPa and 1000 F to approximately 540 C.2-1. bubbling fluidised bed combustion. Relationships between steam pressure. Degradation mechanisms concerning power plant components Figure 3. biomass and a combination of biomass and e. martensitic. oil or peat. waste (industrial. The bed temperature is maintained relatively low. the fuel is loaded on the fire grate and combustion air is fed from below the grate to the primary combustion chamber. Their values give information about the corrosion/deposition processes in the water/steam cycle. Degradation mechanisms concerning power plant components Some typical fuels used in conventional power plants are [12]: coal. In grate-fired boilers. is fastest at 250–500 °C and finishes around 800 °C [12]. Pyrolysis of volatile matters starts at 150–200 °C. the bed is a mixture of inert material. For instance. Iron and copper concentrations in the water/steam are an indicator of the efficiency of conditioning. The typical pressure boundary materials in conventional power plants are ferritic. between 800–900 °C. Cyclone is used to separate flue gas from escaped particles. the silica content will be far below the specified level [14]. black liquor. and coarse fuel particles. When operating with fully demineralized water. oil. A continuous stream of air is fed through the bed to produce turbulence in the bed due to which particles in the bed start to move and the bed becomes fluidised. coal. and respectively in the boiler and turbine. like sand. In the secondary chamber gases evaporated from the fuel bed are incinerated.3. In fluidised bed combustion. municipal). which are continuously circulated back to the bed. austenitic and Ni-based alloys. Most NPPs use steam cycles that differ little from those of conventional power plants except for the source of heat for the steam generator and for steam supply conditions [18]. in order to keep the ash solid and the bed material in the fluidised state. gas. Silica concentrations in some boilers may not exceed certain values due to the specification for turbine operation. Fuel is normally mixed with matter returning from a cyclone [12]. heavy water reactor (HWR). peat. According to the reactor type the NPPs are divided to two categories: light water reactor (LWR). In bubbling fluidised bed furnaces the fluidising particles remain under a defined level. the material of the steam tubes is typically austenitic stainless steel [11]. In the circulating system higher gas stream and finer fluidised particles are used and solid fluidised particles drift throughout the combustor with the gas stream. 34 . The quality of the working fluid in conventional power plants is monitored and it is attempted to maintain such that the concentrations of various impurities stay low enough.g. The most common LWR types are the BWR and PWR. whereas the operating temperatures range from 282 to 288 °C (540 to 550 °F). °C Hot leg temperature. BWR plant operating pressures are typically from 6.5 35 .7 EPR 295.51 Siemens 291. The PWR is currently the predominant nuclear reactor type in the world. The propagation of radioactivity to the turbine-feedwater system is virtually non-existent [18]. The water in the PWR is in slightly higher temperature as in the BWR but is at a higher pressure. boiling water reactor (BWR).5 325.2-1.5 328 15. producing slightly radioactive steam that passes directly to the steam turbines. respectively [19]. has a half-life of only a few seconds.5 15.5 15. Typical operational temperatures for common PWR plant types. The BWRs operate at a pressure that allows boiling of the coolant water adjacent to the fuel elements inside the reactor core. °C Primary circuit pressure. however. Parameter/ Reactor type Cold leg temperature.3 326. water-cooled water-moderated energy reactor (WWER). [20].51 292 329.2–15. Table 3.2-1 are shows typical operational temperatures and pressures for common PWR plant types.3. gas cooled reactor (GCR). so that the reactor coolant remains liquid through the entire reactor coolant loop. fast breeder reactor (FBR). Degradation mechanisms concerning power plant components The LWRs are further divided to the following categories [21]: advanced gas-cooled reactor (AGR).2–12. The radioactivity in the steam. The Table 3. Another difference between these two reactor types is that PWRs have an additional separate water loop that isolates the turbine steam loop from the reactor coolant [18]. The LWRs are designed so that the core is both moderated and cooled by highly purified water and the fuel is enriched uranium that fissions with thermal neutrons [18]. from ref.90 to 7. pressurised water reactor (PWR). MPa Westinghouse Framatome (4-loop plant) 292.24 MPa (1000 to 1050 psi).5 15.1 WWER 440 270 300 WWER 1000 290 325 15.8 12. 3. Degradation mechanisms concerning power plant components The best known HWR is the Canadian Deuterium Uranium (CANDU) reactor. It utilises natural uranium as a fuel and heavy water as a moderator and coolant. These reactors produce a substantial saving due to the absence of fuel enrichment costs, but a large chemical plant is required to supply the quantities of heavy water required [18]. As the vast majority of the operating NPPs are LWRs, the description of further details of HWRs is omitted here. As for local loads, one important issue is welding process induced residual stresses. They are locally confined to the weld regions and can be relatively high, e.g. of the scale of yield stress in tension on and near the inner surface. Thus they have to be taken into account in the structural integrity/safety analyses. The resulting residual stress state in a welded component is determined by welding related parameters, material properties and geometrical constraints. The first issue refers to the local shrinkage, quench and phase transformations caused by the localised thermal cycle. While the latter issue concerns the constraining effect of the surrounding structure, and in case of dissimilar metal welds the unbalance in material properties. It is generally known that there are three primary sources for weld residual stress redistribution or relaxation, which are the following [22]: transient mechanical loads, thermal treatments and irradiation exposure. Mainly weld residual stresses have to be taken into account in NPP environments, whereas due to relatively high operational temperatures of conventional power plants, they can be considered as mostly relieved and thus not necessary to be included in the structural integrity/safety considerations. Even though the LWRs are moderated and cooled by highly purified water, the water still contains chemicals, like boric acid, and impurities, like sulphate, chloride and peroxides that can cause corrosion. Suspended particulate material can also accelerate erosion processes, which can lead to wall thinning. In addition to this the reactor pressure vessels (RPVs) are exposed to degrading effect of the products from the fission reactions [23, 24]. The typical pressure boundary materials in the BWRs and the PWRs are carbon steels and stainless steels. For instance, the material of the steam tubes is typically austenitic stainless steel [24]. 3.3 Fatigue Fatigue is caused by periodic application of stresses by mechanical or thermal loading. The metal subjected to fluctuating stress will fail at stresses much lower 36 3. Degradation mechanisms concerning power plant components than those required to cause fracture in a single application of load [11]. The key parameters are the range of stress variation and the number of its occurrences [6]. The fatigue process can be roughly divided into four stages: cyclic hardening/softening, crack nucleation, crack propagation, and fracture [7]. Low-cycle fatigue, usually induced by mechanical and thermal loads is distinguished from high-cycle fatigue, mainly associated with vibration or high number of small thermal fluctuations. Although, there is no distinctive limit between these two types of fatigue, the traditional approach is to classify failures occurring above 10 000 cycles as high-cycle fatigue and those occurring below that value as low-cycle fatigue. An important distinction between low-cycle fatigue and high-cycle fatigue is that in high-cycle fatigue most of the fatigue life is spent in crack initiation, whereas in low-cycle fatigue most of the life is spent in crack propagation, because cracks are found to initiate within 3 to 10 % of the fatigue life [11]. An example of fatigue fracture of a structural component is presented in Figure 3.3-1. The cyclic nature of the fatigue crack growth through the component section can be recognised in the patterned fracture surface shown in the figure. Figure 3.3-1. Fatigue damage of a shaft component, from ref. [3]. 37 3. Degradation mechanisms concerning power plant components Fatigue failures occur in many different forms. Mere fluctuations in externally applied stresses or strains result in mechanical fatigue. Cyclic loads acting in association with high temperatures cause creep-fatigue. When the temperature of the cyclically loaded component also fluctuates, thermo-mechanical fatigue is induced. Recurring loads imposed in the presence of a chemically aggressive or embrittling environment give rise to corrosion-fatigue. The repeated application of loads in conjunction with rolling contact between materials produces rolling contact fatigue, while fretting fatigue occurs as a result of pulsating stresses along with oscillatory relative motion and frictional sliding between surfaces [25]. In NPP environments fatigue due to temperature fluctuation only, i.e. thermal fatigue, can also in certain cases be identified. All in all, the majority of damage/failures in power plant components can be attributed to one of the above fatigue processes. Such failures take place under the influence of cyclic loads with peak values being considerably lower than the allowed loads according to static fracture analyses. One characteristic of fatigue in metals is work hardening under reversed loading conditions. With continued cyclic loading, the rate of hardening progressively diminishes and a quasi-steady state of deformation, known as saturation, is reached. Once saturation occurs, the variation of the resolved shear stress with the resolved shear strain is not altered by further load cycles and the stress-strain hysteresis loop develops a stable configuration. Obviously work hardening behaviour is a metal specific phenomenon [25]. The lowering of the yield strength upon reloading that follows unloading, even if the reloading is in the same direction as the original loading, is called the Bauschinger effect. This phenomenon decreases with continued cycling. Unsymmetric cycles of stress between prescribed limits will cause progressive increase in the strain response or “ratchetting” in the direction of the mean stress. Typically, transient ratchetting is followed by stabilisation (zero ratchet strain) for low mean stresses, while a constant increase in the accumulated ratchet strain is observed at high mean stresses [25]. The damage process leading to the formation of fatigue cracks is initiated by locally limited plastic deformation in the micro scale range of individiual crystallites at low nominant net stresses. Depending on the lattice orientation and on the dislocation configuration, slip bands can be activated causing irriversible slip processes, see Figure 3.3-2. The repeated loading leads to micro-scale damage accumulation. After a saturation of energy accumulation has occurred, at which state no further slip processes can take place in the crystallite, small micro- 38 3. Degradation mechanisms concerning power plant components cracks form. With an increasing number of load cycles these micro-cracks start to grow and additional regions undergo the process of micro-scale damage. As the crack grows to macroscopic size the stress intensity at the crack tip usually increases so that the crack growth is accelerated until the remaining ligament can no longer bear the stress and complete failure occurs [4]. Three fundamental aspects related to the fatigue crack initiation are: the significance of the free material surface, the irreversibility of cyclic slip, and environmental effects on micro-crack initiation. The micro-cracks usually initiate at the free surface of the material. The restraint on cyclic slip is lower than at inside the material because of the free surface at one side of the surface material. Furthermore, micro-cracks initiate more easily in slip bands with slip displacements perpendicular to the material surface [26], which seems to be logical when looking at Figure 3.3-2. There are two reasons for irreversibility. One argument is that (cyclic) strain hardening occurs which implies that all dislocations do not return to their original position. Another important aspect is the interaction with the environment [27]. Figure 3.3-2. Geometry of slip at the material surface, from ref. [28]. The cyclic growth of a fatigue crack can be seen in the formation of striations, with one striation forming per load cycle, see Figure 3.3-3. The striations are supposed to be remainders of microplastic deformations at the crack tip, but the mechanism can be different for different materials. Because of micro-plasticity at the crack tip and the crack extension mechanism in a cycle, it should be expected that the profile of striations depends on the material type. Striations have not been observed in all materials, at least not equally clearly. Moreover, the visibility of striations also depends on the severity of load cycles. Striations have also shown that the crack front is a curved line and the crack tip is rounded [27]. 39 3. Degradation mechanisms concerning power plant components Figure 3.3-3. Correspondence between striations and load cycles during fatigue crack growth in an Al-alloy specimen, from ref. [27]. The fatigue life of a structural component under cyclic loading can be considered to consist of two phases, which are the crack initiation life followed by a crack growth period until failure. This can be represented in a block diagram, see Figure 3.3-4. Work hardening and other continuum type fatigue phenomena affect the component throughout the fatigue life. Figure 3.3-4. The different phases of the formation and growth of a fatigue crack, from ref. [27]. The fatigue limit is an important material property from an engineering point of view. It can be formally defined as a stress amplitude for which the fatigue life becomes infinite in view of the asymptotic character of the stress versus the number of load cycles (S-N) curve, see Figure 3.3-5. As fatigue tests must be terminated after a long testing time, it appears to be more logical to define a fatigue limit as the highest stress amplitude for which failure does not occur after 40 A more physically based definition of the fatigue limit could be defined as the threshold stress amplitude to take care of microcrack nucleation and subsequent growth to a macro-crack [27]. feedwater piping and nozzles. HWR RPVs. hot reheater piping. ductwork. safe-ends. PWRs. 4]: BWRs.3-5. calandria vessel. from ref. Components/locations in conventional power plants which have experienced fatigue degradation include [29]: headers. tubing. 41 . main steam piping and recirculation pump. steam generator. vessel shell. precipitator. RPV. such as nozzles.3. feed water piping. main steam piping. Components/locations in NPPs which have experienced fatigue degradation include [1. reactor coolant piping. LWRs in general. pressure tubes and calandria tubes. tubes and grid. drums and geometry discontinuities. [25]. downcomers. RPV internals. Figure 3. reducers. transitions and small fillet radii. Degradation mechanisms concerning power plant components high numbers of load cycles. Outlook of a typical stress versus the number of load cycles (S-N) fatigue testing curve. however. are electrochemical in nature and are accompanied with hydrogen production. 16].g. nozzles. the anodic dissolution of the metal can be stimulated. When mechanical stresses or strains are acting in addition to the corrosion impact. The types of corrosion include (but are not limited to) [30]: 42 . geometry discontinuities in general. 15]. 15]. protective oxide layers can rupture or hydrogen absorption can be promoted. main coolant pump. e. Metal ions dissolve in liquid electrolyte (anodic dissolution) and hydrogen is produced in the process. condensor tubes.4 Corrosion Corrosion is the degradation of a material by chemical or electrochemical reaction with its environment. casing. turbine. absorption or interstitial bonding) and can produce secondary damage. In the case of electrochemical processes. corrosion occurs in several steps. This is the process of material loss from the corroded component and of the creation of the corrosion products. and there are generally accepted categories of corrosion based on the appearance and electrochemical processes. either uniformly or locally. blades and shaft. Corrosion can manifest itself in a number of forms. hydrogen embrittlement [15. Degradation mechanisms concerning power plant components steam and water piping vessels valves.3. transitions and small fillet radii. casing and shaft impeller. The combined action of a corrosive environment and mechanical loading can cause cracking even when no material degradation would occur under either the chemical or mechanical conditions alone [4. The corrosion phenomenon is governed by the so called corrosion system which consists of the metallic material and the corrosive medium (the environment) with all the participating matter that can influence the chemical and physical behaviour and the corrosion parameters. Most of the technical corrosion processes. 3. reducers. Corrosion reduces the component wall thickness. The variety of chemical and physical variables of environments and materials leads to a large number of types and appearance of corrosion [4. The free hydrogen can metallurgically interact with the metal (by adsorption. since the related material loss is very small [4. by species such as chlorides. among others. temperature and flow rate. and environmental cracking.g. local corrosion and selective corrosion. pitting. for all hot water and steam pipes of unalloyed or low alloyed ferritic steel. An example of selective corrosion is the intergranular corrosion of sensitised austenitic stainless steels and nickel alloys. This process is not considered as an ageing mechanism. In case of inhomogeneities at the metal surface and/or local differences in the electrochemical reactivity of the environment. chlorides. Since the corrosion attack propagates along the grain boundaries. 15]. intergranular corrosion. Pitting is commonly caused by the breakdown of the passive film on a metal in local areas. Uniform corrosion rates vary with fluid oxygen content. the attack is concentrated on distinct material phases. galvanic (two-metal) corrosion. etc. In selective corrosion. Degradation mechanisms concerning power plant components uniform attack. oxygen. erosion corrosion. 43 . crevice corrosion. 16]. regions adjacent to the grain boundaries or specific alloying elements. the material damage proceeds from the surface into the bulk according to a grain disintegration process. Localised corrosion includes pitting. which occurs e. Crevice corrosion results from local environment conditions in the restricted region of a crevice being different and more aggressive than the bulk environment [15. the corrosion rate decreases considerably or may even stop. Environments with liquid that can cause intergranular corrosion in practice can include.3. crevice corrosion. The corrosion proceeds into the depth of the material without changing the overall geometry of the component. Uniform corrosion refers to a uniform attack over surfaces of the material and results in thinning of the material. 15]. sulphates and hydroxides [4. the creation of local cells is possible which results mostly in local corrosion attack. Corrosion without mechanical loading Corrosion without mechanical loading can be divided to uniform corrosion. When a protective Magnetite layer forms to the metal surface under uniform corrosion. Degradation mechanisms concerning power plant components Stress corrosion cracking (SCC) SCC is a localised non-ductile failure which occurs only under the combination of three factors: tensile stress. primary water stress corrosion cracking. Figure 3. IASCC. Generally. For some combinations of metals and corrosive environments even very low stresses are sufficient to cause SCC. However.3. the level of mechanical stresses is of minor importance compared with the reaction of hydrogen. are also identified [4. The phase of first macroscopic cracking is often preceded by a long incubation phase [4]. IGSCC. In this case. In case of hydrogen induced SCC. 44 . and there may be more than one process that causes SCC [32]. aggressive environment and susceptible material.4-1. [3]. many different mechanisms have been proposed to explain the synergistic interactions of stress and corrosion that occur at the crack tip. and irradiation assisted stress corrosion cracking. In NPP. An example of SCC is presented in Figure 3. The SCC failure mode can be intergranular. PWSCC. 17]. the additional effect beyond typical SCC exists in an interaction of hydrogen with the metal lattice.4-1. A prerequisite for hydrogen induced SCC is a small uniform or local corrosion attack of the material or a reaction between reaction layers and the electrolyte with subsequent hydrogen production [4]. TGSCC. Microscopic detail of SCC in a steel component. the mechanisms can be divided to anodic dissolution mechanisms and cathodic hydrogen induced cracking mechanisms. Actual SCC mechanisms are not clearly known yet [31]. from ref. or transgranular. e. 2. As with SCC. the fluid temperature. in the case of IASCC. making these boundaries vulnerable to corrosive attack. The corrosion resistance of unalloyed or low alloyed ferritic steels in water and steam depends on the formation of protective oxide layers. So PWSCC can occur without the presence of additional aggressive species. crack initiation. That is.g. the oxygen content in the fluid and the component geometry. When these layers are removed. TGSCC is caused by aggressive chemical species. cracks initiate and propagate at a slow rate (for instance. and 3. the mechanical action of a fluid on a metal surface) [1. also called corrosion/erosion mechanism. Flow accelerated corrosion (FAC) FAC. However. The result of FAC is an increased rate of attack on metal because of the relative movement between a corrosive environment and the metal surface. Degradation mechanisms concerning power plant components SCC is a delayed failure process.e. 10E-9 to 10E-6 m/s) until the stresses in the remaining ligament of metal exceed the fracture strength. the fluid velocity. steady state crack propagation. 6]. PWSCC is a form of IGSCC and is defined as intergranular cracking in primary water within specification limits. especially if coupled with oxygen and combined with high stresses. Sensitisation of unstabilised austenitic stainless steels is characterised by a precipitation of a network of chromium carbides with depletion of chromium at the grain boundaries.3. 15]. a normally non-susceptible material is rendered susceptible by exposure to neutron irradiation [4. 45 . IASCC refers to intergranular cracking of materials exposed to ionising radiation. An example of PWSCC is IGSCC of Inconel 600 alloy in primary water. 15]. aggressive environment and a susceptible material. sensitised austenitic stainless steels are susceptible to IGSCC in an oxidising environment. IGSCC is associated with a sensitised material. IASCC requires stress. Carbon and low-alloy steels are susceptible to FAC [4. The severity of erosion varies with the material type. The sequence of events involved in the SCC process is usually divided into three stages [33]: 1. final failure. refers to the combined action of corrosion and erosion (i. the corrosion attack on the metal surface can occur in the form of metal dissolution. The flow conditions shift the equilibrium between protection layer removal and formation which determines the material consumption. pressurizer. ductwork. Degradation mechanisms concerning power plant components The FAC process is an extension of the generalised carbon steel corrosion process in stagnant water. calandria vessel. nozzles and reactor coolant pump. PWRs. feedwater piping. condensor tubes. Microbiologically influenced corrosion (MIC) MIC is the accelerated corrosion of materials resulting from surface microbiological activity. vessel shell. control rod drive mechanisms. tubes and grid. main steam pipe. The major factors increasing the growth of MIC are temperature. water content and oxygen [15]. steam generator. MIC is characterised by the formation of microbial colonies and associated scale and debris on the surface of the metal. pressure. to a lesser extent. 4]: BWRs. turbine. RPV. The difference between generalised corrosion and FAC is the effect of water flow at the oxide-feedwater interface [1]. feedwater pipe and emergency diesel generators. tubing. pH. Components in NPPs which have experienced corrosion degradation in one or more of its various forms include [1. steam and water piping vessels valves. Components in conventional power plants which have experienced corrosion degradation in one or more of its various forms include [29]: headers. RPV internals. FAC takes place at low flow velocities and the corrosion rate is constant.3. casing and blades. stainless steels and nickel alloys. recirculation pipe and recirculation pumps. safe ends. downcomers. precipitator and drums. MIC affects carbon steels and. Any buried system or system using untreated water is susceptible to MIC. the carbon steel corrosion rate is low and decreases parabolically with time due to the formation of a protective oxide film at the surface. In stagnant water. 46 . depending on the total stress as well as the deposition of the eroded particles at locations where there is a low flow velocity (wastage) [4]. LWRs. Consequences of the corrosion/erosion mechanism are thinning of component walls until leakage or rupture occurs. HWR RPVs. [3]. softening processes. two processes occur in the micro-structure that affect the material properties [11. Degradation mechanisms concerning power plant components 3.3. 6].5-1. Of the above mentioned reactions. even below the elastic limit. including [11]: strain hardening. 47 . Basically. strain softening and precipitate overageing. Figure 3. the final failure may be limited either by deformation or by fracture [11]. An example of creep crack in a piping component is presented in Figure 3. Depending on the component.5-1. whereas the other reactions tend to increase the creep rate [11]. The propagation of creep is determined by several competing reactions. Creep reaches a significant level above 0. from ref.4 times the melting point of the material [4. recrystallisation. recovery. 4]: plastic deformation processes and changes in the microstructure. damage processes. cavitation and cracking. materials can slowly and continuously deform even under constant stress and eventually fail. strain hardening tends to decrease the creep rate. Due to thermal activation. Creep induced crack that has propagated through a pipe wall.5 Creep Creep is the deformation that occurs over a period of time in a material subjected to a stress. 5-2. secondary and tertiary. Components are usually designed to be operated only in the primary and secondary regimes. Crack growth due to creep. see Figure 3. With increasing exposure time and accumulation of micro-structural damage the ability for creep deformation decreases and thus ruptures occur in the part that very low ductility. then more unfavourable conditions have to be expected concerning the long range creep behaviour [4. If the thermally induced process (thermal ageing) leads to a decrease in strength. Nucleation of creep pores near the end of the second regime. since they are the consequence of thermally activated mechanisms [4]. An aggressive environment can have an influence on the creep behaviour and the degree of damage resulting from it. Degradation mechanisms concerning power plant components Both of these processes are time dependent and they show certain interactions. Coagulation of the creep pores to form micro-cracks. In metallic materials. primary. These regimes are controlled by different mechanisms. an internal oxidation process can occur and accelerate the creep process [4]. Creep rupture. The changes in the micro-structure can be caused by high temperatures alone. 11]. The initial creep strength depends on the initial strength of the material at the considered temperature. The time dependent creep damage processes occur according to the following steps [11. Deformation due to creep is usually divided into three regimes. in which the occurred damage is not irreversible.3. 48 . 4]: Creep damage in the primary and secondary regimes. In this regime spontaneous failure can occur when large sections of the material have already suffered from this process. from ref. [11]. secondary and tertiary. Helium production does 49 . primary. piping as well as superheater and reheater tubing. The effect of neutron irradiation on metals is mainly to increase the yield and ultimate strength and to reduce the fracture toughness. 6]. Helium diffuses easily under elevated temperatures and forms voids [4. Degradation mechanisms concerning power plant components Figure 3. In addition to the lattice defects caused by the interaction of fast neutrons with the material atoms. 4]: RPV internals of PWRs and in HWR RPVs calandria vessel. flux spectrum.3. Components in conventional power plants which have experienced creep degradation include [29]: superheater and reheater headers. 3. This damage mechanism occurs at neutron fluences above 10E+22 1/cm2 and temperatures usually above 400 C. respectively. fluence and irradiation temperature.5-2. Here r is rupture strain and tr is rupture time. Components in NPPs in which creep degradation has occurred include [1. helium can be produced as a result of nuclear reactions. neutron flux. The extent of the changes depends on the material.6 Irradiation embrittlement Materials exposed to neutron irradiation undergo changes in microstructure and properties. pressure tubes and calandria tubes. Schematic illustration of creep damage curve showing the three different creep regimes. hardness.3. For steels the neutron irradiation is considered to cause significant ageing degradation only for neutron fluences above 10E+17 1/cm2. too. ductility and toughness. 3. heat treatment and initial mechanical properties. thermal ageing is a damage process typically causing decrease in the material strength properties. however. Irradiation embrittlement is the major degradation mechanism associated with the ageing of RPVs of PWRs. One of the dominating factors is the chemical composition of the material and its thermo-mechanical pre-service treatment [4]. chemical composition. In addition to these phenomena. The dominating parameters responsible for thermal ageing processes are [4]: level of temperature. RPV internals. 4]: BWRs. 50 . control rod drive mechanisms and emergency diesel generators. PWRs. Fracture toughness is the property that governs the material resistance to fast fracture. RPV internals. It is generally accepted that the effect of irradiation is to shift this fracture toughness curve to higher temperatures. Under certain conditions.7 Thermal ageing Thermal ageing refers to gradual and progressive changes in the micro-structure and properties of a material exposed to an elevated temperature for an extended period of time [6]. HWR RPVs. The irradiation degradation sensitivity of the materials depends among other things on [4]: material type. Degradation mechanisms concerning power plant components not only lead to changes in material properties but is also combined with an increase in volume (swelling) [4]. pressure tubes and calandria tubes. the shape of the curve being only slightly affected with a small decrease of the upper shelf toughness in the ductile regime [4. material state (micro-structure) and exposure time. It is less important for BWR RPVs because of the lower fluence experienced in the BWR vessel environment [32]. calandria vessel. RPV supports. and increases with the temperature until the material becomes ductile. an increase in ductility can be observed. 6]. which for ferritic steels is small at low temperature where the material behaves in a brittle manner. In general. RPV. LWRs. the environment can have an effect to the material state as well. Components in NPPs that have experienced irradiation degradation include [1. Concrete components may be affected by several degradation mechanisms. The local chemical composition and the diffusion coefficient of the different phases and atoms play an important role [4]. water hammer. concrete degradation and differential settlement. environment effects. 3. Atmospheric carbon dioxide can cause concrete carbonation reducing the structural properties of the concrete. Severe differential settlement can cause concrete cracking and/or misalignment of equipment and can lead to high stresses within the structure [6]. Here environment effects concern processes or phenomena like wetting-drying and freeze-thaw cycling and chemical attack [6. The rate of the above mentioned reactions is strongly dependent on the level of temperature. wear. Such threshold temperatures can exist below which some of the processes are not being activated. Cast austenitic-ferritic stainless steels (duplex structures) and martensitic stainless steels are susceptible to thermal ageing in the normal operating temperature range of PWRs [34]. transformation of phases. Degradation mechanisms concerning power plant components The mechanisms occurring in the micro-structure during thermal ageing are of the following types [4]: precipitation of particles.9 Interaction of degradation mechanisms It has been noticed that some degradation mechanisms can provide a joint effect so that it is more severe than the arithmetically added result of their separate 51 . growth of precipitated phases and the grains of the matrix. 17].8 Other degradation mechanisms Other mechanisms or loading phenomena degrading power plant components include erosion-cavitation. 3. Wear has occurred in the steam generator tubes due to contact with anti-vibration bars.g.3. aggressive environment can increase porosity. Examples of NPP components that have experienced these degrading mechanisms or loading phenomena are presented in the following. e. loss of prestressing. dissolution of precipitates. Flowing water over concrete surfaces can remove significant amount of concrete. and permeability as well as reduce concrete strength. Typical components subject to a loss of prestressing are the prestressed cables in the primary containment. The generalised corrosion fatigue cracking mechanism involves the single or mutual occurrence of hydrogen induced cracking and/or anodic dissolution at the crack tip. In addition to crack initiation. two cases of interacting degradation mechanisms are covered here. The interaction of mechanical alternating stresses and corrosion attack usually leads to transgranular cracking with a low associated deformation. as good results concerning their assessment have already been achieved. No threshold conditions exist for corrosion fatigue with respect to the corrosion system and stress amplitude [4]. 52 . Several corrosion fatigue mechanisms have been proposed to explain the enhanced crack growth rates with varying degrees of success.3. the resulting degradation phenomenon is often quite complex. temperature. loading frequency and wave form. in corrosive environments the number of load cycles needed to crack initiation is considerably smaller than that needed in inert environment. However. those being corrosion fatigue and creep fatigue. sulphur content of steel. selective dissolution (dealloying). the corrosive environment also influences the ensuing cyclic crack growth rate. Corrosion fatigue In case of fatigue. electrochemical potential or oxyen content in water. Due to this and also to the fact that the interaction of various degradation mechanisms is still a quite recent research topic. In corrosion fatigue. Since the corrosion mechanism is mainly time dependent and the fatigue mechanism is controlled by the number of cycles. the physical mechanisms of most interaction phenomena are not clear. such as pH value. the crack growth rate may be influenced by the following features and conditions [4]: characteristics of the corrosive environment. mean load and load amplitude. brittle film rupture. corrosion tunnelling. Degradation mechanisms concerning power plant components effects. a complex interaction between these mechanisms occurs. According to current understanding there appears to be at least four possible mechanisms of anodic dissolution [7]: slip dissolution. In these cases. turbine. steam and water piping vessels valves. However. the damage caused by both fatigue and creep has to be taken into account [11]. creep begins to play a significant role in the degradation pro- 53 . Components in conventional power plants in which corrosion fatigue degradation can occur are those mentioned earlier experiencing fatigue. tubes. pressure. steam generator. casing. main coolant pump. both processes (dissolution and hydrogen evolution) can occur simultaneously at the crack tips over ranges of potential that have been measured or that are suspected to occur [7]. so that only one of them makes the major contribution to cracking and the others can be ignored. In case of such components. blades and shaft. changes in prevailing conditions during operation cause transient temperature gradients. and the flaws initiate near the specimen surface and propagate transgranularly. 4]: BWRs. deformation and brittle hydride. Components in NPPs having experienced corrosion fatigue degradation include [1. Creep fatigue In power plant components which operate at relatively high temperatures. The extent of the resulting fatigue damage depends on the nature and frequency of the transient. casing and shaft impeller. on thermal gradient in the component and on material properties. fatigue dominates the degradation process. Degradation mechanisms concerning power plant components Hydrogen embrittlement appears to be divided into the following five types of mechanisms [7]: decohesion. With lower frequencies and longer hold times. If these transients are repeated. A common approach is to assume that dissolution and hydrogen induced cracking processes are competitive. HWR RPVs. but which are also exposed to corrosive environment [29]. calandria vessel. pressure tubes and calandria tubes. RPV internals.3. The principal method of studying creep fatigue interactions has been to conduct strain controlled fatigue tests with and without a holding period (hold time) during some part/phase of the test. adsorption. With higher frequencies and shorter hold times. the differential thermal expansion during each transient results in a thermally induced cyclic stress. The balance between various phenomena is difficult to sort out. condensor tubes. long hold times. In case of ferritic steels and austenitic stainless steels. from ref. the lower the ductility.4 times the melting point of the involved materials. creep fatigue mainly relates to conventional power plants. 54 . Degradation mechanisms concerning power plant components cess. and the initiating flaws are of a mixed mode involving both fatigue cracking and creep cavitation.9-1. small strain ranges and low ductility favour creep dominated failures. In addition.9-1 in the following. the effect of tensile hold time on fatigue endurance of type 316 stainless steel is shown in Figure 3. intermediate stress ranges and high ductility favour creep fatigue interaction failures [11]. the lower is the creep fatigue endurance. With prolonged hold times and only seldom occurring load cycles. [43]. creep dominates the degradation process [11]. whereas short hold times. As an example.3. Due to operational temperature range in NPPs being in general below the creep range of the metallic pressure boundary components. where it can degrade those components mentioned earlier which have experienced fatigue. but which are also operated at temperatures that are near or exceed 0. Figure 3. Effect of tensile hold time on fatigue endurance of type 316 stainless steel. There is ample evidence to show that rupture ductility has a major influence on creep fatigue interaction. as well as reinforced concrete structures. Analysis methods based on other fields of science are mainly beyond the scope of this thesis. Deterministic analysis methods 4. This is divided to structural mechanics based analysis methods. In case of power plants. some relevant computational parameters and procedures are briefly presented as well. which issue is also dealt with further in this thesis. pipe T-joints and pressure vessels.2 Analysis methods based on structural mechanics Structural mechanics mainly deals with mathematical modelling of structures. Whereas in case of the fracture mechanics based analysis methods. and to some considerations concerning analysis methods based on other fields of science. the connection between structural mechanics and fracture mechanics is such that the results obtained with the former procedures provide input data for computations carried out with those of the latter. including fracture mechanics. however some relevant connecting topics are pointed out. such as slabs and walls. but instead some relevant approaches and analysis tools are considered.4. typical considered structures include straight pipe components. pipe bends. In this connection. 4. to compute stresses/forces and deformations they experience. The basic function of any structure is to carry loads and transmit forces. 55 . A description concerning the structural mechanics based analysis methods is provided with emphasis and an example on power plant components and conditions. Deterministic analysis methods 4. Often. no actual computation equations/procedures are presented.1 Introduction A brief overview of deterministic analysis methods is presented in the following. together with associated loading and supports. which often support the aforementioned components. the self-weights of process components. When including supports to a structural model. such as ORNL model [35] that covers both elastic-plastic material behaviour and thermal creep. more advanced material models exist. being either water or steam. the model can have one. pressure and flow rate. Depending on the capability and scope of the structural model. elastic ideally-plastic. and elastic non-linear plastic. for instance linearelastic. should be considered as dead loads as they are not movable. Live or imposed loads are movable or actually moving loads. Also. with some notable examples of these being dead loads. these mainly include temperature. all caused by the process fluid. some idealisations are performed. Whereas the NPP RPVs often have a support skirt. such as start-up and shutdown. mechanical loads can be included as pressures against surfaces. so as to allow thermal deformations to occur during load transients without causing excessive restrained thermal stresses. loads on structures fall into several categories. Generally.4. Thermal loads are typically set to act on component surfaces. and as point specific forces. a steel structure that is welded around them at some height in the upper half. with typical modelling options being then to use isotropically or kinematically evolving yield surface. Also translations and rotations can be used as loads. as well as shape and supports of the considered structure. live loads and accident loads. When including loads to a structural model. the support conditions of power plant piping systems are often relatively flexible. some idealisa- 56 . line loads distributed along a surface line. however during anticipated/budgeted controlled load transients. As power plant components are mainly exposed to mechanical and thermal loads. Deterministic analysis methods The structural models describe the geometry and material properties of the considered structure. loads can be given as time dependent or independent. The structures carrying these loads form in power plants the pressure boundary. For instance. Further. they vary as well. Depending on the strived accuracy. the stress/strain hardening behaviour can be history dependent. At operational conditions these load parameters have most of the time constant values. only they are considered in the following. Depending on the approach. or their combination. In power plant environments. even though at time intervals component replacements may have to be carried out. The stiffness of support conditions can vary from rigid to very flexible. with the aid of which the RPVs firmly rest on a reinforced concrete structure. two or three dimensional geometry. such as those mentioned above. Also. There are also several possible material models. There is a large variety of possibilities how a structure can be supported. In power plant environments the dead loads typically cover self-weights of the load bearing concrete and steel structures. such as finite difference method. some pressure vessels. Typical support modelling options include limiting at one or several locations translations and/or rotations in/around one. In the modelling process. RPV. Deterministic analysis methods tions are performed. In case of power plant components. and in time dependent analyses small enough time increments. and points. beam models. Such FEM codes also allow using elements of differing types in the same model. case specifically combined methods can be used. In the simple case. the model can be a set of analytical equations. These are typically prepared using finite element method (FEM) based analysis codes. as well as provide tools for sub-modelling. solid three dimensional models. Other numerical techniques exist as well. In addition. pipe bends. to describe behaviour of simple auxiliary support structures. dimensioning and stress distribution analyses of piping systems. FEM solutions supplemented with those computed using analytical equations. Support conditions can be time dependent or independent. truss models. In the following are structural mechanics modelling approaches used for analysing power plant components/systems. dense enough computation grids are to be used. two or three directions to zero. Depending on the scope of the structural model. and some application examples: analytical equations.4. There are numerous advanced commonly used general purpose FEM codes. When carrying out a structural mechanics based analysis. To achieve sufficient accuracy using numerical methods. 57 . numerical structural models are often used. lines. Then convenient mathematical methods are sought for solving the unknown model parameters. called elements.e. i. The basic principle concerning this methodology is to provide approximate solutions for the governing set of partial differential equations in a finite number of nodes using suitable interpolation functions. shell models. piping T-joints. the considered geometry is divided to a finite number of sub-regions. to which in turn the solution values are interpolated from the integration points. dimensioning of single components. within and on the edges of which are nodes. support conditions can be set on surfaces. computing stresses caused by inner pressure for straight pipe components. a model for the considered structure must be prepared first. with which locally more dense element meshes can be realised. single or fully pre-determined values are used for all input data parameters. Such distributions are formed by fitting a suitable distribution function to applicable experimental or simulated data. combined sequential or coupled heat transfer and stress/strain analyses. Deterministic analysis methods The types of analyses often performed for power plant components/systems include: linear or non-linear time dependent or independent static stress/strain analyses. The results from deterministic analyses provide no data concerning their uncertainty/reliability. dynamic eigenvalue and/or eigenfrequency analyses. As for the dynamic analyses. are also considered as distributed in the probabilistic analyses applying structural reliability methods. such as yield strength.4. 