methods, appls and software for structural reliability

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Methods, Applications and Softwarefor Structural Reliability Assessment

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21 August 2001Date

SL/WEM/R/M8663/5/01/CReport No.

CONTENTS

Page

SUMMARY 1

1. INTRODUCTION 31.1 Background 31.2 Historical Development of Methods 41.3 Scope of Review 4

2. BASIC CONCEPTS 42.1 Definitions and Acceptance of Risk 42.2 Failure Modes 5

3. QUANTIFICATION OF RELIABILITY 63.1 Hierarchy of Structural Reliability Methods 63.2 Limit States and Definitions 73.3 Types of Uncertainties 73.4 Types of Analysis 8

4. REVIEW OF APPLICATIONS IN DIFFERENT INDUSTRIES 104.1 Overview 104.2 Nuclear 104.3 Offshore Structures 124.4 Transport 144.5 Bridges and Buildings 164.6 Power, Process and Chemical Plant 174.7 Pipelines 17

5. PROBABILISTIC TREATMENT OF FRACTURE AND COLLAPSE 205.1 Description of Failure Assessment Diagram 205.2 Status of Current FAD-Based Approaches 205.3 Inherent Safety Level of FAD Approach and Use of Partial Safety Factors 215.4 Model Uncertainty in the Failure Assessment Diagram 225.5 Probabilistic Treatment of Failure Assessment Diagram 22

6. TARGET RELIABILITY LEVELS IN DIFFERENT CODES AND INDUSTRIES 236.1 Overview 236.2 Quantifying Societal Consequence 236.3 Treatment of Consequence in Three Major Codes 246.4 Comparison of Target Reliability Levels in Different Industries 27

7. SOFTWARE FOR RELIABILITY ANALYSIS 287.1 Scope 287.2 STRUREL 287.3 ProSINTAP 297.4 CALREL 307.5 PROBAN 307.6 COMPASS 307.7 NESSUS 307.8 ISPUD AND COSSAN 307.9 STAR 6 307.10 UMFRAP 317.11 FORM and MONTE 31

8. FURTHER DEVELOPMENT OF PROBABILISTIC METHODS 318.1 Generic Methods 318.2 Failure Assessment Diagrams 318.3 Distributions of Material Properties 328.4 Reduction of Data Uncertainty 32

9. CONCLUSIONS 32

REFERENCES 34

TABLES 41

FIGURES F1

APPENDIX 1 STRUCTURE OF VARIOUS RELIABILITY SOFTWARE PACKAGES A1/1

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21 August 2001OPEN

SUMMARY

METHODS, APPLICATIONS AND SOFTWARE FOR STRUCTURAL RELIABILITYASSESSMENT

S.E. Webster and A.C. Bannister

Changes in legislation, the trend to life extension and increasing computing power have led to anincrease in the use of reliability methods in many industrial sectors. The advantages of theseapproaches are that overdesign can be avoided, uncertainties can be handled in a logical way,sensitivity to variables assessed and a more rational basis for decision making followed. Themethods have been extensively applied in the Nuclear, offshore, rail, shipping, aerospace, bridge,building, process plant and pipeline industries. Failure processes that can be addressed includefracture, collapse, fatigue, creep, corrosion, bursting, buckling, third party damage, stress corrosionand seismic damage.

In this report, basic concepts of risk, reliability and consequences are first introduced. The types offailure modes that can be addressed probabilistically are then described with reference to global andlocal effects and time-dependency. Types of calculation methods are covered, with emphasis onMonte-Carlo Simulation and First Order Reliability Method, and the sources and treatment ofuncertainty described.

A review of codes providing guidance on target reliability levels related to consequence of failure,and industry practice in defining acceptable failure probability is then presented. The levelsgenerally depend on the reliability of the input data, the consequences of failure and the cost ofreducing the risk. The capabilities of various commercial and development software are assessed; Arange of reliability analysis software is available for general applications, covering any failure mode,and also for fracture specific applications.

Current trends include refinement of calculations of risk throughout a structure's lifetime: 'Reliabilityupdating' coupled with structural health monitoring with sensors enables real-time reliability status tobe defined. Risk consideration as a primary input in component/structure design is becoming morewidespread and the use of the methods for optimisation of materials selection, design and cost isincreasing. Future developments include design and material selection optimisation throughreliability methods, interaction of failure modes to more accurately reflect real materials' behaviour,standardisation of consequence scenarios, increased use of time-dependent reliability analysis andbenchmarking of methods and software.

Corus UK Limited SL/WEM/R/M8663/5/01/CSwinden Technology Centre

1Fax: (01709) 825337Telephone: (01709) 820166

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METHODS, APPLICATIONS AND SOFTWARE FOR STRUCTURAL RELIABILITYASSESSMENT

1. INTRODUCTION

1.1 Background

There are numerous sources of uncertainty in structural design and the absolute safety of astructure cannot be guaranteed due to unpredictability of future loading, variations of materialproperties as they exist in the structure, simplifications to analysis methods for predicting behaviourand human factors. However, the risk of a failure with unacceptable consequences can be reducedto an acceptably low number; estimation of this level of risk is the subject of this report.

The advantages of a reliability approach are twofold: it enables uncertainties to be handled in arational and logical way in design and assessment, in particular it enables the sensitivity ofuncertainty to various design variables to be determined. Secondly, while decisions are seldomclear cut and are never perfect, it provides a more rational basis for decision making than with apurely deterministic analysis.

The fundamental concept for reliability analysis is that resistance and load factors are statisticalquantities with a central tendency (mean), dispersion about the mean (variance) and some form ofdistribution (probability density function, e.g. Normal). When combined together via an expression todescribe the limit state (such as fracture or collapse) there will be a finite probability that the load willexceed the resistance; this defines the probability of failure (Pf) and since reliability is equal to 1-Pf,the inherent reliability of the component against a particular failure mode, and with given resistanceproperties, is defined. The basic definition of this is shown in Fig. 1.

The use of probabilistic methods in structural design and analysis has grown rapidly in thepast five years, in parallel with increasing computing power. There is now a general agreement onthe philosophy behind the use of probabilistic methods in decision making, methods of uncertaintymodelling are accepted and being unified, and numerical techniques have been developed tocompute failure probabilities and sensitivity factors efficiently(1).

Probabilistic methods were originally used for calibration of safety factors in structural codes andtechnical standards. One of the first calibrations was for the 1974 Canadian Standards Associationoffshore code, and since then almost all major codes for land based and offshore structures havebeen developed through a formal calibration process involving some element of probabilisticanalysis. In recent years probabilistic methods have also been used directly in design to account forvarious failure modes for which there was little previous experience, very costly structures or thosewith very large failure consequences. These methods are now being extended to covertime-dependent failure modes and to link component reliability with system reliability.

Recently, probabilistic methods have been further developed to account for new informationbecoming available after the design stage, a process known as reliability updating. Such informationmay be from fabrication, such as control of materials and welding, or from service experience, whereinspection and monitoring provide important additional information. With the additional informationmuch of the uncertainty present at the design stage is removed and improved decisions on repair,strengthening, inspection planning and change in use can be made in a quantitative manner whichwould not be possible if based only on deterministic design information. This has particularrelevance in fatigue loaded structures such as bridges and offshore structures.

1.2 Historical Development of Methods

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While probabilistic methods can be applied to any aspect of structural design or operation, it is theiruse in failure prevention and safety analyses which is the subject of this review. Initial concepts inthis area of probabilistic fracture mechanics were developed in the nuclear and offshore industries inthe 1980s, applications which have associated with them a very high consequence of failure. Morerecently these methods have begun to be used in more conventional structures and guidance nowexists in many design and integrity analysis codes. This may be either in the form of a directreference to such methods, their use to derive partial safety factors or their application tomaintenance and inspection. Public perception and understanding of risk, the associated role ofregulatory bodies and the necessity for a common basis on policy where safety is an issue havefurther strengthened the move to reliability based methods(2).

The advantage of such methods in integrity analyses is that the use of pessimistic assumptions fordata inputs is avoided, and the compounding effects of such assumptions can be minimised. Thiscompound effect makes the results of deterministic analyses often very conservative leading to alack of credibility in their results. The methods can be applied to any mode of failure (e.g. fracture,collapse, fatigue, corrosion, creep and buckling) providing that the limit state can be described by anequation(s) and that one or more of the variables in the equation is statistically distributed.

1.3 Scope of Review

Basic concepts of risk, reliability and consequences are first introduced. The types of failure modesthat can be addressed probabilistically are then described with reference to global and local effectsand time-dependency. Types of calculation methods are covered, with emphasis on Monte-CarloSimulation and First Order Reliability Method, and the source and treatment of uncertaintydescribed.

General treatment in various industries of fracture, fatigue, corrosion and high temperature failureare then covered. Probabilistic definition of the failure assessment diagram (FAD) is addressed insome detail and a review of codes providing guidance on target reliability levels related toconsequence of failure summarised. Finally, the capabilities of various commercial and developmentsoftware is presented, followed by a view on future developments and conclusions.

2. BASIC CONCEPTS

2.1 Definitions and Acceptance of Risk

For structural applications, the probability of failure is assessed in the context of 'consequence offailure' such that 'risk' can be defined where:-

Risk = Probability x consequence

A high probability of failure can be accepted where the consequences of that failure arelow: conversely, a high consequence of failure must be allied to a low probability of occurrence.Societal and governmental acceptance of risk dictates that different industries and structures willhave different combinations of probability of failure and consequence of failure, Fig. 2. There arealso governmental targets of what is considered to be negligible risk, unacceptable risk and a regionin between where risk is treated in terms of 'ALARP' (As Low As Reasonably Practical). High riskcan be treated in terms of mitigating either the frequency of occurrence or reduction of consequence.

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The interpretation of failure probability must be made in the context of the type of structure orcomponent(3). Mass produced components (pumps, valves, electrical devices) can be assessed interms of failure frequency, or time to failure, due to the numbers involved and the fact that theygenerally comprise parts which wear out, rather than fail by some unexpected or complex mechanismwhich may involve human factors. In contrast, engineering structures tend to be unique in theirstructural form and location and are subjected to a range of operating conditions which can causefailure by one failure mode or a combination of many.

The concept used for structures is therefore to sample from the input distributions many times andtheoretically create similar structures under the full range of operating conditions. For the case of anexisting structure, information can be gained on its behaviour and this can be used to refine thecalculations of risk, a form of reliability updating which is not possible with newly designedstructures. The assessed reliability is not solely a function of the structure itself but is alsodependent on the amount and quality of information available for the structure(4).