58 . to see whether the component in question meets the associated structural strength requirements. Such results as nominal stresses. local/global maximum/minimum values are often sought. because they in turn cause thermal stresses. such as yield strength or frequency of load cycle occurrence. From the stress/strain analysis results.g. in inner corner with relatively small radius or where wall thickness changes abruptly. local stresses and stress fluctuations.g. because they form the corresponding stress/strain cycles needed in fatigue analyses often performed for power plant components. The maximum stress values are compared to corresponding material strength. In this approach. e. their results are needed e. to assess the structural response to assumed earthquake loads. and such can take place e. heat transfer analyses. from structural mechanics analyses. Of particular interest of the time dependent stress/strain results are the local stress/strain fluctuations. Whereas in case of thermal results. Most analyses of mechanics are deterministic. whereas those from structural reliability analyses are probabilities or probability distributions. As for the obtained results. The input data parameters often having markedly distributed characteristics in reality. typically they are time dependent or independent stress/strain fields and temperature fields.g. are often needed as input data in fracture mechanics based analyses. in addition to maximum and minimum values often also temporary temperature gradients through wall are sought. as well as to those of the header region. The material of this component is ferriticmartensitic steel X20CrMoV121. The former element type contains one integration point. The overall element mesh of an Abaqus FEM model of a section of the outlet header and the tubes connecting to it is shown in Figure 4. The outlet header is supported by the tubes. in the following is presented a numerical lifetime analysis for an outlet header in a conventional power plant. mainly load transient start-up and pressure variation during operation are considered. and appropriate deformation boundary conditions are given to the cut surfaces. namely that concerning Row5 tube. 59 . An example of the stress result distributions for the most stressed header/tube joint. is shown in Figure 4. For more details concerning the Abaqus code. isoparametric four node linear tetrahedron elements are used. The analysis data are from the article [38].2-1. which are cut on certain length in the model. whereas those used for the connecting tubes are isoparametric eight node linear hexahedron “brick” elements.4.2-2 below for a selected time instant from the duration of load transient start-up [38]. [36. For both the heat transfer and static analyses of the outlet header region of the model. As for loads. see refs. 37]. whereas the latter contains four. whereas its wall thickness and total length are 36 mm and 18 272 mm. The outer diameter of the outlet header is 711 mm. Deterministic analysis methods As an example concerning structural mechanics analyses.6 MPa. respectively. During the former of these load events both temperature and pressure rise in a controlled process within approximately three hours from room temperature and one bar to operational values of 535 °C and 2. from ref. from ref. Row4.4. Distribution of axial stresses [MPa] in the Row 5 joint middle section of the FEM model of the outlet header and the tubes in the middle of load case start-up. Figure 4. The number of elements/nodes is approximately 150 000/46 000. [38].2-2. [38]. Row3. Row2.2-1. Deterministic analysis methods naming of tube rows. The overall element mesh of the FEM model of the outlet header section and tubes connecting to it. 60 . Row5 Figure 4. starting from left and proceeding clockwise: Row1. fatigue. Mode I.4. etc. due to corrosion. fracture mechanics assessments are required throughout the lifetime of a structure or component.3 Fracture mechanics based analysis methods Fracture mechanics concerns the design and analysis of structures which contain cracks or flaws. For other materials. welding flaws. At the top level. A cracked component can be loaded with any of these modes. Materials with relatively low fracture resistance that fall below their so called plastic collapse strength can be analysed on the basis of elastic concept through the use of LEFM. assuming linear-elastic material behaviour. 61 . Mode III refers to out-of-plane shear. as Figure 4. due to cracked inclusions. not just in the beginning of the lifetime. all materials contain flaws. tends to open the crack. fracture mechanics is divided into linear-elastic fracture mechanics (LEFM) and elastic-plastic fracture mechanics (EPFM). Thus fracture mechanics is involved in any detailed design or safety assessment of a load bearing structure.g. the use of EPFM is often necessary. For certain cracked component geometries subjected to external forces. fatigue. which are either microscopic. There are three types of loading that a crack can experience. debonded fibers etc. On some size scale. Westergaard [39] and Irwin [40] were among the first to publish such solutions. where the principal load is applied normal to the crack plane.3-1 illustrates. or macroscopic.. Mode II corresponds to in-plane shear and tends to slide the crack faces with respect to each other. Some theoretical background of fracture mechanics is presented in the following. Deterministic analysis methods 4. or with a combination of two or three modes. As cracks can grow during service due to e. it is possible to derive closed form expressions for stresses. from ref.e. The stress fields ahead of a crack tip in an isotropic linear-elastic material can be written with polar coordinates as: lim r 0 I ij KI 2 r K II 2 r K III 2 r f ij I (4.e. Of the fracture mechanics parameters.4. and fij is a dimensionless function of .3-1c) where ij is stress tensor. Usually K is provided with a subscript to denote the mode of loading. i. [30]. the individual contributions to a stress component are additive [30]: 62 . KII or KIII. r and are the radial and angular coordinates of the polar system. i. K. the stress intensity factor. The three modes of loading that can be applied to a crack.3-1. KI.3-1a) lim r 0 II ij f ij II (4. respectively. completely characterises the crack tip conditions in a linear-elastic material. when more than one loading mode is present.3-1b) lim r 0 III ij f ij III (4. Deterministic analysis methods Mode I Mode II Mode III Figure 4. In a mixed mode problem. An often used fracture parameter in EPFM is called J-integral.3-2) If one assumes that the material fails locally at some critical combination of stress and strain. Deterministic analysis methods total ij I ij II ij III ij (4. The J-integral is defined as [30]: J wdy Ti ui ds x (4. which represents the material fracture resistance. ds is a length increment along the contour .3-3. Thus KIC is a measure of material fracture toughness. whereas x and y are coordinate variables as shown in Figure 4.3-3) where a [mm] is the crack depth (dimension of the crack in the principal direction of the crack growth).3-5) where w [N/mm2] is the strain density. as is shown in Figure 4. and it can be derived from a conservation of energy criterion. J is simply the strain energy rate. Ti [N/mm2] are the components of the traction vector. ui [mm] are the displacement vector components.3-5) is defined as [30]: 63 . In general. only the factor B needs to be derived. Failure occurs when KI = KIC.] is a factor that accounts for the geometry.4. KIC. rapidly occurring brittle fracture is associated with fracture toughness parameter. For many geometries B can be found in handbooks. KIC. The strain energy density w in equation (4. and B = B(a) [ . In EPFM fracture occurs when [30]: J JR (4. S [N/mm2] is the remote stress resulting from the applied load. EPFM is applied. the stress intensity factor (SIF) has the following form [30]: K BS a (4. When fracture is accompanied with considerable plastic deformation. Typically. For any crack in any practical problem.3-2.3-4) where JR [J/mm2] is the fracture energy. then it follows that fracture must occur at critical stress intensity. Consider an arbitrary counter-clockwise path around the tip of the crack. e. Arbitrary contour around the crack tip.4. i.3-3. Deterministic analysis methods ij w 0 ij d ij (4. CTOD.3-6) where ij [N/mm2] and ij [mm/mm] are the stress and strain tensors. Unlike in the case of linear-elastic materials. are illustrated in Figure 4. The opening at the crack tip. stabily and more slowly occurring ductile fracture is associated with fracture toughness parameter JR. Another applicable parameter used in EPFM is crack tip opening displacement (CTOD). the crack tip may experience plastic deformation by blunting in strain hardening materials. Typically. 64 . Figure 4. is a measure of fracture toughness.3-2. denoted here by . respectively. Crack tip blunting and CTOD. 3-8) 65 . Deterministic analysis methods Figure 4. E [N/mm2] is the Young’s modulus and m is a dimensionless constant that is approximately 1.0 for plane stress and 2. Blunting of the crack tip and CTOD.4. ij (4. which is one of most commonly used EPFM models. According to this model. from ref. The result of their work is called the HRR singularity.0 for plane strain. There exist a number of alternative geometric definitions for CTOD. They each assumed a power law relationship between plastic strain and stress.3-7) where y [N/mm2] is the yield strength.3-3. the actual stress and strain distributions are obtained by applying the appropriate boundary conditions as follows: 1 n 1 ij 0 EJ 2 0 Inr ~ n. Hutchinson [41] as well as Rice and Rosengren [42] independently showed that J-integral characterises crack tip conditions in a non-linear-elastic material. [30]. A general form for CTOD can be expressed as [30]: CTOD K I2 m yE (4. these distributed characteristics can be taken into account. where the stress varies as 1/ r. weight/influence function based analysis codes. is dimensionless constant. J-integral completely describes the conditions within the plastic zone. is not predictable. For instance. whereas ~ij and ~ij are dimensionless functions of n and . Most fracture mechanics analyses are deterministic. The J-integral defines the amplitude of the HRR singularity. K. A structure in small scale yielding has two singularity dominated zones: one in the elastic region. i. however.3-9) where 0 is reference stress (most often yield strength). fracture toughness data in the ductile to brittle transition region are widely scattered. The results from deterministic analyses provide no data concerning 66 . r is polar coordinate with origin at the crack tip. These parameters depend also on the stress state. 8 and 12. including both deterministic and probabilistic approaches. Deterministic analysis methods 0 ij E EJ 2 0 Inr n n 1 ~ n. just as SIF characterises the amplitude of the linear elasticity.g. where stress varies as r 1 n 1 . and one in the plastic zone. ij (4. and some application examples: analytical equations. e. J-integral and CTOD values for static crack postulates in single material components with regular geometry and load distribution shape. With structural reliability analyses. see Chapters 7. In the following are analysis approaches used for fracture mechanics assessments of power plant components/systems. general purpose FEM codes.e. plane stress or plane strain. Thus. n is strain hardening exponent and In is an integration constant that depends on n. K and J-integral values for static and growing crack postulates in single material components with regular geometry and arbitrary load distribution shape. K. J-integral and CTOD values for static and growing crack postulates in single and multi-material components with arbitrary geometry and load distribution shape. Fracture mechanics based structural integrity assessment procedures are covered in more detail further in this thesis. Much of what happens in the real world. a single value of fracture toughness is used to estimate failure stress or critical crack size.4. The latter zone often persists long after the linear-elastic singularity zone has been taken over by crack tip plasticity. Namely power plant environments not only include a relatively large number of different materials. Deterministic analysis methods their uncertainty/reliability. but also provide a range of challenging mechanical loading.g. neutron flux. operational temperature range as well as how the inner surfaces of components respond to various kinds of chemical attacks. A more recent development is numerical fluid-structure interaction analyses. For instance. which in turn increases corrosion rate as fresh metal becomes exposed to corrosive conditions. both the yield strength and ultimate strength decrease as a function of temperature. relatively thin protective oxide layers form on the inner surface of power plant components. e. whereas those from structural reliability analyses are probabilities or probability distributions. The extent of these changes depends on the material. Results from these analyses include fluid temperature and pressure distributions as well as distributions for the heat transfer coefficient between the fluid and component inner surface. for instance ferritic and austenitic steels of several types in the pressure boundary.g. Thus it is necessary to clarify how the material properties vary as a function of e. the layers can locally rupture as depending on changes in local chemical conditions. The clarification of these material behaviour/property issues requires experimental research. which was mentioned earlier in Section 3. the extent of which is a material specific issue. Typically. including fracture mechanics.4 Analysis methods based on other fields of science A field of science necessary for any kind of structural analyses concerning power plant systems/components is that associated with material properties/behaviour.7. The results from the fluid mechanics analyses produce input data to structural mechanics based computations. such as turbulent mixing of water flows of differing temperatures in piping T-joints. thermal and chemical conditions. caused by sudden closing of a valve. However. Particle physics is needed when assessing the characteristics of the neutron irradiation that causes changes in microstructure and properties of the materials exposed to it. requires material science associated data as well.4. When an abrupt increase in flow rate occurs. the pressure impulse can be of such magnitude that it may cause temporarily notable deformations to the component containing fluid. Fluid mechanics is needed when simulating the flow patterns in more complex flow conditions. 67 . The assessment of thermal ageing. 4. these phenomena need to be taken into account in the structural integrity analyses. In such cases. 68 . Thus. The effect of neutron irradiation on metals mainly increases the yield and ultimate strength and reduces the fracture toughness.4. In practise. as was explained in Section 3. fluence and irradiation temperature.6. irradiation effects concern only NPP RPVs. it is obvious that these effects need to be taken into account in structural integrity analyses. Deterministic analysis methods flux spectrum. though. This problem is dealt with in the statistical estimation theory. Determining probabilities from scarce and incomplete input data is another problem. Refined Monte Carlo methods provide rather sophisticated procedures. capacity and affordability of available computers keeps improving swiftly. Probabilistic analysis methods 5. a probability 69 . There are two basic philosophical schools in modern probability theory. Monte Carlo simulations become easier and more economical to perform. one based on a frequency interpretation. As the efficiency. The first question that arises is how to handle the data basis in order to end up with input distributions reflecting all the information available. respectively. approximation methods for the calculation of variable probabilities are of great interest. These are also known as the objective and the subjective interpretation. According to frequency interpretation. which is the often remarkable amount of computation work required in order to achieve sufficient accuracy for the results. and one based on a Bayesian interpretation or degree of belief. Due to Monte Carlo calculations being often rather costly and laborious to perform. the probability of interest can be computed in a straightforward manner [76]. This procedure has one major drawback. such as stratified sampling and importance sampling. Once an appropriate statistical model has been set up and the parameters of this model have been determined from the input data. which can help to overcome these difficulties [76]. Probabilistic analysis methods 5.5.1 Introduction There exists a number of methods to determine the probabilities of the variable(s) of interest of a system. This limiting or even prohibiting feature is emphasised especially in the situations where a very low probability is to be determined. Monte Carlo simulation is a simple and most accurate procedure to analyse probability distributions. Examples of these are first-order second-moment methods (FORMs) and second-order reliability methods (SORMs). This probability. R. Bayesian probabilities are also called subjective [45]. it is important to consider the future deterioration rate from all potential mechanisms. and expresses the probability of a component not failing [46]. Pf. The common engineering quantities they represent include: component diameter.5.2 Failure probability and reliability Failure probability. some basic definitions are presented first.2-1) A rigorous structural reliability assessment involves taking into account all sources of uncertainty that may affect the failure of the component or system. The structural reliability analysis procedure is outlined by the following steps [45]: 70 . In terms of failure probability. 5. The rate of degradation may increase in time due to interaction between mechanisms (e. When assessing the probability of failure. wall thickness. maximum operating temperature. or accidental damage that cannot be easily predicted should be assumed to occur at a constant average rate [3]. corrosion and fatigue). Factors. for example. yield strength. depends on the amount of information available.g. [45]. etc. misuse. such as overload. or degree of belief. These terms are referred to as basic variables. maximum operating pressure. a probability is considered as a subjective degree of belief. In Bayesian philosophy. Reliability. and also the uncertainties that arise from lack of knowledge and idealised modelling. is closely related to the failure probability. typically one year. This clearly involves taking into account all fundamental quantities entering the problem. in the chances of a particular event occurring. corrosion rate. is often defined as the mean frequency with which the specified failure event would be expected to occur in a given period of operation. material and contents density. The relation between reliability R and probability of failure Pf is expressed as [47]: R 1 Pf (5. Probabilistic analysis methods is an objective property of some event. it is the expected probability of failure that is reflected. Before going on to describe in more detail the characteristics of various probabilistic methods. basic variables and parameters. model the basic variables and parameters in the failure functions and specify their probability distributions. Epistemic uncertainty can further be divided into subcategories as shown in Figure 5. Aleatoric uncertainty is quantified through the collection and analysis of data. Aleatoric uncertainty refers to the natural randomness associated with an uncertain quantity. and is often termed Type I uncertainty in reliability analysis. 4) Calculate the probability of failure or reliability for each failure event. and is often termed Type II uncertainty in reliability analysis. i. 5) Consider the sensitivity of the reliability results to the input. 3) Identify the sources of uncertainty influencing the failure of the events. and the probabilistic modelling may be interpreted in the relative frequency sense. 6) Assess whether the evaluated reliability is sufficient by comparison with a target.5. 71 . 5. and combine these probabilities where necessary to evaluate the failure probability or reliability of the structural system. Epistemic uncertainty reflects the lack of knowledge or information about a quantity. Probabilistic analysis methods 1) Identify all significant modes of failure of the structure or operation under consideration. 2) Formulate a failure criterion or failure function for each failure event.3 Uncertainty and sensitivity The sources of uncertainty that are relevant to structural reliability analysis can be primarily classified into two categories: 1) aleatoric uncertainties.3-1. being associated with understanding or knowledge [45].e. The observed data may be fitted by theoretical distributions. It can be further divided into subcategories as shown in Figure 5. 2) epistemic uncertainties. being associated with physical uncertainty or randomness and.3-1. and define failure events. the test data should cover uniformly the full range of the applicability of the model. Model uncertainty arises because many of the engineering models used to describe natural phenomena. Tests are often expensive. to analyse stresses and to predict failure of components are imperfect. Models are often based on idealised assumptions. This occurs when the parameters of a distribution are determined from a limited set of data. There are difficulties in quantifying model uncertainty adequately. Two primary types of uncertainty and associated subcategories. Model uncertainty is often assessed on the basis of experimental results. from ref. Probabilistic analysis methods Figure 5. etc. Statistical uncertainty can be considered to arise from [45]: Parameter uncertainty. Ideally. they may be based on empirical fits to test results or observed behaviour. [45]. and variables of lesser importance may be omitted for reasons of efficiency (or ignorance) [45]. and are affected by scale. load rates. but at any level of modelling there are errors relative to the unknown reality. and the data need to be carefully screened to ensure that they are consistent. boundary effects. measurement errors. Tests themselves are idealisations or simplifications of reality. 72 . The smaller the data set the larger the parameter uncertainty. The errors in the model may be known in relation to more advanced models.5. but this too has a number of limitations.3-1. Response function can be a complete FEM model as well. which parameters are insignificant and can be eliminated from the final model. available data are usually limited or non-existent. and then assessing the influences or relative importance of each input parameter on the output variable [48]. and which (group of) parameters interact with each other [48]. and a certain value of it with a small letter. Whereas the cumulative distribution function. A common way to define a random variable in the literature is to denote it with a capital letter. strain or crack growth in a structure. Development of mathematical models for structural analysis purposes consists of several logical steps. Xi. There is a number of reasons for conducting a SA. The distribution of a continuous random variable is controlled by probability density function. corresponds to a definite integral of the probability density function over some selected region. In practise. one of which should be the determination of parameters which are most influential on model output. X. including the needs to determine which input parameters contribute most to output variability. Z [47]. A sensitivity analysis (SA) of the input parameters can serve as a guide to any further use of the model [48]. together with their probability density functions and output variable(s). This uncertainty arises from the choice of a theoretical distribution fitted to empirical data. It is a particular problem when deriving extreme value distributions. 5. It may not be possible to distinguish from each other the two types of statistical uncertainty in practice.e. fx(x). This function describes a structural or mechanical response. is expressed as a function of the random variable. i.4 Response function and performance function A random function can often be described by a number of parameters. the parameters are determined from an infinitely large sample. Z. to quantify the uncertainty in the estimates [52]. SA is conducted by defining the model and its output parameters. Statistical tools can be used to measure the quality of the estimated parameters. however. as follows: 73 . Probabilistic analysis methods Distribution type uncertainty. such as stress. In general. The parameter used in the description of a physical problem is called a response function. since with limited data both the parameters and distribution type may be uncertain. The response function.5. In the ideal situation. x. Fx(x). so a combination of judgement and experience must often be used. 5. X n (5. which issue is covered in more detail further in this chapter.. such as normal distribution. The probability of failure can now be defined mathematically as: Pf Pg 0 g 0 f x x dx (5. this is not generally feasible in probabilistic analysis because of the number of dimensions of the problem. and with dense enough computation grid it can potentially give accurate enough results [45]. and because the area of interest is usually in the tails of the distributions. X 3 . normal or Gaussian distribution. In practice. The region g 0 corresponds to failure.4-3) In some cases. Some of these functions are symmetric... Probabilistic analysis methods Z Z X 1 . X 2 .4-2) where Z0 is the limiting value of Z.5 Commonly applied distribution functions The number of continuous reliability distributions in available handbooks which empirically describe the scatter of material test data is considerable. g. It is defined by: g Z0 Z Z0 Z X 1 .g. this equation is occasionally used in numerical form. 5. X n 0 (5. e. Nevertheless.. Using these functions. probabilistic distributions have been developed e. log-normal distribution. equation (5. Weibull distribution... being one dimension for each basic variable. whereas g > 0 is the safe domain [51]. X 3 . The most commonly applied distribution functions include [46]: exponential distribution. initial flaw size and frequency of occurrence of load cycles. There are procedures to overcome the difficulties encountered in calculating the failure probability.. X 2 . In principle. etc. gamma distribution and Gumbel distribution.4-3) can be integrated analytically. for fracture toughness.g. the probability of failure or reliability can be evaluated using numerical integration.4-1) The performance or limit state function. trapezoidal rule or Simpson’s rule.. whereas others are asymmetric. such as exponential distribution. 74 . gives the failure boundary or critical conditions of a physical problem. FORM methods involve estimating the failure probability by linearising the failure surface at the closest point to the origin in standard normal space. from the identity [45]: u x Fx x u (5. i. A space defined by independent standard normal variables is termed U-space.6-1) (5. Even though the failure surface in standard normal space is rarely planar the curvature at the point closest to the origin is usually so small that the first-order linearisation is valid for most purposes. and basic variable space is termed X-space. The basic variable transformation and the first-order reliability estimate are illustrated in Figure 5. using e.1 First-order second moment method (FORM) To overcome the invariance problem with the failure function. The transformation of independent variables can be undertaken from the cumulative distribution function.6-2) Fx 1 where u is the standard normal distribution function.e. Various techniques and software tools for the transformation of the non-independent variables are available.6 Review of probabilistic analysis procedures 5.6. Iteration is usually necessary to determine the closest point to the origin. The space outside the tangent hyperplane to the failure surface at the closest point to the origin corresponds approximately to the probability of failure.g. or Uspace. it is necessary to transform the basic variables into independent standard normal variables. Probabilistic analysis methods 5. Taylor series of the normalised random variables. 75 . Fx(x). and a number of iterative and optimisation techniques are available.5.6-1 with two basic variables. 5. Probabilistic analysis methods Figure 5.6-1. Illustration of transformation from basic variable space (left) to U-space (right) and first-order reliability estimate, from ref. [45]. The closest point from the failure surface to the origin in U-space is the most probable point (MPP) or design point. The distance between these two points is equal to the first-order reliability index, , denoted as beta in Figure 5.6-1 [45]. This is also referred to as the geometrical reliability index or Hasofer-Lind reliability index [53]. After the first-order transformation, the level curves of the joint density functions become origin centered circles. The limit state surface is a hyperplane in the space of the transferred random variables, meaning the U-space. An approximation of the probability of failure is then given by: Pf 5.6.2 Second-order reliability method (SORM) (5.6-3) Second-order methods improve the accuracy of the first-order probability estimates by including curvature information at the point , and by approximating the failure surface with a quadratic surface. First is necessary to iteratively identify the point . The difference between the first- and second-order estimates 76 5. Probabilistic analysis methods gives an indication of the curvature of the failure surface. If there is a significant difference it suggests that perhaps Monte Carlo methods should be used to confirm the probability of failure estimate. For most practical reliability applications, there is usually a small difference between FORM and SORM estimates [45], only. In many engineering structural problems, the SORM method has satisfactory accuracy, but in some cases, it may lead to erroneous results. A reason for this is that the quadratic approximations may appear to be insufficient when the failure surface is oscillatory in nature or too irregular in the vicinity of the point of maximum likelihood. Moreover, a significant disadvantage of the FORM/SORM methods is that no estimation of the error made is available, and so it is very difficult to validate results [54]. Various methods have been developed to overcome these problems, see e.g. ref. [55]. A general but complicated expression for the failure probability of quadratic forms is derived in ref. [56]. A more convenient but approximate expression has been derived by Breitung as [57]: n Pf i 1 1 0.5 i (5.6-4) where i denotes the principal curvatures of the limit state at the minimum distance point. 5.6.3 Fast probability integration (FPI) In FORM and SORM, the basic random variables are transformed into independent standard normal variables by exact mapping. In some cases, this is not easy to apply and in addition, the original limit state function may become distorted by this transformation. The FPI approach has been developed to overcome these drawbacks. With the FPI, the original distributions are approximated by standard normal variables, instead of transforming them into standard normal. Thus the FPI methods used for linearisation and normalisation are numerical procedures for solving multidimensional integral equations concerning general failure probability or reliability analyses [52]. For instance, the FPI method published by Wu and Wirsching [58] is a SORM, employing a three-parameter normal tail approximation. In this method, the equivalent normals are constructed using a least squares approach. A second 77 5. Probabilistic analysis methods order approximation of the failure surface is used and then transformed into linear one in order to reduce the computational effort. It has been demonstrated, see ref. [59], that the FPI method in the ref. [58] is efficient and provides accurate failure probability estimates. 5.6.4 Mean value methods Mean value method (MV) When the response function is implicitly defined by a computer code, the general failure probability or reliability methods are difficult to apply. The MV method [60] is a first-order method that is suitable to be used under these circumstances. This method is based on the assumption that a first-order Taylor expansion of the response function exists. By omitting the higher order terms, the response function is expressed at the mean values as: n Z X Z i 1 Z Xi Xi n i a0 i 1 ai X i (5.6-5) where parameters a0 and ai are coefficients, and the derivatives are evaluated at the mean value points i. The coefficients can in general be computed by numerical differentiation and the minimum number of performance function evaluations (computer runs) is (n+1), where n is the number of random variables [52]. Based on this linear approximation the cumulative distribution function for response function can be obtained directly, since the distributions for the random variables Xi are fully defined and the response function is explicit [61]. Advanced mean value method (AMV) For non-linear performance functions, the MV solution is often not accurate enough. The AMV method [60, 62] takes the MV solution one step further and is capable of establishing full probability distributions even when the response function is not monotonic, which can result in truncated distributions. In the general failure probability or reliability methods, an iterative optimisation algorithm locates the MPP and then the response function is approximated by a Taylor expansion in the vicinity of this point. In the AMV method, the mean value approximation is used to locate the MPP and then, the corresponding failure 78 5. Probabilistic analysis methods probability is calculated. Then, the analysis is performed again at this point in order to obtain the correct response value. Advanced mean value method with iteration In the AMV with iteration, the accuracy of the MPP location, as obtained with the mean value approximation, is improved with an iteration procedure. A new Taylor series expansion is then performed at the MPP obtained from the mean value solution. Another MPP may then be obtained by using the first-order failure probability or reliability method, after which the analysis is performed again at this point. The analysis process is continued in this iterative manner until sufficient accuracy is achieved [52]. 5.6.5 Monte Carlo simulation (MCS) Often when a mathematical formulation for probabilistic structural/fracture mechanics analysis can be defined, it is not possible to solve analytically. In these cases, a probabilistic solution may be obtained through Monte Carlo simulation (MCS). This simulation method is simply a repeated process of generating deterministic solutions to a given problem. Each solution corresponds to a set of deterministic values of the underlying random variables. The main element of a MCS procedure is the generation of random numbers from a specified distribution. There exist systematic and efficient methods for generating such random numbers from several common probability distributions [63]. Because a MCS solution generally requires a large number of repetitions, in particular for problems involving very rare events, its application to complex problems can be computationally costly. Hence, the MCS should be used with some caution. Often, MCS solutions may be the only means for checking or validating an approximate probability computation method [63]. Plain MCS offers a direct method for estimating failure probability. In essence, the technique involves sampling a set of values of the basic variables at random from the probability density function, and evaluating the failure function for the values to see if failure occurs. By generating a large number of samples, or trials, the probability density function is simulated, and the ratio of the number of trials leading to failure to the total number of trials tends to the exact probability of failure [45]. In order to evaluate the failure probability corresponding to a known performance function, g(Xi), the MCS would consist of the following steps [64]: 79 5. Probabilistic analysis methods 1. Given the predefined probabilistic density functions (PDFs) of the random variables in the performance function, generate a single value of each variable. 2. Assess the performance function: if g(Xi) < 0, a system or component failure takes place. 3. Repeat steps 1 and 2 for N times. 4. Estimate the probability of failure by Pf = Nf /N, where Nf is the number of failures. In order to evaluate the failure probability corresponding to an unknown performance function, the MCS method would consist of the following steps [64]: 1. Given the predefined PDFs of the random variables involved in the deterministic structural analysis (e.g., FEM), generate a single value of each random variable. 2. Perform the deterministic analysis, and record if failure is predicted. 3. Repeat steps 1 and 2 for N times. 4. Estimate the probability of failure by Pf = Nf /N, where Nf is the number of failures. MCS offers a number of advantages [65]: The distributions of the model variables do not have to be approximated. Correlations and other inter-dependencies can be modelled. The level of mathematics required to perform a MCS is quite basic. Commercial software is available to automate the tasks involved in the simulation. Greater levels of accuracy can be achieved by simply increasing the number of calculated iterations. MCS is widely recognised as a valid technique so its results are likely to be accepted. The behaviour of the model can be investigated with great ease. The main drawback with the plain MCS is the computational effort involved. To produce a reasonably accurate estimate of the failure probability, at least 100/Pf 80 although may be expected to pass all standard tests for randomness. Therefore. it is important to realise that all generator types produce pseudo-random numbers that form a long sequence of numbers which. For most applications. MCS seems to be a suitable approach. they can be very efficient. and in favourable circumstances. a semi-analytic technique.6. However. but an innate disadvantage of it is the formidable computational effort for problems involving low probability of failure or the problems that require a considerable amount of computation in each sampling cycle. numerous variance reduc- 81 . successive updating of the sampling density function.6 Response surface methods When the performance function g of a system is not explicitly known. often available as functions in software libraries. there may be problems if a poor generator is used to generate many millions of samples in a problem involving a large number of basic (random) variables [45]. A number of random number generator types are available. Directional sampling. Axis orthogonal simulation. this requires that at least 10E+06 simulations are performed [45]. These techniques include: Importance sampling. involves sampling along random vectors. A variety of techniques have been developed to reduce the number of needed simulations. are acceptable. The advantages of MCS are obvious. MCS methods rely on the use of random numbers. Adaptive sampling. 5. these are called variance reduction techniques. For instance. Knowledge of the failure region can be utilised to significantly improve the efficiency of MCS by tailoring the sampling scheme to the particular conditions. will eventually repeat. To overcome this disadvantage. FORM or SORM are not directly applicable. for failure probabilities around 10E-04. modifying the sampling density function to ‘important regions’ of the failure space. These techniques can also be combined together. standard random number generators. which are most conveniently generated numerically with conventional computers.5. Generally speaking. the sampling space is divided into subspaces from which the sampling is performed. However. Stratified sampling. Probabilistic analysis methods trials are required. IS is a very suitable analysis approach [52].g. therefore. be made to other methodologies which may not be as accurate as the above methods but are nevertheless feasible for a wider spectrum of problems. i. Probabilistic analysis methods tion techniques have been proposed. as MPP is the point in the failure space that is closest to the origin. The crux of the RSM is to fit the RSF to g at the sampling points. which is the distance from the origin to the MPP. No failures occur inside this sphere.6.7 Importance sampling (IS) The main idea of IS is to restrict the sampling domain to the tail parts of the joint distribution of the basic random variables. a convex shape that produces a considerably higher failure probability than the corresponding hyperplane. FORM or SORM can be effectively combined with MCS [66]. However. in particular in the neighbourhood of the design point [66].5. the sampling domain can be restricted so that they remain inside a sphere the radius of which is . For instance. Also. The achieved advantage with the IS as compared to plain MC is the considerably reduced amount of computational work required.e. 5. which may be implicit and/or very time consuming to evaluate. The basic idea of the RSM is to approximate the performance function g. the reliability analysis can then be carried out. The purpose of the IS is often to verify or to improve the accuracy of a solution that is obtained by employing an analytical method [52]. The response surface method (RSM) is such a method [66]. This method has to be combined with an optimisation algorithm or analytical reliability method to find the MPP. After the response surface has been fitted at a sequence of sampling points. e. Recourse must. The RSF typically takes the form of a set of polynomials. if the failure surface is a circle the number of MPPs is infinite. For certain problems. Regression is usually performed to determine the RSF by the least squares method. with a response surface function (RSF) that is easier to deal with. The IS is a particularly useful tool when the failure surface has a disadvantageous shape. there are many practical problems that cannot be dealt with the above mentioned methods. such as those mentioned in the previous section. Various IS methods in- 82 . when there exist singularities at the failure surface or when there are several MPPs. Probabilistic analysis methods clude sphere-based importance sampling. adaptive importance sampling. Latin hypercube simulation (LHS) and stratified sampling. The difference between these two methods is that in the AIS the region that limits sampling is modified in such a way that it stays in the so called safe side during the computational analysis.6-2. Figure 5. the more efficient is the solution. The smaller the searched probability. Adaptive importance sampling (AIS) AIS method is in principle similar to sphere-based importance sampling. as the radius of the spherical surface becomes very small and hence the difference to the MCS remains very small. from ref.5. when compared to the MCS. An example of the application of the sphere-based importance sampling is presented in Figure 5. Sampling techniques employed in the MCS (left) compared to the sampling technique employed in the sphere-based importance sampling (right). 83 . The radius of this spherical surface is defined in relation to the MPP. inside which no sampling is performed. [67]. This is done because the first analysis result of the MPP is always approximate and thus one cannot entirely trust to the correctness of the limiting region that is based on it [67]. The efficiency correspondingly decreases when the number of random variables increases.6-2 [67]. Sphere-based importance sampling The principal in this method is to formulate a spherical surface. there will be 2 to the power of p sampling intervals. One approach to obtaining a representative set of model output data is to use LHS to obtain a representative set of input samples to evaluate using the model. then the order of intervals from which the points are sampled is also selected randomly [68]. The approach is to divide each distribution into n intervals of equal probability. Probabilistic analysis methods 5. LHS is based on stratified sampling of probability distributions. An alternative is to use the median of each interval in the analysis [69]. Having a large number of parameters requires a large number of Monte Carlo simulations to produce defensible results. When a random approach is used for selection. In the simulations.6. This may be of considerable importance when the number of parameters is large [69].5. LHS generally gives better results in calculating the tails of the distribution of risk and requires fewer simulations relative to ordinary Monte Carlo analysis. so that the interval specific location is chosen randomly. one point from each interval is sampled. 84 . The difficulty arises in trying to obtain a representative sample of parameter values from their distributions. Thus if each distribution is divided into two parts and there are p parameters.8 Latin hypercube simulation (LHS) Models that contain a large number of parameters can be difficult to analyse using Monte Carlo methods. etc.1 Introduction Risk analysis is a technique for identifying. In general. which are compared. of that event. The estimation of probability or frequency of hazard occurrence depends greatly on the reliability of the system components. number of fatalities.2 Determination of risk values In most risk assessments. Here. Risk analysis methods 6. number of injuries. a frequency per year or per event (in units of time) may be used. Evaluation of the consequence of this hazardous event. Alternatively. there is a wide range of risk analysis methods and related theories. Pi.g. Consequence. the system as a whole and humansystem interactions [49]. ranked and placed in perspective. is a measure of the impacts of an event. 6.6. characterising. the scope is to present a brief overview of this extensive topic. Ci. The results of risk estimation are then used to interpret the various contributors to risk. This can be in the form of mission loss. The main goals of risk management are to minimise the occurrence of accidents by reducing the likelihood of their occurrence. Thus the risk assessment for an individual 85 . reduce the impacts of uncontrollable accidents and transfer risk (e. The engineering definition of risk is now accepted as being the product of the likelihood and the consequence of the event [2]. It is widely used to support regulatory and resource allocation decisions. payload damage. There are two major parts in risk analysis [49]: Determination of the probability of an undesirable event. damage to property. via insurance coverage). [49]. Risk analysis methods 6. quantifying and evaluating hazards. the likelihood of an event is expressed in terms of probability. Ri. quantitative. Their aim is to help to identify the important safety aspects of the design. i 1 for all possible events in the examined system (6. In a qualitative approach risks are usually presented in a risk matrix as combinations of the likelihood and consequences. quantitative and semi-quantitative risk analysis approaches The risk analysis approaches can be divided to the following three categories [3]: qualitative. Risk analysis methods risk. Qualitative risk analysis approach Qualitative risk approaches are based on assigning subjective scores to the different factors that are thought to influence the probabilities and consequences of 86 .2-2) 6. Both qualitative and quantitative risk assessment techniques are valuable for analysing the safety of nuclear and non-nuclear facilities.6. Qualitative risk analysis is based primarily on engineering judgements. priorities for resource allocation can be established on the basis of risk [70]. and maintenance of a facility by estimating the sources and magnitude of risk. The analysis of accident scenarios should generally be more numerically based and detailed than the qualitative approach. operation. Then. R. the likelihood and consequences of equipment failure are determined for each accident scenario from the underlying distributions of the variables using reliability analysis methods [3]. n for a possible event i in the examined system (6. but may still contain a large element of engineering judgement. Semi-quantitative risk analysis determines single numerical values for the probability of failure and the consequences from every cause and effect. and the total risk.2-1) Pi Ci . In a fully quantitative risk analysis. semi-quantitative. The likelihood and consequences of failure are expressed descriptively and in relative terms.3 Qualitative. can be carried out by applying the following two equations [49]: Ri R Pi Ci . The effect of the failure can be determined at various levels of abstraction starting at the component level. Failures that score high in this rating can potentially be the source of system unreliability. possible.g. it is attempted to predict possible sequences of events that lead to a system failure. in practise. Most of the quantitative risk systems are based on Bayesian decision theory [45]. The effect of failure is then determined as well as the causative agent for the failure. criteria for the descriptive categories of likelihood and consequence of failure should be defined [3]. FMEA is inductive in nature and. In FMEA analysis it is assumed that a failure mode is present and that it causes a failure. reasonably probable and probable for likelihood of failure.4.e. Another limitation is that FMEA is performed for only one failure at a time. The resulting indices for different components (e. For qualitative analysis to be used consistently. The rating is normally based on the probability of the failure mode occurrence. pipe regions.1 Failure mode and effect analysis (FMEA) With the FMEA procedure. The scores are then combined using simple formulae to give an index representing the level of risk. failure modes. One drawback of FMEA is that it provides very limited insight into probabilistic representation of system reliability. Risk analysis methods failure. This might not be adequate for systems 87 . and the corresponding failure modes should be removed [49]. i. the severity of its effect(s) and its detectability. In general. determine their consequences and device methods to minimise their occurrence or reoccurrence. FMEA procedure consists of a sequence of steps starting with the analysis at one level or a combination of levels of abstraction. medium and low for consequences of failure. such as system functions. the failure mechanism. or hazards) can then be ranked to determine components with the highest risk [45]. is used in all aspects of system failure analysis throughout the design process [49]. and high.6. subsystems or components. Examples of the relative terms used in qualitative risk analysis are very unlikely. Quantitative risk analysis approach Quantitative risk analysis (QRA) systems are based on estimating the level of risk by direct assessment of the probability and consequences of failure. A criticality rating can also be determined for each failure mode and its resulting effect. 6.4 Review of risk analysis procedures 6. In the first case. FMEA provides a lot of valuable qualitative information about the system design and operation [49]. Determining the fault tree cut sets involves some straightforward Boolean manipulation of events.4-1. fault trees describe the faults that can happen in a system and relate them to their causes. The top event of this system is labeled as brake system failure. A cut set consists of those basic events the occurrence of which leads to the failure of the system [32]. In the second case. called the top event. The listing and meaning of these symbols can be found from textbooks and handbooks [49. For example. The deduction process is performed so that the fault tree embodies all component failures that contribute to the occurrence of the top event.6. unreliability or reliability associated with the top event can also be determined. Risk analysis methods in which multiple failure modes can occur. An example of a fault tree is shown in Figure 6. The 88 .g. Accordingly. However. such as nuclear energy. It shows an automotive brake system that consists of master cylinder. The fault tree analysis is a deductive process whereby an undesirable event. there are three types of symbols in fault trees [49]: events. at the same time. The fault tree itself is a graphical representation of various parallel and sequential combinations of faults that lead to the occurrence of the top event [49]. brake lining.2 Fault tree Analysis Fault trees represent relationships between a fault in a system and the associated events. wheel cylinder and brake fluid. is postulated. but also the time they occur. The product of FMEA is a table of information that summarises the analysis of all relevant failure modes. a typical top event looks like “failure of control circuit A to send a signal when it should”. There are two ways in which fault trees are used in the life cycle of a safety or mission critical system. process control. 71. applicable failure modes). 6. When postulating events in the fault tree.4. Fault trees are applied in diverse industries. gates and transfers. it is important to include not only the undesired component states (e. 72]. and are not presented here. fault trees are taken as specifications of a part of the system requirements. aerospace and aviation [73]. with reasonable likelihood. In general. and the possible means for this event to occur are systematically deduced. The quantitative evaluation of fault trees involves the determination of the probability of the occurrence of the top event. It is also possible to include individual failure modes of each component as well as human errors during the system operation. where the initiating event is modelled. from ref. The range of applicability of event tree analyses is as wide as that of fault tree analyses. Development of the tree proceeds chronologically. the use of this method may not be necessary. An example of a fault tree for an automotive break system.4-1.4. with the demand on each component or subsystem being postulated. This way the event tree sequences and the logical combinations of events can be considered. Evaluation 89 . but for complex systems. This event describes a situation where a demand for the operation of a system occurs. the upper branch of an event shows the success of the event heading and the lower branch shows its failure. These states can be represented by fault trees.6. 6. wheel cylinders fail to provide adequate braking. insufficient amount of brake fluid to transmit the pressure to the wheel cylinder. For a simple system. [72]. At a branch point. The event trees are horizontally built structures that start from the left end. event trees are particularly useful [49]. Figure 6. The first component demanded appears first. such as NPPs. Event headings in event trees represent discrete states of the systems. Risk analysis methods system will fail in case of the occurrence of any one of the following events [72]: master cylinder fails to produce required pressure. and brake lining fails to provide adequate braking.3 Event tree analysis The event tree analysis is a convenient tool for analysing systems the successful operation of which depends on an approximately chronological but discrete operation of its components or units. An example of a simple event tree. particularly in view of the limited availability of technical data regarding many important uncertainties in risk analyses. the information provided by expert opinion is often treated as probabilistic distributions.4.4 Expert opinion The role of experts is important because their judgements can provide valuable information. Each branch in an event tree is evaluated to determine its probability. Risk analysis methods of event trees is straightforward. When the initiating events in a system have been identified.4-2 [45].4-2. The assumed success or failure of these systems leads to multiple event tree branches that result in specified damage states. Because uncertainties are often represented in terms of probability distributions. Figure 6. The motivation 90 . [45]. A simple event tree example of a failure of the corrosion protection (CP) system in a pipeline possibly leading to pipeline rupture is illustrated in Figure 6.6. The probability of each overall outcome is given by multiplying together the individual probabilities of the branches leading to that outcome [49]. Systems and components used to model the accident sequence progression are placed along the top of the event tree. from ref. event trees and required fault trees are constructed for each group of initiating events. An event tree connects an initiating event with damage states. The event tree branches occur at branch points that specify whether or not the system or component along the top of the event tree is functioning [32]. 6. including nuclear engineering. The combination of probability distributions based on expert opinion summarises the accumulated information for risk analysts and decision makers. A structured approach is needed in order to find a balance between the (often contrasting) arguments of experts representing different disciplines. Risk analysis methods for the use of multiple experts is simply the desire to obtain as much information as possible. According to this guideline the role of an expert panel is to synthesise the views of various experts and identify and characterise the uncertainties in their analyses. The Delphi method [77. The described process covers planning.6. This includes defining the role and responsibilities of the expert panel in RI-ISI as well as its composition. Furthermore. Application areas of expert opinion. Concerning applications for Finnish NPPs. seismic risk and environmental risk from toxic chemicals [74]. as well as the necessary associated documentation to be prepared. Procedures for combining probability distributions are often divided as mathematical aggregation methods or behavioural approaches [74]. the concept of uncertainty about a probability value is both intuitively appealing and potentially useful [75]. Primarily. the expert panel approach has been used to combine the deterministic information on degradation mechanisms and 91 . The purpose and steps of the Delphi method depend on the nature of use. The experts should be those having most knowledge on the issues or questions of concern. In this case. These probabilities may be difficult to estimate even though reasonable engineering judgement is applied. aerospace. are diverse. preparation and realisation of the expert panel. This occurs because expert opinion under incomplete knowledge and limited data is inherently imprecise. Current PRA/PSA methodology uses expert opinion in the assessment of rare event probabilities [75]. the targets of application can be categorized into (1) technological forecasting. 78] is by far the most known method for eliciting and synthesizing expert opinions. The guideline describes a complete approach for using an expert panel. and (2) policy analysis. military intelligence. also called expert judgement. an expert panel is important as it compels the different experts to openly discuss the technical bases of their arguments both among each other and with the decision maker. The issues and/or questions need to be stated by the study facilitators or analysts or monitoring team and high degree of consensus is sought from the experts [76]. A more recent useful guideline concerning the use of expert judgement/panel for NPP applications is “ENIQ Recommended Practice 11: Guidance on Expert Panels in RI-ISI” [80]. The technological forecasting relies on a group of experts on a subject matter of interest. 1990. Methods for the calculation of damage. Then. Studies tend to be wide-ranging and threats to the integrity of pressure systems may only be considered briefly. aviation industry. Typically. 6.5 Hazard and operability studies (HAZOP) HAZOP is a structured brainstorming exercise by a group discussion designed to identify potential variations and deviations from the technical design or operating intent and their consequences. The use of quantitative risk analysis as a foundation for rational decision making is increasing in a number of engineering areas. temperature) before branching out to consider human factors and less likely scenarios [3]. Some risk based regulations.S. Committee for the Prevention of Disasters. EPA/630/R-97/001. Regulations and guidelines: U. “Green book”. the expert panel served both as a critical review of the preliminary results and as a decision support for the final definition of risk categories of piping [79]. Environmental Protection Agency: Policy for use of Probabilistic Analysis in Risk Assessment and Guiding Principles for Monte Carlo Analysis. guidelines and standards for risk analyses are listed below.4. This procedure is a tool for hazard analysis and it aims to ensure that weaknesses in the design intent are detected and acted upon [3]. 92 .5 Risk based regulations. civil and marine engineering [81]. level. HAZOP is a procedural tool designed to highlight and identify hazards and operability problems in industrial plants that could reduce the ability of the plant to achieve productivity in a safe manner. space industry. e. 6. guidelines and standards There is a general trend towards the use of risk based regulations in areas where complex technological systems are in use. nuclear industry. EPA 1997. CPR. otherwise the selection is based on applicability of the documents in a wide range of areas.g. Voorburg: Ministry of social affairs and employment. The main emphasis is on nuclear industry.6. Risk analysis methods probabilistic information on pipe break consequences. these focus initially on credible variations of process parameters (flow. A list of guidewords is sometimes used to stimulate the discussions. 42622 (60 FR 42622). Confédération Européenne d'Organismes de Contrôle. Health and Safety Executive (HSE). requirements and terminology. Methods for the calculation of physical effects.Hazardous Installations Division. 93 . Risk analysis methods CPR. Federal Register. USNRC: An Approach for Using Probabilistic Risk Assessment in Risk-Informed Decisions on Plant-Specific Changes to the Licensing Basis. p. 1997b. 1998. August 16.S. DS-Information DS/INF 85: Risk Analyses. Vol. 1995. Nuclear Regulatory Commission. Radiation and Nuclear Safety Authority (STUK). R 35/CEOC/CR1 87 Def. Methods for determining and processing probabilities. U. 1993. “Red book”. CPR. Management of Health and Safety at Work Regulations 1999. CEOC – Risk Assessment: A Qualitative and Quantitative Approach. 2003. Health and Safety Executive (HSE) .174. Probabilistic Safety Analyses. Second Edition. Committee for the Prevention of Disasters. “Yellow book”. 60. Standards: International Standard ISO 2394: General Principles on Reliability for Structures. Risk Based Inspection (RBI): A Risk Based Approach to Planned Plant Inspection.6. The Hague: SDU. The Hague: SDU. Committee for the Prevention of Disasters. Health and Safety Executive (HSE).8. Guide to Risk Assessment Requirements. 1996. Approved Code of Practice and Guidance L21 (Second edition). 1999. Danish Standards Association. 2000. 1997a. USNRC: Use of Probabilistic Risk Assessment Methods in Nuclear Activities: Final Policy Statement. 1998-06-01. YVL Guide 2. Regulatory Guide 1. and it is often omitted. Australian/New Zealand Standard AS/NZS 3931: Risk analysis of technological systems – application guide. It is important to be aware of the limitations of the used methodologies and the uncertainty or lack of confidence in the results. Canadian Standards Association. International Electrotechnical Commission. Standards New Zealand. when model uncertainty is allowed for it is an important influence on the evaluated probability [45]. Risk analysis methods European Standard EN 1050:1996: Safety of machinery – Principles for risk assessment. British Standard BS 8444: Risk management. However. in many cases. Canadian Standard CAN/CSA-Q850-97: Risk management: Guideline for Decision-Makers. model uncertainty is very difficult to assess and model. The tails of failure probability distributions The sensitivity problem of the tails of the failure probability distributions is a common concern within structural reliability analyses. By its very nature. 1997.6 Concerns with risk assessment In the following are briefly described some of the concerns that have been raised in the literature with risk assessment. 1998. 1995. British Standards Institution. 1996. In a risk analysis. Inclusion of model uncertainty Model uncertainty is caused by the use of simplified or idealised mathematical models that are needed as operational tools in the risk analysis. 6. Part 3 Guide to analysis of technological systems – application guide. Standards Australia. If model uncertainty is for some reason omitted from a risk analysis.Section 9: Risk analysis of technological systems. European Committee for Standardisation. it is the probability content defined by the shape of the distribution tails that most influ- 94 .6. IEC International standard nr 60300-3-9: Dependability managementPart 3: Application guide. 1996. the results should be interpreted with some caution. By definition. an estimation of the variability of such data can be established [3]. valid probability distributions. very few data points at the tail of the distribution influence the modelling of the variable. the sensitivity of the reliability can be examined by using different. However. To do so. in a particularly sensitive situation. Small failure probabilities and lack of data It has been pointed out that typical probabilities of failure often evaluated from structural reliability analyses have no conceivable physical meaning if interpreted in a frequency manner. Unfortunately. In a well constituted problem with well defined basic variable models. Clearly. Validation The small failure probabilities for some components mean that evaluated failure probabilities cannot be properly and completely validated. failures of structural components and components of pressure systems. whilst this is to some extent possible for manufactured items. or where the modelling for the most sensitive variables is limited. a valid distribution should fit the data well. The problem with too little data can be managed by deriving confidence limits that can be determined from statistical analyses. and should comply with any physical constraints or limitations. to obtain sufficient data to assess a failure rate for a specific application. tail sensitivity is rarely a concern. By carrying out this type of analyses on the various data sources. it is necessary to consider very broad categories [45].6. Often. Risk analysis methods ences the evaluated failure probability. By their very nature. Thus. It is often pointed out that statistics based on data at the centre of a distribution carry no information about the extremes. it would either be necessary to observe a small number of similar components for a very long period. even in the fortunate situation where a large data sample is available to define the distribution for a basic variable. or to observe a very large population of structures for a shorter more practical period. it is not possible for most pressure systems which are typically 95 . This is clearly an important concern when physical mechanisms governing the shape of the extreme tails are different from those governing the central part of the distribution [45]. data points located in the tail of the distribution are very unlikely to occur in a population of data. The lack of statistically sufficient amount of empirical data necessary to estimate new parameters is one important reason to apply expert opinion [49]. and so extrapolation is inherently untrustworthy. and corresponding failure data are rare. 96 .6. Indeed. Completeness There can never be a guarantee that all accident situations. thus there is no population of nominally identical structures under nominally identical conditions that might be observed. a number of failures have occurred because the scenario was not and could not have realistically been envisaged. It may also be possible to calibrate the failure models from test data [45]. causes. However. and effects have been considered. for most component types there is a large enough population of components of similar type for at least allowing to obtain some crude comparative values. Risk analysis methods more unique items. Therefore it is important to undertake rigorous hazard identification studies. Deterministic degradation modelling methods for power plant components 7. standards and research articles/reports. In conventional power plants. locally confined. due to relatively high operational temperatures. a fatigue crack will not initiate and propagate. irradiation embrittelement is obviously not an issue.1 Introduction Modelling methods of the major degradation mechanisms concerning power plant components are considered in the following. Fatigue damage is caused by the simultaneous action of cyclic stress. Deterministic degradation modelling methods for power plant components 7. codes. However. and which have been developed by both research institutes and private companies. These include methods that are defined in fitness-for-service procedures. in conventional power plants also creep has to be often taken into account in addition to fatigue and corrosion. Fatigue may lead to emergence of cracks and cause fracture after a sufficient number of fluctuations.7. Of these mechanisms. The plastic strain 97 . the three most often encountered ones in NPP environments are fatigue. However. of the NPP types and components. and permanent structural change that occurs in a material subjected to repeated or fluctuating strains at nominal stresses that have maximum values less than the static yield strength of the material.2 Fatigue Fatigue is the progressive. tensile stress. and plastic strain. If any one of these three is not present. A brief overview is also presented of the various commonly applied component structural integrity assessment methods. SCC and irradiation embrittlement [32]. only the PWR RPVs and LWR internals are susceptible to the last one of these. 7. rather than the crack propagating to some limiting depth. For an overview of the fracture mechanics based analysis methods. A typical example of this approach is the often used rainflow load cycle counting procedure in its different modifications. life curve modification methods. As for fracture mechanics based crack growth computations. and tensile stresses promote crack growth. they strictly require the realistic chronological order of the load 98 . like boiler tubes. a multitude of cumulative fatigue damage models have been developed. dents. microcracks arise. inclusions. All in all.3. One of the most successful applications of the theory of fracture mechanics is in the characterisation of fatigue crack propagation. These models can be divided into six categories [100]: linear damage rules. both published several decades ago. A fracture mechanics based analysis of fatigue flaw growth inevitably requires a thorough understanding of the assumptions. In the process of fatigue failure in an originally intact metal. the principal sites of fatigue crack nucleation include voids.7. non-linear damage curves and two-stage linearisation approaches. That is because most of the other cumulative fatigue models consider fatigue crack initiation or it reaching a macroscopic but still relatively small size as the limiting criteria. through wall or to a size leading to plastic collapse. macroscopic stress concentrations. Since the introduction of damage accumulation concept by Palmgren [98] and linear damage rule by Miner [99]. significance and limitations underlying the development of various crack tip parameters. see Section 4. Deterministic degradation modelling methods for power plant components caused by cyclic stresses initiates the crack. forging laps and folds. coalesce or grow to macro-cracks that propagate until the fracture toughness of the material is exceeded and final fracture will occur [7]. scratches. and energy based theories. continuum damage mechanics models. In case of cumulative continuum fatigue models it is also important to note that the order of loading events in load cycle sequences formed and assembled with commonly used procedures deviate from the actual chronological load history. Approaches based on crack growth concepts and fracture mechanics are covered in more detail here. as well as regions of micro-structural and chemical non-uniformity [97]. there are several computational approaches for the cumulative fatigue damage analysis. approaches based on crack growth concepts. In mechanical components. overloads. If the stress amplitude is changed. as the magnitude of each crack growth increment is dependent both on load amplitude and the current crack size. The procedures for analysing constant amplitude fatigue under small scale yielding conditions are fairly well established. S-N curves. Of the numerous developed cumulative fatigue damage models only some most notable and/or commonly used ones are considered here. The Palmgren-Miner rule asserts that the damage fraction i at any stress level Si is linearly proportional to the ratio of ni. a new partial damage is calculated for this new amplitude level. The S-N curves conveniently display basic fatigue data on a plot of cyclic stress level versus the number of cycles to failure. weld residual stresses and short cracks introduce additional complications that are not yet fully understood. Variable amplitude loading. crack closure. Deterministic degradation modelling methods for power plant components history to be preserved. The total accumulated damage D is then given by [99]: 99 . the number of cycles of operation under this stress amplitude to Ni. Linear damage rules The Palmgren-Miner rule (or Miner’s rule) is a linear damage accumulation rule used to predict the cycles to failure under variable amplitude loading. The damage fraction is computed as follows [99]: i ni Ni (7. is used to predict the number of cycles to failure at a single stress level. where the appropriate Ni is found form the S-N curve. Analytical representation of S-N curves is given in the form [32]: NS b k (7.2-1) where b and k are material parameters estimated from test data obtained using identical specimens.7. although a number of uncertainties remain. the total number of cycles that would produce a failure at that stress level. multi-axial loading. The method relating the stress level and number of load cycles.2-2) where ni N i . large scale plasticity. 7. 100 . i (7. In the two-stage linearisation approaches. Deterministic degradation modelling methods for power plant components D i i i ni Ni (7.2-5a) DII i crack propagation phase. non-linear damage rule procedures have been developed. i (7. Nr is a reference level for N. A representative example of a double linear damage rule is as follows [102]: DI i nI NI n II N II crack initiation phase. they provide improvement to the shortcomings of the linear damage rule.2-5b) where NI = N – NII. A representative and early example of non-linear damage rules is as follows [101]: xi i i i D ni Ni xi (7. the two constants B and are determined from regression analysis of the experimental data. Here. while still retaining its simplicity.2-3) and failure is assumed to occur when D 1 . The presented values for them mainly correspond to high strength steels. Non-linear damage curves and two-stage linearisation approaches Attempting to overcome the inaccuracies of linear damage rules. B = 0. the fatigue damage is divided to two phases. N II B N N r N .2-4) where xi is an exponent parameter related to the ith loading level. load sequence independence and lack of load interaction accountability. those being crack initiation and crack propagation. and = 0. The main deficiencies with linear damage rule used are its load level independence. As for two-stage linearisation approaches.25.65. The experimental verification results show that this model can provide accurate predictions of fatigue lives under repeated block loading. 104]: N 1 i N i´ (7. The model is able to take into account the influence of initial hardening effect by introducing a new internal variable which keeps track of the largest plastic strain range in the prior loading history.2-6) where N is the total accumulated life. An example of this approach accounting for load interaction effect is [103. Continuum damage mechanics models Continuum damage mechanics deals with the mechanical behaviour of a deteriorating medium at the continuum scale. For the one-dimensional case. Nf is number of load cycles to failure. This predictive theory is also expanded to stochastic loading histories. This damage model is highly non-linear in damage evolution and is able to account for the mean stress effect.2-7) where n is number of load cycles. The success of continuum damage mechanics application in modelling the creep damage process has led to extend this approach to ductile plastic damage. whereas i and Ni´ are the frequency of load cycles (ni/N ) and the modified life associated with loading level i. However. respectively. This model allows for the growth of damage below the initial fatigue limit. 111]: D 1 1 n Nf 1 1 1 1 1 1 r1 1 1 1 (7. when the material is subjected to prior cycling above the fatigue limit. Deterministic degradation modelling methods for power plant components Life curve modification methods In the life curve modification methods. the cyclic fatigue damage evolution can be generalized by a function of the load condition and damage state. since a scalar damage varia- 101 . The following non-linear damage evolution equation has been developed as based on measured changes in tensile load carrying capacity and using the effective stress concept [110. the conventional S-N curve is modified to better follow the fatigue test data. the mean stress effect is directly incorporated in the model. creep-fatigue interaction. brittle fracture and fatigue damage. is a material constant and is a function dependent on the stress state. In addition.7. The incremental rate of plastic strain energy. dW/dN. the corresponding change for cyclic strain hardening exponent was negligible. The exponent b is dimensionless. Energy based damage models also allow to include mean stress and multiaxial loads since multiaxial fatigue parameters based on strain energy have been developed [106. 107]. and is the cyclic hardening rate. it was found according to refs. K* and n* are cyclic strain hardening coefficient and exponent determined near failure (r = n/Nf = 1 ). is then derived as: 102 . and fatigue. The expression for is given as: b a p 2 2 (7. [108. p is plastic strain. which in turn led to the development of energy based cumulative damage theories. difficulties will inevitably occur when the model is extended to multiaxial loading conditions.2-8) where is stress range. creep.2-9) where a and b are constants. Energy based damage theories By examining constant strain amplitude test data. This led to the development of the following cyclic stress-strain relation [109]: n* 2 K p 2 r (7. whereas the dimension of the parameter a is such that its product with [MPa] and p [%] is dimensionless.7. It has been realised that an energy based damage parameter can unify the damage caused by different types of loading such as thermal cycling. 109] that while the values of the cyclic strain hardening coefficient changed during the cycling process. Deterministic degradation modelling methods for power plant components ble is employed and the model is written in its uniaxial form involving the maximum and mean stresses. It has been noted that there is a connection between stress-strain cycling loop hysteresis energy and fatigue. mainly considering strain energy [105]. the values for which are obtained as based on experimental data. 2-10) where again r = n/Nf and the energy accumulation is defined by introducing a parameter called the fraction of plastic strain energy. Finally. stable fracture may slow down and stop unless the stress level is further increased [32]. Due to this small plastic zone. the fatigue damage function is expressed as: D 1 n' 1 r1 2b n' (7. JR. Materials with relatively low fracture resistance that fall below their so called plastic collapse strength can be analysed on the basis of elastic concept through the use of LEFM. = W/Wr = r1+ . Under linear-elastic circumstances. and the size of the plastic zone smaller than at the time of fracture. During most part of the cracking process. It accounts for the load interaction effect and the change in strain hardening through the stress response. the crack is much smaller than the one that would cause fracture at the prevailing stress. tends to increase during the fracture process.2-11) is a non-linear. Fracture mechanics based crack growth computation procedures Fracture mechanics can be used to model fatigue of components. The corresponding fracture toughness/resistance. 103 . the stress intensity factor K determines the entire crack tip stress field. If the crack size is close to critical (fracture). the use of EPFM is often necessary. An overview concerning fracture mechanics is presented in Section 4.2-11) where a and n´ is the cyclic strain hardening exponent. until at some point instability occurs causing the fracture to become fast and uncontrollable. Fracture will occur when the stresses at the crack tip become too high for the material to bear. p 4 This model represented by equation (7. This damage model is particularly suitable for materials which exhibit cyclic hardening. load dependent damage accumulation model. LEFM can often be applied. For other materials.3.7. so that fracture will be slow and stable initially. Deterministic degradation modelling methods for power plant components dW dN 1 n 4 1 n 1 n* r K p 2 (7. the crack tip stress field is less severe. Therefore. whereas in case of ductile material accompanied with considerable plastic deformation J-integral applies. Thus. The growth of the crack that eventually leads to fracture occurs by mechanisms entirely different from the fracture itself. However. but because by far the longest time is spent in the growth of much smaller cracks. A fatigue damage propagation model proposed by Forman [113]. as follows [112]: da dN C K m (7.2-1 below. whereas the dimension of the parameter C is such that its product with ( K)m is mm.7. One of the earliest and also most often used fatigue crack propagation models is the Paris-Erdogan equation.2-12) where a [mm] is crack depth. It also assumes that the stress amplitude is constant and that it is small enough so that LEFM is applicable and that the crack growth rate is independent of the previous load history. Failure occurs when the stress intensity factor exceeds the fracture toughness. The exponent m is dimensionless. Region II represents the intermediate crack propagation zone where the length of the plastic zone ahead of the crack tip is long compared with the mean grain size.e. whereas Region I contains the stress intensity factor range threshold below which fatigue cracks do not propagate and Region III is characterised by rapid and unstable crack growth just prior to final failure [114]. Kmax . but much smaller than the crack length.Kmin. These fracture mechanics based computation procedures can be further divided to those intended for constant and variable amplitude loading. the crack propagation equations are models that relate the crack growth rate to the level of cyclic stress. see Region II in Figure 7. Apparently the models of the latter type can also be applied with sufficient accuracy to those of the former. respectively. i. In addition. is: 104 . which covers both the intermediate and high K regions. whereas C and m are material and environment specific constants. The Paris-Erdogan equation assumes that the crack growth depends only on the stress intensity factor range. the Paris-Erdogan equation describes crack growth only at intermediate values of fatigue crack growth curve. For fatigue. Deterministic degradation modelling methods for power plant components this may no longer be true. it is justifiable to use LEFM to obtain the time for crack growth with good accuracy [7]. K [MPa m] is the stress intensity factor range. this should be case specifically checked. It is an empirical formula that relates the cyclic crack growth rate to stress intensity factor range.] is the number of load cycles. N [ . 2-1. log da dN Fracture I II III m Threshold Kth KC log( K) Figure 7. The expression 105 .2-14) is based on a simple physical model rather than a purely empirical fit. crack growth rate da/dN in turn tends to infinity. KC [MPa m] is fracture toughness and R = Kmin/Kmax is stress ratio. Typical fatigue crack growth behaviour in metals. McEvily [115] developed another equation that can be fit to the entire fatigue crack growth curve. as follows: da dN C K K th 2 1 KC K K max (7. Equation (7. it is possible to predict both stable intermediate crack growth and accelerated crack growth rates for various stress ratios.2-13) where CF and my are material and environment specific constants. Deterministic degradation modelling methods for power plant components da dN CF K 1 R KC my K CF K y 1 R K C K max m (7.2-13) indicates that as Kmax approaches KC.2-13). Equation (7.2-14) where Kth [MPa m] is crack propagation threshold.7. With equation (7. a crack only propagates while its flanks are separated. is quite insignificant.2-15a) where U is a factor depending primarily on the stress ratio R. stress state and environment. as follows: K eff K max K cl U K (7. K. Deterministic degradation modelling methods for power plant components distinguishes between two crack advancement modes. Other attempts have included linking U to the specimen geometry.2-15b) Concerning the factor U above. and is then driven by the effective stress intensity factor range. As inconsistencies in assessing the K values for the first contact between the crack flanks could occur due to the fracture surface asperities. The difference between the K values concerning opening. and closure. According to Elber [116].2-12) yields [116]: da dN C K eff m CU K m (7. it is considered that fatigue cracks do not grow when being closed. is introduced. To account for crack growth retardation. Wheeler [120] developed a fatigue damage propagation model based on yield zone concept for tensile overloads. McClung [119] has reported of three distinct K dependent regimes of crack closure and concluded that no single equation could describe the closure in all three regimes. is employed. usually an average closure/opening stress intensity factor. thus using a single value Kcl has proven to be an adequate approximation. whereas Hudak and Davidson [118] attributed the confusion and controversy over the effect of loading variables on closure to experimental factors. due to the gradual nature of the crack behaviour between the points of complete opening and closure. a retardation parameter. Shih and Wei [117] have reported a definite dependence on Kmax.2-15a) in the Paris-Erdogan equation (7. This retardation parameter is dependent 106 . Cp. the use of an average value is recommended. ductile striation and static mode. Keff. As there is no unique definition of crack closure or crack opening. Kop.7. It is noted that for tests in non-aggressive environments the number of adjustable parameters becomes zero when K values are large with respect to Kth [114]. By replacing the stress intensity factor range with the effective stress intensity factor range of equation (7. In general. Kcl. which can be the case when the surrounding stress field has turned to compression. Cp Ry ap a m1 for (a + Ry) < ap. The Wheeler model equation together with definitions for its parameters can be expressed as [120]: da dN C pC K m (7. The procedure includes a multi-parameter yield zone model. ap is the sum of a and largest prior yield zone. and the computation process follows the load cycles. This model allows to use other crack propagation formulations as well.7. The limitation of this model [120] is that the values for the shaping exponent must generally be obtained experimentally. i. m are defined in connection of Paris-Erdogan model. The model considers mainly variable amplitude loading conditions. among the needed input data is initial crack length. m1 is shaping exponent. this model does not predict acceleration caused by compressive overloads (underloads). Deterministic degradation modelling methods for power plant components on the current plastic zone. Johnson [121] has developed a systematic technique for modeling fatigue crack growth under variable amplitude loading to account for interaction effects. Modified Forman type crack growth equation was used in developing the model.e. a0.2-16) where: Cp is retardation parameter. crack grows increment by increment. as follows: 107 . and Cp = 1 for (a + Ry) > ap. thus the right side of equation (7. overload plastic zone and a shaping exponent. acceleration and underload effects by decreasing or increasing the stress ratio. Ry is extent of current yield zone. see equation (7. which accounts for crack growth retardation. C. Further. (ap a) is distance from a crack tip to the boundary of the yield zone caused by the last tensile overload in a sequence.2-16) can be written as: Cpf( K). where f( K) is a function describing the crack growth. When using the model.2-12). K th 1 Reff K th . [122] proposed a simple model for the fatigue crack growth rate which considers the plastic component of the J-integral as a damage factor. t = material wall thickness. eff .2-17) where: m = 1 at R 0. Further. The proposed damage accumulation theory assumes that: the total plastic strain energy density absorbed by the material is prior to reaching its ultimate state constant to be determined experimentally. This model [121] includes the assumption that the load interactions are a result of the residual stress intensity due to plastic deformation at the crack tip. This model is expressed as: 2 da dN 2 4 K max 2 y 1 1 2 1 1 R 2 1 1 2 1 1 R 2 (7. maximum allowable stress ratio. 108 . the elastic strain energy density stored by the material does not cause damage and is released upon unloading. constant amplitude loading data was used for computation of threshold K under variable amplitude loading. Rmax 0. K eff y a. Wang et al. ZOL = plastic zone diameter for the applied Kmax.2-18) where: 2 K max 2 K eff .6 . Reff K min K max K red K red K min. threshold K range for variable amplitude loading.eff K max. and n = 2 at R < 0. Deterministic degradation modelling methods for power plant components da dN C K 1 Reff K C m K (7.2 Z OL t 0.7. The FITNET procedure covers a wide range of materials and takes into account the crack closure effect on the crack growth behaviour in more detail.g. Based on the research to obtain the procedure equation (7. = average local yield strength.2-18).7. it was concluded that the crack growth rate is not only a function of K. On the other hand. but also of average local yield strength. According to this fatigue damage model [122] the accumulated plastic strain energy will cause the material damaging and lead to failure. The test data and equation predictions were then found to be in reasonably good agreement. This plastic damage only occurs after the maximum elastic component has been reached. y y = local yield strength. p and q are parameters for which material specific values are given in the procedure documentation. the procedure contains many parameters the values for which need to be obtained experimentally in order to assure reliable computations. which follows a cycle-by-cycle integration method as using the sigmoidal crack growth rate relationship. Deterministic degradation modelling methods for power plant components = constant determined according to the material and loading conditions. and which model is now also included in the quite recently published European Fitness For Service Network (FITNET) structural integrity assessment procedure/guideline [125]. on stress ratio R. fracture toughness and amplitude of the applied effective stress intensity factor in Regions II and III.2-19) where C. The values for parameters f. n. Kth. Kmax and KC are obtained from associated functions which are dependent e. A more advanced development is the fatigue crack growth rate model by Forman and Mettu [124]. constraint conditions and thickness. The mode1 was verified using the test results published in ASTM STP 789 [123]. 109 . The model equation is given as: da dN 1 f C 1 R n K 1 K th K 1 K max K C p q (7. 2-12) earlier. many new methodologies have been developed. An example of multiaxial fatigue degradation models based on equivalent stress intensity is presented in the following. During recent decades. see equation (7. 128. Marquis and Socie [130] consider methods based on cyclic J-integral as a separate group from the energy based methods. [126]. Deterministic degradation modelling methods for power plant components Discussion on damage caused by multiaxial fatigue loading Power plant components can be subjected to multiaxial fatigue loading. where the stress intensity factor range is replaced by equivalent stress intensity factor range. The multiaxial fatigue degradation models proposed in the literature may be categorised into three groups: stress based. they are not applicable for non-proportional loading due to changing directions and/or ratios of the principal stresses [126]. In their comprehensive more recent review on the subject. and hence. such as the critical plane approach. mesoscopic scale approach. In order to handle nonproportional loading effects on fatigue resistance.25 (7. integral approach. the fatigue crack growth rate may also be expressed by a simple Paris-Erdogan type model. 129]. fatigue damage evaluation of components/structures under non-proportional multiaxial loading has become a growing research interest as conventional multiaxial fatigue criteria are based on proportional fatigue data. For long life fatigue problems. strain based and energy based methods [127. This leads to the following formulation for the equivalent stress intensity factor range: K eq K I4 4 8 K II 4 8 K III 1 0. so that the model equation becomes: da dN C K eq m (7.7. the corresponding principal directions and/or principal stress ratios vary during a loading cycle or loading block. most of the multiaxial fatigue criteria are stress based.2-21) 110 . They are based on various concepts.2-20) The basis for the model by Tanaka [131] is that displacements behind the crack tip reach a critical value. For mixed mode loading. In these circumstances of non-proportional multiaxial loading. etc. Keq. in which the cyclic loads in various directions may be applied at different frequencies and/or with a phase difference. Before computing the strain energy density factor.2-20) are taken from standard mode I crack growth experiments. and is Poisson’s coefficient. It is based on the strain energy density around the crack tip and is expressed as: S a11 K I2 2a12 K I K II 2 a 22 K II 2 a33 K III (7. .2-25) 111 .2-24) S 0 at = 0 Then the crack growth rate can be computed as: da dN CS S m2 (7. a12. As for strain energy density fatigue degradation models. Poisson's coefficient and the inclination angle of the crack. see Section 4.2-23) K K K III This equation includes both stress range and mean stress. For cyclic loading a cyclic strain energy density factor range is defined as: S 2 a11 a 22 0 0 mean II K Imean K I K II a33 a12 0 0 mean III mean K II KI K Imean K II (7. Crack growth under other loading conditions can be obtained with the use of the appropriate equivalent stress intensity factor range given by equation (7. Other forms of the equivalent stress intensity factor range have also been used. The material constants in equation (7. S. as follows: S 2 0 at = 2 0 (7.2-21). Deterministic degradation modelling methods for power plant components where KII and KIII are the ranges for mode II and III stress intensity factors. Sih [132] proposed a criterion for mixed mode loading known as the strain energy density factor.2-22) where the coefficients a11. the direction of crack growth 0 must be determined from the necessary and sufficient conditions for crack growth. Crack extension occurs when the strain energy density factor reaches a critical value in a direction defined by 0. This will be the direction of minimum strain energy density. a22 and a33 under plane strain are dependent on elastic modulus.7.3. crack growth data could be fitted in terms of J to obtain the constants directly. J.2-27) In cyclic J-integral approach.7. KI( ) is often used for elastic-plastic loading conditions. controls the crack growth as [130]: da dN CJ J m 2 (7. Keq( ). where is strain range. Further. the latter two being denoted as FI and FII.2-27) gives the effective strain intensity factor range based on the strain energy release rate as [130]: K eq E FII 21 2 0. G is shear modulus and is shear strain range.2-29) Alternatively. crack growth is analysed by assuming that the cyclic J-integral range.2-20) as follows: CS 2 E C 1 2 1 m2 (7. can be written in terms of the strain ranges and crack geometry factors for modes I and II. any of the equivalent stress intensity models could be used with the appropriate substitution of E or G for the stress. an effective strain intensity factor range. 112 . K and J are related and the constant CJ can be evaluated as: CJ CE m 2 (7. Concerning equivalent strain intensity models.5 FI E 2 a (7. Equation (7. Deterministic degradation modelling methods for power plant components where the values of the constants can be determined from the standard crack growth constants in equation (7.2-28) For elastic loading. The strain intensity factor range.2-26) where is Poisson’s coefficient. For instance.1 General and pitting corrosion General corrosion is the result of a chemical or electro-chemical reaction between a material and its environment. This corrosion degradation is more common in austenitic stainless steels than in carbon steels. 137]. Deterministic degradation modelling methods for power plant components 7. Corrosion rates of carbon and low-alloy steels increase with rising surface temperatures until the electrolyte is evaporated [134]. when humidity exceeds 70 %. The ratio between humidity and temperature determines the thickness of the film. Generally. 