The perception and acceptance of risk depends on the level of understanding of the particularactivity or structure, the level of confidence in the source of information and the freedom of choicethat an individual has; it would normally be expected that if a definite choice was made, then a higherlevel of risk would be tolerated. A summary of certain societal risks and broad indicators of what isconsidered tolerable are given in Tables 1 and 2(5). Failure rates for populations of buildings andbridges(5) are given in Table 3 for comparison.

2.2 Failure Modes

The general concept behind all probabilistic methods is that some or all of the inputs containinherent uncertainty and these can combine to give an uncertainty rating for structural performance.For general structural assessment purposes it is standard practice to assess safety by comparison ofload and resistance effects using established design rules to predict the likelihood of failure. Wherethere are uncertainties in the input variables, or scatter in the material properties, reliability-basedmethods can be employed to determine the probability that the load effects will exceed theresistance effects. Inherent scatter in a material property will affect the failure probability and it istherefore not only the mean value of a property which is important, as in deterministic analysis, butalso its variance and the type of distribution used to represent the dataset.

Depending on the failure mode, material properties, temperature, geometry and loading will influencethe reliability of the component. It is more usual to assess failure modes which contribute to theultimate limit states rather than serviceability limit states. These include yielding, fracture, fatigue,creep, corrosion, stress-corrosion cracking, bursting and buckling.

These can be divided firstly into those which act only at a crack tip as compared to those which actglobally. Secondly, those which have a time element associated with them (time-variant) and thosewhich are time-invariant can also be defined. This leads to a 2 x 2 matrix, Fig. 3. However, sincehuman factors also play a major role in risk, structural reliability should also acknowledge, if notquantify, those factors which are not directly incorporated in the calculation procedure but will affectrisk, this adds a third dimension to the matrix; an example of the complex interrelationship betweenthese different factors is shown in Fig. 4(6).

A schematic of a reliability analysis for failure of a corroded pipe is shown in Fig. 5(7). This shows theinteraction of materials and operating data, together with inspection data and model uncertainties,and illustrates the range of inputs needed for a typical reliability analysis.

3. QUANTIFICATION OF RELIABILITY

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3.1 Hierarchy of Structural Reliability Methods

The underlying principles for reliability analysis were defined in the 1950s by Pugsley andFreudenthal(8,9). The subsequent evolution of methods was initially slow, first order methods beingdeveloped in the 1970s and only in the 1980/90s were the methods extended to structural systems.The reasons for this include(3):

� The traditional approach of overdesign, which carries relatively low cost penalties atthe design stage.

� The priority for understanding modes of failure rather than risk of failure.

� Probabilistic methods were not considered relevant in traditional engineering.

The increased use of risk-based approaches is thought to be due to:

� Change in legislation to safety leading to the need to quantify risk.

� The trend of life extension of existing plant and structures, many of which do not meetthe requirements of current codes.

� Increased experience with probabilistic approaches and increased computing power.

� The potential cost savings which can be made when applying risk-based methods.

It is generally accepted that reliability methods can be characterised into one of 4 levels:

� Level 1 uses partial safety factors to imply reliability and is used in simple codes.

� Level 2 is known as second moment, First Order Reliability, Method (FORM). Therandom variables are defined in terms of means and variances and are considered tobe Normally distributed. The measure of reliability is based on the reliability index β. InAdvanced level 2 methods the design variables can have any form of probabilitydistributions.

� Level 3 have multi-dimensional joint probability distributions. System effects andtime-variance may be incorporated. They include numerical integration and simulationtechniques.

� Level 4 includes any of the above, together with economic data for prediction ofmaximum benefit or minimum cost.

All methods are approximate and the problems become more difficult as the number of randomvariables and the complexity of the limit state function increase and when statistical dependencebetween random variables is present. The asymptotic approximate methods such as FORM are themost suitable for a large variety of structural reliability problems, although simulation methods areuseful as complementary methods. A summary of this hierarchy is given in Table 4, although in thepresent report, most attention is given to the methods of Monte-Carlo Simulation (MCS) andadvanced First Order Reliability Method (FORM).

3.2 Limit States and Definitions

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The fundamental notion is the limit state function which gives a discretised assessment of the stateof a structure or structural element as being either failed or safe(10). The limit state function isobtained from traditional deterministic analysis, but uncertain input parameters are identified andquantified, as shown for a pipeline analysis in Table 5(17). Interpretation of what is considered to bean acceptable failure probability is made with consideration of the consequences of failure, whichcan be societal, environmental or financial.

The general case of reliability, shown in Fig. 6, enables definition of the following parameters:

� Safety Margin.

� Limit state.

� Probability of Failure (Pf).

� Reliability.

� Reliability index (β).

The limit state, M, is a function of material properties, loads and dimensions; M>0 represents safety,M<0 represents failure and M=0 represents attainment of the limit state. The probability of failure isgiven by P(M<0) and therefore the reliability index is related to probability of failure via a uniquerelationship, Fig. 7.

3.3 Types of Uncertainties

Formal uncertainties can be classified into three categories: Physical, knowledge based and human:The first represents natural randomness intrinsic to a variable and is known as objective uncertainty,such as wind loading. The second, subjective uncertainties, can be reduced at a cost by collectingmore data or adopting more realistic models while the third category is hardest to quantify andmodify.

Knowledge based uncertainty can be further subdivided into statistical, model and phenomenologicaluncertainties. Statistical uncertainty arises due to a limited number of observations being used tomake up a sample which is then taken to represent a population. Generally, a sample does notperfectly represent the full population but the degree of imperfection is never known, although it canbe estimated. Modelling uncertainty is caused by the use of simplified relationships betweenvariables to represent real behaviour. Methods use to simplify loads and structural responses aslimit state equations are examples of modelling uncertainties. Typical levels of modellinguncertainties are given in Table 6(11).

Phenomenological uncertainty arises because unimaginable phenomena occur which affectstructural failure. Examples include wind-induced resonant effects and frequencies of earthquakeloading and since such phenomenon have not been previously encountered they are particularlyimportant for novel structures or those which attempt to extend the state-of-the-art. Humanuncertainty accounts for well over 50% of all structural failures and ranges from variability in taskperformance to gross errors but they are not covered in any more detail in this report.

The difference between 'real' experienced risk of structural failure and the modelled or predictedfailure probability (which is lower) is usually referred to as the adjunct probability of failure(11). It ismainly attributable to human error and modelling uncertainty; as long as these remain there will

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always be a gap between predicted and experienced risks, this gap is generally 1 to 3 orders ofmagnitude(11), although accounting for modelling uncertainty alone in fracture analyses gives adifference of one order of magnitude(12) and for these reasons, predicted reliability levels are bestreferred to as notional, rather than absolute, levels and are better suited to comparison purposes.

3.4 Types of Analysis

3.4.1 Simulation v Transformation Methods

In simulation methods, a number of random samples are made and the probability determined bysimple ratios; in transformation methods, the integrand is transformed into a standard type ofdistribution which can then be analysed using the particular properties of the distribution. Decisionof relevant failure modes and their limit states are common to both classes of analysis, as isinterpretation of the consequences of failure. The methods differ in the middle step of determinationof failure probability from distributions of applied and resistance factors.

3.4.2 Monte-Carlo Simulation (MCS)

MCS is a relatively simple method which uses the fact that failure probability can be expressed as amean value of the result of a large number of random combinations of input data. An estimate istherefore given by averaging a suitably large number of independent outcomes (simulations) of thisexperiment.

The basic building block of this sampling is the generation of random numbers from a uniformdistribution. Simple algorithms repeat themselves after approximately 2 x 103 to 2 x 109 simulationsand are therefore not suitable to calculate medium to small failure probabilities.

Once a random number u, between 0 and 1, has been generated, it can be used to generate a valueof the desired random variable with a given distribution. A common method is the inverse transformmethod. To calculate the failure probability, one performs N deterministic simulations and for everysimulation checks if the component analysed has failed. The number of failures is NF, and anestimate of the mean probability of failure is the ratio of NF to N. A schematic of the MCS method isshown in Fig. 8.

An advantage with MCS, is that it is robust and easy to implement into a computer program, and fora sample size tending to infinity, the estimated probability converges to the exact result. Anotheradvantage is that MCS works with any distribution of the random variables and there are norestrictions on the limit state functions.

However, MCS is rather inefficient, when calculating failure probabilities, since most of thecontribution to Pf is in a limited part of the integration interval. In addition, for very low failureprobabilities, a large number of simulations is required for the result to converge to the actual value;in these case FORM is preferred or the method of Importance sampling (MCS-IS) can be used.

3.4.3 Monte-Carlo Simulation with Importance Sampling (MCS-IS)

MCS-IS is an algorithm that concentrates the samples in the most important part of the integrationinterval. Instead of sampling around the mean values, as in MCS, sampling is made around the mostprobable point of failure. This point, called MPP, is generally evaluated using information from aFORM/SORM analysis and as such the MCS-IS has limited application except for cases whereconvergence in FORM cannot be achieved due to complexity of the limit state.

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3.4.4 First Order Reliability Method (FORM)

FORM uses a combination of analytical and approximation methods and comprises threestages: Firstly, independent of whether each parameter has been defined as a Normal, Log-Normalor Weibull distribution, all variables are first transformed into equivalent Normal space with zeromean and unit variance. The original limit state surface is then mapped onto the new limit statesurface. Secondly, the shortest distance between the origin and the limit state surface, termed thereliability index β, is evaluated; this is termed the design point, or point of maximum likelihood, andgives the most likely combination of basic variables to cause failure. Finally, the failure probabilityassociated with this point is then calculated via the relationship between β and Pf. This is shownschematically for the case of a linear safety margin in Fig. 9. For non-linear limit states, the failuresurface is linearised at the design point, Fig. 10, the error in β depending on the non-linearity of thefunction at this point.

By transforming the variables into equivalent Normal variables in standard Normal space (mean = 0and standard deviation = 1). This gives the joint probability density function as the standardisedmultivariate Normal which has many useful properties; This is known as the Hasofer-LindTransformation(14) and by its application the original limit state surface g (x) = 0 then becomesmapped onto the new limit state surface gU (u) = 0. Calculation of the shortest distance between theorigin and the limit state surface, β, requires an appropriate non-linear optimisation algorithm. Amodified Rackwitz and Fiessler(15) algorithm is used as the default algorithm in most reliabilityanalyses, which works by damping the gradient contribution of the limit state function, is a robustalgorithm and converges quite quickly for most cases. Finally, the failure probability is calculatedusing an approximation of the limit state surface at the most probable point of failure, and therelationship shown in Fig. 7 is used for this.

FORM is more efficient than MCS in terms of computing time and accurate results can be obtainedeven when the failure probability is low. All the random parameters must however be continuous andlarge errors can also result if there are local minima in the limit state or high non-linearity at thedesign point(16). Despite these limitations, FORM is the most popular reliability analysis method, canbe easily extended to non-linear limit states and has a reasonable balance between ease of use andaccuracy.

3.4.5 Second Order Reliability Method (SORM)

The approximation of the limit state at the design point as a straight line is a step which leads toerrors in FORM analyses, the magnitude of which depends on the degree of non-linearity of the limitstate equation. In SORM, a parabolic, quadratic or high order polynomial is used to describe thelimit state surface, centred on the design point. This leads to higher accuracy but is not generallyconsidered necessary for the majority of engineering applications.

Examples of calculations made using MCS, FORM and SORM for one analysis case are presentedin Reference (17).

4. REVIEW OF APPLICATION IN DIFFERENT INDUSTRIES

4.1 Overview

In the following sections the application of reliability methods in a number of different industries isreviewed. This is limited to those industries in which the method forms an integral part of design,construction and operation and covers predominantly fracture, fatigue and corrosion failure

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processes. Examples of application to inspection scheduling, life extension, design and change inoperating conditions are described.

4.2 Nuclear

4.2.1 General Characteristics

The application of reliability-based methods in the nuclear industry is widely documented and only aselection of representative literature is reviewed here. The R5(18) and R6(19) methodologies are themost widely applied for high and low temperature failure assessments respectively, and both can betreated probabilistically. R5 reliability analysis is currently at the development stage while examplesof R6 application in reliability are well documented.

Examples of the application of each in a reliability context are given in Reference (20). The use ofthe R6 method in support of safety cases, and determination of acceptable levels of reserve factors,has however demonstrated that it is usually the lack of high quality input data, particularly defect sizedistributions, that limits the usefulness of the approaches rather than any inherent limitation of themethods themselves(21). Similar approaches for AGRs(22) emphasise the application of the NuclearSafety Principles (NSPs) of prevention, protection and mitigation to initiating events in variouscombinations depending on the acceptable probability of occurrence, which in turn is inverselyrelated to the severity of consequences of the event. In order of increasing consequence, the failureprobabilities (Pf) and protection against the event(22) are:

v Frequent: Pf>10-3 per year, protection and mitigation with two lines of protection.

v Infrequent: 10-3>Pf>10-5 per year, protection and mitigation with one line of protectionwith redundancy.

v High Integrity: 10-5>Pf>10-7 per year, Demonstrate that all reasonably practical stepshave been taken to provide a line of protection.

v Incredibility of Failure (IoF): Pf<10-7 per year, need to demonstrate IoF.

An IoF approach, as is used for ultimate limit state events such as fracture, comprises a number ofsteps:

� Build Quality (Materials, welding, pressure testing, inspection, operational history).

� Design Assessment (Code used, operational loads).

� Defect Tolerance (R6, BS 7910).

� Failure Modes (Fatigue, creep, fracture, stress corrosion).

� Degradation (corrosion, radiation embrittlement).

� Forewarning of Failure (Leak detection, observable events prior to total failure).

These steps form the basis of a safety case, with consideration of fault loadings and inspectionstrategies added as the final stage.

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4.2.2 Creep Analysis

The R5 method enables analysis of creep crack growth; The material properties describing creepcrack growth and creep strain responses are usually treated probabilistically, while all otherparameters are handled as deterministic quantities. Monte-Carlo simulation is generally used due tothe complexities of the equation describing the limit state.

An example of the application of this is given in Reference (20): MCS was used with the creep crackgrowth properties and the creep strain being defined as probability density functions and all otherinputs as deterministic. The example showed how an initial distribution of defects would change bycreep crack growth in 1 year increments over a ten year period. Conditional probabilities were alsoaddressed since ductile materials tend to have higher creep strains and lower crack growth rates; byaccounting for this interrelationship of material properties the failure probability at the end of the tenyear period was reduced by an order of magnitude.

4.2.3 Fracture and Collapse Analysis

The R6 method uses the well-known Failure Assessment Diagram (FAD) which enablessimultaneous analysis of fracture and collapse for a component with a flaw, Fig. 11. Materialproperties and flaw sizes are usually treated probabilistically, while applied and residual stress isdeterministic. MCS is relatively straightforward with the R6 method and other codes using the FAD,although FORM analysis can also be applied but with some limitations to ensure convergence ofsolutions. The FAD approach and its treatment from a reliability aspect is covered in more detail inSection 5.

Ideally, full conditional probabilities for materials properties should be established since strength andtoughness are related but alternatively realistic lower tails can be imposed on the distributions toreduce the level of pessimism. This approach is also described in the context of corrosionperformance in relation to steel composition(23): By using the actual composition from a testcertificate, and based on knowledge of the performance of different compositions, a more realisticestimate of failure probability can be obtained than if the minimum or maximum allowable limits ofeach element had been assumed.

Dependence of failure probability on quality of flaw and fracture toughness data is emphasised indetail in many publications(15), and an example of effect of quality of NDE data on resultant failureprobability is shown in Fig. 12(13), while in Fig. 13 the effect of increasing scatter of fracturetoughness on resultant Pf is highlighted(24), an increase in COV in toughness from 8 to 10% leadingto an increase in Pf of two orders of magnitude. Time dependence is also relevant since irradiationembrittlement over time can lead to reduced toughness and hence increased failure probability; Aschematic of the effect of this is shown in Fig. 14 for the case of reducing toughness (e.g. due toirradiation) and increasing stress (e.g. due to loss of area by corrosion) with time.

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4.2.4 Sensitivity and Benchmark Studies

A further issue in most safety assessments, and in nuclear components in particular, is that there ishigh reliability and a relatively small number (in statistical terms) of components or plant. Theconcept of actual failure probabilities is therefore better interpreted as relative, or notional, failureprobability(25). Round robin analyses have been carried out to investigate the reproducibility of theseapproaches(25,26). The study in Japan(26) indicated that while seven different computer programmesgave failure probabilities which were within a factor of 2-5 of each other, the sensitivity of the resultto assumed fracture toughness was such that the degree of neutron irradiation greatly influences thejudgement on plant life extension. Such sensitivities are further studied in Reference (25) whereFORM (First Order Reliability Method), SORM (Second Order Reliability Method) and a round robinMCS (Monte-Carlo Simulation) were compared, Fig. 15. The treatment of failure probabilities asrelative values, indicating where greatest sensitivity to inputs lies, and the fact that only 'orders ofmagnitude' of Pf values are of interest seem to be the main conclusions of comparative studies.

The US Heavy Section Steel Technology (HSST) Programme(27) concentrated on the effects ofmaterials and flaw data distributions on calculated failure probabilities. The input flaw sizedistribution has a dominant influence on Pf, a detailed analysis of inspection capabilities(28) showsthat the probability of detection (POD) and probability of correct sizing (POS) vary significantly withinspection method and quality, and type and size of flaw, Fig. 16. The advantages of applyingconstraint corrections for the cases of shallow cracks have also been identified(27). Inclusion ofductile tearing prior to cleavage fracture was seen as a mode that should be treated probabilisticallyand the possibility of including both constraint and tearing analysis has been demonstrated based onthe Weibull Stress concept(28). As well as the influence of the actual flaw distribution on failureprobability, the effects of using varying qualities of inspection on resultant failure have beendemonstrated probabilistically and specific use of reliability-based methods for definition ofinspection and maintenance schedules are also widely documented. It is predicted(30) that reliabilityapproaches will play an increasing part in structural integrity safety cases.

4.3 Offshore Structures


Probabilistic methods have also been applied for many years in the offshore industry, although dueto the loading regimes there is more emphasis on linking fatigue and fracture than in the nuclearindustry. Safety cases are now required for offshore structures in the North sea, and inspectionplans are linked to these. The ALARP principle is a cornerstone of reliability analysis of offshorestructures, most analyses are linked to the optimisation of inspection plans in terms of location,frequency and reliability of detection methods. Floating Production, Storage and Offloading (FPSO)structures are covered under Section 4.4 of this report.

4.3.2 System Reliability

Due to the nature of welded jacket structures one of the key issues is to link individual memberfailure to overall system failure(31): The degree of redundancy, consequences of failure, warning timeof impending collapse, accessibility of joint for inspection and limits of the probabilistic model areusually linked to define the potential probability of failure by different paths. Redundancy must alsobe defined in terms of ultimate strength, fracture and fatigue limit states. These interrelationshipsinevitably lead to complex analysis with the number of failure paths increasing factorially with thenumbers of members and joints: Methods for assessing this complexity are described in Reference(32).

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4.3.2 Reliability Updating and Inspection Scheduling

Since fatigue is the dominant damage mechanism in offshore jacket structures, there have beennumerous examples of reliability assessments linking crack growth with inspection requirements.Most approaches involve a reliability-based interpretation of the S-N curve in conjunction withMiner's Rule. This method can be used to define the required inspection interval to maintain aspecific level of reliability(31), Fig. 17, or for reliability updating to refine the original crack growthcalculations based on knowledge obtained by inspection of the actual, or similar, joints. An exampleof this second application(33), shows the potential advantages for carrying out remnant lifeassessments of existing structures and applies the following steps:

(a) Estimate of fatigue reliability, β, for each joint.

(b) Identification of critical joints where β is less than the target value (Pf higher).

(c) Identification of a subset of critical joints to be inspected.

(d) Carry out inspections.

(e) Reliability updating for all joints.

(f) Planning of following inspection surveys.

Activity (c) requires a correlation to be established between joints so that the results of the inspectioncan be extended to the non-inspected joints; this is achieved by the following;

� A reliability model based on S-N to account for fatigue failure and crack detectionduring inspection.

� A rational correlation between joints based on type, geometry and response to loads.

� A Bayesian updating of the reliability estimates for all joints, based on the results of theinspected joints. This enables system reliability updating using FORM/MCS in real timeand uses all information made available by the surveys.

� A method for accounting for inspection uncertainty by modifying the PDFs for cracksizes.

This method has proved successful for life extension, significant improvements in reliability beingobtained due to the updating method, Fig. 18.

Reliability approaches have also been used to demonstrate the suitability of inspection techniques.Flooded Member Detection (FMD) uses the presence of water in a normally air-filled structuralmember to indicate through-wall damage. In order for this method to be viable the structure musttolerate the damage without an increase in the overall probability of failure and risk-based methodshave been used to demonstrate this(34). The required size of an inspection sample needed to updatereliability based on the original safety index of an uninspected joint(33) can also be determined, Fig.19.