7.3.3 Corrosion Of the various identified forms of corrosion. and as the mechanism continues it propagates deeper into the metal forming a pit [134].4. in LWRs piping components of stainless steel are susceptible to pitting corrosion [4] A number of computation procedures for general corrosion rates using electrochemical and thermo-dynamic models exist both for carbon steels and austenitic stainless steels. the easier the diffusion of oxygen through the film that drives the corrosion reaction.g. as the computed results mainly concern corrosion potential parameters. [135. For instance. The thinner the film. an electrolytic cell is formed with the anode at the minute area of active metal. refs. Pitting corrosion is a localised corrosion attack in aqueous environments containing dissolved oxygen and chlorides. When passivity breaks down at a spot on a surface. see e. a moisture film forms on the steel surface providing an electrolyte [133]. The large electric potential difference between the two areas accounts for considerable flow of current with rapid corrosion at the anode. modelling of those which mainly concern power plant components are considered here.7. At ordinary temperatures and in neutral or near neutral media. see Section 3. such as current density at 113 . These corrosion mechanisms are general corrosion. only some notable ones are presented here. The anode does not spread because it is surrounded by passive metal. and the cathode at the larger area of passive metal. both oxygen and moisture must be present to corrode steel. wall thinning due to corrosion. However. It is typically characterised by uniform attack resulting in material dissolution and sometimes corrosion product buildup. pitting and stress corrosion cracking (SCC).g. Of the numerous models that have been developed for these corrosion mechanisms. in LWRs the steam generator tubes are susceptible to general corrosion [4]. such models are not suitable for mechanical analyses of e. which is possibly limited by the corrosion and/or the diffusion rate of impurities [32]. The model for assessment of pitting propagation rate.3-1) where t is time since the start of corrosion process. This is an unstable process. see e. 7. thus leaving more accurate assessment of the dependence between these two parameters unclarified. and n is empirical material and environment specific constant.g. and n is power law exponent for non-linear behaviour (where applicable). V. often the scatter of SCC data plotted as crack propagation rate against KI is remarkable. A straightforward equation for assessment of general corrosion propagation depth. t0 is constant. A number of electro-chemical and thermo-dynamic models exist for pitting corrosion. V(t) V0 = constant. is [138]: dt Ct n (7.7. C is effective corrosion rate in absence of coating. Deterministic degradation modelling methods for power plant components the corrosion site. The data resulting from SCC tests have been presented in both S-N curves and fracture mechanics formats [32]. [140]. by Engelhardt and Macdonald [139] is: V t V0 1 t t 0 n (7. is valid can be comparable with the observation time (or even with the service life of the component/system). t is time since the start of corrosion process. In SCC. It is important to note that in many cases the period of time over which the approximation. the high chemical corrosion rate at the crack tip is driven by the fast diffusion rate of embrittling impurities towards the crack tip.3. The surface energy is thereby reduced and the critical stress intensity factor decreases. and by the energy dissipated in the plastic zone ahead of the crack. The computation models describing SCC and fatigue induced crack growth for intermediate region are often similar. 114 . ref. with C and n being empirical material and environment specific parameters. d(t). which is mainly determined by the surface energy on the plane of crack propagation.3-2) where V0 is initial pitting propagation rate.2 Stress corrosion cracking (SCC) Crack propagation is the favoured action once the stress intensity factor exceeds a critical value. However. the time variable of interest is measured in number of load cycles. Deterministic degradation modelling methods for power plant components In simple fracture mechanics based treatments. SCC and fatigue crack growth computation procedures it is assumed that a stress field created at the crack tip causes the crack propagation. 115 . da dt 1 2 3 da dt plateau subcritical crack propagation total failure Kth KC K Figure 7. t is time and KI is the crack tip value of type I stress intensity factor. which in turn are assessed based on experimental data.3-1. that for fatigue. An illustration of SCC propagation rate as a function of K for metallic materials is shown in Figure 7. showing the stages 1. 2 and 3 as well as the plateau velocity. An illustration of SCC propagation rate as a function of K for metallic materials. The stress intensity factor K for SCC computations is based on a single tensile stress.3-1. Note. whereas the stress intensity range K is based on an alternating stress for fatigue crack propagation [32].7. whereas C and n are material and environment specific constants.3-3) where a is crack depth. whereas for SCC it is measured in time units. The mathematical expression for intermediate (Stage 2) SCC rate equation is [136]: da dt CK In (7. The popularity of this SCC equation is based on its simplicity and availability of values for constants C and n. is crack growth rate coefficient = 2. 142]. is exponent = 1.67E-12 at 325 °C for da/dt in units of m/s and K in units of MPa m.3-4) has been developed for Alloys 182 and 82 which are also commonly used NPP component materials. K is crack tip stress intensity factor [MPa m]. 144]: 116 . two SCC modes are identified. According to one approach the crack tip strain rate increases rapidly with K. According to another approach the crack tip corrosion rate and transport of species within the crack increase rapidly with increasing crack volume and hence with K [33]. This equation is [143. A further developed and in some applications useful modification of the Stage 2 SCC rate equation that also takes into account K threshold is [141. which is a commonly used material for NPP pressure boundary components [141. Kth is crack tip stress intensity factor threshold = 9 MPa m. T is absolute operating temperature at crack location [K].3-4) is mainly applicable to alloy Inconel 600. as mentioned earlier in Section 3. A similar approach as equation (7. Values for material and environment specific constants C and n are presently available only for IGSCC. However. Equation (7. Tref is absolute reference temperature used to normalise data = 598. Deterministic degradation modelling methods for power plant components When considering smaller scale structure of metallic materials.15 K (325 °C).4.3-4) where: Q g is thermal activation energy for crack growth = 130 kJ/mol. R is universal gas constant = 8.7. 142]: da dt exp Qg 1 R T 1 Tref K K th (7.314E-03 kJ/mol·K. A comprehensive explanation for Stage 1 behaviour has this far not been developed. this SCC mode is encountered much more often in power plant environments than TGSCC. namely intergranular SCC (IGSCC) and transgranular SCC (TGSCC).16. [145]: m m y da dt Mi0 z F1 m t0 f n n 1 E m n 1 2dK dt K da dt r (7. z is charge change at electrolysis (dimensionless). except for crack propagation that is clearly perpendicular to the dendrite solidification direction. F is Faraday constant [Coulomb/mol]. and otherwise the parameter explanations/values are the same as for equation (7. m is decay constant of current at newly exposed material surface (dimensionless). t0 is starting time of current decay at newly exposed material surface [s]. where forientation = 0.3-4). f is strain caused by break of protective film at crack tip (dimensionless).0. An example of these models is the SCC propagation equation by Shoji et al. forientation = 1. Deterministic degradation modelling methods for power plant components da dt exp Qg 1 R T 1 Tref f alloy f orientatio n K (7.7. i0 is initial value of decay curve of electric current on newly exposed surface [A/mm2].385 for Alloy 82.3-5) where: falloy = 1.3-6) ln K2 r 2 y where: M is atomic mass [kg/mol]. 117 .0 for Alloy 182 and 0. y are constants (dimensionless).5. is yield stength [MPa]. is density [kg/m3]. . Such more advanced SCC propagation models have been developed that take also into account electro-chemical characteristics and consequent behaviour of protective oxide layer. r is radial distance from crack tip [mm]. The SCC model expressed by equation (7. it is possible to approximately estimate an adequate value by considering the solution conductivity. newly exposed surface corrodes following the Faraday’s law for electrolysis to form a new film. As for m. break of protective film break of next film Figure 7. Deterministic degradation modelling methods for power plant components E is Young’s modulus [MPa]. Current decay after break of protective film followed with formation and break of new film.7.3-6) contains the uncertain variables r and m. [145]. A similar derivation of crack growth rate can be made based upon the slip-oxidation mechanism [146]. equation (7.3-6) contains the assumption that the SCC propagation is regarded as a cyclic corrosive process assisted by strain relaxation. This model consists of combined kinetics of the plastic de- 118 . A quite recent predictive methodology for SCC propagation using a mechanochemical model based on a slip formation/dissolution was presented by Saito and Kuniya [147]. corrosion potential and the degree of chromium depletion on the grain boundaries. from ref. When protective oxide film on the surface of steel is broken by operational load.3-2 below. n is work hardening exponent in Ramberg-Osgood equation (dimensionless). This process is depicted in Figure 7. However.3-2. which is a function of for example K and KISCC.3-7c) is crack tip strain rate.7. 119 . m is density of the metal. z is number of electrons involved in the reaction rate. Nslip is the number of active slip bands. i0 is initial dissolution current density at the bare metal surface. is angle between the direction of the slip plane and tensile stress. and b is Burgers vector. F is Faraday’s constant. The model equations are: da dt n A0 ct Cm (7. Crack tip strain rate ct is calculated from a separate equation. t0 is short time constant. environments and stress conditions. M is atomic weight of the metal. [147] the SCC propagation model enables to predict quantitatively the crack growth for a wide combination of materials.3-7a) A0 n M i0 t 0 zF m 1 n (7. which is the threshold stress intensity factor for SCC. water conductivity and corrosion potential.3-7b) Cm where: ct 2 d cos N slip n d b 2 (7. n is numerical constant correlated with the degree of sensitisation. According to the ref. d is dislocation density. Deterministic degradation modelling methods for power plant components formation process as a mechanical factor and the slip dissolution-repassivation process as an environmental factor at the crack tip. nd is the number of dislocations contributing to formation of a slip band. in structural integrity assessments concerning SCC the crack postulates are often located to these regions. Some remarks concerning the application of SCC propagation computation models are presented in the following. in water with temperature of 288 °C equation (7. the available values have mostly/invariably been conservatively developed as upper bound solutions based on the experimental data. due to SCC tests being relatively expensive to carry out.3-8) The results presented in ref.1 10 07 2. Firstly. in this case conservatively on the tensile side. due to welding process the weld regions and adjacent heat affected zones (HAZs) are often sensitised. Mostly/invariably SCC model equations are limited to concern only sensitised materials. the variation of the K values along the crack front is typically relatively large. being a conservative choice. most/all experimental SCC data are proprietary and thus not available. for austenitic stainless steel of type 304. [147] are in good agreement with the experimental SCC measurements performed for the same conditions. As the scatter of SCC data sets is typically wide. Furthermore. However. Often.5 10 10 exp 3 10 19 1. Such components are at most weakly susceptible to SCC. all these models contain from a few to several material and environment dependent parameters. as in most cases it is also the maximum value. such parameter data are at least partly missing in the available technical literature. describes quite poorly the overall/average growth 120 .74 10 21 2 13 (7.7. whereas in the operating NPPs.5 10 20 K I 7. there are numerous components of SCC susceptible materials that have not experienced sensitising treatments/conditions. the upper bound approach can in this connection lead to excessively conservative SCC equation parameter values. However. As for the published parameter data. which is a commonly used NPP piping component material. Consequently. In commonly used fitness-for-service handbooks. typically only that from the crack tip is used in the incremental SCC propagation computations. the values for which have to be assessed based on experimental data. Deterministic degradation modelling methods for power plant components For instance. in weld and HAZ regions locally confined weld residual stresses have to be taken mostly/fully into account as well.3-7) is written after incorporation of associated parameter data as [147]: da dt 1. On the other hand. On the other hand. of the K values over the crack front. these stresses have also been developed as upper bound solutions based on the underlying experimental data. and consequently the crack tip value. and time to rupture. Thus the SCC propagation computations are often burdened with conservative assumptions coming from several sources at the same time.4-2) where PLM is Larson-Miller parameter. M and N are material dependent parameters and C log nA . When performing computational SCC analyses. resulting with the following equation: PLM T log t r C M N log (7. Other time tempera- 121 . Garofalo [150]. The approach combines the Arrhenius law for the thermally activated process rate with the constitutive equation by Norton (7. . tr is time of rupture. Later on. Larson and Miller developed a time temperature parameter approach [154] that relates the stress state to the time of rupture. tr. constitutive equations relating these parameters to stress and temperature have been developed. See Section 3. more advanced and also more complex constitutive equations for modelling creep have been developed. and thus it appears that further procedure and test/input data treatment development would be necessary. those by Graham and Walles [149].g. both being material specific independent constants for which values are obtained experimentally. it is quite difficult to avoid ending up with excessively conservative results.4-1) where A is the rate coefficient (considered temperature dependent) and n the creep exponent. To obtain these.5 for a more detailed description of the issue. Obviously. this calls for contributions from several fields/sources of science. T is temperature. the most relevant parameters with regard to creep degradation are the strain rate. which expresses the relation between strain rate.7. Deterministic degradation modelling methods for power plant components potential of the crack front. e. 7. . One of the earliest and still commonly used constitutive equations for modelling creep is that by Norton [148].4-1). as follows: A n (7.4 Creep Creep degradation occurs in power plant components that operate at high temperatures relative to the material melting point. . Dyson and McClean [151] as well as Merckling [152]. This equation describes creep behaviour mainly in the secondary regime. and prevailing stress. From a continuum point of view. most often all time temperature parameters are fitted as polynomial stress functions. called mixed rule. called strain fracture rule. As according to commonly used notation. is expressed as [156]: i ri 1 (7.7.4-3) but based on strains. One commonly used cumulative creep damage equation. is expressed as [157]: ti t ri 12 i ri 12 1 (7. a further developed combining model as compared to that expressed by equation (7. the number of all considered conditions. n. such as that by Manson and Haferd [153]. a creep damage approach combining time and strain. is not shown in the equation. However. Deterministic degradation modelling methods for power plant components ture parameter equations have been developed as well.4-5) Finally. A similar creep damage model as that expressed by equation (7. is expressed as [155]: ti t ri 1 (7.4-3) where ti is time spent at condition i and tri is life to rupture at the same conditions. called life fraction rule. When the fractional damages add up to unity.4-2) indicates that the Larson-Miller parameter is linear against logarithmic time.4-4) where i is strain cumulated at condition i and ri is rupture strain at the same conditions. The most common approach to calculation of cumulative creep damage is to compute the amount of life expended by using time or strain fractions as measures of damage. as obtained from associated standard/normative documents.4-6) 122 .4-5). Cumulative damage equations for creep exist as well. Equation (7. is [158]: k ti t ri 1 k i ri 1 (7. Then. then failure is postulated to occur. such as grain boundary cavitation. time scale and temperature range in question. During the growth of a macroscopic crack at high temperatures. local creep degradation modelling procedures are more on focus than those considered this far. Here. Time dependent fracture mechanics approaches are required when creep failure is controlled by a dominant crack in the structure. secondary and tertiary creep. Numerous constitutive equations for modelling creep exist.7. all four types of creep response can occur simultaneously in the most general case. Components that operate at high temperatures relative to the melting point of the material may fail by slow and stable extension of a macroscopic crack. nucleate during tertiary creep. however. Fracture mechanics based methods can also be applied for modelling creep degradation. creep failure is postulated to occur when the fractional damages add up to unity. being based on continuum approach corresponding to more globally occurring creep degradation. and in the primary and secondary stages of creep at moderate distances from the tip. The material at the tip of a growing crack is in the tertiary stage of creep.4-3) to (7. Deterministic degradation modelling methods for power plant components where k is constant. The material may be elastic remote from the crack tip. 123 . For the equations (7. Deformation at high temperatures can be divided into four regimes: instantaneous (elastic) strain. Traditional approaches to design in the creep regime apply only when creep and material damage are uniformly distributed.4-6). The elastic strain occurs immediately upon application of the load. and for application they all require a sufficient amount of experimental creep data for the material. and the earlier mentioned creep regimes primary. see Figure 7. since the material is obviously failing locally. Whereas microscopic failure mechanisms.4-1. Typically at least partly/mostly such data are available on the technical literature. A formal fracture mechanics approach to creep crack growth was developed soon after the J-integral was established as an elastic-plastic fracture parameter. see equation (4. [161] independently proposed what became known as the C*-integral to characterize crack growth in a material undergoing steady-state creep. defined as: kl w 0 ij d ij (7.4-8) The analogy by Hoff [162] implies that the C* integral is path independent.3-5) in Section 4. [160] and Nikbin et al. The C* integral is defined by replacing strains with strain rates. Landes and Begley [159].3. When strain rate in secondary creep follows a power law: 124 . [30]. and displacements with displacement rates in the J-integral. as follows: C* wdy ij nij ui ds x (7. from ref.4-1.4-7) where u i are displacement rate components and w is the stress work rate (power) density. Creep zones at the vicinity of the crack tip. Deterministic degradation modelling methods for power plant components Figure 7. Ohji et al.7. because J-integral is path independent. 3-9) in Section 4. ij (7. the C*-integral defines crack tip conditions in a viscous material.4-11a) 125 . secondary regime creep. ~ij . C* is the appropriate characterizing parameter. the crack growth rate is as fast that it overtakes the creep zone. as: t1 K I2 1 2 . n 1 EC (7.4-10b) where the constants In. the time dependent crack growth rate in a viscous material should depend only on the value of C*. provided steady state creep. When the crack grows with time. t1.3. In brittle materials.4-1). Deterministic degradation modelling methods for power plant components ij A n ij (7.3-8) and (4. n is a creep exponent rather than a strain hardening exponent. Note that in the present case. Experimental studies [159–162] have shown that creep crack growth rates correlate well with C*. and then crack growth can be characterized by KI because the creep zone at the tip of the growing crack remains small. if the crack growth is sufficiently slow for the creep zone to spread throughout the structure. or. Just as the J-integral characterizes the crack tip fields in an elastic or elasticplastic material.4-9) where A and n are material constants from the Norton equation (7. i.7. At the other extreme. the behavior of the structure depends on the crack growth rate relative to the creep rate. then it is possible to define a HRR type singularity for stresses and strain rates near the crack tip as: ij C AI n r 1 n 1 ~ n. see equations (4. Thus. Riedel and Rice [163] defined a characteristic time for the transition from short time to long time behaviour. C* applies to long time behavior.4-10a) and: ij C AI n r 1 n 1 ~ n. ij (7. Explained in another way. is the dominant deformation mechanism in the specimen.e. and ~ij are identical to the corresponding parameters in the HRR model. The C* parameter applies only to crack growth in the presence of global steady state creep. Both parameters approach C* in the limit of steady state creep.4-13) where (Ct)SSC is separately computed small scale creep limit for Ct.7.4-11b) When significant crack growth takes place over time scales much less than t1.0. The advantage of Ct is that it can be measured relatively easily. is total displacement rate and t is creep displacement. but this ratio is less than unity for small scale creep and transient behaviour. Saxena proposed the following interpolation between small scale creep and extensive creep: Ct Ct SSC 1 t C (7. t2. The latter quantity appears to be better suited to materials that experience relatively rapid creep crack growth. Riedel [166] introduced a new parameter Ch*. Deterministic degradation modelling methods for power plant components t1 J n 1C (7. which is the primary creep analog to C*. Saxena [165] defined an alternate parameter Ct which deviates to some extent from C(t). is defined by: t2 Ch 1 pC p 1 p (7. In the limit of long time behavior. C*/Ct = 1. The characteristic time that defines the transition from primary to secondary creep. the behavior can be characterized by KI. Based on a finite element analysis. while Ct is related to the rate of expansion of the creep zone. respectively): Ct C t1 t 1 (7. The primary creep may have an appreciable effect on the crack growth behaviour as well if the size of the primary zone is significant.4-14) 126 . The C(t) parameter characterizes the stresses ahead of a stationary crack. Unlike KI and C*.4-12) where t is time. a direct experimental measurement of C(t) under transient conditions is usually not possible. while C* is the appropriate parameter when significant crack growth requires times >> t1. Ehlers and Riedel [164] suggested the following simple equation to interpolate between small scale creep and extensive creep (short and long time behaviour. Webster et al.7. When primary creep strains are present also the interpolation scheme for C(t) is modified [166]: Ct t1 t t2 t p 1 p 1C (7.4-18) 127 . this parameter data may be missing in the available technical literature. da0/dt. Nikbin. can be approximated in relation to that in the steady state as: da 0 dt da s dt n 1 (7. This range describes the effects of constraint on crack growth due to both material properties and size/geometric factors. see ref.4-17) f where das/dt is the steady state crack growth rate and f is taken as the uniaxial failure strain. the steady state creep crack growth rate.4-16) where D0 and are material constants which can be measured experimentally using the NSW model. For plane stress conditions this failure strain is f f . [167]. 168. the initial crack growth rate. At least partly. da/dt. Deterministic degradation modelling methods for power plant components where p is material specific constant for primary creep. the values for which have to be assessed based on experimental data. As for crack growth rates during creep. [167. whereas for plain strain conditions it is f f 30 . 169] have presented a model which is based on fracture mechanics to determine the time taken for the creep damage to accumulate at the crack tip starting from first initial elastic loading. can be correlated with sufficient accuracy to the C* as follows: da dt D0 C (7.4-16) can be approximated as: da s dt 3C 0. Then. For most engineering metals equation (7. Essentially.4-15) The creep damage models presented here contain from a few to several material and environment specific parameters.85 (7. Deterministic degradation modelling methods for power plant components For most engineering materials the values of n usually fall within the range of 5 to 10. the redistributions of stresses occurring with creep must be taken into account. however. If the Charpy-V properties do not meet the above criterion.S. no accepted standard approach exists. the description concerning computation of irradiation embrittlement of NPP RPVs corresponds mainly to that in the U. Most European approaches for evaluating irradiation effects in reactor pressure vessels are based on the ASME code [173. In some countries. In design and for predicting remaining life in operation of components in elevated temperatures. It is therefore necessary to establish effective stress criteria governing creep and rupture under multi-axial stress conditions and to be able to interpret them in terms of uniaxial test data. ASME code Section XI [173. like U. Many countries apply the ASME code directly. which suggests that da0/dt is approximately 10 % of the corresponding steady state value. Here. 174] together with the U.99.. In addition.S. Revision 2 [171]. but a case by case best estimate analysis is carried out [177].5 Irradiation embrittlement The material changes caused by neutron irradiation that may affect the use of the NPP reactor pressure vessels (RPVs) are connected to the fracture resistance of materials. respectively. components in service generally experience multi-axial loading conditions. If the minimum impact energy and lateral expansion in the Charpy-V test (3 specimens) are at least 68 J and 0. 174] procedure.7. RTNDT is taken as the tem- 128 . irradiation reduces the fracture toughness of the materials and therefore it is important to know the rate of reduction as a function of irradiation dose. [170]. at a temperature equal to NDT + 33 C. RTNDT. extensive use is made of test data obtained with specimens subjected to uniaxial loading [11].K. However. 176] the approaches have. see the paper by Roberts et al. This temperature is determined from the nil-ductility temperature (NDT) and the Charpy-V impact test. Regulatory Guide 1. The NDT temperature is determined in accordance with the ASTM Standard E 208 [172]. but in some countries like France [175. 7. the NDT temperature is taken to represent RTNDT. As a rule. For a brief review of the former of these two aspects. experienced a national variation.9 mm. The reduction in the fracture toughness is evaluated with the aid of material specific reference nil-ductility temperature. 7. however. For welds. to account for possible plasticity effects. the margin is 31 C and for base metal. but the safety factors are different and also the definition of upper limiting plateau. e. Both approaches assume that the temperature dependence of fracture toughness is not affected by irradiation. “upper shelf”. The ASME code Section XI [173] includes both a static fracture initiation reference curve and a crack arrest reference curve. for modern steels RTNDT is equal to NDT. This margin is put on top of the experimental RTNDT . differs from that in ASME. where RTNDT is as explained above. minus 33 °C. The French approach originates from the ASME methodology. Reference [171]. the margin is not.5-1) where CF is the radiation embrittlement coefficient and f is the neutron fluence (×10E-19 n/cm2) with E > 1 MeV. It is additionally assumed that the difference between static initiation and crack arrest is constant. Reference [173] provides also equations for both of these curves. It is assumed that the true static fracture toughness shift and crack arrest shift is equal to or less than TK41J ( RTNDT). Normally. For the fracture mechanical assessment of normal operation conditions. the crack arrest curve also describes crack initiation. When two or more credible surveillance data 129 . requires the use of an additional margin in the determination of RTNDT . more than RTNDT /2. The ASME code is based upon linear-elastic fracture mechanics. the irradiation induced shift of the RTNDT is defined as the shift in the 41 J impact energy transition temperature. In the ASME code approach. TK 41J .10 log f (7. and the static fracture initiation reference curve emergency and faulted conditions. i. but it is more flexible. enabling the fracture toughness temperature dependence to be described by a single curve. In addition to the different reference curves. 19 C.g. In both cases. Based upon RTNDT the fluence dependence can be calculated as [171]: RTNDT CF f 028 0. Deterministic degradation modelling methods for power plant components perature at which the requirements are reached. The crack arrest reference curve is intended to describe normal operation conditions. The shift is either determined based upon Charpy-V impact tests or from the chemical composition of the material. applying a chemistry factor. also safety factors are applied.e. The French approach applies essentially the ASME fracture toughness reference curves. Both curves are given as a function of effective temperature (T-RTNDT). 9 mm lateral expansion transition temperature. RTNDT is generally controlled by TK 0. RTNDT0 and RTNDT in units of F. the European approaches also apply that concept in an effort to determine the irradiation shift directly from the steel chemistry. The French codes [175. The lateral expansion corresponding to a certain energy level is affected by the material yield strength. The French codes contain chemistry factor equations which are dependent on phosporus.5-3) where RTNDT0 is the initial NDT temperature for unirradiated material. [171] gives different chemistry factors for welds and base metal in a tabulated form. Deterministic degradation modelling methods for power plant components sets are available. 130 . Even when RTNDT is calculated with the chemistry factor. the radiation embrittlement coefficient is determined from the experimental data by least square fitting. The irradiation induced shift of the RTNDT is defined as the shift in the 56 J impact energy transition temperature. and T. The effects of irradiation on the crack initiation and arrest fracture toughness can be estimated by applying the shift in the RTNDT to shift the ASME lower bound curves by moving the curves by the same shift amount.02 T RTNDT0 200 26. The French approach does not apply additional safety margins. whichever is greater.9 mm .0145 T 200 RTNDT0 RTNDT 100 (7.233 exp 0. [171]. TK 56J . The French approach differs from the ASME methodology. 176] allow the fluence dependence of RTNDT to be determined experimentally.806 exp 0.78 1. ref. Definitions of critical fracture toughness for the lower bound crack initiation value KIC and the lower bound crack arrest value KIa are presented in Appendix A of ASME Section XI [173]. but leaving the shapes unaltered. 178]: K IC min 33.7. one is required to use the additional safety margin prescribed in ref.5-2) K Ia min RTNDT 160 (7. or the shift in the 0. TK 0. but do not give any recommendations for the type of expression to be used. For the ASME methodology type steels. Therefore. Increasing the yield strength makes plastic deformation of the specimen more difficult. copper and nickel. These relationships are expressed as [32.2 2. As for the chemistry factor.9 mm . while KIC and KIa are expressed in units of ksi in. joint damage by crack growth is considered. more general cumulative damage is covered as well. all environments will reduce the fatigue life of a component. In almost all corrosion fatigue cases. Since fatigue failures usually occur at stress levels below the yield strength after a number of cycles. corrosion fatigue is the reduction of the fatigue resistance of a material due to the presence of a corrosive environment. The nominal fatigue limit. cracking is transgranular [138].6 Modelling of interaction of degradation mechanisms Modelling of interaction of degradation mechanisms is considered in the following. is eliminated. the number of possible combinations for interacting degradation mechanisms is relatively high. Corrosion fatigue Corrosion fatigue is fatigue enhanced by corrosion reactions. Concerning corrosion fatigue. environmental fatigue correction factor approach for incorporating effects of environment into fatigue evaluations is briefly described as well.. competition models.7. the presence of a corrosive environment reduces the number of cycles to failure as well as the stress level at which failure occurs. More than two degradation mechanisms can participate in the interaction. and models for environmentally modified material deformation and fatigue properties [180–183]. These models can be divided into three categories: superposition models [179]. and (2) the mechanical fatigue fracture process in a corrosive environment is identical with the process in an inert environment. In the first case. These cases of interacting degradation mechanisms are corrosion fatigue and creep fatigue. Although corrosion fatigue is characterized by cracking like SCC. Thus. 131 .e. i. The superposition models and the competition models are developed based on the following assumptions: (1) an environmental fracture process is independent from the mechanical fatigue fracture process. Theoretically and in several cases in practise. if any. However. corrosion fatigue is not environment specific. whereas in the second case. Deterministic degradation modelling methods for power plant components 7. Many models which combine fatigue and stress corrosion effects have been suggested for use in predicting the corrosion fatigue crack growth rate as a function of fatigue loading frequency. the scope is here limited to cover the major degradation mechanism interaction phenomena encountered/experienced in the power plant environments. 6-3). Load cycle dependent corrosion fatigue normally occurs in environments that do not result in significant environmentally assisted cracking (EAC) under static loading and where mass transport and electrochemical reactions that contribute to fatigue acceleration are very rapid. . The time dependent corrosion fatigue is sensitive to the frequency. and f is the loading frequency. Deterministic degradation modelling methods for power plant components Corrosion fatigue can be load cycle dependent.7. Combining equation (7.3. the crack growth rate approaches the inert rate 132 . At high frequencies.2.6-2) EAC where da dt EAC is the average environmental crack growth rate over a loading cycle.6-2) indicates.2 and the EAC terms with a suitable procedure in Section 7.6-3) EAC Concerning the equations (7. Most material/environment combinations display both cycle dependent and time dependent behaviour. multiplied by the inert growth rate [30]: da dN aggressive da dN (7.6-2) gives the following more general expression for corrosion fatigue [30]: da dN aggressive da dN inert 1 da f dt (7. as follows [30]: da dN aggressive da dN inert 1 da f dt (7. the inert terms are computed with a suitable fatigue crack growth procedure in Section 7. Time dependent corrosion fatigue can be modelled by a simple superposition of the inert fatigue crack growth rate with the environmental cracking rate. time dependent. respectively. Of these the load cycle dependent corrosion fatigue corresponds to a simple acceleration of the fatigue crack growth that is insensitive to the loading frequency.6-1) inert This expression can be applied for K values above the fatigue threshold Kth for the inert environment.6-1) to (7. By definition. as equation (7.6-1) and equation (7. The acceleration factor may be a constant or it may vary with K. The crack growth rate can be represented by an acceleration factor. the growth rate is independent of frequency for cycle dependent corrosion fatigue. or dependent on both of these. Fen. the environmental crack growth per cycle dominates over fatigue.nom.nom N air. When there is a combination of time dependent and cycle dependent acceleration of crack growth rate. A saw-tooth waveform might be expected to produce less environmental cracking per cycle. for example. wrought and cast austenitic stainless steels. presently it is included in the JSME codes for nuclear power generation facilities.6-4) For the above mentioned four major categories of structural steels the nominal environmental fatigue correction factor has the following form: 133 . see refs. and Ni-Cr-Fe alloys. the cyclic loading followed a square wave form instead of a sine wave.RT. Deterministic degradation modelling methods for power plant components because the EAC growth per cycle is negligible. which is defined as the ratio of fatigue life in air at room temperature. the former dominates at low frequencies and the latter dominates at high frequencies. Another variable that can affect the corrosion fatigue crack growth rate is the wave form of the cyclic loading. as follows [184]: Fen. Nair.7. those being carbon steel. Environmental fatigue correction factor approach The report NUREG/CR-6909 [184] by USNRC provides an environmental fatigue correction factor (F en) methodology that is considered acceptable for incorporating the effects of reactor coolant environments on fatigue usage factor evaluations of metal components for new reactor construction. [185. In those instances. one might expect the average EAC growth rate to be greater because the maximum load is held for a sustained period with a square wave form. If. all else being equal. In the ref. The effects of reactor coolant environments on the fatigue life of structural materials are expressed in terms of a nominal environmental fatigue correction factor. a square wave form might actually result in less environmental cracking per cycle because a sustained load is less damaging than a continually rising load. RT N water (7. A mitigating factor is that environmental cracking is often faster during periods when K is increasing. 186]. The methodology reflects the earlier development on the subject carried out in Japan. Nwater. At low frequencies. to that in water at the service temperature. respectively. low-alloy steels. and the rate is proportional to 1/f. [184] the environmental fatigue correction factor methodology for performing fatigue evaluations is described for the four major categories of structural materials. U2. see Paragraph NB-3200 [174]. For carbon and low-alloy steels.e. as determined in ASME Section III [174] for Class 1 fatigue evaluations.2 U 3 Fen. see Paragraph NB3600 [174]. U3. nom exp C1 C 2 S T O (7. The values for parameters such as strain rate. the partial fatigue usage factors for a specific stress cycle or load set pair is multiplied by the environmental fatigue correction factor: U en.6-5) where S*. dissolved oxygen in water. and for carbon and low–alloy steels sulphur content.7.1 U 2 Fen. all transformed input data parameters are defined as a function of their nominal values. rising ramp with increasing strain or stress). considering the effects of reactor coolant environments is then calculated as: U en U1Fen..1 U1Fen. this calculation is performed only for the tensile stress producing portion of the stress cycle constituting a load pair.3 . the same applies otherwise but S* = 1. in Appendix I of ASME Section III [174]. Un.. …. corresponding to conditions below the strain amplitude threshold. Deterministic degradation modelling methods for power plant components Fen. O* and are transformed sulphur content.6-6) where the fatigue usage factor values are determined according to the current code fatigue design curves. For the two latter steel categories also O* is constant. For these four transformed input data parameters. To incorporate environmental effects into the Section III fatigue evaluation. as given e. level of dissolved oxygen and strain rate. though with steel category and in some cases water chemistry specifically differing values.i is the nominal environmental fatigue correction factor for the “i”th stress cycle. whereas C1 and C2 are constants. temperature. are also needed to calculate Fen for each stress cycle or load set pair..i . The necessary input data for this fatigue evaluation procedure includes partial fatigue usage factors U1. the definitions are given steel category specifically.nom is one. temperature. whereas for wrought and cast austenitic stainless steels and Ni-Cr-Fe alloys. Uen. Because environmental effects on fatigue life occur primarily during the tensile loading cycle (i. U i Fen. The minimum value of Fen.1 (7. T*. 134 . The values of the two constants C1 and C2 vary case specifically between 0 and 1.6-7) where Fen. The cumulative fatigue usage factor. respectively.n (7. or load set pair. U n Fen..g. Then. However. The correction factor. k k .7. The modified rate approach has been proposed to predict fatigue life under changing test conditions. Figure 7. and the stress/strain responses are in general three dimensional. It allows calculating Fen under conditions where temperature and strain rate are changing. is assumed to increase linearly from 1 with increments of strain from a minimum value min [%] to a maximum value max [%]. the actual loading histories encountered during service of NPPs are far more complex. and strain amplitude. T . Tk max k min (7. 135 . [184]. temperature. and k is the subscript for the “k”th incremental segment. Application of the modified rate approach to determine the environmental fatigue correction factor Fen during a transient. Fen .6-8) where n is the total number of strain increments. from ref.6-1 below. Fen for the total strain transient is computed as: n Fen k 1 Fen. Deterministic degradation modelling methods for power plant components Nearly all of the existing fatigue -N data concerning ref. [184] are obtained under uniaxial loading histories with constant strain rate. An illustration of the application of the modified rate approach is presented in Figure 7.6-1. which remarkably limits its feasibility to actual plant conditions where the stresses/strains experienced by components are in general three dimensional. and that the rules for determination of strain rate for NPP piping components are in the course of preparation. 136 . [187].7. the description of the modified rate approach appears insufficient for computing the strain values and consequent F en values for actual NPP piping components concerning typical load transients with varying load parameter values. More recently. i. [174]. [184] for carbon steel. which is mostly the same as that presented in the report NUREG/CR-6909 [184]. Namely. it is also mentioned in ref. On the other hand. that the presented strain rate computation approach has interim status. and neither is the choice of the associated strain component. a strictly linear approach. as caused by the LWR environment during service. The new developments include general guidance on combining plant load transients to load cycles for fatigue analysis purposes. Namely. ASME has provided Code Case N-792.e. see equation (7. the choice of sign for the cycle specific stress intensities computed according to the Paragraph NB-3200 of ref. [174] for load cycles as defined therein. Fatigue Evaluations Including Environmental Effects [187]. the provided approach for assessment of strain rate is more limited than e. the modified rate approach described in the NUREG/CR-6909 [184]. the partial usage factors Ui are computed according to the Paragraphs NB-3200 and NB-3600 of ref. [174] is not unambiguous. low-alloy steels. [187]. wrought and cast austenitic stainless steels and Ni-Cr-Fe alloys. Moreover. also presented are slightly updated versions of the ANL S-N fatigue design curves in air given in ref. it is simply given as the stress intensity divided by Young’s modulus multiplied by the time from the start of load transient for the stress difference corresponding to the stress intensity to reach the maximum value. This document [187] contains Fen based methodology for incorporating the effects of reactor coolant environments on fatigue usage factor evaluations of metal components. and the correspondence of these cycle definitions and the rising ramps with increasing strain or stress considered by the modified rate approach is unclear for the various possible kinds of load cycles encountered in the actual NPP environments. to supplement fatigue assessment procedures in ref. Deterministic degradation modelling methods for power plant components However. However. that for the four mentioned material types it can be used for evaluation of thermal and mechanical fatigue. Other gaps concerning the conduct of application of the Fen approach in practise exist as well. The main shortcoming of the NUREG/CR-6909 [184] Fen approach appears to be its uniaxial nature. It is also mentioned in ref.g.6-8) above. the damage rate is given by the sum of the following two equations: 1 da FA a FA dt T C s p s p p p k2 k2 for tension. daCA/dt. failure is assumed to occur. daFA/dt. where t is the time at a given stress and tr is the time to rupture at that stress. Due to restrained thermal expansion these transients cause thermally induced stress cycles. in which N is the number of cycles at a given strain range and Nf is the pure fatigue life at that strain range.6-9) where N/Nf is the cyclic portion of the life fraction. The stress relaxation period is divided into time blocks during which an average. and creep induced cavity growth. The time dependent creep life fraction is t/tr. A strain based damage rate approach which takes into account the rate of damage accumulation has been proposed by Majumdar and Maiya [189-192]. If the two damage mechanisms are additive. and for each time block t/tr is computed and summed up. is expressed as [188]: N Nf t tf D´ (7. Deterministic degradation modelling methods for power plant components Creep fatigue interaction Changes in loading conditions during operation cause transient temperature gradients to power plant components operating mostly at relatively high temperatures. Components which are subject to thermally induced stresses generally operate within creep range so that damage caused by both fatigue and creep has to be taken into account [11]. When D´ = 1. They view the total damage as consisting of fatigue induced crack growth. D´ is the cumulative damage index. combining the damage summation of Robinson for creep [155] and that of Miner for fatigue [99]. constant value of stress prevails.6-10) for compression. (7. 137 . The extent of ensuing fatigue damage depends on the severity and frequency of occurrence of these transients as well as of the material properties.7. The most common approach to take into account the simultaneous effect of creep and fatigue is based on linear superposition. This procedure. 7. Deterministic degradation modelling methods for power plant components and: 1 daCA aCA dt G G s p s p p k3 k3 p for tension, (7.6-11) for compression. where T, C, G, k2, k3 and s are material parameters dependent on temperature, environment and the metallurgical state of the material, p and p are the current and absolute values of plastic strain and plastic strain rate, respectively, whereas aFA and aCA are the crack and cavity size at time t. The parameters T and C are included to account for differences in growth rates in tension and compression. Equation (7.6-10) describes the crack growth induced damage due to fatigue, whereas equation (7.6-11) describes the crack growth induced damage due to creep, respectively. The parameters T and C are included to account for differences in growth rates occurring in tension and compression. The parameter G is given the appropriate sign for the tensile or the compressive stress regime. Final failure is computed as the reciprocal of the sum of the crack and cavity damage. For cases where the fatigue and creep damages are interactive (not additive), Majumdar and Maiya have proposed the following slightly modified equation [193]: 1 da FA a FA dt T C 1 ln aCA aCA,0 s p p k2 (7.6-12) Here it is assumed that cavities of size larger than aCA,0 interact with fatigue induced crack and increase the crack growth rate. As for parameter , when it equals zero the fatigue crack growth is unaffected by the cavities and reverts to continuous cycling as according to equation (7.6-10). The equation for the creep induced cavity growth does not alter, i.e. equation (7.6-11) is applied as such. The damage rate approach allows taking into account the effects of various wave shapes on fatigue life. This model has been successfully applied e.g. to 1Cr-MoV steels [194] and austenitic stainless steels [189-192]. The crack growth rate computation equations proposed Majumdar and Maiya contain several parameters the values for which have to be determined experimentally. Nikbin, Webster et al. [169, 195] concluded that a more straightforward model containing less parameters to be assessed experimentally can be 138 7. Deterministic degradation modelling methods for power plant components derived for computation of crack growth rate due to joint effect of fatigue and creep. This model equation for computing the total/combined crack growth rate, (da/dN)TOTAL, is: da dN C K TOTAL m da dt f (7.6-13) where the first term on the right side is the Paris-Erdogan equation for fatigue induced crack growth, see equation (7.2-12), f is frequency and da/dt is computed according to equation (7.4-16), respectively. The creep fatigue interaction damage models presented here contain from a few to several material and environment specific parameters, the values for which are obtained as based on experimental data. At least partly this parameter data may be missing in the available technical literature. 