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4.3.3 Application of LEFM Crack Growth Methods

Although the above methods apply mainly S-N approaches for fatigue, LEFM-based crack growthmethods have also been applied: In Reference (35) the da/dN approach was used mainly to calibrateS-N design lines which were not available for the particular joint configuration, and environmentaleffects were also incorporated. In Reference (36) the crack growth law was treated deterministicallybut with local ’hot-spot' stress ranges determined from FE, factored by probabilistically defined waveexceedance data, giving a Weibull distribution for the hot-spot stress range. The method used issummarised in Fig. 20, the mean fatigue life of each joint was characterised as one of four states:

� Time to first detectable crack growth.

� Time to 20% through-thickness cracking.

� Time to through-thickness cracking.

� Time to joint failure, defined as when joint stiffness is reduced to 50% intact value.

4.3.4 Other Failure Modes

Although fatigue crack growth parameters show inherent scatter, the loading and characterisation ofrealistic load spectra dominate most fatigue analyses and this is where most probabilistic effort tendsto be focused in structures where fatigue is the dominant limit state. Hence, this review has beenconcentrated on these effects.

probabilistic methods have also been applied to offshore corrosion problems, fire and blast and seastate modelling but these are not addressed here.

4.4 Transport


In most transport applications the variable which is most difficult to define with any level of certaintyis the loading and most reliability work is focused on fatigue rather than fracture. In manyapplications reliability methods are used to set inspection regimes optimised for safety and cost, butunlike nuclear and offshore structures, many transport applications directly involve public use andthe aim target reliabilities reflect the duty of care that this entails.

4.4.2 Aircraft

Aircraft structures are subjected to a wide variety of live loadings and the principal issue is one offatigue life estimation and the linking of this to inspection schedules. Consequently, the loadingspectrum forms the largest single parameter to be treated probabilistically in aircraft applications(37).The approach used for fatigue estimation often involves full scale testing of components which,coupled with reliability analysis enables optimum inspection scheduling to be made. The issue ofengine reliability is beyond the scope of this review.

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4.4.3 Rail

In rail applications, probabilistic methods are being increasingly used to set track and vehicleinspection intervals. The former tend to be based on traffic composition (axle load and speed), thefatigue crack growth characteristics of the rail and the interaction with other degradation modes suchas wear and corrosion. The identification of factors that increase the risk of failure, coupled with thedefinition of high risk joints enables rail NDE to be applied where it is most needed(38). Theoptimisation of inspection strategies for safety critical rail vehicles has also been made based onfatigue analysis, but linked to a fracture approach for defining critical flaw size. One approach(39) isto use MCS with the fatigue crack growth parameters in the Paris Law, the threshold stress intensityfactor and the loadings defined as probability density functions and all other parameters defineddeterministically. A risk-cost benefit analysis is then used to set optimum inspection intervals as afunction of mileage.

4.4.4 Ships

In ships, the issue again is complexity and unpredictability of loading, coupled with the presence ofhuge distances of weld runs, largely uninspectable regions and failure mode interaction. Loadingspectra in ships are dependent on factors such as trading patterns, cargo loading arrangements,speed, heading angles and time at port. A recommended method for estimation of wave loadings,frequencies and structural response from a fatigue perspective is given in Reference (40), whileestimation methods for statistical characteristics of random load variables in ships(41) providesguidance on treating wave-induced bending loads and provides recommendations on COVs forthese, Table 7. A prototype code for probability-based design requirements for fatigue of shipstructures has been developed(42), in which the S-N curve is described probabilistically. Thestrength modelling error, in this case the uncertainty in Miner's Rule, is quantified by letting thedamage at failure being a log-Normally distributed random variable with median 1.0 and a COV of30%. Four levels of sophistication of calculation are presented, with target safety levels definedaccording to a three-level ranking based on consequence of fatigue cracking.

4.4.5 Floating Production, Storage and Offloading Vessels

The use of converted tankers or new-build vessels as Floating Production Storage and Offloading(FPSO) units offshore is an increasing trend. The integrity of such units is usually assessed on acase-by-case basis as the environmental loadings are site-specific. Rules for tankers involved inworld trade were previously used for this although reliability-based codes for FPSO assessment hasbeen published by both Lloyd's Register and ABS. A Probability of Exceedance (POE) approach isoften used for quantifying wave-induced dynamic loads(44,45) which are then applied in probabilisticfatigue analysis. Reliability analyses for assessment of safety levels of FPSO hulls involve fivefailure modes(44): hull girder collapse, hull girder yield, failure of unstiffened panels, failure of stiffenedpanels and fatigue. FORM analyses have been used(44) taking all inputs (dimensions, flaw sizes,material properties, loadings) as random variables with specific distributions and coefficients ofvariation from various studies(41). Deterministic analysis was first carried out to identify the mostcritical regions and then a probabilistic analysis carried out on the failure mode with the lowestreliability index. The resulting reliability indices were compared with Recommended target reliabilityindices for ships and FPSOs and indicated that current codes have adequate, but not excessive,safety margins.

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4.5 Bridges and Buildings

4.5.1 Fatigue Loading

Reliability methods applied to bridges usually address fatigue issues and life extension, but increasein failure probability throughout life due, for example to increased traffic load and loss of load bearingarea through corrosion, must also be considered. A comprehensive description of the developmentof a method for assessing the risk of fatigue failure in highway bridges is given in Reference (46).Initial crack shape, fatigue crack growth data and fracture toughness were treated as statisticaldistributions. Over 100 stress range histograms were obtained from 40 bridges and equivalentconstant amplitude stress ranges determined. An MCS approach was used with Variance ReductionTechnique (similar to importance sampling) to reduce the required number of simulations when therisk of failure is small. The simulation output gave the risk of fatigue failure for specific bridgedetails, the total system failure probability then being calculated by the total number of details in thesystem and the correlation coefficients between them. The model enabled definition of maximumlength of service life extension, specified inspection intervals and maximum fatigue failure risk atwhich the bridge must be inspected, Fig. 23.

Methods for incorporating random variable amplitude traffic loading with data from inspection havebeen combined into a probabilistic fatigue analysis to identify the statistical properties of damagecontribution(47). The fatigue lifetime of each critical detail was then assessed, the method haspotential for assessing future damage due to increased traffic growth and truck weights. Themethods have also been applied to general bridge management programmes(48), and bridgedeterioration(49).

4.5.2 Seismic Loading

Assessment of structural failure due to seismic activity, both for bridges(50) and buildings(51,52) hasbeen carried out probabilistically. In Reference (50) a force based seismic code with dynamicreliability theory was applied and showed the complex nature of the problem of integrating structuralresponse with seismic response, although probability of failure generally increased as periodbetween the vibration peaks increased, Fig. 24. The effect of variability of material ductility wasstudied from the viewpoint of vertical deflection capacity of a cantilever beam, as would occur in anearthquake(51). Probabilistic models for yield stress and hardening capacity (as indicated by theinverse of the yield/tensile ratio) were introduced into a FORM analysis and studied for a wide rangeof steel beam and column geometries. The results can be applied in design by the definition ofgeneralised parameters involving cross-sectional properties and material uncertainty coefficients.

A FORM method has also been used for determining the required fracture toughness of columnsused in direct-welded moment connections in seismic areas(53). The aim target reliability was thatquoted in the design code for a severe earthquake with low return period, stresses were determinedfrom a finite element analysis and fracture toughness requirements evaluated using a FORManalysis of the Failure Assessment Diagram.

4.5.3 Static Strength Modelling

Methods have been applied to the static strength modelling of structures which are highly sensitiveto geometrical variation. The dominant failure mode in these cases is buckling and a variety ofstiffened steel structures have been analysed(54,55). The application is however not limited to steel,concrete structures have also been assessed from the aspect of material property variations and theeffect of loading cases on the buckling of GFRP cylinders has also been analysed(56).

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4.6 Power, Process and Chemical Plant

The main application of reliability methods in fossil power, process and chemical plant appears to bein the area of risk-based maintenance and inspection (RBMI). The types of plant to which RBMI isapplied is vast but the majority of failures are associated with pressure equipment(57) such as tanks,storage vessels, pumps, towers, heat exchangers and boilers, Fig. 25. Formal approaches detailingsuch methods have been, or are soon to be, adopted in the USA(58,59,60). An extensive review ofindustries applying RBMI was carried out in the preparation of the RIMAP proposal(61), which aims toprovide a unified approach for a probabilistic lifetime and consequence analysis method, Fig. 26.

One criticism of current quantitative methods for RBMI, and therefore limiting their wider use, is theircomplexity and the unrealistic nature of such analyses being carried out by plant engineers in shorttimescales. A complicating factor is the vast number of damage mechanisms (thinning, fatigue,stress corrosion cracking, metallurgical damage, mechanical damage) and the complex interactionsbetween them. Furthermore, while the definition of consequences of failure will always besemi-qualitative and open to individual interpretation, RBMI is an expanding area because of thebenefits it confers and current thinking is that computational methods and individual experienceshould go together to form a workable RBMI strategy(57). A move towards this has already beenmade with the development of appropriate software to formalise the processes of consequenceanalysis to target maintenance effectively(62).

4.7 Pipelines


As for nuclear applications, there is a large amount of literature relating to risk and reliabilityassessments of pipelines and only a brief overview is given here. Probabilistic analysis of pipelinesis introduced in Reference (63), risk analysis approaches for these applications being pioneered byBritish Gas and now applied widely in this field. Despite there being over 100000 km of pipelines inwestern europe alone, the failure rate is extremely small. Extensive legislation governs themanufacture, installation, operation and inspection of pipelines and the aim of this regulation is tolimit the likelihood and consequences of any failure. Consequently, pipelines are the safest methodof transporting energy(64). Where flaws are detected, fitness-for-purpose must be demonstrated andthis is the main aspect of pipelines in which risk-based methods are applied. FAD-type approaches(R6, BS 7910) form the backbone of such assessments, although obtaining representative data isusually the limiting factor.

The major cause of damage in onshore pipelines is third party interference, although groundmovement can also affect some locations and fabrication flaws may be an issue in ageing olderpipelines; 70% of pre-1968 pipelines would be classed as unacceptable to current standards,whereas this value reduces to 10% for those fabricated between 1968 and 1972(63).

Particular emphasis is placed on consequence analysis, a function of the stored energy in thesystem and the human population density in the vicinity. The probabilistic methods are applied tothe assessment of damage or flaws, the setting and optimisation of inspection frequencies, settingmaintenance schedules, life extension and pressure uprating of existing pipelines. Examples of eachare described briefly below. It is noted in many of the references cited that the codification of limitstates and reliability-based pipeline design in the USA has trailed behind that of Europe. Corporateinterest in this issue in the USA varies widely, evidently due to concerns regarding public perceptionand liability.

4.7.2 Third Party Damage

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Use of the FORM method for estimating pipeline failure frequencies has demonstrated that thismethod represents a suitable compromise between accuracy and useability(65). The probability offailure, given the presence of third party damage, can be estimated using this method, although thefrequency of occurrence of impacts can only be determined through the use of historical data.Interrelationships between dents, gouges and failures are determined from existing 'models'(66,67) anda two-parameter Weibull distribution fitted to damage data. However, the fitting of lines to thesedistributions is itself subject to confidence limits, and by different fitting methods thefailure probabilities were found to change by a factor of 1.5-3, highlighting sensitivity to distributionfitting.

A reliability-based limit state approach has also been developed for the design of pipelines to resistthird party mechanical damage. This involved statistical quantification of damage, Fig. 27, coupledwith strength properties defined as random variables which were then applied to existing puncturemodels in a probabilistic manner and compared with test results. This enabled sensitivity studies tobe made which characterise the failure probability as a function of each input variable. Applied loadand excavator tooth contact length were found to be the most significant variables, mainly due to thehigh COV of the load (45%). Examples of reliability levels for various design factors are shown inFig. 28.

4.7.3 Pressure Uprating

The effect on reliability of pressure uprating of pipelines has been demonstrated in several studies inorder to justify the safe use of higher pressure levels. An extensive study for justification of apressure uprating of a sub-sea pipeline is reported in Reference (69). This involved a probabilistictreatment of all credible failure modes: these are yielding (serviceability limit state), and, for ultimatelimit states, bursting, external corrosion, internal corrosion and fatigue crack growth of weld defects.Using such an approach it was demonstrated that only for the fatigue limit state was the probabilityof reaching a failure condition greater than 'negligible'. At the time of the study, the pipeline wasoperating at 100 bar g, uprating to 130 and 135 bar g gave a calculated increase in failure probabilityof 12 and 156 times respectively, demonstrating the sensitivity of certain limit states to small changesin input values, Fig. 29. Based on experience of the pipeline the uprating to 130 bar g wasconsidered acceptable.

The application of limit state reliability methods to pipe operation above 80% SMYS has beendemonstrated using FORM analysis with limit state equations to describe rupture of new pipe, fromcorrosion damage and from dent-gouge damage(70). Yield strength, Charpy impact energy, pipediameter, wall thickness, operating pressure, flaw/gouge size and corrosion rate were considered tobe random variables with all other parameters treated deterministically. This showed that the burstof defect-free pipe was not a credible failure mode and that it is important to consider damage andtime-dependent deterioration. The failure rates for corrosion defects were expressed as atime-dependent function, Fig. 30(a), while those for dent-gouge failure were represented as afunction of wall thickness, Fig. 30(b). Overall, the suitability of the FORM method for justifyingpressure uprating was demonstrated, although the importance of linking this with a pipelinemanagement system for maintaining, monitoring and controlling structural integrity through the full lifewas highlighted.

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4.7.4 Fracture and Collapse

In a study on reliability of pipeline girth welds, both plastic collapse and unstable fracture limit stateswere assessed(71) using the Failure Assessment Diagram with respect to specific target reliabilitylevels (β = 1-5), although an industry-wide acceptable target was not thought to be presentlyfeasible.

The necessity for accurate flaw sizing and the definition of realistic COVs for material properties isemphasised(71), fracture toughness being treated as a mean and 50% of this value in orderto account for variability of CTOD not revealed due to a limited number of test data being available.Once the target reliability levels had been established the FORM method was again used, but withthe aim of deriving appropriate partial safety factors for application to the main data inputs (flowstress, toughness, stress, flaw size) in a limit state approach. The dominant overall uncertainty forboth limit states was found to be flaw size and a high degree of conservatism was noted in thefracture analysis as a simple LEFM approach was used. Furthermore, the difficulty in predicting theoccurrence and magnitude of ground movement meant that the contribution of this to overall failureprobability could not be quantified. The analysis is applicable on a 'per-weld' basis, and the conceptof system reliability and time-dependency were not addressed.

FORM/SORM have also been applied in a study to define the upper limits of yield/tensile ratio forreliable pipeline operation(7). Following review of ultimate and serviceability limit states, and stressand strain controlled cases, the three main failure modes in which Y/T played a major role weredefined as pipe burst, local buckling and axial rupture. Models describing each of these failuremodes are being assessed probabilistically, with particular emphasis on the definition of flow stress,after first demonstrating suitable model accuracy by comparison with existing test data.

FORM and MCS methods based on the failure assessment diagram have been used to justify safeuse of pipes containing girth welds with low weld metal fracture toughness(72). A system reliabilityapproach was adopted and the total reliability determined from the product of individual probabilitiesfor a flaw giving a failure prediction, a flaw existing in a tension zone and a flaw existing in the weld,multiplied by the number of welds considered in the pipeline.

A similar approach has also been used to investigate the probability of pipe failure duringhydrotesting(73), using TWI sofware(74).

Although not limited to pipeline applications, a reliability-based approach has also been used toassess HAZ fabrication hydrogen cracking(75). Nomograms for preheat requirements as a function ofCEV, heat input, combined thicknesses and hydrogen scales were combined with a probabilistictreatment of HAZ hardness to derive probability of cracking values as a function of heat input. Themethodology and typical output are shown in Figs. 32 and 33 respectively.

4.7.6 Inspection Planning

Since third party damage is the principal cause of pipeline failure, reliability-based methods havealso been applied to evaluate the effectiveness of different design and maintenance practices usedto protect pipelines(76). Mechanical damage assessment statistics were collated from numeroussources and a two-component model based on a fault tree applied. The first component relates tofrequency of mechanical interference, the second to probability of puncture when interferenceoccurs. Probability of occurrences were determined from historical statistics, while cost implicationsof pipeline outage were compared with the costs of mitigation measures to give a cost-benefitanalysis for maintenance/repair measures compared to preventative measures. Similar approachesare also used for scheduling inspection for corrosion damage.

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5. PROBABILISTIC TREATMENT OF FRACTURE AND COLLAPSE

5.1 Description of Failure Assessment Diagram

The Failure Assessment Diagram (FAD) gives a graphical representation of the potential effect of adefect on the integrity of a structure. The FAD is a two dimensional plot and indicates the propensityof the defect to cause failure by plastic collapse and brittle fracture. The basic FAD has two axes, Kr

and Lr where:

Kr = Applied stress intensity/fracture toughness.

Lr = Applied stress/yield stress.

Kr is known as the brittle fracture parameter and Lr the plastic collapse parameter. The threeprincipal inputs which are necessary for a basic deterministic calculation to be performed are cracksize, stress and fracture toughness. If all three are known the safety of a structure can be evaluated,while if any two are known the critical level of the third parameter can be determined. The brittlefracture parameter can also be defined in terms of J or CTOD-based fracture toughness.

Once the co-ordinates of the analysis point has been evaluated and plotted on the FAD furtherinformation can be gained depending on the relative position of the analysis point in FAD space.The FAD locus divides this space into 'safe' and 'unsafe' regions, the shape of the locus allowing forthe interaction of yielding and fracture. Furthermore, depending on where the analysis point falls themost likely failure mode can be estimated; the regions of 'fracture-dominated', 'collapse-dominated'and 'intermediate' behaviour are divided up according to the ratio of Kr/Lr. Another feature of theFAD is that some element of work hardening is allowed for since the Lr cut-off level of 1.0 representsan allowable maximum stress equal to the mean of yield stress and UTS.

5.2 Status of Current FAD-Based Approaches

The main analysis methods using the FAD are R6(19), BS 7910(77), SINTAP(78) and API579(79).BS 7910 is a new standard which replaces the former BS PD 6493. While changes in the scope ofthe standard have been made and the treatment of data inputs revised, the basic concept of the FADremains unchanged. The plastic collapse parameter in the FAD is now defined in terms of yieldstress, with a subsequent change to the shape of the FAD and definition of the cut-off in terms ofyield stress/ultimate tensile stress ratio. The SINTAP method also applies a FAD method based onyield stress but has alternative options for addressing cases such as weld strength mismatch and thetreatment of constraint.

Establishing the significance of a result in FAD space can involve one, or a combination of, thefollowing concepts:-

� Sensitivity analysis.

� Definition of reserve factors.

� Use of partial safety factors.

� Probabilistic analysis.

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The definition of reserve factors and sensitivity analysis can be linked by determining how sensitivethe final result is to variations in the input data: A higher reserve factor is needed for those situationswhere the result is very sensitive to realistic variations in these data.

Similarly, if a high reserve factor is deemed necessary, this can also be achieved by applying partialsafety factors to those variables which are shown to have the greatest sensitivity on the analysispoint.

5.3 Inherent Safety Level of FAD Approach and Use of Partial Safety Factors

The results of analyses using the above methods are based on the assumption that failure will occurwhen an assessed defect gives rise to a point which falls on the failure assessment diagramwhereas in practice it is often found that the FAD gives safe predictions, due to its inherentconservatism, rather than critical ones. Data from wide plate test programmes to validate the failureassessment diagram approach were used to investigate the effects of the conservatism inherent tothe failure assessment diagram approach and to derive appropriate partial safety factors(80,81), Fig.34.

A relatively high level of safety can be observed in the FAD, indicating that the method is inherentlysafe. By expressing the distance from the origin to the failure locus as a ratio of the distance fromthe origin to each data point, the inherent safety factor can be determined for each test result atdifferent angles around the FAD, Fig. 35. The region of the FAD is expressed as the angle Thetawhere θ = 90° equates to pure brittle fracture (Kr = 1) and θ = 0 corresponds to pure plastic collapse(Sr = 1). The resultant plot, Fig. 36, shows that in all except two cases (R/r 1) the method is safeand that the highest safety factor is obtained in the brittle fracture region of the FAD. In BS 7910 andSINTAP allowance has not been made for the inherent level of conservatism of the FAD, furtherstudies would be needed before this could be included in fitness-for-purpose analyses, discussed inSection 5.4.

The resulting recommendations for partial safety factors to be applied to the best estimate (mean)values of maximum tensile stresses and flaw sizes, and to the characteristic (i.e. minimum specified)value of toughness and yield strength, are given in Table 8. It is emphasised that the partial safetyfactors will not always give the exact target reliability indicated but should not give a probability offailure higher than this target value, although this premis has been questioned(82).