7.7 Commonly applied component degradation assessment procedures One convenient way to categorise various structural integrity/degradation assessment procedures is to divide them into low and high temperature procedures [196]. Low temperature procedures can be further divided into assessment procedures based on failure assessment diagram (FAD) calculation procedures and fitness-for-service guides, assessment procedures based on crack driving force (CDF) techniques and other procedures. More recently, also procedures that apply for both high and low temperatures have been developed. Low temperature procedures Low temperature structural integrity/degradation assessment procedures can be further divided to the following subgroups [196]: assessment procedures based on FAD approach; calculation procedures, fitness-for-service guides, assessment procedures based on CDF techniques, others. The FAD methodology is based on the use of an integrated graphical representation where fracture failure and plastic collapse are simultaneously evaluated by 139 7. Deterministic degradation modelling methods for power plant components means of two non-dimensional variables (Lr and Kr). These two variables represent the ratio between the applied value of either stress or stress intensity factor and the resistance parameter of the corresponding magnitude (yield strength or fracture toughness). Once the plotting plane is determined by the variables Lr and Kr, each procedure defines a critical failure line which establishes the safe or acceptable zone. This zone is the area bounded by the mentioned line and the axes. The procedures which apply FADs have been divided into two different groups according to differences in structural applications and objectives [196]. The following documents contain a FAD calculation procedure: R6 Method, Revision 4 [200], BSI PD6493 [202], SSM; A Combined Deterministic and Probabilistic Procedure for Safety Assessment of Components with Cracks [203]. The following documents, in addition to containing a FAD procedure, contain a fitness-for-service Guide: EXXON, Fitness-for-service Guide [204], MPC, Fitness-for-Service Evaluation Procedures for Operating pressure vessels, tanks and piping in refinery and chemical service [205], API 579, Recommended Practice for Fitness-in-Service [206]. The assessment methodology of the procedures based on the use of CDF diagrams is different from the philosophy of FAD based documents. The evaluation of fracture failure and plastic collapse are not computed simultaneously. Therefore, two independent steps are needed. On the one hand, a direct comparison between applied stress or load and flow stress or limit load, and on the other hand, a diagram should be plotted where the applied stress intensity factor, Jintegral or crack tip opening displacement (CTOD) is compared with the corresponding toughness values. The CDF diagrams are easier to interpret in a physical sense than the FADs. Moreover, if the CDF technique is used together with the J-R curve, the full crack propagation history is described [196]. The following documents present a CDF technique: GE-EPRI Approach [207, 208], ETM, The Engineering Treatment Model for Assessing the significance of crack-like defects in engineering structures [209] and The En- 140 7. Deterministic degradation modelling methods for power plant components gineering Treatment Model for assessing the significance of crack-like defects in joints with mechanical heterogeneity (strength mismatch) [210]. Documents that present other procedures than FAD or CDF: ASME Section XI [173] code, which contains figures and tables to which the real situations can be compared to, RSE-M Code [211], which is very similar to ASME code, KTA Code [212], which is very similar to ASME code. High temperature procedures The early approaches to high temperature life assessment show methodologies which are based on defect free assessment codes. The fracture mechanics based procedures basically followed the low temperature assessment procedures that have been developed within Europe and elsewhere [196]. The high temperature procedures include: ASME Code Case N-47 [213], RCC-MR Code [214], R5 Method [215], BS PD 6539 [216]. The early approaches to high temperature life assessment methodologies include ASME Code Case N-47 [213] and the French RCC-MR [214]. These procedures have many similarities, and both of them are based on lifetime assessment of uncracked structures. ASME Code Case N-47 [213] is the first design code to formally include linear damage summation as a method of predicting material failure at high temperature, consisting of a fatigue (Miner cycle summation) component and a creep (Robinson time summation) component. The French Code RCC-MR [214] incorporates many of the concepts behind ASME Code Case N-47 [213] but with modified stress analysis procedures. One advantage of the more recent R5 Method [215] approach is that a defect may be postulated and the subsequent behaviour may be predicted for likely service cycles. Whereas a step-by-step high temperature creep crack growth computation procedure is given in the British Standard document PD 6539 [216]. 141 7. Deterministic degradation modelling methods for power plant components Combined high and low temperature procedures During recent years, four European structural integrity/degradation assessment procedures that apply for both high and low temperatures have been published. These are SINTAP (Structural Integrity Assessment Procedures for European Industry) [197], FITNET (European Fitness For Service Network) [125], HIDA procedure [198] and FKM guideline [199]. These structural integrity/degradation assessment procedures cover all types of power plants as well as most of other types of industry plants. They also include compendia of analytical structural and fracture mechanics parameter solutions for most common industry plant component types, such as plates, straight pipes, pipe bends, T-joints and pressure vessels. 142 Different probabilistic approaches to model ageing phenomena are used depending on the degradation mechanism in question. it is assumed that the occurrences of successive events are exponentially distributed with the same parameter. there is a lot of failure data available concerning such components and thus statistical analyses can be performed.e. A brief description concerning some probabilistic approaches to model ageing of power plant component is presented in the following. The behaviour of degradation mechanisms is often stochastic and thus probabilistic techniques should be applied in the modelling. information of the degree of component degradation is obtained by surveillance and condition monitoring. This mainly concerns active components. The ageing of components which have a mean time to failure that is significantly shorter than the assumed plant lifetime is called short-term ageing. the components are often assumed to have a constant failure rate. 143 . Probabilistic degradation modelling methods for power plant components 8. such as pumps and valves. the component failures occur according to a homogeneous Poisson process. In straightforward reliability calculations. In the Poisson process. This mainly concerns passive components. The evaluation of the expected remaining lifetime is based on monitored parameters. In the case of long-term ageing. Probabilistic degradation modelling methods for power plant components Time dependent probabilistic models can be used to identify possible trends and to predict the future behaviour of components. Typically. Then. The component failure rates generated from large amount of data in reliability databases are usually based on the assumption of constant failure rate [217].8. the amount of failure data is very limited. i. such as pipes. with the result that it is not possible to estimate the most likely strength of the structure.2-12). initial crack distribution. Crack initiation and growth in these components can be caused by various phenomena. 144 . The most commonly used type involves randomisation of an appropriate crack growth equation. the scope of described probabilistic modelling methods mainly concerns initiation and growth of cracks in passive power plant components. see equation (7. Thus. e. For instance.2 Probabilistic fracture mechanics based methods 8.g. Probabilistic degradation modelling methods for power plant components 8. By doing so. There are three basic classes of probabilistic models for crack propagation. the variability in the stress range. one approach to modelling random fatigue is to treat the parameters of the ParisErdogan equation.1 Probabilistic structural mechanics based modelling methods Here. the crack growth rate should be modelled probabilistically [32]. C and m. such as fatigue and corrosion. Many of the shortcomings of traditional deterministic analysis methods can be alleviated or overcome by applying probabilistic analysis methods. a fracture criterion that relates the stress amplitude and the crack length to failure must be applied if the stresses are varying. Because there is much scatter in empirical crack propagation data.8. failure due to crack growth is the dominant degradation mechanism for most power plant components that are made of metal(s). The main disadvantages of all deterministic analysis methods are that [45]: a) Properties and partial safety factors (where used) are often not given as best estimates or most likely values. material properties and environmental factors can be characterised [32]. four factors must be explicitly considered. Other types of models include evolutionary probabilistic models (Markov or diffusion models) and cumulative jump models [32]. The randomisation is carried out by introducing appropriate quantities characterising random effects into the empirical crack growth law.1 Introduction Of the many possible degradation mechanisms. as random variables. These are load history. 8. Because the critical crack size depends on the magnitude of the applied load. the Paris-Erdogan model.2. or their combination. crack growth history as a function of the load history and a failure criterion. to estimate the fatigue reliability of a component. since fracture toughness data in the ductile to brittle transition region are widely scattered. and different types of structure. or the overload necessary to cause failure or collapse. Probabilistic degradation modelling methods for power plant components b) The risk of failure or collapse. Recently. For example. is not predictable. and thus being able to perform sensitivity analyses of the factors affecting this process. Probabilistic fracture mechanics (PFM) provides an approach for probabilistic modelling of initiation and growth of cracks in components. the most common object of application has been pressurised components. primarily RPVs and piping in NPPs. Because of these complexities.g. However. For instance. Much of what happens in the real world. The advantages of PFM include also the possibility of modelling clearly the uncertainties related to the material degradation process. orientations and locations. initial crack size is typically one of the rarely well known variables. Other factors also introduce uncertainty into fracture analyses. however. it is not appropriate to view fracture toughness as a material constant having a single value. this parameter can be projected over a range of sizes with probabilities of occurrence 145 . the emphasis in the first efforts was more on crack initiation than fracture mechanics aspects. but considers one or more of the input variables to be random instead of having a single deterministic value. for example. The earliest PFM developments include aircraft applications. Most fracture mechanics analyses are deterministic. c) The assumption that most design parameters are known constants rather than statistical variables is in most cases a gross simplification. PFM is based on deterministic fracture mechanics (DFM) procedures.8. d) The safety factor approach is not so easy to apply in assessment of existing structures and for making maintenance decisions. e. to evaluate how changes in operating conditions can affect the failure probability of the structure [87]. For ageing management purposes it is of interest. such as earthquakes and accidents can result in stresses significantly higher than the intended design level. Weld joints have received a particular interest. as fabrication defects and cracks tend to concentrate in those regions [90]. Rather than assuming a fixed given initial size. a single value of fracture toughness is used to estimate failure stress or critical crack size. Extraordinary events. A structure may contain a number of flaws of various sizes. may vary widely for different structural members and components. crack growth should be viewed probabilistically rather than deterministically [30]. Service conditions. The results can then be used to ascertain the suitability of the component for service. length. temperature. In the simplest case. The failure probability at any time is then equal to the probability of having a crack larger than the critical size (which can also be a random variable) [90]. uncertainty in crack size for a given level of indication.8. Material properties. location. depending on the characteristics of the em- 146 .2. Not all of these parameters are considered random in every application. Only those variables with uncertainty or scatter producing the greatest effects on life or reliability calculations need to be recognised as such. First. and many of the input variables should be considered random. tensile properties. Probabilistic degradation modelling methods for power plant components and detection estimated for each size. 8. load cycle rate. a DFM analysis would then be performed after inspection for each size to provide the crack size distribution as a function of time (or stress load cycles) in service. Fracture mechanics input parameters typically considered to be random include [90]: Initial crack size. environment. depth. Crack detection probability. stress levels.2 PFM analysis procedure A typical PFM analysis is described in the following. Often one or more of the fracture mechanics variables is not known with sufficient certainty. fracture toughness. detection probability as a function of size. one or more parameters or variables are randomised. subcritical crack growth characteristics. The failure probability at any given time may be determined by combining conventional fracture mechanics calculations with an appropriate statistical approach. Then crack growth simulations are performed. Parameters or variables that can be considered as random were presented in Section 8. Cumulative failure probabilities are calculated as a function of time in operation [44]. A flow chart example of a PFM analysis is shown in Figure 8.g. During the crack growth simulation.2.8.and in-service inspections are considered and failure judgements of failure states. Figure 8.2. weight and/or influence functions. Fracture mechanics models employed are based on LEFM. pre.2-1.g. A flow chart of a PFM analysis.1.3 Engineering models and PFM Many components. Most of the applications of PFM for power plant components lead to very low failure probabilities which have to be determined by sophisticated numerical procedures from rather scarce and incomplete input data. leak or rupture. Newman-Raju solutions. Probabilistic degradation modelling methods for power plant components ployed analysis model. e. e. which could be otherwise rejected. can be saved by removing such arbitrary and unrealistic worst case assumptions as simultaneous occur- 147 .2-1 below. from ref. 8. are performed. [44]. or on EPFM or on a combination of LEFM and EPFM. e. particularly for applications in pressure boundary components. but not as pronounced as those concerning deterministic analyses. in which case the model becomes a probabilistic one. In addition.8. It may also be assumed. It is well established that EPFM models provide more realistic measures of fracture behaviour of cracked structures with high toughness and low strength materials compared to LEFM models. The primary reason for this consequence is that PFM in any form terminates the assumption that all conceivable deleterious events will occur simultaneously. Currently. Then. PFM models are based on conventional DFM principles. for example. Any of the components of a DFM analysis. PFM is applied to remove such unrealistic conservatism in component reliability analyses. The resulting conservative pure PFM analysis represents a notable improvement compared to deterministic model on which it is based. though in their pure form they have limitations similar to. the fracture response is essentially ductile. The second method consists of calibrating the PFM model against all available relevant experience. and allowing the model to account probabilistically for actual variation in critical parameters which are based in test results and/or experience. probabilistic engineering models contain a number of inherent limitations with respect to reliability prediction. When considered individually. probabilistic analyses based on EPFM models are not widespread and are only currently gaining notice. In contrast. Cracked components composed of these materials. Hence uncalibrated tools cannot treat problems for which fracture mechanics tools are not available. can be treated as a random variable. there are many methods and applications for PFM in various manufacturing industries. Here calibrated PFM essentially represents a combination of the database and pure PFM approaches [90]. nearly all of which are based on LEFM models. Probabilistic degradation modelling methods for power plant components rence of improbable events. that only one crack will exist and that the crack will be situated in a weld [90].g. These same limitations are present when models are constructed as a component of a pure PFM analysis [90]. crack size and stress history. and the material is 148 . the operational temperature region of power plants is above the brittle-to-ductile transition temperature of the component materials. when used in power plants. pose a serious threat to structural integrity. The first is to use conservative bounds for input variable probability distributions where accurate distributions are unavailable. The basic assumptions inherent for DFM are naturally included in the respective PFM models. Two methods are available to alleviate these shortcomings in pure PFM. Additional assumptions are usually made when a PFM model is constructed for a particular application. EPFM should be applied in fracture analyses of these components [91]. material properties and environment. skewness.4 Numerical PFM methods Numerical techniques for generation of failure probabilities from PFM models are generally required for more complex problems. The probabilistic aspects of crack repair are drawing attention nowadays as well. estimating these distributions can be difficult.) of the dependent variable from the moments of the independent variables and the partial derivatives of the functional relationship between the independent and dependent variables. as workable techniques are readily available.2. Still. It is often assumed that all detected cracks are repaired and that no new cracks or adverse conditions are introduced during the repair process. variance/covariance. This technique is also known as the method of system moments [92]. aspect ratio. convolution and MCS. a separate numerical generation technique would have to be employed to calculate the individual stress and strength distributions. The stress/strength overlap can be calculated once the statistical distributions of both parameters are known or estimated. The numerical techniques include: stress/strength overlap. Application of these techniques does not necessarily lead to a large amount of computations or complicating factors. Probabilistic degradation modelling methods for power plant components capable of considerable inelastic deformation. Only the moments of the distribution of the dependent variables are provided [90].8. In all but the simplest cases. Typical assumptions might include consideration of crack depth. variance. 149 . this technique is only approximate. It provides estimates of the moments (mean. In the case of fatigue crack growth. As such. even though this assumption may not be realistic. fracture toughness and subcritical crack growth characteristics to be independent [90]. fracture toughness. assumptions have to be made in the development of a probabilistic treatment. etc. Variance/covariance analysis by perturbation techniques is based on the assumption that some variables will deviate by relatively small amounts from their respective means. 8. Numerical results from the construction of PFM models are produced in a variety of ways. the time dependent strength distribution would at least depend on the following input parameters and their interactions: initial crack size. However. However. It increases in accuracy as the variances of the independent variables decrease. Then. A single simulation computation is therefore precisely equivalent to a single application of the corresponding DFM analysis. the probability of failure will be small. and the effective crack size. a0. Similarly. many simulations would be required for accurate enough results. thus drawing from the tails of the distributions controlling the failure probabilities. initial crack size. Sampling procedures are discussed in Section 5. being a general numerical technique for determining the distribution of the dependent variables. crack growth rate and nominal stress values are randomly sampled. MCS. In many instances. This can be alleviated by selective sampling from the distributions of the input variables. is computed after a number of load cycles. 94]. a. The proportion of times that a exceeds ac is equal to the probability of failure with N cycles.8. N. the derivation of the multidimensional integrals is often a complicated task. is broadly applicable to the generation of numerical results from PFM models [93. a value for KIC is sampled and used along with the sampled values of stress to provide the critical crack size value. In the context of a typical PFM problem. The inclusion of dependencies between input variables is a straightforward procedure with MCS. This is because the vast majority of sampled cracks would be from the much more likely portion of small cracks. it is not necessary to randomly sample from the initial crack size distribution. Difficult numerical integrations are usually required and care must be taken to obtain accurate results. Hence. importance sampling would involve sampling only from the large crack end of the initial 150 . In principle. In the following is described an example concerning importance sampling.6. derives the probability density function of the dependent output random variable from the functional relationship between the dependent and independent variables and the probability density functions of the independent variables. Selectivity is then compensated for during the analysis of the numerically generated results. this method is capable of providing analytical results. If it is known that large initial crack sizes associated with small probability are required for the failure to occur. However. As described earlier in Section 5.6. with the exception that some of the input values have been selected randomly [90]. MCS consists of selecting a value for each input variable at random from its assumed statistical distribution for substitution into the underlying deterministic model. Probabilistic degradation modelling methods for power plant components The convolution method. The incorporation of relatively simple statistical and physical dependencies among the input variables is also very difficult [90].5. ac. which is sometimes referred to as “direct synthesis”. The stratification of sampling space. and aspect ratio. The cumulative failure probability up to time t is computed as: m Pf T t j Nf t Nj Pj (8. called cells. Nj is the number of samples taken from the j:th cell. and Pj is the probability that an initial crack exists in the j:th cell.2-2 illustrates a typical stratification of an initial crack sampling space composed of normalised crack depth. Probabilistic degradation modelling methods for power plant components crack size distribution. from ref. Figure 8. a/c.8. Nf (t) is the number of failed samples in the j:th cell up to time t. [44].2-1) where m is the total number of cells. a/t. The results would then be compensated by the probability of having an initial crack in the portion of the sampled distribution. Stratified sampling is another sampling method that can be used to reduce the large amount of computational work related to assessment of small probabilities. Within each cell. The sampling space is divided into a number of mutually exclusive small subspaces.2-2. 151 . an individual sampling is carried out according to a postulated initial crack distribution [44]. Figure 8. A predetermined number of samples are then taken from each of those cells. where t is wall thickness and c is half of the crack length. of a semielliptical surface crack. the crack samples located in the upper part of the sampling space seem more likely to fail than those in the lower parts. a parallel processing algorithm combined with the stratified MC method is very useful [95. The failure rate is also called the hazard rate.3-1) where t is time.3-2) If the probability of a failure within a short time interval (t. The reliability of the component at time t can be defined as [32. (t). the failure times are exponentially distributed and 152 . considerable reduction in the amount of computational work can be expected by ignoring these cells in a sampling plan. Since the samples taken from the “No failure” region would never lead to failure. 217]: t f t Rt (8. Modelling of ageing with probability distribution functions The time dependence in failure occurrence can be expressed with the failure rate. defined as the probability of failure given the component has been in operation for a certain time [32.3. Computation time would increase as a function of the number of cells and samples employed. t + t) is independent of the age of the component. 217]: t Rt exp 0 d (8. many cells and samples would be selected from the whole sampling space to ensure computational accuracy.8.3 Statistical modelling methods 8. f(t) is the probability density function of component failure and R(t) is reliability function. To overcome this problem. 8. t.1 Short-term ageing models The models discussed in the following are applicable to estimate component ageing from failure data. Since the assessment of the region of uncertainty cannot be known before calculation. Probabilistic degradation modelling methods for power plant components As shown in Figure 8. A region of uncertainty may exist between “Failure” and “No failure” regions. 96].2-2. The choice of the form of the prior distribution is generally based on computational reasons. and is usually referred to as the burn-in or infant mortality period. When the failure rate is increasing. the failure rate of components changes over time as is shown in Figure 8. The probabilistic short-term ageing models of power plant components presented here are divided to three groups [217]: component replacement models. ageing models of tested stand-by components.3-3) is increasing when x 0 for each t 0 . With respect to ageing. the failure rate is constant. Bayesian methods are of special interest since the amount of failure data of power plant components is often very sparse. The second interval. These methods are used to include all relevant information to the estimation of model parameters. In the Bayesian approach. the Bayesian approach requires the definition of the prior distributions. where the failure rate increases as the component deteriorates. [219]. The part of the curve after t2 represents the so called wearout failures. Modelling approaches to short-term ageing In general. These include Weibull.8. It is often referred to as the useful life of the component or system. In reliability engineering. is the period of random failures and is usually approximated by a constant hazard rate. from time t1 to t2. from time t0 to t1. represents early failures due to material and/or manufacturing defects. ageing models of repairable components. Several distributions commonly used in reliability engineering are suitable to describe events with an increasing failure rate. The first interval.3-1. The basic principles of Bayesian approach are described for example in ref. The parameter values of prior distributions may be based on the existing data and/or expert opinions. 153 . A distribution is an increasing failure rate distribution if [218]: x t Rtx exp x d (8. In this case. which depicts the so called bathtub curve. the probabilities are always conditioned to the background information. this is the period of the component lifetime that is of concern [32]. gamma and log-normal distributions. Probabilistic degradation modelling methods for power plant components the failure occurrence is independent of time. it is said that component is ageing. Thus. Failure rate Burn-in period Useful life Ageing (t) The linear approximation to the Ageing portion t0 t1 Time t2 Figure 8. Component failure rate as a function of time. M(t). The expected number of renewals (i. A renewal process. F. the successive intervals between failures can be treated as random variables of a renewal process. identically distributed non-negative random variables. the failure histories can be described with renewal processes [220]. of which all do not disappear. The uncertainty of the times between failures is described with a distribution which is called the intensity of the process [217]. If the failed components are replaced with new ones.8. If the time for the replacement of a component is negligible and the time to failure of each component is distributed identically.3-4) where F k t is the k-fold convolution of F. failures) during the time interval (0. which can be considered as identical to the failed components in the beginning of their lifetime.3-1. 154 .e. is a sequence of independent. Probabilistic degradation modelling methods for power plant components The repair or replacement time of components is assumed to be very short as compared to the failure time and is thus neglected. t) is called the renewal function. It can be expressed in terms of the underlying distribution as [220]: M t k 1 F k t (8. In Case (c) the maintenance is assumed to have a fractional effect on the failure rate [217]. the mean times between component failures remain also constant.3-2 can be modelled as a non-homogeneous Poisson process. t) is [217]: pn t t exp n! n t (8. It may also be assumed that the increase in failure rate begins in the “Ageing” region of the bathtube curve. Some hypothetical examples of the behaviour of the failure function are shown in Figure 8. The probability pn(t) that failures have occurred during time interval (0.e. This is because the times between events cannot be straightforwardly considered as independent and identically distributed random variables. after repair the condition of the components is assumed to be the same as just before the failure.8. where a basic failure rate is assumed to be increasing but after the failure a decrease in the failure rate describes the effect of maintenance. A more realistic model is illustrated in Case (b). The renewal models cannot be applied in the case of repaired or maintained damaged components. Probabilistic degradation modelling methods for power plant components As the successive failure times are distributed according to the same distribution in a renewal process. which in turn is defined as: t t 0 td (8. called the cumulative failure function.3-2 illustrates a situation where the failure rate is increasing in time and is not affected by the failures. The failure rate mode presented at Case (a) of Figure 8.3-6) 155 . The statistical estimation of renewal processes reduces to the estimation of the parameters of the underlying life length distribution.3-2. i. it becomes dependent on the history. if the subsequent life lenghts have been observed. it is assumed that a baseline failure rate is increasing due to the environmental conditions during the component service [217]. In ageing analyses.3-5) where t is the integral of the intensity. Case (a) of Figure 8. If the development of the hazard function is assumed to be influenced by component repairs. 3-1.3-1. In a linear ageing model. in this order.3-7) where 0 is the constant random failure rate and is called the ageing acceleration rate. The failure function is defined in this case as [221]: t 0 t (8. as shown in Figure 8.3-8) t 0 t t0 (8. Probabilistic degradation modelling methods for power plant components When (t) is constant the model reduces to the homogeneous Poisson process. The second term of equation (8.3-9) where t0 is time for start of operational lifetime. They are defined. The most commonly used non-homogeneous Poisson process is the Weibull process. see Figure 8. (t) is increasing with time. For ageing components. Other frequently used ageing models are the exponential failure rate model and the Weibull failure rate model. by [222]: t 0 exp t (8.3-7) describes the “Ageing” region of the total failure rate. 156 . the effects of the ageing process on the component failure probability are expressed as a simple linear equation that relates the component failure rate to exposure time of the component to the ageing mechanism(s).8. 3-2. Realisations of failure rates of some ageing models with imperfect repair. [217].8. from ref. failure rate failure rate failure rate 157 . Probabilistic degradation modelling methods for power plant components Figure 8. t Tk . [32]. it is suggested that the size of a data sample should be at least 100 in the case of Weibull rate model. the Weibull failure rate being the most commonly used of the three. [224] observed that the linear failure rate performed poorly in more instances than the other two models and due to its nonexponential form was most difficult to work with analytically. Tk are maintenance times and ai is the maintenance specifically constant effect of i:th maintenance.3-9) has its historical roots in machine and material lifetime analyses. Probabilistic degradation modelling methods for power plant components Each of the three models presented in equations the (8. t Ti . the failure rate is defined as a function of time and number of failures/maintenances by [217]: k t t t j 1 ai . failure data pertaining to component ageing is scarce and justification for parametric models is sometimes doubtful. which is the simplest of these models. as is illustrated in Case (c) of the Figure 8. Now the failure rate can be defined as [217]: t ai t t . It is not certain how much data are exactly needed for accurate definition of model parameters.3-11) The decrease may be expressed for instance with the fractional learning model. as mentioned earlier. The linear failure rate. as is in Case (b) of the Figure 8.11. the data will most likely fit these three models adequately. If there is a sample of recorded component lifetimes that is large enough and the parametric constraints do not greatly contradict the true underlying lifetime distribution. In ref. The maintenance effect ai must be defined so that (t) remains positive. Tk 1 (8. Ti 1 (8.3-7) to (8. When it is assumed that maintenances cause a decrease in the otherwise increasing failure rate. It may also be assumed that the maintenance affects the failure rate in a multiplicative way. as [217]: t Ac i . t Tk .11.8. 223]. Wolford et al. it is suggested that good approximate tests and confidence intervals can be obtained with somewhat smaller samples. In ref.3-10) where t(t) is increasing failure rate. [225]. has been used in determining age dependent risk [221. where the failure rate is a function of events. Tk 1 (8. However.3-12) 158 . 8. Probabilistic degradation modelling methods for power plant components where c is the reduction factor and A is the event rate parameter before the first event. More generally, the repair may be assumed to have an effect on some parameter of the failure rate as [217]: t t t, i (8.3-13) where i is the maintenance specifically constant value of at i:th maintenance. The models above present only some specific cases of repairable component ageing models, several other corresponding models may be developed. Besides the models of repairable components, the ageing of tested stand-by components may be modelled in terms of failure probability in the test situation. In such models, the time may be discretised to describe for instance i:th test interval which may facilitate the computations. The probability of failure is assumed to be constant during the test interval. The increase of the number of failures in a test may be due to the fact that components wear between the tests, and degraded but not yet failed items are not identified in the test. Furthermore, the repaired component may not correspond to a new one after the repair. An example of an ageing model for tested stand-by components is the modified binomial model. In this model, the successful cases in tests are binomially distributed but the distribution parameter is allowed to evolve in time. This kind of model can be used to detect a trend in the results [217]. Unless the ageing model suits perfectly the underlying failure rate for the examined components, the model results may not be directly applicable. To overcome this drawback, it is suggested [32] to use a non-parametric inference based procedure. This includes a test for increased failure rate, a test for increased failure rate average, and test for “new better than used”. 8.3.2 Long-term ageing models Information about long-term ageing of components is obtained via condition monitoring. This data can be used e.g. to assess the remaining lifetime of the components. Examples of long-term ageing mechanisms include crack growth in metals, which issue is discussed in Chapter 7, and the combined thermal and radiation ageing of polymeric insulating materials in cables. The evaluation of long-term ageing is often based on the expected physical phenomena causing the component degradation. 159 8. Probabilistic degradation modelling methods for power plant components The use of probabilistic models instead of deterministic ones is needed, when the experiments show a significant scatter in the results. Examples of such uncertainties are variations in material properties and changes in environmental conditions. Modelling approaches to long-term ageing The probabilistic long-term ageing models of NPP components presented here are divided into three groups [217]: randomised deterministic models, stress-strength-time models, cumulative damage models that apply Markov chains. In the randomisation of deterministic models, parameters describing uncertain phenomena are treated as random variables, or a random noise term is added in the model to account for variations. The scatter in the results of experiments is used to estimate the distribution parameters. One approach to randomisation of deterministic models is PFM, see Section 8.2. The advantage of PFM is the possibility of modelling clearly the uncertainties related to the ageing mechanism, and thus being able to perform sensitivity analyses for the factors affecting it. The time evolution of structural reliability can be described with a time dependent model, where the applied stress or load and the strength of the structure are stochastic processes. Stresses and strengths may vary during long operation intervals as a function of time or the number and/or severity of stress applications. Ageing denotes the decrease in strength as a function of time, cyclic damage denotes the decrease in strength as a function of the number of stress cycles and cumulative damage denotes the case in which the decrease in strength is by both the number and severity of stress applications. Component failure occurs when strength has decreased below a stress dependent level. Failure can also occur when the cyclic or cumulative damage has exceeded a certain limit corresponding to the strength. The reliability of the component can be defined as [32]: Rt PY X , 0, t (8.3-14) where Y(t) denotes the strength of the component and X(t) denotes the applied stress. 160 8. Probabilistic degradation modelling methods for power plant components The stress-strength-time models may be classified according to the uncertainty about stress and strength [226]. Both stress and strength can basically be classified as deterministic, random-fixed or random-independent. Illustrations of some stress-strength-time modelling cases are presented in Figure 8.3-3. Case (a) of Figure 8.3-3 describes conditions where strength is assumed to be constant and stress is an increasing random function. If the degree of degradation corresponding to the accumulated stress can be described with a Wiener process [227], the failure time distribution has the form: f t 1 y exp 2 t t y 2 t 2 2 (8.3-15) where y is the constant value for strength, denotes mean, denotes standard deviation and t is time. Case (b) of Figure 8.3-3 illustrates a cumulative damage model, where load peaks occurring at random weaken the strength a discrete amount. If the times between the occurrences of loading events are exponentially distributed with the same parameter and the strength decrease is constant, this case can be modelled as a Poisson process. Additional uncertainty arises if the strength of the component is not necessarily reduced by the shock [217]. Case (c) of the Figure 8.3-3 illustrates a case where the occurrences and durations of loads are random, but the level of stresses is constant. It is assumed that the decrease in strength during each loading depends on the duration of the loading. If the occurrences of the loads are Poisson distributed and the duration of each load is exponentially distributed, the reliability of the structure can be computed [217]. Let us assume that the stress rate and the duration are exponentially distributed with parameters and . The failure occurs when the strength has decreased below a critical level, which is called d. The reliability of the structure is the probability that during the time interval (0, t) the accumulated time in the stress state (T(t)) is less than the critical value d [228]: d PT t d e t d 1 t d 0 e y y I1 2 t d y dy (8.3-16) 161 8. Probabilistic degradation modelling methods for power plant components where I1(x) is the modified Bessel function of order 1 [229]: 1 i 2 1 i 2 I1 x e J 1 xe (8.3-17) where J1 is the Bessel function of the first kind of order 1. Case (d) of the Figure 8.3-3 illustrates circumstances in which there are two sources for stress, one being constantly in effect (e.g. temperature or radiation) and the other consisting of a variety of discrete shock loads. The accumulated stress is then [217]: t X t 0 X c t dt aN t (8.3-18) where Xc is the continuous load and N(t) is a point process describing the occurrence of shocks. The failure occurs when the accumulated stress X(t) exceeds the strength. The strength may be assumed as random but independent of time. 162 8. Probabilistic degradation modelling methods for power plant components Figure 8.3-3. Some simplified stress-strength-time modelling situations, from ref. [217]. The simplified cases above are applicable only to very limited situations. In practise, the formulation of the functions Y(t) and X(t) causes problems. There 163 3-19) where pi are the initial probabilities of various damage states. here defined as n. is fatigue crack growth. the cumulative damage can be modelled with Markov chains [230]. that of a pipe.g.8. The crack growth can be modelled with a transition probability matrix. Discrete time Markov models are applicable to describe situations where the damage is caused by cyclic loadings. The final state. The model becomes continuous if the cycles are assumed to occur randomly according e. which can be described with a stationary. is described with the following state vector: 0 p0 p1 pi pn (8. the amount of damage is discretised to specific damage states.3-20) where P is the transition probability matrix. discrete time and state Markov process. In this case. In this example. when k < m. 164 . which contains the transition probabilities from each damage state to all considered more severe damage states. Probabilistic degradation modelling methods for power plant components are almost invariably difficulties in obtaining stresses and especially in obtaining strength. can be computed as: m 0 Pk D Pm k (8. e. 0.g. where the states represent the depth of the crack and the discrete time describes the fatigue cycles. the inspection procedure can be described with matrix D.3-21) This approach concerns the cyclic loading and the Markov property is assumed only between the crack tip at the end of the cycle. the crack size distribution k is: k 0 Pk (8. the crack propagation is not continuous in time. in addition. which contains the detection probabilities of cracks. it was assumed that at the time instant t the number of loadings needed to determine the distribution of the crack depth is known. If. An example of an ageing mechanism. If a detected crack is repaired and no new cracks initiate during the inspections. After k cycles. corresponds to the situation where the crack has propagated through the component wall. to a Poisson process. The crack depth distribution after m cycles with one inspection between the cycles k and k+1. The initial size of the crack. In the United States the method is referred to as PRA. aerospace. for car manufacturing. it can be used for analysing of almost any complex technical system.9. the method is referred to as PSA. Risk analysis applications for power plant systems 9. In most countries. 165 . A detailed description of these approaches would be beyond the scope of this thesis. the technique is practically the same. An accident sequence typically includes an initiating event followed by any number of failures or successes of control and mitigating functions. chemical and process industries.g. wide scope and importance in power plant applications. Risk analysis applications for power plant systems 9. Due to their complexity.1 Introduction Risk analysis applications presented here are used in several industries. Despite having two names. they are presented here in this separate chapter. The Finnish Radiation and Nuclear Safety Authority (STUK) has adviced/announced that also in Finland this method should be referred to as PRA. thus instead an overview is presented here. The scope of applicability of PRA/PSA is wide. and the responses of systems or equipment to developing conditions. Recently. enabling conditions that retard or allow the accident sequence.2 Probabilistic risk/safety assessment Introduction PRA/PSA is a technique used to identify combinations of events that compose accident sequences and estimate their frequency of occurrence together with consequences. The considered risk analysis applications are PRA/PSA and RI-ISI. 9. In addition to power plants the technique is also applicable e. logical. manufacturing or assembly and operation. nuclear power. or the adverse consequences that the technological entity may be eventually subjected to as a result of the occurrence of the initiator? 3. In general. Different paths through each event tree represent different accident sequences beginning with the same initiating event but having a different outcome because of system failures or conditions that occur during the course of the accident. a PRA/PSA is needed when decisions need to be made that involve high stakes in a complex situation. or what are their probabilities or frequencies? Motivations to perform PRA/PSA In many modern technological applications (e. Branches of an event tree that represent failures of complex systems or processes are often developed using fault trees. Accident sequences for complex systems or processes can then be ranked according to the severity of their consequences and the estimated frequency of their occurrence [82]. PRA/PSA usually answers three basic questions [1. each event in the sequence is assigned a probability or frequency from operational data. or what are the initiators or initiating events that lead to adverse consequences? 2. chemical processing industry. To estimate the probability of an accident sequence. PRA/PSA has proven to be a systematic. and comprehensive tool to assess risk (likelihood of unwanted consequences) for the purpose of [84]: increasing safety in design. Fault trees are used to help the analyst systematically deduce the possible causes of a failure or fault.9. What can go wrong with the studied technological entity. The frequency for an accident sequence is estimated using probability mathematics to combine the probabilities of the events which in turn combine to create the sequence. How likely to occur are these undesirable consequences. etc. Uncertainty in the probability or frequency estimates is included in the frequency estimates of the accident sequences using simulation and other mathematical techniques. What and how severe are the potential detriments. 