Additionally, there is no unique solution for partial safety factors and even when a preliminaryseparation is made into load and resistance groups there are still many alternative combinations ofpartial factors which could be applied to the separate input variables to give the same required targetreliability. The most appropriate solutions are those for which the partial safety factors remainapproximately constant over a wide range of input values. The ratios between the different factorsshould be primarily dependent on the relative COVs of the input data but, as noted in Reference(80), it was found that there was some effect of the absolute values of the some input variables. Ithas been suggested that a probabilistic assessment should be used in conjunction with the partialsafety factor approach since the latter may not always give the target failure probability and wherethis is the case the results are likely to be unconservative(82). This is because the relationshipbetween probability of failure and reserve factor depends on standard deviation of the variable andthe position of the point in the FAD: Fig. 37 shows how the position of constant failure probabilitywithin the FAD varies with standard deviation in flaw size.

It should be noted that the partial safety factors on fracture toughness are applicable to mean minusone standard deviation values as an approximate estimate of lowest of three. It is recommended that

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sufficient fracture toughness tests should be carried out to enable the distribution and mean minusone standard deviation to be estimated satisfactorily.

5.4 Model Uncertainty in the Failure Assessment Diagram

The relationship between partial safety factor and overall failure probability is linked directly to thevariability and uncertainty of specific random data inputs and was studied in detail in Reference (80).Furthermore, the consequence of failure may also affect both the target reliability and the weightinggiven to the partial safety factors, given in Table 8 for BS 7910 recommended values withoutmodelling uncertainty, to achieve this target reliability. These conservatisms may arise from anumber of effects but under loading conditions similar to the dataset, the inherent conservatism canbe considered as a modelling error. Including these modelling uncertainties in the calculations ofpartial factors leads to a modified set of safety factors where it is desired to remove theseuncertainties and where they are known to be represented by conditions of the wide plate tests,given in Table 9(80).

Typically, removal of the modelling uncertainty allows a reduction in the general recommendedpartial safety factors of the order of 0.05 to 0.1 on stress, and 0.2 to 1.0 on fracture toughness. It isnoted in Reference (81) that incorporating the modelling uncertainty into such assessments reducedthe failure probability by generally less than one order of magnitude and that any improvements aremost likely at low failure probabilities and in the elastic-plastic (knee region and higher Lr values)region of the FAD, Table 10. Furthermore, the modelling uncertainty can be characterised by a threeparameter Weibull distribution, although where there is confidence regarding the dominant region ofthe FAD then the appropriate factor for that region can be used.

5.5 Probabilistic Treatment of Failure Assessment Diagram

Probabilistic fracture mechanics is based on the concept that all or some of the input parameters foran FAD analysis contain inherent uncertainty, for example due to lack of detailed information, testingvariation or material variability. The uncertainty in the data inputs manifests itself as an uncertaintyin the resulting analysis point, and those due to uncertainty in flaw size and material toughness aregenerally considered to have the greatest effect on uncertainty of the final result. The most likelyanalysis point on the FAD, its associated statistical distribution and the relationship between thesetwo aspects and the failure/no-failure boundary of the FAD enables the probability of failure to bedetermined for a given set of inputs and their distributions.

MCS and FORM are the two most widely used methods for a reliability-based interpretation of theFAD. Various programs are available for automating the analyses, including TWI's FORM/MONTEprogram, British Energy's STAR6 program and the SINTAP consortium’s proSINTAP software:Within proSINTAP, the following parameters are treated as random parameters:

� Fracture toughness.

� Yield strength.

� Ultimate tensile strength.

� Defect size given by NDE.

These random parameters are treated as not being correlated with one another and can follow aNormal, log-Normal, Weibull or exponential distribution. This and other software is covered in moredetail in Section 7.

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6. TARGET RELIABILITY LEVELS IN DIFFERENT CODES AND INDUSTRIES

6.1 Overview

Target reliability levels depend on the consequence and the nature of failure, the economic losses,the social loss or inconvenience, environmental consequence and the amount of expense and effortrequired to reduce the probability of failure. Target levels are usually calibrated againstwell-established cases that are known from past experience to have adequate reliability, althoughnovel types of structure require formal approaches to define appropriate levels. The reliability indexof a structure is often quoted rather than failure probability since there is a substantial differencebetween the notional probability of failure in the design procedure and the actual failure probability.Most codes apply the ALARP principle (As Low As Reasonably Practical) which recognises that inreliability the level of return of incremental safety improvement diminishes with increasing reliability.The process is therefore one of optimisation of safety and cost.

6.2 Quantifying Societal Consequence

One of the more qualitative aspects of reliability analysis is the estimation of consequences.Attempts to formalise this have only been partially successful due to the difficulty in assigning 'typical'scenarios and an unwillingness in some industries to be seen to assign any fatality as being anacceptable condition. The current convention of the HSE(2) is a benchmark value of ~£1m for the'Value of a Statistical Life' (VOSL): This concept is usually interpreted as that which people areprepared to pay to secure a certain averaged risk reduction and equates to a reduction of individualrisk of 1 x 10-5 being worth ~£10, it is not the value assigned to compensation for loss of life.Structural reliability is important first and foremost if people may be killed or injured as a result ofcollapse. In ISO 3294(83) it is suggested that an acceptable maximum value for the failure probabilityin those cases might be found from a comparison with risks resulting from other activities. Takingthe overall individual lethal accident rate of 10-4 per year as a reference, a value of 10-6 seemsreasonable to use. The maximum allowable probability of failure of the structure then depends onthe conditional probability of a person being killed, given the failure of the structure(83):

P(f | year) P(d | f) <10-6 year-1 . . . (1)

The probability P(d | f) is the probability that a person present in the structure at the time of failure iskilled. If a building is seldom visited by human beings, a further reduction factor may be introducedin Equation (1) which gives a minimum requirement for human safety from the individual point ofview. Since authorities explicitly want to avoid accidents where large numbers of people may bekilled, the additional requirement in Reference (83) is of the type:

P(f | year)<A N-α . . . (2)

where N is the expected number of fatalities. The number A and α are constants, forinstance A = 0,01 or 0,1 and α = 2. Modifications of the numerical values are possible in specialcases.

A number of alternative methods have been proposed(5) for quantifying acceptable failure probabilityin the context of consequence, as defined by design life, number of people at risk, social criteria andpotential of prior warning of failure. One such expression is given in as(84):

Pft=1E-4 (Ks x nd)/nr . . . (3)

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Where Ks is a social criterion varying from 0.005 for structures which pose a threat to generalsociety to 5 for structures which do not affect the general public, nd is the design lifetime in yearsand nr is the number of people at risk. Alternatively, one expression has been developed whichtakes account of activity type (e.g. normal, high exposure) and warning factor(85):

Pf t= 1E-5 (A/W)nd x n-0.5 . . . (4)

Where A is an activity factor varying from 0.3 to 10 for low and high exposure structures respectively,and W is a warning factor varying from 0.01 for fail-safe conditions to 1.0 for failure modes whichhave no prior warning.

It is however noted in Reference (5) that the use of such expressions is open to wide interpretation,they do not account for many other relevant issues and comparisons are difficult without specificinformation and the context of the calculation.

For loading situations which occur with low frequency, such as earthquakes, the aim reliability levelis generally lower. If this was not the case the cost of guaranteeing very low failure probabilities forevents which are unlikely to occur would be prohibitively high. It is therefore recognised that theoccupancy and functionality of buildings should be considered with the frequency of damaging eventwhen defining safety levels for individual elements of buildings(86).

6.3 Treatment of Consequence in Three Major Codes

6.3.1 ISO 2394: General Principles on Reliability for Structures

ISO 2394(83) is a recently introduced (1998) standard, the remit of which is to provide a commonbasis for defining design rules relevant to the construction and use of the wide majority of buildingsand civil engineering works. It is emphasised in this code that structural reliability is an overallconcept comprising models for describing actions, design rules, reliability elements, structuralresponse and resistance, workmanship, quality control procedures and national requirements, all ofwhich are interrelated. The standard provides a full description of methods to be applied includingmodels, limit state designs, probability-based design, partial safety factors approach and theassessment of existing structures.

In the context of ultimate limit states, the following points are stated in ISO 2394:

� Consequences of failure are defined at three levels and incorporate economic, socialand environmental consequences.

� Failure is considered to occur by four methods:

(i) an unfavourable combination of circumstances within normal use.

(ii) exceptional but foreseeable actions (e.g. climatic).

(iii) consequence of error or misunderstanding.

(iv) unforeseen influences.

� Any foreseeable scope of damage should be limited to an extent not disproportionate tothe original cause.

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� A consideration is given to durability by classification of the structure into one of fourclasses which have notional design working lives of 1-5 years, 25 years, 50 yearsor 100+ years.

� The composition, properties and performance of materials, the shape and detailing ofmembers and the quality/control of workmanship (fabrication) are considered keyissues from a durability view.

� From a probabilistic point of view, an element can be considered to have one singledominating failure mode. A system may have more than one failure mode and/orconsist of two elements, each one with a single failure mode.

Probabilistic structural design is primarily applied to element behaviour and limit states (serviceabilityand ultimate failure). Systems behaviour is of concern because systems failure is usually the mostserious consequence of localised component failure. It is therefore of interest to assess thelikelihood of systems failure following an initial element failure. In particular, it is necessary todetermine the systems characteristics in relation to damage tolerance or structural integrity withrespect to accidental events. The element reliability requirements should depend upon the systemscharacteristics.

Properties of materials should be described by measurable physical quantities and shouldcorrespond to the properties considered in the calculation model. Generally, material properties andtheir variability should be determined from tests on appropriate test specimens, based on randomsamples which are representative of the population under consideration. By means of appropriatelyspecified conversion factors or functions, the properties obtained from test specimens should beconverted to properties corresponding to the assumptions made in calculation models, and theuncertainties of the conversion factors should be considered.

The recommended target reliability levels are a function of the relative costs of safety measures andthe consequences of failure and are summarised in Table 11. For 'great consequences' themaximum acceptable probability of failure for the cases of high, moderate and low costs ofimplication of safety measures are 10-3, 7 x 10-5 and 10-5 respectively. The middle of these values iscomparable with that implied in Eurocode 3.

6.3.2 Eurocode 3

Eurocode 3(87) for steel structures was published in 1993 although it is not yet in widespread use. Areview of the partial safety factor and reliability levels associated with the material toughnessrequirements of this code(80) has demonstrated that a reliability approach has been used althoughthere is disagreement within the EU on the underlying inputs used for this. Partial safety factors forlive loads are higher than for permanent loads due to the increased uncertainty of the former. Thetarget reliability index of EC3 Annex C (Fracture Avoidance) is 3.8, corresponding to a failureprobability of 7 x 10-5.

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6.3.3 British Standard BS 7910

In BS 7910(77) the consequence of failure are defined as moderate, severe and very severe which, incombination with two levels of structural redundancy this gives six levels of target failure probability,Table 12: All values refer to probability of failure of individual components, the overall objective is toprotect the complete structure against failure, accepting that it may be possible to tolerate localdamage in some locations of redundant structures.

In redundant structures, failure of a single component may be accommodated by alternative loadpaths and, although undesirable and expensive, it may be possible to make a case for a highertarget probability of failure for such a component compared to a critical one which would causecomplete failure. 'Moderate' consequences are interpreted as potential financial costs without threatto life. If failure is predicted to be by brittle fracture, which will by its nature occur without warning,the consequences should be interpreted as 'severe' or 'very severe'. In other respects, 'severe'consequences should be interpreted as any potential threat to human life and 'very severe'consequences as a potential threat to multiple lives. If failure is expected to be by plastic collapseand provided that there is no threat to human life, the consequences may be interpreted as'moderate'. In order to achieve these reliability levels a system of partial safety factors is used in BS7910 which, when used in combination for the data inputs, are intended to give a specific reliabilitylevel in the FAD analysis, Table 8.

As scatter in material properties increases, COV increases, and a higher safety factor must be usedto maintain the same failure probability. In addition to this type of material variability, the assessedstate may be close to a mode change that could drastically alter material properties. In particular,the ductile-brittle transition may induce cleavage in an otherwise ductile process, and higher factorsmay be required in these conditions. One level of reliability can also be achieved through the use ofdifferent levels of partial safety factor and reliability is therefore not a unique value in this respect.

There are many other circumstances listed in BS 7910 that might lead to the requirement forincreased reserve factors:

� The true loading system has to be simplified or assumptions have to be made in orderto analyse the component.

� The non-destructive examination capabilities are indistinct.

� Flaw characterisation is difficult or uncertain.

� The assessed loading condition is frequently applied or approached.

� Little pre-warning of failure is expected, forewarning being likely in cases of ductilefailure.

� There is a possibility of time dependent effects (fatigue, creep, corrosion).

� Changes of operational requirements are possible in the future.

� The consequences of failure are unacceptable.

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6.4 Comparison of Target Reliability Levels in Different Industries

In the context of what constitutes an acceptable level of risk, it is accepted(88) that the likelihood offailure due to the coincidence of under strength material, constructional inaccuracies, andoverloading is acceptably small, and by 'acceptable' it is meant that the frequency of occurrenceshould not be greater than it has been in the recent past.

The public acceptance of risk over which they have a choice is different to that over which they donot, just as a service which is paid for (e.g. air travel) carries with it a duty of care which voluntaryrisk taking (car travel, leisure) does not. The definition of appropriate target reliability levels istherefore a difficult area and must be made with consideration of:

� Level of choice over whether to take risk.

� Consequences (societal, environmental, financial).

� Structural redundancy level.

� Prior warning of failure.

In addition, the appropriate measure of failure probability must be considered, for example, reliabilityover planned life span, reliability per year, per inspection interval or per operating unit (/km/year inthe case of pipelines). The conventional approach to reliability-based code development is tocalibrate them against existing practice and implied levels of structural safety. This is summarised inFig. 38(5), following an iterative procedure to define the required combination of partial safety factorsdeemed sufficient to achieve a specific target reliability index (β), and hence maximum aim failureprobability. However, this approach has limitations in terms of accounting for human factors, and inthe past has been subject to a certain amount of fitting (known as the 'gap factor') to ensureconsistency with existing codes(88).

For new or novel structures, the method may not be appropriate and a more structured analysisaddressing all credible failure modes may be needed. An example of this was the move to floatingoffshore structures where neither existing codes for fixed platforms, nor classification society rulesfor ships, were considered appropriate(89).

Similarly, target reliability can also be re-defined for existing structures in circumstances including:

� Change of use, including increased load requirements.

� Concern about design or construction errors.

� Concern over quality of materials and workmanship.

� Effects of deterioration.

� Damage following and extreme loading event (storm or earthquake).

� Concern over serviceability.

In these cases, load factors may have increased but very often the additional information gatheredduring the life of the structure can be used for reliability updating using Bayes' theorem(90), thusoffsetting the effect of increased loadings or decreased resistance on calculated reliability.

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A comparison of target reliability levels and corresponding maximum acceptable failure probabilitiesdefined for various consequences in different structures is given in Table 13, and summarised in Fig.39. For the three codes ISO 2394, BS 7910 and EC3, there is reasonable agreement that the aimacceptable probability of failure for a structural element is 7 x 10-5 for 'severe' consequence and1 x 10-5 for 'very severe' consequences. For ships, failure probabilities vary between 10-5 and 10-3

depending on failure mode and consequence, while for FPSO and TLP (Tension Leg Platform)floating structures 10-4 is generally adopted. The UK offshore target is 10-4, and is similar to meanaim values for API and DNV offshore codes. Building codes have variable target levels dependingon materials of construction, occupancy and loading modes (dead, live, wind, snow, earthquakeloads) but are as low as 5 x 10-2 for survivability in earthquakes. Pipeline reliability depends onnature of medium (gas or oil), whether the line is off- or onshore, and in the latter case on populationdensity. For onshore gas lines, target maxima are typically 10-4 to 10-6 per km per year. The nuclearindustry has one of the highest general target reliability levels of any industry (β = 5.2), giving atarget maximum probability of 10-7.

7. SOFTWARE FOR RELIABILITY ANALYSIS

7.1 Scope

Reliability analysis software is available for general applications in which any failure mode can beaddressed using the appropriate limit states, and for fracture-specific applications. Validation ofsuch software is usually carried out by benchmark exercises between different programs since it isnot feasible to compare results with real failure statistics.

A summary of all the software reviewed is given in Table 14, and structure of the different programswhere available is given in Appendix 1.

7.2 STRUREL

STRUREL (STRUctural RELiability) is a general purpose reliability software series that has beendeveloped to perform computational tasks in a windows environment and using the most recenttheoretical findings. It is owned and developed by Reliability Consulting Programs GmbH, based atthe University of Munich (http:www.strurel.de).

It 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 best fitting method to be used. COMRELcomprises 44 models and limit state equations can be input for failure modes not addressed. Itincludes MCS, FORM and SORM methods, and in the case of the time-variant version includesmethods for incorporating random and point-in-time events. SYSREL enables multiple failure criteria

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for parallel and series systems to be evaluated, including conditional events. It links directly withCOMREL, making it straightforward to check individual failure criteria before combining them in asystem analysis.

STRUREL has applications in many fields and many examples of its use for fracture, fatigue,collapse, corrosion and general strength problems exist. A joint industry project is planned for 2001in which a variety of reliability problems will be assessed using STRUREL as a benchmarkingexercise in comparison with other software(97).

7.3 ProSINTAP

ProSINTAP (PRObabilistic Structural INTegrity Assessment Procedure) automates MCS and FORManalysis of the failure assessment diagram and is only applicable to fracture and collapse failuremodes(13,98). It consists of five input decks:

� Geometry.

� Loading.

� Material.

� NDE (Non-Destructive Evaluation).

� Analysis.

The Geometry section comprises stress intensity factor solutions for a range of plate and cylindergeometries with surface and through-thickness cracks.

The load module enables through-thickness distributions of applied and welding residual stress to beincorporated.

In the material module, yield strength, UTS and fracture toughness and their associated statisticaldistributions are input. This requires the mean and standard deviation of each parameter as definedby the Normal, Log-Normal or Weibull distribution.

The NDE module enables defect sizing data to be input, but also allowing for treatment as anexponential distribution.

The Analysis module enables the user to select MCS or FORM methods and to apply partial safetyfactors if required in order to achieve a specified target reliability method.

Validation of the software was carried out within the SINTAP project by comparison with softwarefrom British Energy (STAR 6 program) and UMIST (UMFRAP program).

Pro SINTAP currently has the following limitations:

� Embedded defects are not included.

� Probability Density Functions for input parameters are considered to beindependent; i.e. there is no facility for cross-correlating a yield strength distribution anda fracture toughness distribution.

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� Only element failure probabilities are calculated: system reliability and the effect ofelement failure on failure of a subsequent element is not included.

7.4 CALREL

CALREL (CALibration of RELiability) is a general purpose program for structural reliability analysisusing MCS, FORM and SORM for components and systems. It was developed and is owned by theUniversity of Berkeley California (http://socrates.berkeley.edu/~otl/CALREL.html). It contains a largelibrary of probability distributions for independent and dependent variables and has facilities forsensitivity analysis, limit state functions are input by a user-defined subroutine.

7.5 PROBAN

PROBAN (PROBabilistic ANalysis) is a general reliability program developed by DNV and forms partof their hydrodynamic and structural analysis software SESAM (http://www.dnv.com/software).FORM, SORM and MCS are included, as is sensitivity analysis and reliability updating. It can beused to analyse several different failure modes simultaneously, to determine conditional probabilities,for cost optimisation and to derive partial safety factors.

7.6 COMPASS

COMPASS (COMPuter Methods for Probabilistic Analysis of Structures and Systems) is a generalpurpose program developed by Martec Ltd. (Canada). It includes a library of failure modes, datadistribution characterisation, component and system analysis and a link to finite element outputs.Limit states not covered can be input as user-defined sub-routines. FORM, SORM, MCS andMCS-IS are included.

7.7 NESSUS

NESSUS (Numerical Evaluation of Stochastic Structures Under Stress) was originally developed forthe US defence industry by NASA. It automates FORM, SORM and MCS, but its main strength liesin its ability to interface with finite element and boundary element codes (Including ABAQUS andPatran). It includes a number of 2 and 3 dimensional elements in the geometry library and canaccount for large non-linearity of material behaviour. Random variables include geometry, loads,material properties and motions such as vibration and oscillation. It is currently being developed bythe Southwest Research Institute in Texas (http://www.swri.org) although emphasis is on aerospaceapplications, particularly creep performance of engine components.

7.8 ISPUD AND COSSAN

ISPUD (Importance Sampling Procedure Using Design Points) is based on importance samplingmethods, with time-dependency handled via definition of extreme values. COSSAN (ComputationalStochastic Structural ANalysis) uses third-party finite element cases with MCS and MCS-IS. Bothare owned and developed by the University of Innsbruck, Austria (http://info.uibk.ac.at).

7.9 STAR 6

STAR 6(99) is British Energy’s reliability software for automating reliability analyses of the fracture andcollapse analysis procedure R6(19). It is similar in approach to proSINTAP and includes FORM andMCS, but a more extensive library stress intensity factor solutions for different componentgeometries is included. proSINTAP has been validated against STAR 6(19).