49. operation and upgrade. Appropriate resource allocation depends on a 166 . calculations and/or expert judgements.. saving money in design.). 71]: 1. Risk analysis applications for power plant systems Accident sequences are usually developed using event trees.g. Concerning NPPs. development of an inventory of possible techniques for the desired analysis.9. given that a core damage state occurs. plant. In addition to quantifying the likelihoods of the releases. presently PRA/PSA is applied mainly for NPPs. These are described as follows [83. three PRA/PSA levels are usually defined.g. e. the physical layout of the system or process (e. Level 2 PRA/PSA builds on the results of level 1 PRA/PSA. Level 2 PRA/PSA: Containment failure and radionuclide release frequencies are usually estimated in this level. also the actual amounts of fission products released are estimated in this level. are estimated in this level. facility. Scope of PRA/PSA As for power plant environments. Level 3 PRA/PSA: The offsite consequences from a release. Developing a comprehensive scenario set is a special challenge. In general. design). familiarisation and information assembly. early and latent cancer fatalities. maintenance and test procedures as well as protective systems. administrative controls. and systematic methods are essential [85].g. 86]: Level 1 PRA/PSA: Conditional core damage (CDF) is usually estimated in this level. collection of general data. 167 . given a radionuclide release occurs. identifying the ways in which releases of fission products can occur from the plant. estimating the public health risks and environmental consequences. in this level the events and event sequences that can lead to the exceeding of plant safety limits are identified and their probabilities of occurrence are quantified. Steps in conducting a PRA/PSA The steps in conducting a PRA/PSA are listed in the following [49]: methodology definition. This level provides insight into the accident mitigation and emergency response provisions. Risk analysis applications for power plant systems well formed risk model. Level 3 PRA/PSA builds on the results of a level 2 PRA/PSA. sequence or scenario development. component failures and human errors. predicting future availability of equipment based on past experiences and test data. and their inclusion to both event trees and fault trees. Risk analysis applications for power plant systems identification of initiating events. The most widely used method for determining the uncertainty in the output risk assessment is to use a sampling process. It is important that both natural variability of physical processes and the uncertainties in knowledge of them are properly accounted for [85].9. failure data analysis. internal events external to the process. fault tree and event tree sequences are quantified to determine the frequencies of scenarios and associated uncertainties in the calculation. such as those due to coupling between their failure mechanisms. definition of a complete set of scenarios that include all the potential propagation paths that can lead to the loss of confinement of the hazard following the occurrence of an initiating event. quantification. These probabilities are then combined through the risk expression to determine the value of the risk assessment for that sample. correspond to events that originate within a system. in which values for each basic event probability are derived by sampling randomly from the probability distribution of each event. The development of scenarios introduces model assumptions and model parameters which are based on what is currently known about the physics of the relevant processes and the behaviour of systems under given conditions. external events. This 168 . application of event tree and fault tree techniques to considered systems. Uncertainty and sensitivity in PRA/PSA Randomness (variability) in the physical processes modelled in the PRA/PSA imposes the use of probabilistic models. identification of events (abnormal events) that could result in a hazard. system analysis. quantification of initiating events. identification of dependencies between event paths. correspond to events that originate outside of the system. dependent failure considerations. etc. because of their complex nature. Summary of PRA/PSA approach PRA/PSA is generally used for low probability and high consequence events for which insufficient statistical data exist. operational (resulting from enhanced procedures. These changes can be physical (resulting from plant modifications. [84]. component ageing.2-1. An illustration of the main steps of a PRA/PSA analysis is shown in Figure 9. In addition. if the PRA/PSA is 169 . implementation of data collection systems. PRA/PSA analyses are carried out to support decisions. there are also changes in our understanding of the plant. from ref. use of some of the PRA/PSA tools may not be necessary [84]. Risk analysis applications for power plant systems sampling process is repeated many times to obtain a distribution on the risk assessment [85]. All in all. are subject to change with time. A diagram of the main steps of PRA/PSA analysis approach.2-1. due to the analysis of operational experience.) and organisational. Sensitivity analyses are also frequently performed in a PRA/PSA to indicate analysis inputs or elements whose value changes cause the greatest changes in partial or final risk results. They are also performed to identify components in the analysis to whose quality of data the analysis results are or are not sensitive [85].).9. Figure 9. Living PRA/PSA Nuclear facilities. etc. If enough statistical data exist to quantify system or subsystem failure probabilities. development of improved models. Therefore. etc. which are given in Guide YVL 2. The Finnish Radiation and Nuclear Safety Authority (STUK) has also defined a set of numerical design objectives for Finnish NPPs. This has led to the concept of a “living PRA/PSA”. commercial airliners [85]. which can be defined as a PRA/PSA which is updated as necessary to reflect the current design and operational features with providing detailed documentation of the updating process [88]. e. Because of the large uncertainties in the estimates of fatalities. 170 . the PRA/PSA must be updated or modified when necessary to reflect the above changes. These levels may change as technology improves or emphasis changes with regard to environmental issues [50].g. The individual latent cancer fatality risk in the region between the site boundary and 10 miles beyond this boundary will be less than 2E-06 per year (one thousandth of the risk due to all other causes). has established safety goals for NPPs as follows [236]: The individual early fatality risk in the region between the site boundary and one mile beyond this boundary will be less than 5E-07 per year (one thousandth of the risk due to all other causes). the individual latent cancer fatality risk due to all causes was taken to be 2E-03 per year. Establishing the boundaries and guidelines for risk acceptability is a government responsibility. The latter refers to the release of radioactivity to the environment.S. According to STUK the following numerical design objectives cover the whole NPP: The mean value of the probability of core damage is less than 1E05/year. Thus. the NRC is using subsidiary goals in its daily implementation of risk informed regulation. Risk analysis applications for power plant systems to be of continuing use in the enhancement and understanding of plant safety. Acceptable risk levels The acceptability of risk by individuals depends on the degree of control over the risk producing activity that they perceive they have. Typically. These goals were established using as a criterion the requirement that risks from NPPs should be smaller than other risks by a factor of 1000. Probabilistic Safety Analyses [89]. people demand much lower risks from activities over which they have no control.A.8.9. These are the reactor core damage frequency and the large early release frequency. The Nuclear Regulatory Commission (NRC) of U. Design flaws and hardware related. such as RPV and reactor coolant piping. event tree and fault tree analyses of current PRAs/PSAs do not include passive SSCs. Risk analysis applications for power plant systems The mean value of the probability of a release exceeding the target value defined in section 12 of the Council of State Decision (359/91) must be smaller than 5E-07 per year. 171 .9. According to STUK the risks associated with various accident sequences of the PRA/PSA are to be compared with each other to ensure that no dominant risk factors deviating from the common risk level remain at a plant [89]. Technical bases to request and receive exemptions from unnecessarily conservative regulatory requirements. Approaches to reduce operation and maintenance costs while meeting or exceeding safety requirements. Some benefits and drawbacks of PRA/PSA After completing the compulsory PRA/PSA efforts. The current PRA/PSA models usually include the following drawbacks [32]: As a general practise. Although this argument may be reasonable when SSCs are new. These have included new insights into and an in-depth understanding of [1]: Design flaws and cost effective ways to eliminate them in design prior to construction and operation. The argument for doing so is that the failure rates for the passive SSCs are negligibly small. performing organisations have usually discovered benefits beyond mere compliance with regulation. Normal and abnormal operation of complex systems and facilities even for the most experienced design and operating personnel. operator related and institutional reasons impacting safety and optimal performance at operating facilities and cost effective ways to implement upgrades. However the containment has to be designed in such a way that its integrity is maintained with a high likelihood in case of both low and high pressure core damage. The traditional PRA/PSA approach assumes a constant rate for each component failure mode. a critical evaluation of this assumption is needed when the subjects of the investigation are older. 1 Introduction Presently. whereas risk-based inspection (RBI) approach is applied for conventional power plants and other industry plants involving structural systems bearing relatively severe loads and involving periodic inspections. could be different for older plants than for new plants. the current PRA/PSA methodology does not model other kind of dependencies. operations and safety.9. In-service inspection (ISI) is an essential element of the defence in depth concept. The current PRA/PSA methodology does not make such a distinction. the understanding of degradation mechanisms and the experience gained from the operating experience of NPPs. chemistry. RI-ISI reflects recent developments in PRA/PSA methodology. measured by risk. Risk analysis applications for power plant systems Except for failure rates of identical components that have been covered by common cause failure analyses and by complete state-of-knowledge correlation. PSA. stress analysis. ISI helps to confirm that basic nuclear safety functions are preserved and that the probability of radioactive materials breaching containment is reduced. ISI consists of non-destructive examination (NDE) as well as pressure and leakage testing. materials. Before resorting to a risk-informed pro- 172 . systems. design. The fundamental idea is to identify high-risk locations where the inspection efforts should be concentrated. such as human error rates and component unavailability because of testing and maintenance. RI-ISI aims at rational plant safety management by taking into account the results of plant specific risk analyses. inspection result analysis and final feedback to the risk analysis. in order to maintain a living ISI programme. maintenance. Failure probabilities of events other than hardware failure. together with reduced irradiation doses for the inspection teams [237]. The development of a RI-ISI programme requires expertise from a number of different disciplines including inspection. It also requires a longterm co-ordinated management commitment through inspection qualification.3 Risk informed in-service inspection (RI-ISI) 9. 9. The objective is to provide an ongoing improvement in the overall plant safety.3. the risk-informed in-service inspection (RI-ISI) analysis approaches are applied to NPP piping systems. A best result is achieved with a combination of all these approaches. and conditional large early release probability. assessment of degradation potential. CCDP. assessment of risks and the consequent forming of a risk informed inspection program. When NPP 173 .g. RRW).g. In the top level. statistical estimation from experience data. RBI requires a wide range of information in order to assess the probability and consequences of equipment failure and develop an inspection plan. the foremost of which typically being the amount and quality of available and applicable degradation data. based on deterministic structural models. use of formal expert judgement. it is essential. have become commonly used consequence measures. conditional core damage probability. In connection to RI-ISI. RAW). how much each of these approaches weighs as compared to others under various conditions depends on several aspects. RI-ISI increases the chances that mitigating actions will be taken. A plant database containing an inventory of the equipment and associated information is a useful way of managing the relevant data [3]. therefore. to obtain the backing and commitment of the utility and plant management [237]. such as risk increase factor. RDF (known also as risk reduction worth. CLERP. The plant specific PRA/PSA tool is used for the quantification of piping failure consequences. However. and risk reduction factor. e. The PSA allows the calculation of several risk importance measures. the RI-ISI analysis is divided into the following three parts: assessment of consequences. Risk analysis applications for power plant systems gramme. RIF (known also as risk achievement worth. and thereby reduces the frequency of failure [3]. There are basically the following three options/approaches for quantitative assessment of leak and rupture probabilities/frequencies of piping components: use of PFM and/or other structural reliability approaches. or a change to the operating conditions. e.9. By identifying potential problems. Inspection is an initiator for actions such as the repair or replacement of deteriorating equipment. large databases. An impending failure and its consequences are not prevented or changed by RI-ISI unless additional mitigating actions are taken. Risk analysis applications for power plant systems piping degradation data are available only scarcely.3 Process of risk-informed inspection planning The main elements constituting the process of risk informed inspection planning are as follows [237]: 1) 2) 3) 4) Assurance of the long term commitment of senior management to the risk informed methodology. 174 . 2) The identification of an inspection programme that will decrease the risk from selected sites as far as it is practical and with due consideration of the costs and accumulated radiation dose to plant workers. Collection and analysis of the information required to carry out the risk assessment.3.9. Definition of the scope of the equipment/structures to be considered in the application. The fundamental principles of any risk informed inspection programme are [237]: 1) The ability to define the consequence. Formation of the RI-ISI assessment team. The term optimise in this context is seen as a process that maintains defence in depth. and to a considerable extent that of expert judgement as well. b) minimising the radiation dose to personnel involved in the inspection activities. the role of structural reliability approaches is strongly emphasised. The advantage of such an approach is the optimisation of the inspection efforts.2 On principles of RI-ISI RI-ISI aims at a rational plant safety management strategy by taking into account the results of plant specific risk analyses. whilst [237]: a) improving or at least maintaining the overall plant safety.3. probability and risk associated with structural failures so that the ISI programme can focus on an integrated defence in depth strategy. 9. c) providing improved plant reliability. 9. which is a quantitative approach. Assessment of the consequences of failure for all the components included in the scope of the application. quantitative and semi-quantitative methods for RI-ISI In principle. the semi-quantitative and the purely qualitative approach. taking into account only the involved degradation mechanism(s) and some factors affecting their severity. and thus a purely qualitative approach to RI-ISI that does not make use of PRA/PSA insights would be difficult to justify. corresponding to the assumed levels of severity/threat. Both in quantitative and qualitative approaches. 9. In the nuclear industry. low and none. both failure probabilities and consequences are assessed with physical models and expressed in physical units. 238. 239]. 11) Assessment of the implication on inspection qualification. 12) Feed back of the obtained information (after completing the inspection). 240. Risk analysis applications for power plant systems 5) 6) 7) 8) 9) Definition of the level of the evaluation.9. three different approaches to the use of risk concepts in developing ISI programs can be considered: the purely quantitative. a notable example of a qualitative approach is the EPRI RI-ISI procedure [17. 175 . Assessment of the probability of failure for all the components included in the scope of the application. The use of PSA is the foundation of risk informed approaches in the nuclear industry. whereas a representative example of a quantitative one is the PWROG Methodology [242. the risks are assessed with PRA/PSA. Of the commonly applied RI-ISI methods. the failure probabilities are assessed qualitatively. with simply dividing these mechanisms to classes having such descriptive names as high. medium. 10) Choice of the components to be inspected according to chosen criteria.4 Qualitative.3. Performing sensitivity studies to determine the impact of changes in key assumptions or data. In the qualitative approach. The semi-quantitative approach combines features from both quantitative and qualitative approaches. Ranking of the risks associated with all the components. In the quantitative approach. 241]. the consequences are considered and expressed with the same quantitative measures. 3-2 in the following. Some European countries have adapted these methodologies to their particular context. Thus. based on a plant specific PSA and with due recognition of the current mechanistic understanding of any degradation mechanisms. Flow charts presenting the methodologies of these two RI-ISI methodologies are presented in Figures 9. However. Likewise. it is recognised that the current state of knowledge and understanding of some of the degradation mechanisms and the availability of the required plant data can be insufficient to accurately quantify the probability of failure.9. and thus the consequence analysis must be complemented by some degree of qualitative assessment. as only in such framework. many current PRA/PSA analyses may not be detailed enough for the level required in an ISI application. 241]. 9. it is possible to quantify the risk change achieved when implementing the RI-ISI program. 239] and EPRI methodology [17.3.5 Brief overview and comparison of two main RI-ISI methodologies The two main RI-ISI methodologies that are also widely applied in Europe are: PWROG methodology [242. issues and development of RI-ISI methodologies. Risk analysis applications for power plant systems Ideally. a purely quantitative approach should be chosen whenever feasible. 176 . 238. the most feasible approach to the development of any RI-ISI program today would arguably be the semi-quantitative one. 240. but have not added developments that differ from the original methods.3-1 and 9. Recently has been published a notable IAEA report [243] on current status. 9. Risk analysis applications for power plant systems Figure 9. 239]. 177 .3-1. 238. from refs. Flow chart of PWROG RI-ISI methodology. [242. 3-2. 241]. from refs. 240. 178 . [17. Flow chart of EPRI RI-ISI methodology. Risk analysis applications for power plant systems Figure 9.9. 9. The quantification of the POD is not a straightforward task. In the PWROG methodology. Risk analysis applications for power plant systems The main steps or structure in both the EPRI and PWROG methodologies are rather similar: scope definition. The detection probability is typically presented in the form of POD functions. which describe the detection probability as a function of the flaw size. The main differences arise from the way of performing the failure probability analyses.g. A quantitative measure of inspection effectiveness is needed in order to calculate the reduction in risk associated with inspection. the risk ranking is based on RRW. In the EPRI methodology. There are also other minor differences related e. The output from the European Network for Inspection Qualification (ENIQ) qualification process [252] is generally a statement concluding whether or not there is high confidence 179 . The impact of these differences has been studied in an OECD-JRC co-ordinated RI-ISI benchmarking project RISMET [251]. to the segmentation. the construction of a POD function requires a considerable amount of data before statistical confidence is achieved. while EPRI methodology uses CCDP (and CLERP). the approach is basically qualitative. risk ranking and finally selection of inspection sites.g. In the PWROG methodology. segmentation. For a recent notable document on POD issues. The use of risk importance measures is somewhat different. and quantification is basically done by using bounding values.3.6 Probability of detection of cracks The effectiveness of inspection is an important input parameter in RI-ISI analyses. [254]. The reliability of NDT techniques is usually quantified using flaw detection reliability in terms of the probability of detection (POD).9. it is often expensive and time consuming to produce such a large amount of data. Quantitative estimates of the inspection capability enable a better optimisation of ISI. It contains a literature review of some important papers and reports. probability of failure analysis and failure consequence analyses. a review of the statistical models that have been proposed for quantification of inspection reliability as well as descriptions and recommendations on statistical best practices for producing POD curves. flaw depth or length. e. In case of NDT methods. and the final selection of inspection sites follows different rules. However. a specific structural reliability analysis tool is used to quantify the failure probabilities. see ref. As a result. [252]. that within a segment. This is similar to the segmentation in the EPRI methodology.g. the ENIQ methodology is not designed to provide a quantitative measure of inspection capability of the type which can be used by quantitative RI-ISI. Then only a single POD value is used. In RI-ISI approaches.3. The POD curves used in U. Risk analysis applications for power plant systems that the required inspection capability will be achieved in practice. without accounting for the detection limit. The development of full POD curves seems both impossible. a more accurate and refined modification of the qualitative EPRI RIISI risk matrix procedure was suggested at VTT. Simplified estimates would be of sufficient accuracy in RI-ISI applications. The modifications include [245]: changing the degradation category in the risk matrix from qualitative to quantitative.7 RI-ISI approach developments by VTT VTT has aimed at developing a RI-ISI approach which uses quantitative classification for both failure potential and consequence assessment. 180 . The VTT approach also includes an expert panel to review the initial risk ranking.9. However. in the SRRA code of the PWROG RI-ISI methodology [239]. If the failure probability is not evaluated using a PFM approach. in ref. It also means it is difficult to “benchmark” the confidence associated with any given inspection qualification. a simplified approach can be used for the evaluation of the impact of inspections. the segmentation follows the basic principle. applications. as is shown e. This is in line with the ENIQ Framework Document [237] and the Finnish regulations.g.S. Studies on the sensitivity of the risk reduction to POD have been documented e. The simplified POD step functions can then be implemented in the analysis process and used for calculating the effect of inspections. in recent VTT reports [244. but an estimate is given based on expert judgement or a statistical approach from experience data. 253]. 9. component and defect range. In the VTT approach.g. have been subject to criticism due to sometimes unrealistically high detection capability assumptions. NPP piping systems are for risk analysis purposes typically divided into regions/parts called segments. for the specified inspection system. but recognising the above mentioned limitations in failure probability quantification [244]. e. and for RI-ISI applications also impractical. the degradation mechanism(s) and the consequence(s) of a piping failure remain unaltered. from both BWR and PWR units. 4. 3. and assessing and optimising the piping segment risks with a Markov system based analysis tool developed at VTT. 4 [200]. obtaining pipe rupture consequences from plant specific PSA. 249. 5. The PFM analysis tools were applied to a small but representative set of NPP pipe components. [247]. instead of evaluating them roughly based just on operational/process conditions as is done in the EPRI procedure.9. model for inspection quality. comparison of results for different inspection strategies. and a discrete time Markov process analysis). The overall method of the applied discrete time Markov procedure for piping risk analyses can be summarised in six steps as follows [245. construction of degradation matrix transition probabilities from PFM simulations and database analysis of crack initiation frequencies. yearly core damage probability and average values for both of these over plant lifetime. they indicate that the use of a simplified POD could be justifiable in RI-ISI applications. where measures of interest include yearly rupture probability. A good correspondence between the PFM results computed with these three tools was achieved. whereas the considered degradation mechanism was SCC. 181 . 250] and JRC approach (which is based on a probabilistic implementation of the R6 Method. Rev. 246]: 1. see ref. 2. and 6. which is used to construct inspection matrix transition probabilities. The PFM part of the VTT RI-ISI approach has also been benchmarked against some of other PFM analysis tools in a recent international project called PODRIS. Markov model to calculate pipe rupture probabilities for chosen inspection schemes. In PODRIS. As for other significant results. the main interest was to investigate whether it is justifiable to use a simplified POD curve. These other PFM/RI-ISI analysis tools were NURBIT by Det Norske Veritas (DNV) [248. crack growth simulations based on PFM. Risk analysis applications for power plant systems assessing the piping segment failure probabilities with a PFM based analysis tool developed at VTT. A. with the aim of benchmarking several RI-ISI methodologies. Canada and Japan.3. The applied RI-ISI methodologies were: SKIFS [255]. U. The RI-ISI applications were compared among each other and to the deterministic ASME XI ISI selection procedure. The riskinformed methodologies showed some significant differences and resulted in slightly different risk ranking and selection of inspection sites. The scope of the benchmark was limited to four systems.9.. PWROG-SE (an adaptation of the original PWROG methodology to the Swedish regulatory environment). representing utilities. Also VTT participated in RISMET [251]. and promote the use of RI-ISI [257]. various RI-ISI methodologies were applied to the same case. 239]. The results of the benchmark exercise RISMET improve the knowledge on differences in approaches and their impact on plant safety. In RISMET. and the RI-ISI approaches identify safety important piping segments that are ignored by approaches not using the PSA. Risk analysis applications for power plant systems 9. but the variety regarding safety class. regulators and research organisations. 238. the results of the benchmark indicated that the risk impact of these differences is small. potential degradation mechanisms and pipe break consequences ensured a good coverage of issues for a comparative study.8 RISMET project – benchmarking of RI-ISI methodologies The RISMET project was launched in 2005 by the Joint Research Centre of the European Commission (JRC) together with the Nuclear Energy Agency of the OECD (NEA). However. PWROG [242. EPRI [17] and ASME Code Case N-716 [256].S. It featured more than twenty participating organisations from Europe. 182 . consisting of four selected piping systems at the Swedish Ringhals 4 PWR unit [251]. 10. especially for components which are not safety critical. pressuriser. recirculation pump. expert opinions and probabilistic techniques for prioritising and determining risk significance of ageing. After the components are selected. in the U. Ageing research programmes have been conducted in several countries. main primary pump. steam generators. 234]. containment.g. Power plant component ageing analyses 10. cables and concrete structures.1 Selection of components for analyses The evaluation of safety significance and ageing sensitivity of components can be based on the analysis of operating experience.10. cables and concrete structures. When the ageing is detected. the most safety significant components are [231]: RPV. the analysis methods for ageing detection and prediction are chosen. vessel internals. Japan and France [32. containment. For most PWRs. the most safety significant components are [231]: RPV. pressuriser.. Power plant component ageing analyses Ageing of various SSCs affects the safety of power plants. vessel internals. steam generators. main primary pipes. For most BWRs. This chapter deals mainly with ageing analyses concerning NPP components. The objective of component ageing analyses is to identify the degradation mechanisms of components and the increase in failure occurrences. e. The 183 . The ageing analyses should be focused on those components that are the most significant from the safety point of view.S. 233. 232. The suitability of a procedure depends on the selected component and on the available data. suitable techniques are applied to prevent or mitigate the effects of ageing. are availability and costs. Other significant selection criteria. main primary pipes. to assess the remaining lifetime of components and to find suitable means to prevent or mitigate the effects of ageing. Current maintenance. If the collected degradation and failure data is properly analysed and reported. Some of the safety related components contribute more than others towards ensuring plant safety and the extent to which these components are susceptible to ageing also varies considerably [235]. IAEA has proposed a component selection process for NPPs [235]. which is based on a systematic examination of all systems and structures from a safety perspective. 184 . The approach presented here is based on a systematic selection and prioritisation of safety important NPP components for more detailed ageing analyses. It is neither practical nor necessary to evaluate and quantify the extent of the many thousands of individual power plant components. Safety important components may experience varying degrees of degradation due to ageing under different loading and environmental conditions. with or without severe consequences. the operating experience data can be effectively used to identify systems and components which are susceptible to ageing.10. The failure of safety important components can result in different degrees of loss of system safety functions. Power plant component ageing analyses results from PRA/PSA can (and usually are) be used in association with the probabilistic techniques [4]. Safety important components in a NPP have already been classified. Therefore. A NPP has a large variety of components. This examination is based on the following precepts: Required safety margins for NPP components have to be maintained at all times. a more rational and cost effective approach is required. Existing databases pertaining to ageing effects in NPP components are not sufficiently comprehensive. testing and inspection programmes in NPPs include different strategies of varying effectiveness to deal with ageing effects. Methods used for monitoring ageing effects in specific components and structures may vary significantly. many of which are essential for overall plant safety. Thus the basic management methods are controlling and slowing down ageing as well as replacement of components. expert elicitation of knowledge of structural integrity and materials. decisions on management of ageing should be made on the basis of identified degradation mechanisms and their severity.10. The application of these methods is component specific. the possibilities of replacement and early failure detection possibilities.g. use of check-lists and mechanism descriptions. 185 . re-evaluation of surveillance and condition monitoring methods. The effects of ageing on SSCs are detected through different mechanisms in plant operation. review of experience across similar industries or plants. 10. the ageing management means following the efficiency of current maintenance procedures and reviewing the test programmes. The selection of an efficient method for mitigating ageing in component material depends on an accurate determination and evaluation of the degradation mechanism that caused the ageing [6]. In the case of repairable or replaceable components which will age relatively fast compared to the plant lifetime. the expected ageing rate and mechanisms. In the case of components which are difficult to replace. depending on e. mitigate or restore the effects of ageing [6.2 Identification of ageing mechanisms The process used for identifying deterioration mechanisms for pressure systems and other systems containing hazardous materials should be conducted in a wide ranging and systematic manner. The mechanism that has given rise to the degradation of the behaviour or characteristics of component material can be determined through the subsequent analysis of these effects. the efforts may be focused on the reduction of environmental stressors. It is more effective if it involves experienced staff from different disciplines rather than being the work of a single person. such as preventive/corrective maintenance. Power plant component ageing analyses 10. monitoring and performance monitoring of SSCs. computer based expert systems.3 Mitigation of ageing effects In the last phase of ageing analyses. inspections. 217]. The primary goal in this phase is to find suitable methods to prevent. Acceptable processes could include combinations of the following [3]: review of specific plant history and information from previous inspections. availability or maintenance costs. 186 . Power plant component ageing analyses One principal concept in ageing and maintenance management is the concentration of effort on reduction of the failure probability of the most important components [87]. Commonly applied ageing management methods for SSCs in NPPs can be grouped into three main categories [6]: change of SSC design.10. change/recovery of material characteristics. change of operating parameters. The RCM approach intends to optimise the use of maintenance resources by identifying the most critical components with respect to safety. Reliability centered maintenance (RCM) is a method for establishing a scheduled preventive maintenance programme resulting in improved component reliability and minimised costs [258]. and selecting for these components the most appropriate maintenance procedures with the aid of decision logic. Validation of such software is usually carried out by benchmark exercises between different programs since often it is not feasible to compare results with real failure statistics. The overview ends with a summary of the presented analysis tools. In the general analysis applications. PRAISE considers the initiation and/or growth of crack-like defects in piping welds. The code was originally developed at Lawrence Livermore National Laboratory (LLNL) for the assessment of seismic events on the failure probability of PWR piping. An overview of some of the noteworthy of these tools is presented in the following. The emphasis of this presentation is on the component analysis tools. fracture or other. Some of the presented analysis tools are commercially available and some are so called in-house programs. following with a more brief presentation of the system analysis tools. the availability of which may be limited or non-existent. PRAISE is an acronym for Piping Reliability Analysis Including Seismic Events [259].1 Component ageing and risk analysis software PRAISE PRAISE is a PFM computer code for estimating probabilities of leaks and breaks in NPP cooling piping.11. Probabilistic ageing and risk analysis tools 11. can be addressed using the appropriate limit states. Latest version of the code is named WinPRAISE. This meaning that both fabrication cracks and those induced by service 187 . any failure mode. which is a Windows version of the original PRAISE code [260]. The probabilistic analysis approaches in the presented software vary from FORM and SORM to MCS. due to their scarcity. 11. The emphasis is on component analysis tools which are presented first. Probabilistic ageing and risk analysis tools There are many software tools that can be used for system and component risk and ageing analyses. PRO-LOCA can also predict failure probabilities for primary water SCC (PWSCC). sub-critical crack growth models for fatigue and SCC for both initiated and pre-existing circumferential defects. and thereby to provide the successor to the PRAISE code. MCS. and in summary the code has now the following features [263]: crack initiation models for fatigue or stress corrosion cracking for previously unflawed material. Both fatigue and intergranular SCC (IGSCC) are addressed. In the case of SCC. The considerable scatter in such results is quantified and incorporated into a probabilistic model. the background of PRO-LOCA is that in 2003 the USNRC began its development [262] with the intension to adopt and apply advances in fracture mechanics models. The initiation analyses are based on the results of laboratory studies and field observations in austenitic piping material operating under BWR conditions. the growth model is divided into two parts: the early growth of initiated crack and fracture mechanistic crack growth. in which some of the input parameters. with stratified sampling on initial crack size. 188 . The fracture mechanistic SCC growth rate model parameters are based on data on growth rates in fracture mechanics specimens. The crack growth analysis is based on deterministic fracture mechanics principles. PRO-LOCA has been developed further. is used for probabilistic computations [261]. In the recent MERIT project. Probabilistic ageing and risk analysis tools are covered. In brief. the PRO-LOCA code addresses the failure mechanisms associated with both pre-existing cracks and service induced cracks. Presently PRAISE contains models for the analysis of both SCC and fatigue crack growth. PRO-LOCA has also been incorporated an improved basis for simulating weld residual stress (WRS) distributions. Like PRAISE. are considered to be random variables. PRO-LOCA The PFM based PRO-LOCA analysis code has been developed for predicting break probabilities for loss-of-coolant-accidents (LOCAs) in NPP piping systems. Other improved capabilities are in the areas of leak rate predictions and the prediction of critical/unstable crack sizes.11. In addition. The code has features for fatigue crack growth analysis in both ferritic and austenitic steels. The probability of detection (POD) of cracks is based on a model where the POD is a function of the crack area [260]. such as initial crack size. which includes the features in PIFRAP for the failure frequency evaluation. Note that the level of quality assurance of the PRO-LOCA code is such that the code in its current state of development is considered to be more of a research code than a regulatory tool. The PIFRAP code is intended for calculation of the failure probability of a specific pipe cross section with a certain stress state and possibly containing a circumferential crack growing due to IGSCC. In order to perform the probabilistic evaluations. subcritical crack growth until wall penetration and leakage. NURBIT was developed [250]. The results obtained with the PRO-LOCA code are a sequence of failure frequencies which represents the probability of a surface crack developing. Here risk means the core damage frequency. a computer code named PIFRAP (PIpe FRActure Probabilities) was developed. and instability of the through-wall crack (pipe rupture) [263]. The PIFRAP code is based on the model described by Nilsson et al. Probabilistic ageing and risk analysis tools models for flaw detection by inspections and leak detection. Later. and crack stability. PIFRAP contains the analysis programs LBBPIPE and SQUIRT for the evaluation of crack growth and leak rates. NURBIT version 2. A more thorough description of the PIFRAP model with sensitivity analyses can be found from the SKI report describing the application to establish risk informed ISI priorities for Oskarshamn 1 piping [265].0 also includes a cost-benefit analysis where the balance between inspection costs and failure costs are made. NURBIT NURBIT developed by Brickstad and Zang [250] is a RI-ISI/PFM software. commonly used in PSA studies. a through-wall crack developing. and six different sizes of crack opening areas corresponding to different leak flow rates or LOCA categories.11. [266]. PIFRAP calls upon these programs for different values of initial crack length and required quantities for the integration are obtained to give the leak and rupture frequency. NURBIT also performs a complete RI-ISI selection based on a risk ranking procedure including the selection of the most appropriate inspection interval. The PRO-LOCA code can thus predict the leak or break frequency for the whole sequence of initiation. A number of assumptions were made for the probabilistic analysis [267]: 189 . In addition. which enables 190 . the code requires deterministic data on pipe geometry. to a calculated length when the depth is 1. The probabilistic data to the PIFRAP code are [267]: the probability that a crack with an assumed depth is initiated during a certain time interval. The deterministic part of the software ProSACC may be used both for assessment of detected cracks or crack like defects and for defect tolerance analysis. ProSACC The analysis software ProSACC [268] developed by Det Norske Veritas (DNV) contains both deterministic and probabilistic analysis capabilities. the probability of not detecting a crack at an ISI.0 mm). Probabilistic ageing and risk analysis tools stresses are deterministic. Due to the assumptions made. the growth of the crack will be deterministic and will only depend on the initial crack length for a given geometry and given stresses. and above this limit there is a probability of not detecting a leakage flow. The procedure.0 mm. but a possible dependence on length is retained in the general equations). could be used to calculate possible crack growth due to fatigue or stress corrosion and to calculate the reserve margin for failure due to fracture (if wanted. loading conditions and material properties [267]. crack growth is deterministic. including a limited amount of stable crack growth) and plastic collapse. In addition. the probability of not detecting a leak rate for a given leak rate detection limit. there is a detection limit for leakage flow. initial length of the crack is random (based on Swedish distribution data). the probability of not detecting a crack at an ISI depends on crack length and crack depth (the actual function used was not dependent on the crack length. which is based on the R6 Method [201]. initial crack depth is fixed to 1. ProSACC also has an option.11. initial length of the crack (initial crack length/depth is calculated from statistical data backwards. austenitic piping and ferritic piping. For each analysed case. other crack growth analysis input data parameters are considered to have single deterministic values. In VTTBESIT analyses. The simulation ends either when the crack depth reaches the outer pipe surface. 272]. frequency of load cycle occurrence. Section XI [173]. 269] and Fraunhofer-Institut für Werkstoffmechanik (IWM. thousands of separate simulations are calculated. ProSACC mainly considers surface crack postulates in plates. Appendices A.11. hollow cylinders and spheres. length of initial cracks. Probabilistic VTTBESIT The analysis software VTTBESIT developed by both VTT [244. contains both deterministic and probabilistic analysis capabilities. 246. For the time being. The probabilistically treated crack growth analysis input data parameters are as follows: depth of initial cracks. Both cracks caused by fabrication and those induced by service are considered. The probabilistic part of the software ProSACC is based on a PFM procedure that calculates two different failure probabilities: probability of failure. C and H for assessment of cracks in ferritic pressure vessels. values of the above mentioned distributed input data parameters/variables are sampled at random from the respective probabilistic 191 . The covered component geometries are hollow cylinders and straight plates. With VTTBESIT it is possible to quickly compute mode I stress intensity factor values along the crack front as well as crack growth. respectively. the probabilistic features are considered here. The covered crack growth mechanisms are fatigue and SCC. or the time cycles reach the end of plant lifetime. but its scope has been extended by adding probabilistic capabilities to the code. Mainly. Germany) [270. 271. 247. probability of failure. the crack growth increment in each time step is computed from the respective crack growth equation. Probabilistic ageing and risk analysis tools assessment of cracks according to the 1995 edition of the ASME Boiler and Pressure Vessel Code. defect not detected by NDT/NDE. The code was originally intended for deterministic fracture mechanics based crack growth analyses. and for each of these. defect size given by NDT/NDE. finite element code for structural analysis.2 for a more detailed description concerning Markov models. This method is a stochastic process in which the probability distribution of the current state is conditionally independent of the path of past states.g. the states of the Markov process correspond to crack penetration depths through component wall. intact one. conditional on the existence of an initial flaw. in RI-ISI analyses. STRUREL STRUREL (STRUctural RELiability) is a general purpose reliability software series that has been developed by Reliability Consulting Programs GmbH [273] for performing different computational tasks. being either fatigue or SCC. The annual crack depth information for each simulation is transferred to the second phase of the analyses. The program comprises several independent but interrelated programs: STATREL. time-invariant and time-variant analysis of component reliability. and the transition probabilities those from a lower state to higher states (deeper cracks). Probabilistic ageing and risk analysis tools distributions. the analyses are based on Markov process. In the VTT application. including also the case of no inspections. Goodness of fit tests are also included to demonstrate the 192 . For the whole duration of the assumed operational plant lifetime the results obtained from a VTTBESIT analysis cover yearly piping component failure probabilities for different inspection strategies. module combining COMREL with NASCOM. a characteristic called the Markov property. STATREL enables appropriate distributions to be derived for datasets input from e.e. The effect of inspections is taken into account with POD functions. spreadsheets. as each time a crack is detected it is assumed that the component in question is repaired or replaced with a new i. distribution fitting and analysis of time series. reliability analysis of systems. SYSREL.11. simulation. In the second phase. NASREL.3. see Section 8. statistical analysis of data.g. Typically each simulation run spans several decades of assumed plant operation with the crack depth calculated yearly. COMREL. These results can then be used further e. NASCOM. On the other hand. for the considered degradation mechanism. the effects of inspections are included in the model as transitions from a cracked state to the flawless state. The code is designed to work on its own or to operate as a shell program in conjunction with other structural analysis programs. It consists of five input data decks: Geometry. FORM and SORM methods. Material. The load module enables through thickness distributions of applied and welding residual stress to be incorporated. COMREL comprises 44 models and limit state equations can be input for failure modes not addressed. The Analysis module enables the user to select MCS or FORM methods and to apply partial safety factors if required in order to achieve a specified target reliability method. yield strength. fatigue. rupture stress and fracture toughness and their associated statistical distributions are input. SYSREL enables multiple failure criteria for parallel and series systems to be evaluated. corrosion and general strength problems [274]. log-normal or Weibull distribution. ProSINTAP ProSINTAP (PRObabilistic Structural INTegrity Assessment Procedure) automates MCS and FORM analysis of the failure assessment diagram and is only applicable to fracture and collapse failure modes [275.11. making it straightforward to check individual failure criteria before combining them in a system analysis. California [277]. including conditional events. including nuclear industry. Structural failure criteria are defined in terms of one or more limit state functions. Loading. NDE and Analysis. This requires the mean and standard deviation of each parameter as defined by the normal. In the material module. Probabilistic ageing and risk analysis tools best fitting method to be used. The NDE module enables defect sizing as input data. The Geometry section comprises stress intensity factor solutions for a range of plate and hollow cylinder geometries with surface and through-thickness cracks. 276]. and there exists many examples of its use for fracture. CALREL CALREL (CALibration of RELiability) is a general purpose structural reliability analysis program developed by the University of Berkeley. The program includes MC simulation. and for the case of the time-variant version it includes methods for incorporating random and point-in-time events. but also allowing for treatment as an exponential distribution. 193 . It links directly with COMREL. The specification is by the user through user defined subroutines. collapse. STRUREL has applications in many fields. 194 . employing first or second-order fittings of the limit state surfaces. It is similar in approach to ProSINTAP and includes FORM and MCS. NESSUS NESSUS is a general purpose tool for computing the probabilistic response or reliability of structures. Additional distributions may be included through a user defined subroutine. NESSUS provides a built-in finite element structural modelling capability [280]. CALREL is available for purchase from UC Berkeley as both object and source code. Probabilistic ageing and risk analysis tools CALREL is capable of computing the reliability of structural components as well as systems. The framework of NESSUS allows the user to link traditional and advanced probabilistic algorithms with analytical equations. first-order reliability bounds for series systems.11. external computer programs including commercial finite element codes and general combinations of the two. The finite element codes NESSUS has interface to include ABAQUS. directional simulation for components and general systems. but a more extensive library of stress intensity factor solutions for different component geometries is included [200]. first-order reliability sensitivity analysis for independent and dependent variables with respect to distribution and limit state function parameters. Specific macro commands are available for the following types of analyses: first-order component reliability analysis. MCS for components and general systems. second-order component reliability analysis by both curvature fitting and point fitting methods. STAR 6 STAR 6 [278] is a reliability software developed by British Energy for automating reliability analyses of the fracture and collapse analysis procedures in R6 Method [200]. CALREL has a large library of probability distributions for independent as well as dependent variables. developed by Southwest Research Institute (SwRI) [279]. Also. NASTRAN and PRONTO. The results for a component analysis in NESSUS are reliability or cumulative distribution function and importance factors. and thus it accounts for possible correlation between the stress and strength parts of the performance measure. FORM. The importance factors indicate which variables contribute most to the unreliability and may be used for design changes or manufacturing process modifications [280]. NESSUS allows linking of different analysis packages or analytical functions to allow a general relationship of physical processes to predict the uncertainty in the performance of the system [280]. NESSUS provides also a general formulation of function g that allows linking of different analysis codes and analytical functions. The objective in developing NASGRO was to improve an earlier computer code to accommodate more recent advancements in fracture mechanics and crack propagation theory and to meet the needs for damage tolerance and durability analyses [282]. NESSUS licences are available for purchase. a stress or strain from a finite element analysis can be used with a fatigue life equation or S-N curve to define the performance of a structure. In NESSUS function g is formulated such that g = g(Xi. stress computation. stress or strain) or include the strength of the system to compute the reliability. For example. MV. NESSUS can compute the cumulative distribution function of system performance (e. These include MCS. developed by SwRI.11. or P[R < S] or P[g < 0]. SORM. R and S may be complex models involving other random variables such that R = R(Xi) and S = S(Yi). Yi). where g is the limit state function. The capabilities of NASGRO include: fatigue crack growth and fracture analysis. AMV and LHS. structural life assessment. 195 . S. R. FPI.g. exceeding the strength. fatigue crack growth properties processing and storing. Probabilistic ageing and risk analysis tools Eleven probabilistic algorithms are available in NESSUS. NASGRO Deterministic analysis software NASGRO. is a suite of computer programs comprised of analysis modules linked together by graphical user interfaces (GUIs) [281]. Traditional reliability analysis involves computing the probability of the stress. In addition. It has been in use in every French NPP unit since 1994.g. 196 . When the 1E-03 fractile estimate appears to be higher than the critical value. plane MCS is applied. MC variance reduction technique is applied. influence of temperature and geometry. The software is commercially available. Probabilistic ageing and risk analysis tools When using NASGRO. The probability for thickness being less than critical is evaluated after this. the main parameters of influence that are taken into account are: effect of the water chemistry: pH. If not. Due to its proprietary nature. Material properties for crack growth can be chosen from a large database by selection from a menu. Mo). In the modelling of FAC. Cu. Analysis in complex two dimensional geometries with or without cracks to obtain stress intensity factors and stresses can be performed using the boundary element module of NASGRO. The material property module can also be used to enter. The probabilistic computations allow to: follow the evolution of the fractiles of the residual thickness distribution as a function of time. a detailed documentation of BRT-CICERO or the code itself are not available for purchase [1]. steel composition (Cr. BRT-CICERO provides e. The probabilistic module calculates the evolution of the 1E-03 fractile of the residual thickness distribution until it reaches the critical thickness. the following results: wear and wear rate of the pipe wall. evolution of residual thickness determined from the nominal thickness and the tolerances of the nominal thickness. BRT-CICERO Electricité de France (EDF) has developed software BRT-CICERO [283] for both deterministic and probabilistic evaluation of pipe wall thinning due to flow accelerated corrosion (FAC). various types of analyses can be performed in an interactive mode using the GUIs for each module.11. oxygen. edit and curve fit crack growth data. influence of mass transfer in single phase flow. follow the evolution of the probability that the residual component thickness is less than the critical thickness as a function of time. evaluate the residual lifetime of selected component. According to PROPSE. e. constant crack aspect ratio is assumed. In PROPSE. It is developed by Westinghouse to support piping RI-ISI analyses. In PROPSE. SRRA is applicable for probabilistic integrity assessment of RPVs and RPV internals. SRRA SRRA (The Structural Reliability & Risk-Assessment) [285.11. Besides pipe components. and application of FORM with sensitivity factors using the most probable point of failure in a standard normal space. Through-wall cracks are not considered in the present program version. NDE. operating conditions. where a conservative assumption is to use a high value. The SRRA applies PFM approach and MCS to simulate the effect of time dependent degradation mechanisms and infrequent loading events and to derive the probability of piping failure for a variety of failure modes. fracture toughness is assumed as normally distributed. PROPSE contains two different algorithms to calculate the probability of failure: simple MCS with an error estimate on probability of failure. the user is required to provide best estimate (median) values for various input parameters related to pipe material. between those components which have an active damage mechanism and those which have not. material yield strength is assumed as normally distributed. the following assumptions are made: crack depth is assumed as log-normally distributed. failure is predicted when the surface crack reaches a critical condition.g. and covariance for tensile strength is assumed to be the same as for the yield stress. Each parameter is associated with a statistical distribution and standard deviation which are based on industry data and detailed analyses are not 197 . Probabilistic ageing and risk analysis tools PROPSE DNV has developed software PROPSE (PRObabilistic Program for Safety Evaluation) [284] for computation of probability of failure when the NDT/NDE have found/missed a defect in NPP piping system inspections. geometry. service stresses and cycles. 30. In SRRA. the resulting failure probabilities are usually very small for shallow defects in moderately stressed components. in terms of failure probabilities and risk for core damage. This illustrates the general observation that there is a distinct difference. 242] is an analysis software for assessing pipe component failure probabilities. waterhammer or a pipe break.11. before such a flaw would develop into a large leak or break. it is assumed that mitigating actions would be taken on a detected through-wall flaw. credit may also be given to the reactor coolant leak detection system in the calculation of large leak and break probabilities. SRRA computes the lifetime failure probability for three different failure mode types concerning piping pressure boundary: small leak (i. By entering best estimate leak detection capability. Probabilistic ageing and risk analysis tools required to be entered by the user. 198 . by FAC). Based on user defined input data for ISI interval and POD model. Again. such as earthquake. full break (i. a through-wall flaw that leads to leakage beyond a userdefined value). wall thinning due to material wastage. assuming that flaws detected by ISI will be mitigated before failure can occur.e.(e. the code simulates the effect of the modelled degradation over time and at each time step the pipe stability against rupture is checked with respect to design limiting loading conditions.e. large leak (i. for the applicable NDE technique SRRA also calculates the (reduced) failure probability. assumptions for initial flaw distributions are built-in to the program based on industry correlations with respect to weld geometry and pre-service inspection.g. Design limiting stress input data may also be attributed with a frequency corresponding to the frequency of occurrence of the corresponding loading event. complete severance of the pipe cross-section). through-wall flaw with minor leakage). Similarly.e. These mechanisms include: low-cycle fatigue crack growth of an existing (fabrication) flaw. In each MC simulation trial. crack growth of an existing flaw due to SCC. high-cycle fatigue induced stresses exceeding the fatigue crack threshold. The SRRA code has the capability to simulate the effect of a variety of time dependant material degradation mechanisms for components of carbon steel and stainless steels. The second part treats flaw depth and fracture toughness of vessel materials as random variables. flaw (crack) size. For each pass through the simulation loop. the crack is extended 0. These sampled values are eventually used to determine the fracture toughness. (c) material properties. it is possible to assess the failure probability of a PWR RPV under pressurised thermal shock (PTS) conditions. This analysis estimates values of crack tip temperatures and applied stress intensity factors for several crack depths. If crack arrest occurs.25 in. the software performs time history analysis for a given transient. the simulated value of the fracture toughness is compared with the applied stress intensity factor at the crack tip. otherwise. The deterministic part defines the stress model and the probabilistic part defines the strength model. (b) surface heat transfer coefficient. If crack initiation occurs.11. Effects of cladding are included in the heat transfer and stress analysis. If the fracture toughness is less than the applied stress intensity factor. fluence at the inner wall. The proportion of through wall cracks in a large number of passes through the simulation loop is an estimate of the conditional probability of through-wall cracking (vessel failure). a deterministic fracture mechanics analysis is performed for a temperature and pressure transient entered by the software user. It uses influence coefficients for calculating applied stress intensity factors. and the crack arrest toughness KIa is simulated. These values are used by the probabilistic analysis performed in the second part. The sampled values of fracture toughness are compared with the stress intensity factors (estimated in the first part) at a sampled flaw depth to determine crack initiation and growth. flaw location. The software provides a selection of initial crack distributions. In the first part. At the end of each time step. simulated values of initial RTNDT. Probabilistic ageing and risk analysis tools VISA-II With VISA-II analysis software [286] developed by Oak Ridge National Laboratory (ORNL). and copper and nickel contents are selected from their respective distributions. crack initiation occurs. the simulation moves to the next time step. and (d) wall thickness and radius of the vessel. The VISA II code is divided into two parts to define stress and strength models needed for the following MC analyses. The VISA II code uses closed form solutions for heat transfer and stress calculations. otherwise the 199 . the simulation moves to the next time step. The necessary input data for the deterministic analysis include: (a) pressure and temperature of the reactor coolant as a function of time for a given PTS transient. With these values fixed for a given pass. DANCER and ISIBREAK evaluate the probability of failure of a welded joint. DANCER does not use MCS to obtain the failure probability. The model simulates the effects of both radiographic and dye penetrant surface inspections. This process continues until either the crack grows through wall or the transient is completed.. a conditional probability of failure follows directly from the distributions of Paris-Erdogan 200 . rather than measure. both the technical expertise of welding process engineers and mathematical modelling are used to build a model that will simulate the weld manufacture and the errors that lead to different types of defects. The calculated value of the shift is then added to the sampled initial RTNDT to obtain the adjusted RTNDT. Then.11. Its output is a histogram or frequency plot of the defects that may formate during the normal build of a weld. Instead. which includes a period of defect initiation prior to crack growth to failure. Probabilistic ageing and risk analysis tools crack is extended another 0. The program is designed for multi-pass welds only and can handle ‘Single’ and ‘Double Vee’ weld constructions. 288]. This fracture toughness is compared with the applied stress intensity factor determined in the first part of the analysis. and a new value of KIa is simulated. since the failure criteria are deterministic. RR-PRODIGAL The structural reliability software RR-PRODIGAL [299] by Rolls-Royce consists of the following three program parts: DANCER. With this methodology. Various welding processes are addressed including submerged metal arc welding. a defect distribution and density for a given type or family of welds [287. slag. such as an area of stress concentration. The objective of the program DANCER is to simulate the number and size of defects generated during the welding process. the sampled value of fluence at the crack tip is calculated. lack of fusion. the value of the shift in RTNDT is calculated using sampled values of copper and nickel and the attenuated fluence. ISIBREAK and SNOWBREAK. whilst SNOWBREAK evaluates the probability of failure for an area. For each time step. The conditional probability of through wall cracks is obtained by dividing the number of simulations (passes) that resulted in through wall cracking by the total number of simulations that were made. pores with tails and cracks in heat affected zones. In the software methodology. The defects that may initiate in weld beads include center cracks. the model attempts to build up. which is used to estimate the simulated fracture toughness.25 in. The first function of ISIBREAK is to utilise the stress input data to estimate the conditional probability of failure for defects of varying size and position within a weld. the crack growth to failure follows basically the same procedure as that used in ISIBREAK except that only one conditional situation. a set of conditional failures can be determined and hence the distribution of the conditional failure probabilities over the range of initial defects is determined [299]. one important assumption is that there is a positive correlation between the crack initiation and the crack growth. Of degradation mechanisms. Its primary use is for areas of stress concentration where defects could initiate at the surface by fatigue and then grow on through the wall of the component. whereas the applied failure criteria are based on procedures in the R6 Method. leak and break probabilities from pre-existing semi elliptical shaped inner surface cracks driven by cyclic loading conditions (fatigue) can be estimated. only fatigue induced crack growth is considered. The conditional probability of failure of a defect is the probability that it will cause failure given that it exists. with the conditional probability of failure to calculate an overall probability of failure for the weld. Rev. When a defect has initiated. that of a 2 mm surface breaking defect. 4 [200]. The deterministic fatigue crack growth rate is calculated by slightly modified Paris-Erdogan equation. The probabilistic nature is determined by the uncertainties of the input data entering the deterministic routines. Probabilistic ageing and risk analysis tools crack growth rate equation parameters. ISIBREAK then obtains the calculated weld defect distribution from a DANCER file and combines it. The effect on the probability of failure of one or several in-service inspections (ISI) during the life of a component can also be included to the analyses. With PROST. Thus. It is the task of the software user to enter the PODs as ISIBREAK does not contain POD data. The calculation of the subcritical crack growth and the final instability are based on deterministic fracture mechanics principles. using a simple mathematical technique. The SNOWBREAK routine in RR-PRODIGAL is an attempt to estimate the probability of failure from non-weld areas.11. However. A graphical user interface supports the necessary data input. 201 . PROST PROST (PRObabilistic STructure Analysis) [300] is a PFM analysis software to evaluate leak and break probabilities of piping systems in nuclear power plants. requires analysis. The user can determine the POD curve by specifying ten crack depth values and their corresponding detection probabilities in percent together with the time inspection interval. The second are stochastic load sets. stress intensity factors for the maximum and minimum load at deepest and surface point and the combination probability. Two probabilistic computation procedures are available in PROST. this data are used to identify if during an inspection a crack is detected or not. the accumulated failure probability is given as a function of time. For this method. The first are cyclic load sets recurring with user defined frequencies. failure mode. like seismic events or transients. In a second output file. 50 % and 95 % failure probability curves as a function of time are given too. User defined weld residual stresses can be included as well. log-normal. 202 . The first is the plain MCS. Then for all combinations that fail are listed the actual parameters. processed 5 %. Probabilistic ageing and risk analysis tools Uncertainties in the data on geometry. failure time. If the stratification method was selected. Internally. which is realised by entering a POD curve. The 50 % curve represents the expected failure probability that is equivalent with the results of a MCS run. For this method. weibull and exponential distribution function. The regular output file contains the information about all given input parameters including the run time variables.11. The second option is a stratification method based on a variance reduction technique. a combination of the distributed parameters will be selected randomly and the crack propagation for a given time period is computed. which are dominant for fatigue problems. For each parameter the selection possibilities includes the normal. Additionally. crack depth. depending strongly on the mean values of the distribution functions of the input parameters. The user can choose if he wants to consider ISI in the computation. The functions can be truncated on request by input of a minimum and maximum value. the user can select two out of all distributed parameters (usually the crack depth and the crack length to depth aspect ratio) to be integrated. Two different types of loading conditions are distinguished in PROST. it is possible to consider the parameter to be deterministic with a fixed value. which occur with user defined probabilities. material properties and crack size of a given fatigue problem can be considered via different distribution functions. crack length. consultants. CAFTA is developed by Data Systems & Solutions. and throughout the world. industry contractors.A. One result of this effort is CAFTA. Using CAFTA one can build. The Cutset Generator for generating minimal cut sets from the Fault Tree Editor. 203 .2 System ageing and risk analysis software CAFTA CAFTA is an integrated tool to perform PRA. SAPHIRE is developed by the Idaho National Engineering and Environmental Laboratory (INEEL). Event Tree and Piping Instrumentation Diagram (FEP) and Graphical Evaluation Module (GEM).11. Models And Results Database (MAR-D). a joint venture between Rolls-Royce and Science Applications International Corporation (SAIC). national laboratories.g. incorporating linking event tree/fault tree methodology [292]. the funding for which is provided by USNRC [289]. the following integrated modules: The Fault Tree Editor for building. Since 1992. The functions and modules in SAPHIRE include: functions for creating event trees and fault trees. Data Systems & Solutions has worked with the Electric Power Research Institute (EPRI) to develop a suite of Windows based PRA/PSA software tools and applications. universities. Presently. vendors. and The Cutset Editor for viewing probability and structural changes. System Analysis and Risk Assessment (SARA). CAFTA includes e. utilities. other government agencies and government contractors [72].S. These users include USNRC staff. which has over 1500 users worldwide [292]. SAPHIRE SAPHIRE (Systems Analysis Programs for Hands-on Integrated Reliability Evaluations) is a software application developed for performing a complete PRA/PSA using a PC. SAPHIRE has been distributed to thousands of users in the U. analysing and maintaining fault trees. The three level Reliability Database Editor for controlling model data. Fault Tree. quantify and analyse event tree/fault tree models of considerable size or complexity. Probabilistic ageing and risk analysis tools 11. architectural engineering firms. The software provides a user interface for modelling from the basic fault tree with AND and OR gates to advanced fault tree and event tree integration of sequences in linked event trees with boundary conditions and common cause failure (CCF) events. for their minimum cut sets. RiskSpectrum PSA RiskSpectrum PSA [291] developed by Relcon Scandpower Inc. uncertainty and sensitivity analysis facilities. sensitivity. WinNUPRA is used in several NPPs and chemical process facilities in the U. The latest program version was released in 2010. FaultTree+ provides CCF analysis. FaultTree+ has been used to perform systems reliability analysis by a wide range of different industries for nearly three decades.g. The software allows analysing large and complex fault and event trees producing the minimal cut representation for fault tree TOP events and event tree consequences. FaultTree+ FaultTree+ is an interactive graphics and analysis software for performing a PRA/PSA using integrated fault tree. WinNUPRA consists of six major analysis modules designed to generate and analyse minimal cut set solutions of various fault trees and cut set equations for accident sequences. and for Boolean manipulation (merging) of cut set equations.S. event tree and Markov analyses.A. Operations are provided for direct solution of fault trees. integrated. The WinNUPRA software package is developed by Scientech Inc. importance and time dependent analysis. and other countries. importance analysis. event trees originating from different initiating events. [293]. The program allows constructing a single project database containing generic data and event tables. Fault tree TOP events may be used to represent specific 204 . fault trees with multiple TOP events. the following analysis options: minimal cut set.11. Probabilistic ageing and risk analysis tools WinNUPRA WinNUPRA is a self-contained. and the NUPRA User’s Group. The code is developed by Isograph Inc. The integrated analysis tool is specially designed for solving large PSA models and covers e. CCF tables and consequence tables. PSA/PRA software package for Level 1 PRA/PSA analyses [290]. is a fault tree and event tree software tool licensed for use in many NPPs worldwide. labour availability. the effects of the resulting non-independence are accounted for. If the fault trees contain common events. Probabilistic ageing and risk analysis tools columns in the event tree. Users may feed the end branches of event trees into secondary event trees eliminating the need for the user to reproduce identical event tree structures leading to identical consequences. Fault trees can be linked to event trees by specifying a fault tree which calculates a branch probability. Severity values may be assigned to a given consequence allowing safety. By allocating safety. The system lays out the diagrams automatically. for the construction. This allows the updating of their historical data records observing the effects on predicted system performance. When using fault trees. LOGAN Fault and Event Tree Analysis The LOGAN Fault and Event Tree module is a program developed by Reliass Inc. These models could be simple failure and repair models or they could represent complex dependencies including ageing. AvSim+ AvSim+. AvSim+ provides means to construct fault tree or network diagrams (reliability block diagrams) by using drag and drop facilities. environmental and operational consequences to selected system failures one is able to determine the frequency and duration of each type of consequence.11. AvSim+ automatically organises the diagram after receiving logical connection data. evaluation and printing of fault and event trees and is widely used for quantitative risk assessment [295]. provides a MC simulation package for analysing systems availability and reliability problems using fault trees or reliability block diagrams [294]. environmental and operational criticality values to be determined over the lifetime of the system. also developed by Isograph Inc. etc. spares requirements. When using networks. Once a logical fault tree or network structure is defined one can define failure and maintenance models to represent the performance of components within the system. operational phases. 205 .. Historical data (times to failure and times to repair) are automatically analysed using the Weibull Analysis facility and connected directly through to component failure models. Multiple branches are also handled to allow for partial failures. AvSim+ automatically deduces the failure logic of the system after receiving block structure and logical connection data. standby arrangements. various forms of sensitivity analysis. developed presently by ExproSoft. In addition. including e. minimal cut set analysis. frequency of TOP event and failure frequency distribution. mean time to failure. One can analyse the extrapolated failure times as times-to-failure data. According to ReliaSoft [296]. Other notable computed results include: importance measures at cut-set level. Performance measures calculated by the code include: unavailability. non-parametric analysis and analysis of variance. one can use the competing failure modes analysis option to assign a separate distribution to each failure mode identified in the data set. improvement potential and order of smallest cut-set. systems and projects. and optimisation of proof test intervals to minimise testing while achieving safety/reliability targets. warranty analysis. Probabilistic ageing and risk analysis tools The fault and event tree capabilities of LOGAN Fault and Event Tree module include: fault/event tree integration. unavailability at cut-set level and uncertainty analysis. Weibull++ Weibull++ 6 software [296] developed by ReliaSoft Corporation allows performing life data analyses utilising multiple lifetime distributions. print-out in conventional top-down format or LOGAN horizontal format. ABB. criticality importance.11.g. survival probability. is a tool for fault tree analysis and construction. integrated reliability database. Birnbaum's structural. their software products and services are currently used by hundreds of companies. CARA-FaultTree CARA-FaultTree [297]. Computed component importance measures include: Birnbaum's reliability. with an active reliability engineering program. It uses globally recognised standards and methodologies to analyse components. Vesely-Fussell. The Degradation Analysis utility allows extrapolating the failure times of a component based on its performance over a period of time. degradation analysis. ITEM ToolKit 206 . including all forms of the Weibull distribution and generalised gamma distribution. The capabilities of the code include analysis of data from various censoring schemes. ITEM ToolKit ITEM ToolKit software [298] is a suite of prediction and analytical modules developed by Item Software Inc. Siemens and TÜV. average availability. ITEM ToolKit is widely used throughout the world by companies in defence. increased safety and reduced liability. Level 1 part of this code was taken into use in 1991. According to software developer [298]. FinPSA produces cut set files classified by consequence only. On the top level. Any number of intermediate levels can be defined. aerospace. asymmetric common cause failures. and its efficiency and robustness have been demonstrated in many benchmarks and in practical PSA use. Probabilistic ageing and risk analysis tools allows taking a total system approach in analysing individual systems and components. the algorithm has been continuously developed. fault trees. and can be accessed and edited simultaneously by several users. The analysis capabilities of the software include: FMECA. FinPSA is a new version of SPSA for Microsoft Windows [301]. Since then. FinPSA STUK started to develop living PSA computer code SPSA in 1988. PSA models can be shared or private models. The PSA team can solve minimal cut sets together with several computers utilizing parallel computation. After the search. Shared models reside in a server. The development of FinPSA will continue with new features like: task-oriented model for control systems (from RELVEC code). computation of unreliability and unavailability. Results are available in several formats to Windows clipboard. Each of the several modules of the software is designed to analyse and calculate the failure rates of components and systems in accordance with the appropriate standard. electronics and other industries. additional CCF models. This allows the optimisation of design targets with respect to component selection. printer and files. level 2 (dynamical containment event trees already implemented in SPSA). One can analyse reliability and availability at the component or system level and view the entire project. analysis of CCFs and creation of minimal cut sets. Preliminary version of FinPSA level 1 is available. FinPSA searches for cut sets for each event tree sequence. the cut sets of individual sequences are automatically combined and grouped according to userdefined hierarchy of consequences. and almost every item in the model. events trees.11. 207 . FinPSA works together with common Windows programs for importing and exporting data base tables. analysis of uncertainty and sensitivity. FinPSA is developed for teamwork. ProSACC. Thus. PROPSE against STAR6. It would have been way beyond the scope of this thesis to go through all or even most of them. aerospace. chemical. These include e. Probabilistic models that predict rare events cannot be validated by comparing the results to observed failure and degradation data. SAPHIRE and PSA Professional. the main sources of difference can be identified. based on which the models can be developed further. e. e. NURBIT against WinPRAISE. process and transportation. Thus. as the number of failure mechanisms encountered in NPPs is much greater. it appears that for the time being several analysis codes are needed if one wishes to cover the degradation of all safety significant NPP systems and components. This is a drawback. Comparisons of this kind have been made e. PROST and WinPRAISE against each other.11. can be applied within many industries. those with fault and event tree analysis capabilities. 208 . Probabilistic ageing and risk analysis tools 11. It appears that most of the available ageing degradation analysis tools consider only one or two degradation mechanisms.g. 302.g. only some of the most noteworthy examples of these analysis tools are covered in this thesis.g. 303]: SRRA against PRAISE. Thus. RR-PRODIGAL. probabilistic VTTBESIT against NURBIT. There exists. In several cases these comparisons have resulted in identification of errors and model improvements. One way to verify these models to some extent is to benchmark the independently developed models describing the same phenomena against each other.g.3 Summary of ageing and risk analysis software There exists a large number of software for ageing and risk analysis of systems and components. a number of system analysis tools that are specifically developed for power plant industry purposes. Many of the system analysis tools presented in this thesis. for the following models [247. ProSINTAP against STAR 6. nuclear. 264. however. three different initial crack distribution assumptions.1 Introduction Two analysis codes are used in the lifetime analysis application for piping components. With these analysis tools. it is possible to calculate e. These sizes are assumed to correspond to representative small. yearly failure probabilities for the considered piping cross-sections.g. To allow performing the present computational analyses. and the second one is PIFRAP. 0 published in 1997 is used. The examined characteristics in the computations cover e. The last issue means the service time in years from the start of operation. three pipe sizes and three considered time spans. 209 .12. and of some PIFRAP computations for comparison purposes. As in the NPP piping components SCC typically occurs as a circumferentially oriented inner surface crack in the heat affected zone (HAZ) adjacent to weld. As for PIFRAP.0 rev. such postulates are also considered in the present analyses. The first and obvious one of them is the probabilistic VTTBESIT.g. the analysis code version 2. medium and large pipes in Finnish BWR NPP reactor circuit piping systems. respectively. Probabilistic lifetime analysis application for piping components 12. The considered degradation mechanism is SCC. The analyses consist of a set of Monte Carlo simulations with probabilistic VTTBESIT. the probabilistic VTTBESIT was developed further to some extent. both of which are described in more detail in Chapter 11. developed at VTT. All examined pipe cross-sections are similar butt welds.2 Examined piping components and associated input data The computational probabilistic analyses are performed for three pipe sizes. 12. Probabilistic lifetime analysis application for piping components 12. 73 9. for the operational temperature of 286 C of Finnish BWRs the material property values were interpolated.0 Outer radius/wall thickness [-] 7. Pipe Small Medium Large Outer diameter [mm] 60 170 510 Wall thickness [mm] 4. The presented yield strength values correspond to 0. as for that particular temperature the material property values were not given in the ref.50 7. the needed geometry input data are quite limited. The cross-section dimensions of the three pipe component sizes considered in the probabilistic lifetime analyses are presented in Table 12.2. the commonly used option to assume the material properties of the adjacent base material for the weld material is also applied here. and presented here for a representative range of temperatures. The obtained data are in the form of tables. 12.12. this is a conservative assumption. which is a commonly used material type for NPP piping system components. As the considered pipe cross-sections are assumed to be located in straight pipe components. 210 . Table 12. which is gratefully acknowledged. Representative Small.2 Material properties The base material of the considered pipe components is assumed to be austenitic stainless steel TP 304.81 12. as typically the strength properties of the weld materials are to some extent better than those of the base materials.2-1. In most cases.2-1. Material property values for various metallic NPP component materials are taken from Section II of the ASME code [305]. As the material property data of welds are often more difficult to obtain. Probabilistic lifetime analysis application for piping components and they were provided by expert personnel from TVO [307].2.2 % of strain in uniaxial tension test.1 Geometry The geometry input data concerning all computations are presented in the following.0 11.0 26. On the other hand. Medium and Large BWR reactor circuit pipe sizes selected for probabilistic lifetime analyses. [305]. Temperature [ C] 20 100 150 200 250 275 286 300 Yield strength [MPa] 172 146 132 121 114 111 109 108 Design stress [MPa] 115 115 115 110 103 Tensile strength [MPa] 483 452 421 406 398 102 98 397 393 Table 12.1 15. Probabilistic lifetime analysis application for piping components Table 12.2 19.2 15.2-2.6 186 17 17.12. U and Y1 in Section II of ASME code [305]. Temperature [ C] 21 38 66 93 121 149 177 204 232 260 286 316 Elastic modulus [GPa] 195 Thermal conductivity [W/m°C] 14.2 190 16.6 Specific heat [J/kg°C] 484 486 496 506 516 520 529 535 539 544 548 551 Coefficient of thermal expansion [(1/°C)*10 -06 ] 15.2-3.0 17.4 16. Some mechanical properties of type 304 steel as a function of temperature.3 178 176 174 18.4 15.8 16 16. Strength properties of type 304 austenitic stainless steel as a function of temperature.9 15.7 16.5 16. see Tables 2a.6 15.9 17.9 19.6 211 .1 16. see Tables TCD.5 183 18 18.2 16. TE-1 and TM-1 of Part D of Section II of ASME code [305]. 3 Loads and failure mechanisms The loads considered in the analyses are mechanical and thermal. [KI] = MPa m. and for the base material: KIC > 350 MPa m. Under operational conditions. The considered crack growth type is mode I. According to ref. Material SS TP 304 C 4. the fracture toughness data for this welding type are: KIC = 182 MPa m. when the base material of the pipe is austenitic stainless steel and temperature is 288 C. [203]. [308] 12. the considered mechanical loads are operational process pressure. Probabilistic lifetime analysis application for piping components The assumed welding type is shielded metal arc welding (SMAW). stresses acting perpendicular to crack faces.5E-12 3. i. with corresponding stress intensity factor KI [MPa m] as the governing load parameter.e. allow considering two degradation mechanisms. as that region is more susceptible to SCC than the base or weld material. Thus. respectively. The axial membrane 212 . and residual stress distribution in and near the weld.2-4 presents the values of parameters C and n used in the SCC rate equation for the considered austenitic stainless steel (SS) TP 304. Due to the choice of examined location. The dimension of C is [(mm/s)/((MPa m)n)] whereas n is dimensionless. Here. As all considered crack postulates are circumferentially oriented and located in the inner surface. only axial stresses are taken into account. VTTBESIT and PIFRAP. and for most of the time the temperature outside the insulation is of the scale of room temperature. Table 12. which is a commonly used method for fabricating NPP piping welds. Thermal loading is also caused by the water inside the pipe components.2. the residual stress distribution caused by the welding process must be considered as well. which are intergranular SCC (IGSCC) and fatigue (in case of PIFRAP limiting to high cycle vibration).00 n Environment water Refs.0 MPa and 286 C. The dimensions used in the crack growth equation are: [da/dt] = mm/s. The values of parameters C and n used in the SCC equation for the considered material. JIC = 168 kJ/m2. the only considered degradation mechanism is IGSCC. JIC > 620 kJ/m2.12. The crack postulates are assumed to be located in the HAZ. as caused by the water inside pipe components. The outer surfaces of NPP piping components are typically thermally insulated. Both applied analysis tools. the values of inner pressure caused by and temperature of the fluid are 7. Table 12.2-4. but to ensure that appropriate WRS distributions are applied to the Small pipe size of austenitic stainless steel and with wall thickness of 4. Medium and Large pipe components are presented in Figures 12. respectively. 312]. and local minimum values of the same scale in compression. When actual WRS measurement data are not available. as is in the present case. 213 . in NPP piping component welds. 310]. Mainly.g. Thus. but for Small pipe size according to the ASME recommendations [309. Probabilistic lifetime analysis application for piping components stresses. The WRS distributions could also be simulated more accurately by FEM.2-1) where p [N/mm2] is pressure. as such simulations require input data for a relatively large number of parameters. these limits concern ferritic steels. to NPP piping component welds. The dominant role of the WRSs in the total axial stress distributions can clearly be seen in these figures. namely the ASME recommendations [309.2-3. The distributions of the axial WRSs in the weld centre-line and HAZ are assumed according to two commonly applied procedures. D [mm] is outer diameter and t [mm] is wall thickness. but again due to lack of data concerning the actual welding process. Typically. The former referring to the mechanical and thermal processes executed during the whole production sequence and the latter to the elastic-plastic behaviour of the structure. the WRS distributions were assumed according to the SINTAP procedure [311. 310] and the SINTAP procedure [311. The axial stress distributions applied in the probabilistic lifetime analyses of the Small. These two procedures provide conservatively defined WRS distributions e. For Medium and Large pipe sizes.0 mm.12. AXIAL [N/mm2]. 312].2-1 to 12. handbook WRS distribution assumptions are often used. as the former procedure is limited to cover wall thicknesses from 9 to 84 mm. the local maximum values of WRSs are of the scale of material yield stress in tension. The residual stress distributions present in a structure are the result of the manufacturing history and the elastic-plastic properties of the structure. this option was not resorted to. the locally confined WRSs often provide the dominant component to total stress distribution within and near the weld region. The horizontal axis in all these figures corresponds to radial coordinate through wall with the origin in the inner surface. 310] were used. the WRS equations in the ASME recommendations [309. caused by the inner pressure are computed with the following analytical equation [306]: AXIAL pD2 t 2t (12. mostly the NPP piping components are supported quite flexibly. The colours of the curves correspond to: green. stress caused by pressure.0 t wall [mm] 2.2-1. light red.5 2.0 1. One potential location for thermal stresses is pipe bends.5 1. blue.12.0 pressure ASME-Au p+WRSs-Au Figure 12. axial [MPa] 214 . This can happen. The axial stress distribution through wall used in the probabilistic lifetime analyses of the Small pipe component. to ASME recommendations. Probabilistic lifetime analysis application for piping components In some piping components. 300 200 100 0 -100 -200 -300 0. However. when the piping component supports resist thermal expansion. no thermal stresses are included in the computations. Due to lack of specific data. thermal expansion can cause thermal stresses.5 3.0 3.0 0.5 4. WRSs acc. total stress. Totally rigid support conditions can cause very high thermal stresses to the piping component. 0 15. to SINTAP procedure. light red.0 t wall [mm] 20.0 6.2-3.0 10.0 pressure SINTAP-Au p+WRSs-Au Figure 12.0 pressure SINTAP-Au p+WRSs-Au Figure 12. The axial stress distribution through wall used in the probabilistic lifetime analyses of the Large pipe component.0 4.0 10.12. axial [MPa] 215 .0 5. axial [MPa] 400 300 200 100 0 -100 -200 0. to SINTAP procedure.0 2. stress caused by pressure. blue. WRSs acc.0 t wall [mm] 8. light red. The colours of the curves correspond to: green. The colours of the curves correspond to: green.2-2. total stress. total stress. blue.0 25. Probabilistic lifetime analysis application for piping components 400 300 200 100 0 -100 -200 0. WRSs acc. stress caused by pressure. The axial stress distribution through wall used in the probabilistic lifetime analyses of the Medium pipe component. The semielliptical surface cracks had depths between zero and (with a low probability) the full wall thickness. three different initial crack distribution assumptions are used in the probabilistic analyses. The log-normal distribution equation used in defining depth probabilities. y is the scale parameter and X is the median of x.2.25 inch wall thickness) were inspected using radiographic inspection. Existing fabrication cracks The assessment of probabilistic distributions for depth and length of the existing fabrication/manufacturing cracks in stainless steel piping used here are from ref. Log-normal distributions were used to characterize the crack depths. [313] by Khaleel and Simonen.2-2b) is the shape where x is the variable for which the probability is calculated. ~ parameter. it was assumed that the welds for all pipe sizes used the manual metal arc process and that the welds (except for the 0. The parameters of these log-normal distributions are listed in Table 12. These distributions are briefly described in the following. In developing these to some extent conservative distributions. for existing manufacturing cracks in stainless steel piping is [313]: fx x 1 2 ~ ln X yx exp ln x 2 2 y 2 y . Surface lengths were between a semi-circular shape and (with a low probability) the full pipe circumference.2-2a) y (12.12. and were conservatively placed at the inner pipe surface. fx(x). The initial cracks were circumferential. y 216 . Probabilistic lifetime analysis application for piping components 12.4 Initial crack size distributions As mentioned earlier.2-5 in the following. and (12. 0555 Shape parameter [-] 0.136.500 The probability distribution for lengths of existing manufacturing cracks in stainless steel piping is defined using crack aspect ratio. [304].5382.12.0440 0. SCC induced initial cracks in VTTBESIT The assessment of probabilistic distributions for depth and length of SCC induced initial cracks in stainless steel piping used here are from ref.419. which may affect the applicability of these distributions for the present analyses to some extent.3672 0.2669 0. all of which are circumferentially oriented cracks at the inner pipe surfaces. [313].0480 0.0991 0. Probabilistic lifetime analysis application for piping components Table 12. The development of these distributions was based on the flaw data from nine Swedish BWR units. f0 C 2 12 1 1 ln 2 2 m 2 exp .2-2c) where = 0. [245] by Cronvall et al.250 0.1063 0. Pipe wall thickness Parameters Median flaw depth [inch] 0. The probability distribution for aspect ratio is given by [313]: 0. C = 1.0047 0. and m = 1. 1 (12.0960 2.000 2. from ref. This data consists of 98 detected SCC cases.2-5. which is given here as = b/a. For wall thicknesses falling between the presented ones the corresponding parameter values are to be interpolated. Parameters for log-normal crack depth probability distribution for existing manufacturing cracks in stainless steel piping.562 1. 217 . not all of the above mentioned 98 crack cases were considered. This aspect ratio distribution was assumed to be independent of the flaw depth. In the assessment of probabilistic distributions for initial crack sizes.0256 Flaws per weld [-] 0. see ref.0035 0. where b is the total length of the surface crack and a is the crack depth.1784 0.0892 0.4993 Flaw density per inch of weld [1/inch] 0.0028 0.0508 [inch] 0. 070*exp(-0. In the legend “f” stands for probability density and “x” for relative initial crack depth. Probabilistic lifetime analysis application for piping components The fitted exponential probabilistic density functions for estimated depth and length distributions of SCC induced initial cracks are presented as probability density against relative crack dimension in Figure 12. from ref. The fitted exponential probabilistic density function for estimated depth distribution of SCC induced initial cracks. 0.062*x) Figure 12.08 0.04 0.00 5 15 25 35 45 55 65 75 85 95 initial crack depth/wall thickness [%] f(x) = 0. probability density [1/%] 218 .2-4.02 0. [245].2-4a.10 ' 0.12.06 0. 10 ' 0.062*x) Figure 12.08 0. [245]. the original data of detected cracks were first modified so as to better correspond to assumed initial sizes. In PIFRAP. SCC induced initial cracks in PIFRAP The assessment of probabilistic distributions for depth and length of SCC induced initial cracks in stainless steel piping used in PIFRAP is based on the same flaw data from the nine Swedish BWR units [304]. A truncated exponential function was fitted to thus obtained modified set of crack lengths.2-4b.0 mm.12. In the legend “f” stands for probability density and “x” for relative initial crack length.02 0. [245]. This treatment was such that the measured length of each detected crack was reduced two times the distraction of the measured crack depth and 1. as the above described distributions developed by Cronvall et al.04 0. For developing a distribution for initial crack lengths. from ref.06 0. the depth for SCC induced initial cracks in stainless steel piping is assumed to be fixed to 1.070*exp(-0.00 5 15 25 35 45 55 65 75 85 95 initial crack length/inner circumference [%] f(x)=0. The fitted exponential probabilistic density function for estimated length distribution of SCC induced initial cracks. probability density [1/%] 219 . Probabilistic lifetime analysis application for piping components 0.0 mm. 40 years and 60 years. [313] and SCC induced cracks from ref. fabrication cracks from ref. 25 %. [245]. 20 years. the last one corresponding to pipe failure.3 Analysis characteristics The scope of the probabilistic analyses is described below. thus the scope of analyses is all in all 9 computed probabilities. the earlier described Small.12. altogether two different distributions for both initial crack depth and length. as corresponding to pipe failure. i. Medium and Large pipe components. Probabilistic lifetime analysis application for piping components 12. Medium and Large pipe components. This is followed by some relevant details concerning the analysis flow of the two applied analysis codes.0 rev. probabilistic VTTBESIT and PIFRAP. SCC induced cracks from ref. considered crack depths in relation to wall thickness. considered initial crack size distributions. 75 % and 100 %.3. 12. considered initial crack size distributions. 50 %.e. 40 years and 60 years. note that with PIFRAP version 2. the earlier described Small. The scope of the probabilistic analyses with PIFRAP: considered pipe sizes. 220 . considered crack depths in relation to wall thickness. 20 years.1 Scope of probabilistic analyses The scope of the probabilistic analyses with VTTBESIT: considered pipe sizes. [304]. 0 it is not possible to compute probabilities for cracks with depth less than 100 %. considered time spans from the start of operation. 100 %. considered time spans from the start of operation. thus the scope of analyses is all in all 72 computed probabilities. 12. 50 % and 75 % of wall thickness were is to the analysis code. Probabilistic lifetime analysis application for piping components The effect of periodic pipe component inspections is not considered here. both applied analysis codes do allow the inclusion of inspections. [246. More precisely. the crack growth equation and its parameters are assumed to be deterministic. as mentioned earlier. both depth and length of initial crack postulates are assumed to be random with exponential probability density functions.2 Details concerning the analysis flow of the two applied analysis codes Details concerning the analysis flow of probabilistic VTTBESIT The probabilistic VTTBESIT calculates the probability of failure for a loaded pipe with circumferential or axial cracks that may propagate due to SCC or low/high-cycle fatigue. 221 .3. The theory and procedure behind the probabilistic VTTBESIT are given in refs. which are typically performed in NPPs yearly. most often the outer surface. In all analysed cases the operational lifetime was assumed to be 60 years. Account is taken to the probability that the crack remains undetected during successive inspections. to allow performing the computational analyses the probabilistic VTTBESIT was developed further to some extent. 247]: for SCC analyses the stresses are assumed to be deterministic. The following assumptions are made for the probabilistic analyses [246. 12. As mentioned above. crack detection probabilities are assessed with POD functions. so that the larger it is the more probable is its detection. the possibility to compute the probabilities for growing crack postulate to reach the depths of 25 %. However. the growth of the crack due to SCC will be deterministic and will only depend on the initial crack depth and length for a given geometry and given stresses. The probability of detecting a crack in inspections is dependent of its size. for fatigue induced crack growth analyses their cyclic frequency of occurrence is assumed as Poisson distributed. Typically. inspections are not taken into account here. Due to these assumptions. Failure is taken to mean the event when the crack postulate has grown to just reach the opposite surface of the pipe component. As mentioned earlier. 247]. the growth of the crack will be deterministic and will only depend on the initial crack length for a given geometry and given stresses.] is the number of times the Monte Carlo simulations have reached or exceeded the selected degradation/failure state in terms of crack depth a [mm]. As mentioned above.12. and fi0 [ . see ref. [248]. the initial crack depth is assumed to be fixed to 1. N [ . N a a FS .i . seismic loads etc. t [years] is time. water hammer loads.3-1) where p FS . The value of the J-integral is calculated according to the R6 procedure.] is the total number of Monte Carlo simulations. The following assumptions are made here for the probabilistic analyses [304]: the stresses are assumed to be deterministic. Failure is taken to mean a complete break due to J –integral controlled fracture or plastic collapse.tj t [ . the fracture toughness properties used in the analyses were those of the weld material. Probabilistic lifetime analysis application for piping components The basic failure probability equation of VTTBESIT can be expressed as: p FS . In order to be on the safe side.] is the rate of crack initiation due to SCC. was here based on the mentioned Swedish IGSCC data with the value of this parameter being 4.0 mm. Details concerning the analysis flow of probabilistic PIFRAP PIFRAP calculates the probability of failure for a loaded pipe with circumferential cracks that may propagate due to IGSCC or IGSCC in combination with high cycle fatigue. Other used material properties were those of the base material. 222 . [304]. the crack growth law and its parameters are assumed to be deterministic. inspections are not taken into account. The rate of crack initiation due to SCC. the probability that a crack with the assumed depth is initiated during the time interval (ti. The theory and procedure behind PIFRAP are given in reference [315]. Account is taken to the probability that the crack remains undetected during successive inspections. Due to these assumptions.tj t Na N a FS .i .08E-04. fi0. The growth of the above mentioned crack postulate is calculated with the procedure presented in ref.i [ .] is failure probability corresponding to selected degradation/failure state. ti + dt) is given by the function fi(ti)dt.i f i0 (12. and from that point the growth of a through thickness crack to final failure of the pipe. The first way is by providing the crack growth rate equation coefficients. l0 f al l 0 dl 0 (12. In order to be on the safe side. The crack growth rate concerning IGSCC can be defined in two ways. measured per reactor year as a mean value for the remaining operating time of the plant.t). In PIFRAP the crack growth calculations are performed by the PC program LBBPIPE. pf represents the probability for a rupture.] is conditional probability of rupture given the initial crack length l0. Values for these constants are presented in Section 12. the fracture toughness properties used in the analyses were those of the weld material. [304]. as obtained from refs.3. is: da dt CK I n (12. T . 314].2.2. T [years] is service life.T lc p f 0 t .3-3) which was presented earlier in Section 7. In both the probabilistic and deterministic modules of VTTBESIT.3. the fracture mechanics based crack growth rate equation used in the simulations depicting the intermediate (stage 2) SCC. i.3 Sub-critical crack growth In the sub-critical crack growth analyses. [136. Crack opening areas and mass leak rates are also calculated for the leaking crack. 12. and the rest of the material properties used were those of the base material. pf0 [ . pf is divided by the time (T . which is a program for leak-before-break analysis of circumferential cracks in pipes subjected to IGSCC or fatigue.e. the propagation of an initial surface crack through the pipe wall is assessed.12. The authors of the program use the same approach in ref. LBBPIPE estimates the growth of an initial surface crack through the pipe wall to leakage. The crack growth rate equation used in PIFRAP is [304]: 223 . Finally.2. Probabilistic lifetime analysis application for piping components The failure probability equation of PIFRAP can be expressed as [304]: 2 Ri p f t.3-2) where t [years] is time. where tF [years] is time of rupture. The lower length limit lc [mm] is the solution of the equation tF(lc) = T. 07E-04 1.4 Results and discussion The numerical results from the probabilistic VTTBESIT and PIFRAP analyses for to the considered three pipe sizes are presented in the following. 12.00E+00 3.15E-04 0.34E-04 1. Probabilistic lifetime analysis application for piping components da dt C0 K I C1 n (12.00E+00 4. fi0. equation (12.00E+00 224 .53E-05 1.12.00E+00 0.36E-04 2.e.00E+00 0.08E-04 [304].3-4) is upon application the same as equation (12.3-3). The rate of crack initiation due to SCC. Time from start of operation [years] 20 20 20 20 40 40 40 40 60 60 60 60 Reached crack depth through wall [%] 25 50 75 100 25 50 75 100 25 50 75 100 Probability. see Tables 12.67E-06 0.00E+00 0. i.4-2.07E-04 1.00E+00 0.63E-05 0.4-2 as well as Figures 12.22E-06 0.41E-05 5.4-1. no threshold for propagation of SCC is considered.00E+00 Probability. Medium pipe [1/year/weld] 3. The results from the probabilistic SCC analyses with VTTBESIT for pipe sizes Small.00E+00 9.08E-04 4.3-4) where the input data parameters are otherwise the same as in equation (12.4-1 and 12. Thus here.08E-04 4.27E-04 0. is the earlier mentioned 4.00E+00 4. with the value of additional parameter C1 being here zero. Small pipe [1/year/weld] 4.05E-04 1.00E+00 4.4-1 and 12. Large pipe [1/year/weld] 1. Medium and Large with initial crack size distribution as fabrication cracks.00E+00 0. the assumed operational lifetime is 60 years.22E-04 0.01E-04 1.08E-04 4.00E+00 0.00E+00 2.02E-06 0. Table 12. As mentioned earlier.34E-06 0.18E-05 0.00E+00 Probability.86E-04 8.3-3). 0E-04 1. Medium and Large pipe sizes. 50% 20y. 225 .0E-03 probability of failure [1/year/weld] 1. 100% 60y. Medium and Large with initial crack size distribution as fabrication cracks.0E-07 SVTB MVTB LVTB Figure 12. 25% 20y.0E-05 1.12.0E-06 1. 75% 40y. The results from the probabilistic SCC analyses with VTTBESIT for pipe sizes Small. 25% 40y. 50% 40y. MVTB and LVTB correspond to Small. 100% 1. Probabilistic lifetime analysis application for piping components 20y. 75% 60y.4-1. 50% 60y. 75% 20y. Here SVTB. 25% 60y. 100% 40y. 91E-04 3.53E-07 3.79E-04 3. Small pipe [1/year/weld] 3.23E-04 1.95E-05 3.61E-04 3.71E-04 2.94E-04 3.43E-06 6.87E-04 2.89E-04 3. Reached crack Time from start of opera.80E-04 3.75E-04 4.58E-04 2.82E-04 Probability.65E-04 3.19E-04 3.18E-06 3. Large pipe [1/year/weld] 3.67E-06 1.56E-05 3.04E-06 1.02E-04 2.97E-04 1.69E-04 2.90E-04 3.22E-06 226 .94E-04 Probability.02E-04 2.97E-04 3.12. Probabilistic lifetime analysis application for piping components Table 12.82E-04 3. The results from the probabilistic SCC analyses with VTTBESIT for pipe sizes Small.41E-04 3.98E-04 3.28E-04 3.depth through wall [%] tion [years] 20 20 20 20 40 40 40 40 60 60 60 60 25 50 75 100 25 50 75 100 25 50 75 100 Probability. Medium pipe [1/year/weld] 3.4-2.89E-04 3.56E-04 2.37E-04 3. Medium and Large with initial crack size distribution as SCC induced initial cracks. 00E+00 227 .0E-05 1. Medium and Large with initial crack size distribution as SCC induced initial cracks [304] are presented in Table 12.01E-04 3. Probabilistic lifetime analysis application for piping components 20y. As mentioned earlier. Medium pipe [1/year/weld] 2. Small pipe [1/year/weld] 1.0E-04 1. Time from start of operation [years] 20 40 60 Reached crack depth through wall [%] 100 100 100 Probability.0E-03 probability of failure [1/year/weld] 1. the assumed operational lifetime is 60 years.12. Table 12. 75% 20y. Here SVTB. as well as in Figure 12.84E-04 2.4-3. The results from the probabilistic SCC analyses with VTTBESIT for pipe sizes Small.26E-05 0. 50% 20y. Large pipe [1/year/weld] 3.0 rev.00E+00 Probability. MVTB and LVTB correspond to Small. 25% 20y. 100% 60y. 75% 40y.4-3. 25% 60y. Medium and Large with initial crack size distribution as SCC induced initial cracks. 50% 60y.4-2.50E-04 3. 100% 1. Note that with PIFRAP version 2. Medium and Large pipe sizes. The results from the probabilistic SCC analyses with PIFRAP for pipe sizes Small.0E-06 1.33E-05 0. 75% 60y.4-3. The results from the probabilistic SCC analyses with PIFRAP for pipe sizes Small. 25% 40y. Medium and Large with initial crack size distribution as SCC induced initial cracks [304]. 50% 40y.0E-07 SVTB MVTB LVTB Figure 12.00E+00 Probability. 0 it is not possible to compute probabilities for cracks with depth less than 100 % of wall thickness. 100% 40y.19E-05 0. Concerning fabrication cracks. whereas for Large pipe size these probabilities were zero. varying 228 .4-3. there was notable variation in the results. varying approximately between 10E-06 to 10E-04. 100% 40y. Probabilistic lifetime analysis application for piping components 20y. 40 and 60 years. 100% 1. the probability to reach 25 % of wall thickness was greater than zero in all cases.0E-03 probability of failure [1/year/weld] 1. As for SCC analysis results obtained with probabilistic VTTBESIT. As for the analysis result diagrams. whereas in others they were quite conformable. the scaling of them has been selected so that most of the results could be included visibly enough together with the associated scatter. Here SPFP. MPFP and LPFP correspond to Small. for all pipe sizes the probabilities for crack postulates to reach 50 and 75 % of wall thickness was greater than zero.0E-04 1.0E-06 1. Concerning SCC induced initial cracks.0E-07 SPFP MPFP LPFP Figure 12. The results from the probabilistic SCC analyses with PIFRAP for pipe sizes Small. in some cases. Medium and Large with initial crack size distribution as SCC induced initial cracks [304]. the cases with result value of zero are not included in them. Within the considered time spans of 20. 100% 60y. however these and all other result cases can be found in the corresponding result tables. Medium and Large pipe sizes.0E-05 1. As the scaling of the vertical axis in the diagrams is logarithmic.12. in case of Small and Medium pipe sizes the probabilities for crack postulates to reach 50 and 75 % of wall thickness varied approximately between 10E06 to 10E-04. 12. When concerning the effect of pipe size to VTTBESIT results. they were in all cases quite conformable as compared to VTTBESIT results. whereas with SCC induced initial cracks they are considerably more conformable. Within the time spans of 20 and 40 years. With fabrication cracks the scatter of probability results is of several decades for these pipe sizes. Concerning both fabrication cracks and SCC induced initial cracks.01E-04. However. for all pipe sizes the probability for crack postulates to reach 100 % of wall thickness was zero. the higher the failure probabilities. however with the mean value being to some extent smaller than that corresponding to probability to reach 25 % of wall thickness. This difference is explained by initial depth of fabrication cracks being limited on average to approximately 229 . 40 and 60 years was 3. with PIFRAP version 2. for the time span of 60 years this probability was in all cases zero. whereas concerning SCC induced initial cracks this probability varied for all pipe sizes approximately between 10E-07 to 10E-04. as only the wall thickness matters here. Probabilistic lifetime analysis application for piping components again approximately between 10E-06 to 10E-04. the maximum probability for crack postulates to reach 100 % of wall thickness within the considered time spans of 20. 40 and 60 years was 2. there are notable differences between the probability result sets concerning fabrication cracks and SCC induced initial cracks.0 rev. with fabrication cracks the probability results are for these pipe sizes in general lower than with SCC induced initial cracks.94E-04. From the results it can be seen. whereas the corresponding minimum probability value was zero. Further. However. those remaining within the range from 10E-04 to 10E-03. for Medium and Large pipe sizes. in all cases the probability for SCC induced initial cracks to reach 100 % of wall thickness was greater than zero. whereas the corresponding minimum probability value was zero. As for SCC analysis results obtained with PIFRAP. As in all cases here the tensile stresses are at maximum in the inner surface. varying approximately between 10E-05 to 10E-04. which consequently leads to the highest crack growth and failure probability results. Concerning fabrication cracks. The most severe total stress distribution is associated with the Small pipe size. The maximum probability for crack postulates to reach 100 % of wall thickness within the considered time spans of 20. As mentioned earlier. 0 it is not possible to compute probabilities for cracks with depth less than 100 % of wall thickness. the outer diameter plays no role in them. that the higher the total tensile stress. and consequently the initial crack postulates that all open to that surface are prone to propagation. Here the cases with fabrication cracks and SCC induced initial cracks lead to quite matching probability results. the resulting failure probabilities are highest. concerning probability against size. This is reflected in the results so that the failure probabilities obtained with PIFRAP reside mainly between those obtained by VTTBESIT. This partly explains why in case of this latter pipe size the resulting failure probabilities are also the lowest. The analysis results obtained with probabilistic VTTBESIT are more scattered than those obtained with PIFRAP. The Medium pipe size cases result with the second highest failure probabilities. the distribution of SCC induced initial cracks in PIFRAP falls mostly somewhere between the distributions of fabrication cracks and SCC induced initial cracks in VTTBESIT. being associated with the most severe total stress distribution. as the dominant stress component.5 mm. whereas that for the Large pipe size is non-linear. being WRSs. this leads to lower failure probabilities for pipe components with thicker walls. again in case of the Small pipe size. Namely. All in all. all being in the form of probabilistic density functions. Similar effects as those described for VTTBESIT results above can be recognized also in the corresponding PIFRAP results. the slower is the ensuing crack growth. This is obviously realistic. this turning point is approximately in the middle of the wall. for the Small and Medium pipe sizes the shape of the total stress distribution is linear. but the shapes of through wall stress distributions need to be considered as well. This can be mainly attributed to the differences in the used initial crack assumptions. there are regions both in tension and in compression. The nearer to the inner surface the total stress distribution turns from tension to compression. It could be assumed that with exactly same initial crack distributions these two codes will provide quite accurately matching results for the same set of analysis cases. Namely. respectively. For the Small and Medium pipe sizes. In the light of the obtained probabilistic SCC analysis results.12. has to be self-balancing in the axial direction. whereas the SCC induced initial cracks can be deeper than that (with a small probability). Namely. the generally less severe the WRSs are and consequently also the total stress distribution. Not only the maximum total tensile stresses through wall affect the crack growth and failure probabilities. Probabilistic lifetime analysis application for piping components 2. and the Large pipe size cases with the lowest failure probabilities. the thicker the pipe wall. In both stress distribution shape cases. and with larger wall thicknesses this limitation leads to comparably lower crack growth and failure probabilities. whereas for the Large pipe size it is approximately 25 % from the inner surface. 230 . As the actual WRS distributions were not available. those being for VTTBESIT the depth and length of initial cracks as well as frequency of occurrence of cyclic loads.12. Probabilistic lifetime analysis application for piping components The presented probabilistic results are supposed to be conservative. due to which it is easy to perform sensitivity analyses. Compared to purely statistical modelling tools. This viewpoint also serves the needs of RI-ISI analyses for NPP piping systems. Further. fracture toughness is a distributed parameter. the to some extent conservative distributions given in refs. the PFM approach of VTTBESIT and PIFRAP describes much more realistically component degradation. As or more significant than looking at single pipe failure probability values. 310. For instance. when SCC has been detected after some number of years in operation. However. These are features which cannot be changed or modified by the program user. As a single analysis typically covers several decades of plant operation. is to compare these values against each other. 311. as both codes accept only constant values for SCC equation parameters. and typically the strength properties of the materials of the latter type are lower than those of the former.g. [309. and for PIFRAP only the length of the initial cracks. Some characteristics of both probabilistic VTTBESIT and PIFRAP as modelling tools are discussed in the following. where the given values are invariably lower than those obtained from the actual material test result data. The advantages of both VTTBESIT and PIFRAP include relatively short analysis run times. which in this case were not available either. [305]. 312] were used. 231 . The formation of component groups according to risk level in RI-ISI is preceded by formation of component groups according to failure probability and failure consequences. Moreover. As the material property data of the weld material were not available. those of the base material were used. the material strength data of the base material were obtained from a normative ref. Neither VTTBESIT nor PIFRAP do allow considering a change in water chemistry during a single analysis run. It is a drawback that quite few input data parameters are allowed to be random. the values of many material properties are distributed and therefore a single value cannot correctly describe them. this limitation unrealistically does not allow taking into account the improvements of water chemistry carried out in many NPPs. the inclusion of physical models is another advantage of both VTTBESIT and PIFRAP. The inclusion of realistic load histories consisting of typical NPP load transients of various types is possible with VTTBESIT but is not with PIFRAP. due to the assumptions made with input data. e. which completely omit the modelling of physical degradation phenomena. in order to achieve more accurate/realistic analysis results. which would give the quantified order of proneness to degradation/failure. Arguably.12. Probabilistic lifetime analysis application for piping components In the light of the discussion above concerning the scope and characteristics of VTTBESIT and PIFRAP. More precisely. the shortcomings concerning input data and analysis scope of these two analysis codes should be improved. the best use of the single result values can be achieved by comparing them against each other. All in all. this would provide for considered single failure probabilities such interpretation as how many times more/less a particular pipe weld is prone to fail than the others. the probabilistic analysis results presented in this section are at least to some extent suggestive. 232 . e. Ageing degradation can be observed in a variety of changes in physical properties of metals. erosion. especially passive components. Summary and suggestions for future research Lifetime analyses of structural systems and components are discussed in this thesis. In general. Ageing models of all these kinds are presented. the relevant ageing mechanisms and prevailing environmental conditions. In practise. When the resulting failure probabilities from probabilistic analyses are combined with knowledge of system or component failure consequences. age related degradation may affect the dynamic properties. fatigue capacity. the ageing analyses are usually performed with a suitable analysis tool. These materials may undergo changes in their dimensions. Summary and suggestions for future research 13. wear. it is also possible to perform a risk analysis. structural resistance/capacity. reactive chemicals and synergistic effects [5]. cracking. These analyses necessitate a thorough knowledge of structural properties. with analysis software. In addition.13. embrittlement. i. ductility. These ageing mechanisms act on components due to a challenging environment with high heat and pressure. loads. The power plant components have generally substantial safety margins when properly designed and constructed. concrete and other materials in a power plant. mechanical or dielectric strength. physical or chemical processes such as fatigue. probabilistic or a combination of these two types. However. A selection of probabilistic system and component ageing and risk analysis tools together with an application example are also presented in this thesis. Ageing degradation presents itself as a variety of ageing mechanisms. radiation. A better understanding of the effect of ageing degradation on structures and components. 233 . corrosion and oxidation. structural response. failure mode and location of failure initiation. the available margins for degraded structures are not well known. is needed to ensure that the current licensing basis is maintained under all loading conditions [10]. the nature of ageing models of structural systems and components can be deterministic. principles of RI-ISI. risk analysis methods. brief overview and comparison of three main RI-ISI methodologies. process of risk-informed inspection planning. deterministic degradation modelling methods. Presently. the emphasis here is on the component analysis tools. In the general applications. RI-ISI is a very topical subject in Finland. Some of the presented tools are capable for analyses of both individual components and systems comprising several components. 234 . Concerning structural as well as risk analysis methods for power plant components. and benchmarking of RI-ISI methodologies in the recent RISMET project. qualitative. The RI-ISI associated issues considered in this thesis include: introduction and basics of RI-ISI. Some of the presented analysis tools are commercially available and others are proprietary. identification of ageing mechanisms and mitigation of ageing effects. any failure mode. Keeping this in mind. As for structural and risk analysis applications. it has been attempted to cover RI-ISI here from a number of viewpoints to get a wider description of this subject. probabilistic degradation modelling methods. quantitative and semi-quantitative methods for RI-ISI. RI-ISI approach developments by VTT. Validation of such software is usually carried out by benchmark exercises between different programs since it is not feasible to compare results with real failure statistics due to their scarcity. ageing and risk analysis applications. because the Finnish Radiation and Nuclear Safety Authority (STUK) now requires RI-ISI analyses for all piping systems of both existing and planned/future NPPs [316]. Component ageing analyses are discussed as well. can be addressed using the appropriate limit states. In the top level they consist of selecting components for analyses. fracture or other. those presented and discussed in this thesis include: deterministic analysis methods. The probabilistic analysis approaches of the presented software vary from FORM and SORM to MCS.13. conduct and process of ageing degradation analyses. Summary and suggestions for future research Both physical and probabilistic ageing models for power plant components are presented in the thesis. probabilistic analysis methods. the knowledge that is represented by data based on too few statistics. Living PRA/PSA is one way with which problems related to time dependent ageing can be approached. These include structural mechanics.g. and PIFRAP developed by DNV. Some drawbacks and concerns of risk assessment that would need further research/development are presented in the following: model uncertainty.13. The matching of the obtained analysis results is at least good. fracture mechanics. In the probabilistic failure analysis computations. also data concerning relevant degradation mechanisms and failure modes are needed. or unintentionally because of a lack of data. in risk analyses. Summary and suggestions for future research The computational application part of this thesis concerns probabilistic failure and lifetime analyses to a relatively small but representative set of NPP piping components. All in all. In addition to data concerning structural properties. As lifetime and risk analysis procedures are in a developing stage. The results showed that this task is cumbersome. and material science. the sensitivity of the tails of the failure probability distributions. supports and environment. lifetime and risk analyses of power plant components and systems are disciplines that require combined knowledge from several fields of science. and not all questions were answered. whereas the applied PFM analysis procedures are quite similar. Essential sources for piping degradation and failure data are international and plant specific databases. the small failure probabilities for some components mean that evaluated failure probabilities cannot be properly and completely validated. or full understanding of the system. In a fairly recent and thorough study [1]. probability mathematics. human errors. primary and secondary loads. Incorporation of unavoidable ageing effects of SSCs into PRAs/PSAs is a subject that would benefit from further research/development. as a living modification PRA/PSA is simply just updated as necessary to 235 . two analysis codes were used. The main source causing in some cases deviations in the results is the differences and limitations in the treatment of input data parameters. quantities are omitted either intentionally. those being probabilistic VTTBESIT developed by VTT and IWM. it was attempted to apply linear ageing failure rate model to incorporate ageing effects into PRA/ PSA. However. e. there are many aspects and features that call for further research. Since both techniques have the same goal.g. there are still degradation phenomena the physical mechanisms of which are not clear. which include scope. and thus being able to perform sensitivity analyses for the factors affecting this process. This is a problem that can be overcome through the development of computational procedures. in primary systems of NPPs. Through improved knowledge of various degradation phenomena. range of validity. For instance. any comparison between components that have been analysed with different methods will be at least to some extent questionable. the strictness in the upper bound approach used in the treatment of the underlying data concerning experimentally defined model parameters characterising environment and material as 236 .13. Concerning this issue. This could include further developing the computation procedures as well as providing more realistic estimates for certain model input parameters. ENIQ has quite recently published a notable report [317] on ensuring the applicability of PFM and other structural reliability approaches. the considerably high level of conservatism in the current SCC propagation computation approaches could be relieved. While PFM procedures lack standardisation. A difference between PFM and PRA/PSA that causes difficulties is on which level the input data are based on actual observations. it has been used extensively to analyse entire complex systems in nuclear industry and elsewhere. However. An example of such degradation phenomena is SCC. more accurate models of them can be formulated. This report summarises the verification and validation requirements that structural reliability models and associated analysis codes should satisfy in order to be suitable for such purposes. In general. accuracy and realistic consideration of the underlying physical phenomenon or phenomena. Namely. physical degradation models need development in many areas. PFM has had a more limited area of application and has been used for more specialised purposes concerning certain single components e. The advantage of PFM is the possibility of modelling clearly the uncertainties related to the structural degradation process. Even though most degradation phenomena are quite well known. These degradation phenomena need further research. a better utilisation of them could potentially be beneficial. it does not have the capability to predict how time dependent SSC ageing affects failure probabilities and risk. there are still methodological problems and there is no standard format for how to perform PFM analyses. Summary and suggestions for future research reflect the current design and operational features. As for PRA/PSA approach. It is highly desirable to obtain standardised methods for PFM analyses like the one that has been developed for PRA/PSA. as the failure probability results given by various analysis codes may not be comparable due to differences in analysis models and required input data. The approaches to compute strain rates. Another topical example is the environmental fatigue correction factor approach. since statistical ageing methods omit the modelling of the underlying physical phenomenon or phenomena.13. Improved degradation models would also be beneficial from the viewpoint of RI-ISI. as the number of failure mechanisms encountered in NPP environments is much greater. On the other hand. their applicability is limited. each of which would cover one or two degradation phenomenon or phenomena and the failure probability results of all of which would be comparable. Statistical ageing models of SSCs are not hampered by the above mentioned problems. it appears that for the time being several codes are needed. a more realistic proposal of a new inspection program could be prepared. it would be necessary to develop either one analysis program that considers all relevant degradation phenomena or a family of analysis programs. The aim would be to obtain a reasonably conservative SCC propagation computation approach. With more accurate and realistic degradation models. The analysis programs should also be able to consider the distributed nature of at least all those parameters the actual nature of which is markedly distributed.e. which is a drawback. Often. These results could in turn be used to perform a more accurate quantitative risk analysis. One problem that would also benefit from further research is the modelling of several degradation phenomena acting simultaneously. Thus. This causes problems. if one wishes to cover the degradation of all safety significant NPP systems and components. modelling problems are caused by scarce and unreliable input data. the scope and accuracy of this approach is not sufficient for application concerning NPP environments. Presently. Summary and suggestions for future research well as WRS distributions could be to some extent relaxed. i. the Fen procedure [184]. remarkable benefits could be achieved in the forms of financial savings and shorter times which NDT personnel have to work under radiation exposure. It appears that most of the available NPP SSC lifetime analysis codes consider only one or two degradation mechanisms. Based on the results from such risk analysis. Thus. more reliable failure probabilities could be calculated. combine load transients to cycles as well as to select and treat strain and stress components should be clarified. Another capability that is needed when considering power plant environments is 237 . Thus. The number of locations included in the inspection program is often considerably smaller in new RI-ISI programs than in older and more conservative inspection programs developed with deterministic approach. e. However. i.g. could be alleviated or even overcome. The development and better availability of component degradation/failure databases will hopefully provide means for how the problems related to small failure probabilities. On the other hand. One possibility for further research would be to collect all available good quality NDT data and develop simple POD functions environment. The amount of NPP piping component degradation data in the international databases has increased and quality improved. [318. In the case of NDT methods. Summary and suggestions for future research that the applied degradation modelling tools should be able to consider plant specific features and characteristics. 319].e. the inclusion of plant specific features and characteristics in the degradation analyses necessitates the availability of plant specific databases. to very rare events.13. Other capabilities that degradation modelling tools of components should be able to consider are small failure probabilities and quantification of the effectiveness of NDT. it is often expensive and time consuming to produce such a large amount of data. degradation mechanism and material type specifically for a representative set of power plant pipe sizes. which consequently allows improving accuracy in the statistical piping degradation estimates. which describe the detection probability as a function of the flaw size. see refs. the construction of a POD curve requires a considerable amount of data before statistical confidence is achieved. 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A procedure for estimation of pipe break probabilities due to IGSCC. Sweden. et al.S. Helsinki. 30. In practise.e.Series title. with analysis software. the lifetime analyses in question necessitate a thorough knowledge of structural properties. fracture mechanics. Computational application of probabilistic failure and lifetime analyses to a representative set of NPP piping components with probabilistic codes VTTBESIT and PIFRAP are presented after that.fi/publications/index. number and report code of publication VTT Publications 775 VTT-PUBS-775 Author(s) Otso Cronvall Title Structural lifetime. Box 1000.) 978-951-38-7761-3 (URL: http://www.g. Degradation models of all these kinds are presented here. A selection of probabilistic system/component degradation and risk analysis software tools is presented in the latter part of this thesis. RI-ISI VTT Technical Research Centre of Finland P. risk analysis. Description of probabilistic analysis methods covers both physical and statistical approaches. physical probabilistic methods are based on deterministic analysis methods with the modification that one or more of the model parameters are considered as probabilistically distributed. This is followed with a description of deterministic structural analysis methods. +358 20 722 4520 Fax +358 20 722 4374 .fi/publications/index. reliability. reliability and risk analysis approaches for power plant components and systems Abstract Lifetime. FI-02044 VTT. lifetime and risk analyses are usually performed with a suitable analysis tool. structural mechanics and fracture mechanics based analysis methods as well as the disadvantages of the deterministic analysis approach. When the system/component failure probabilities are combined with knowledge of failure consequences.vtt. probabilistic or a combination of these two types.vtt. PFM. fracture mechanics. reliability and risk analysis methods and applications for structural systems and components of power plants are discussed in this thesis. material science and fluid mechanics. Finland Phone internat. These methods are needed in the lifetime analyses of structural systems and components of power plants. The nature of degradation models of structural systems/components can be deterministic. i. loads. The thesis ends with a summary and suggestions for future research. An overview of power plant environments and a description of the various degradation mechanisms damaging the power plant systems and components are presented first. Finnish abstr. 264 p. the relevant degradation mechanisms and prevailing environmental conditions. it is possible to assess system/component risks. In general.jsp) Date Language Pages December 2011 Keywords English. Some important risk analysis applications are described.g. These include probabilistic risk/safety assessment (PRA/PSA) and risk informed in-service inspections (RI-ISI). Often.O. ISBN 978-951-38-7760-6 (soft back ed. such as structural mechanics. Several risk analysis methods are presented as well as some limitations and shortcomings concerning to them.jsp) Series title and ISSN Project number VTT Publications 1235-0621 (soft back ed. probability mathematics. covering e. These analyses involve many fields of science. Several probabilistic analysis procedures are presented. Modelling methods for various degradation (or ageing) mechanisms are presented.) 1455-0849 (URL: http://www. Publisher Structural mechanics. Monte Carlo Simulation (MCS) and importance sampling. e. . 02044 VTT Puh. Työssä esitetään kaikkiin näihin tyyppeihin lukeutuvia vaurioitumismalleja.ja riskianalyysit (PSA ja PRA) ja riskitietoiset tarkastusohjelmat (RI-ISI). probabilistisia tai näiden yhdistelmä. 264 s. risk analysis. 020 722 4520 Faksi 020 722 4374 . kuten Monte Carlo -simulaatio ja tärkeysperusteinen otanta. Todennäköisyysperusteisia analyysimenetelmiä koskeva kuvaus kattaa sekä fysikaaliset että tilastolliset lähestymistavat.jsp) Avainnimeke ja ISSN Projektinumero VTT Publications 1235-0621 (nid.ja riskianalyysit tehdään yleensä jollakin sopivalla analyysityökalulla. sekä eräitä niitä koskevia rajoituksia ja puutteellisuuksia. Ensin esitetään yleiskatsaus voimalaitosympäristöistä sekä kuvaukset erilaisista voimalaitosten rakennejärjestelmiä ja -komponentteja koskevista vaurioitumismekanismeista. tiiv. fracture mechanics.vtt. sekä erittelyä deterministisen lähestymistavan puutteista. PFM.jsp) Julkaisuaika Kieli Sivuja Joulukuu 2011 Avainsanat Suomi. Sitten käsitellään deterministiset rakenneanalyysimenetelmät. varmuuden ja riskien analysoimisen lähestymistapoja voimalaitosten komponenteille ja järjestelmille Tiivistelmä Tämän lisensiaattityön aiheina ovat voimalaitosten rakennejärjestelmien ja -komponenttien käyttöiän.) 978-951-38-7761-3 (URL: http://www. Työn lopuksi esitetään yhteenveto sekä jatkotutkimusaiheita koskevia ehdotuksia. kuten lujuusopin ja murtumismekaniikan menetelmät. Rakennejärjestelmien/-komponenttien vaurioitumista kuvaavat mallit voivat olla tyypiltään deterministisiä. numero ja raporttikoodi VTT Publications 775 VTT-PUBS-775 Tekijä(t) Otso Cronvall Rakenteellisen eliniän.Julkaisun sarja. Näihin analyysimenetelmiin liittyy usea tieteen ala.ja käyttöikäanalyysit.fi/publications/index. Kun rakennejärjestelmien/-komponenttien todennäköisyydet yhdistetään tietämykseen seurausvaikutuksista. kuormista. merkittävistä vaurioitumismekanismeista sekä vallitsevista olosuhteista. Työn jälkipuoliskolla esitetään valikoima rakennejärjestelmien/-komponenttien vaurioitumisen ja riskien analyysisovelluksia. RI-ISI VTT PL 1000. voidaan arvioida vastaavat riskit.vtt. engl. kuten lujuusoppi. Työssä esitetään useita probabilistisia analyysimenetelmiä. luotettavuuden ja riskitarkastelujen analyysimenetelmät ja sovellukset. Näihin lukeutuu todennäköisyyspohjaiset turvallisuus. Työssä esitetään useita riskianalyysimenetelmiä. Työssä esitetään valikoima erilaisia vaurioitumismekanismeja koskevia mallinnusmenetelmiä. Näitä menetelmiä tarvitaan rakennejärjestelmien ja -komponenttien käyttöikäanalyyseissa. ISBN 978-951-38-7760-6 (nid. todennäköisyysmatematiikka. Julkaisija Structural mechanics. eli sovellusohjelmalla. murtumismekaniikka. Usein fysikaaliset probabilistiset menetelmät perustuvat deterministisiin vastaaviin sillä muunnelmalla. Esitetään muutamia merkittäviä riskianalyysin sovelluksia. Sen jälkeen edustavalle valikoimalle ydinvoimalan putkistokomponentteja esitetään analyysiohjelmilla VTTBESIT ja PIFRAP tehdyt todennäköisyysperusteiset vikaantumis. että yksi tai useampi malliparametri on asetettu probabilistisesti jakaantuneeksi. materiaalitiede ja virtausmekaniikka. Käytännössä käyttöikä. Yleisesti ottaen kyseiset käyttöikäanalyysit edellyttävät perusteellisia tietoja rakenteellisista ominaisuuksista.fi/publications/index. reliability.) 1455-0849 (URL: http://www. probabilistic fracture mechanics is an often used approach.VTT CREATES BUSINESS FROM TECHNOLOGY T  echnologyandmarketforesight•Strategicresearch•Productandservicedevelopment•IPRandlicensing •Assessments.vtt. ISBN 978-951-38-7760-6 (soft back ed. STRUCTURALLIFETIME.jsp) .testing. Probabilistic structural analysis procedures can be used to assess their effects. with which one can assess the remaining lifetime and structural integrity of systems/components of power plants. The last objective is to provide an application example.vtt.) ISSN 1235-0621 (soft back ed. There are various degradation mechanisms to which power plant systems and components are susceptible to.. reliability and risk of structural systems and components of power plants. When the system/component failure probabilities are combined with knowledge of failure consequences. it is possible to assess system/component risks.inspection.certification•Technologyandinnovationmanagement•Technologypartnership •••VTTPUBLICATIONS775 Lifetime. The scope and characteristics of such tools are described.fi/publications/index. the lifetime analyses of power plant systems/components necessitate a thorough knowledge of structural properties. Another objective is to collect information concerning the relevant existing probabilistic degradation and risk analysis tools. This was pursued by making an extensive literature survey. reliability and risk analysis methods and applications for structural systems and components of power plants are considered in this thesis. One objective is to collect both the methods and background theories for evaluation of operational lifetime. the relevant degradation mechanisms and prevailing environmental conditions. In general. loads. Several objectives were set for this thesis..RELIABILITYANDRISKANALYSISAPPROACHESFORPOWERPLANTCOMPONENTS..fi/publications/index. This was carried out by making a failure probability analysis to a representative set of nuclear power plant piping components.jsp) ISSN 1455-0849 (URL: http://www.) ISBN 978-951-38-7761-3 (URL: http://www. Examples of important risk analysis applications for power plants include probabilistic risk/safety assessment (PRA/PSA) and risk informed in-service inspections (RI-ISI). For power plant components.