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7.10 UMFRAP

UMFRAP is an in-house code developed by UMIST (Manchester, UK) for reliability assessment ofthe failure assessment diagram. It has been used for the determination of partial safety factors foruse in BS 7910 but is not used outside UMIST at present.

7.11 FORM and MONTE

FORM and MONTE automate probabilistic level 2 FAD analysis based on the predecessor toBS 7910, using FORM and MCS methods respectively. They were developed as part of aTWI group-sponsored project and were made available to the project sponsors. Both programs areDOS based and have now been largely superseded. Figure 31 was constructed using the MONTEprogram.

8. FURTHER DEVELOPMENT OF PROBABILISTIC METHODS

8.1 Generic Methods

� Interaction of failure modes to more accurately reflect real materials' behaviour.

� Use of methods for optimisation (of material properties, cost etc.) rather than definitionof failure probability(100).

� Standardisation of consequence scenarios and definitions to enable uniformity of riskanalysis in different structures.

� More use of time-dependent reliability analysis: e.g. definition of parameters in thefatigue crack growth or stress corrosion crack growth laws as random parameters andprobabilistic definition of time to failure.

� Use of reliability updating for analysis of service-proven structures throughout life.

� Benchmarking of methods, software and industry approaches; Round-robin activitiesinvolving different establishments analysing the same set of problems with differentmethods, distribution fits and software; Industrial networks on reliability analysis suchas Asranet(101).

8.2 Failure Assessment Diagrams

� Facility to assess embedded flaws.

� Facility to account for material degradation with time (e.g. irradiation embrittlement,hydrogen effects).

� Realistic defect distributions related to quality and capability of NDE used.

� Secondary stresses defined as random variables, accounting for variation in measuredvalues or effects of material properties (e.g. weld metal yield stress) used to estimatethem.

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� Fracture toughness distributions determined from distributions of Charpy impact energyusing the Master-Curve correlation method.

� Facility to account for effects of weld strength mismatch as random distribution basedon distributions of yield strength of parent plate and weld metal.

� Facility for determining significance and relevance of MOTE (minimum of threeequivalent) fracture toughness values for datasets of varying size.

8.3 Distributions of Material Properties

� Covariance of property distributions, particularly strength and toughness since theseusually show a negative correlation and the probability of low toughness occurringsimultaneously with low strength for a given material is low.

� Facility to use truncated property distributions, which remove the lower tail of e.g. yieldstress distributions where a Normal fit would give a yield stress below the minimumspecification value for a given grade.

� Increasing demand will be placed on materials suppliers to provide information relatingto the statistical distributions of mechanical properties.

8.4 Reduction of Data Uncertainty

� Linking of structural health monitoring (strain, fatigue, pressure gauges) to providereal-time snapshot of instantaneous reliability rating; over time, the reliability trend canbe established showing effects of e.g. increased loading, increased fatigue damage.

� Use of so-called 'Smart Techniques' (Neural networks, fuzzy logic, genetic algorithms)to extract structural response profiles from noisy health monitoring data obtained fromactual structure.

� Risk consideration as a primary input in structure/equipment design will become morewidespread and designing for inspectability to assess degradation by many differentfailure modes will become increasingly important.

9. CONCLUSIONS

Changes in legislation, the trend to life extension and increasing computing power have led to anincrease in the use of reliability methods in many industrial sectors. The advantages of theseapproaches are that overdesign can be avoided, uncertainties can be handled in a logical way,sensitivity to variables assessed, materials' selection optimised and a more rational basis fordecision making followed.

Risk is most commonly defined as the product of probability of occurrence and consequence:Probability of failure is usually quantified per year or over the full lifetime, whereas consequence isusually qualitative and defined as societal, environmental or financial. Levels of acceptable risk aredefined by safety regulatory bodies and are usually classified as 'tolerable', 'intolerable', or a regionin between where risk must be reduced to 'As low as reasonably practical'.

The fundamental concept for reliability analysis is that resistance and load factors are statisticalquantities with a central tendency (mean), dispersion about the mean (variance) and some form of

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distribution. When combined together via an expression to describe the limit state there will be afinite probability that the load will exceed the resistance; this defines the probability of failure (Pf) andsince reliability is equal to 1-Pf, the inherent reliability of the component against a particular failuremode, and with given resistance properties, is defined.

First Order Reliability Method (FORM) and Monte-Carlo Simulation (MCS) are the most commonbasic techniques and are applicable to all probabilistic problems. Of these, FORM is usually thepreferred method as it does not depend on the number of simulations carried out. For complex limitstates, FORM may not converge and an answer not be obtained. In these cases MCS is used but alarge number of simulations must be made when failure probability is low, thus requiring extendedcomputing time.

The methods have been extensively applied in the Nuclear, offshore, rail, shipping, aerospace,bridge, building, process plant and pipeline industries. Failure processes addressed includefracture, collapse, fatigue, creep, corrosion, bursting, buckling, third party damage, stress corrosionand seismic damage. Advantages of a reliability approach have been demonstrated in the areas oflife extension of reactor pressure vessels, inspection scheduling of offshore structures, cost-benefitanalysis for NDE inspection of rails, design of fatigue sensitive ship structures, definition ofinspection intervals for steel bridges, design of buildings and bridges in earthquake zones andpressure uprating of pipelines.

The Failure Assessment Diagram (FAD) remains the standard approach for deterministic andprobabilistic structural integrity assessment. This method forms the basis of the R6, BS 7910 andSINTAP methods. It contains an inherent safety factor which varies depending on position within theFAD and although this has been quantified it has not yet been incorporated in any procedure.

A number of standards provide recommended maximum aim probability of failure levels (i.e. a targetreliability). The levels generally depend on the reliability of the input data, the consequences offailure and the cost of reducing the risk. Acceptable maximum failure probabilities in accordance withmany different structures, industries and consequences have been reviewed: An aim maximumfailure probability of 7 x 10-5 for 'severe' consequences and 1 x 10-5 for 'very severe' consequencesseems to be common to several types of structure.

Reliability analysis software is available for general applications in which any failure mode can beaddressed using the appropriate limit states. STRUREL (STRUctural RELiability) is a generalpurpose reliability software family that has been developed to perform computational tasks in auser-friendly windows environment and using the most recent theoretical findings. Owned anddeveloped by Reliability Consulting Programs GmbH, STRUREL appears to be the most likely useddue to its modular format and balance between cost and functionality. Other general purposesoftware either forms part of more expensive suites or is developed by academic institutions withoutconsidering commercialisation.

For fracture-specific applications, ProSINTAP (PRObabilistic Structural INTegrity AssessmentProcedure) automates MCS and FORM analysis of the failure assessment diagram and is applicableto fracture and collapse failure modes. It has been fully validated but has limitations in terms ofrange of geometries and flaw types. Star 6 (British Energy) performs similar calculations but with amore extensive geometry library.

Refinement of calculations of risk can be made throughout a structure's lifetime: 'Reliability updating'coupled with structural health monitoring with sensors enables real-time reliability status to bedefined. the use of so-called 'Smart Techniques' to extract structural response profiles from noisyhealth monitoring data obtained from the actual structure will facilitate this.

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Risk consideration as a primary input in component/structure design will become more widespreadand designing for inspectability to assess degradation by different failure modes, and repair wherenecessary, will become increasingly important. Materials suppliers will be increasingly required toprovide data on distributions of mechanical properties.

Future developments include design and material selection optimisation through reliability methods,interaction of failure modes to more accurately reflect real materials' behaviour, standardisation ofconsequence scenarios, increased use of time-dependent reliability analysis, benchmarking ofmethods and software, and more advanced treatment of property distributions (covariance andtruncation).

Finally, it is noted that by carrying out reliability analyses, the chances of failure are not altered butthey are clarified.

A.C. BannisterKnowledge Group Leader

S.E. WebsterManagerTransport Applications Department

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49. V. Sarveswaran and M.B. Roberts: 'Reliability Analysis of Deteriorating Structures - theExperience and Needs of Practising Engineers', Structural Safety, December 1999, Vol.21, (4), pp357-372 (16).

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51. M.K. Chryssanthopoulos, G.M.E. Manzocchi and A.S. Elnashai: 'ProbabilisticAssessment of Ductility for Earthquake Resistant Design of Steel Members', Journal ofConstructional Steel Research S2, 1999, pp47-68.

52. M.K. Chryssanthopoulos, C. Dymiotis and A.J. Kappos: 'Probabilistic Calibrationof Behaviour Factors in EC8-Designed R/C Frames', Engineering Structures, 2000, Vol.22, pp1028-1041.

53. A.C. Bannister: 'Reliability Basis of Toughness Requirements for Sections UnderSeismic Loading', Corus UK Limited, Asranet 2nd Annual Collarium, 9 July 2001, RINA,London, 2000.

54. M.A. Bonello and M.K. Chryssanthopoulos: 'Buckling Analysis of Plated StructuresUsing System Reliability Concepts', Proc. 12th Offshore Mechanics and ArcticEngineering Symposium, 1993, Vol. II.

55. P.K. Das, D. Faulkner and R.A. Zimmer: 'Efficient Reliability Based Designs of Ring andStinger Stiffened Cylinders under Combined Loads', Proc. 6th Intl. Conf. on 'Behaviourof Offshore Structures', BBP Technical Services Ltd., London, Vol. 1, pp180-193.

56. M.K. Chryssanthopoulos, A.Y. Elghazouli and I.E. Esong: 'Validation of FE Models forBuckling Analysis of GFRP Cylinders', Composite Structures, 2000, Vol. 49, pp355-367.

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64. Anon: 'Interstate Natural Gas Pipelines - Delivering Energy Safely:' Interstate NaturalGas Association of America.

65. D. Linkers, M. Bilio and N.K. Shetty: 'A Probabilistic Approach to the Fracture ofOnshore Gas Transmission Pipelines: Practical Applications', in Proceedings ofReference (21).

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68. R. Espiner, A. Edwards and A. Francis: 'Structural Reliability Based Approach toUprating a Subsea High Pressure Gas Pipeline', Source Unknown.

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71. R.W. Warke, C. Ferregut, A.G. Glover and D.J. Horsley: 'A Reliability-Based Method forAssessing the Fitness-for-Service of Pipeline Girth Welds', PRCI-EPRG, 11th BiennialJoint Technical Meeting, Arlington, Virginia, 1997.

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TABLE 1SELECTED RISKS FOR CERTAIN ACTIVITIES IN SOCIETY(5)

0.160000.02Structural Failuresc

8 - 2480001 - 3Building Firesc

40200020Manufacturing

150 - 440220070 - 200Construction Work

3001500210Coal Mining (UK)

1520080Train Travel

200300700Car Travel

24201200Air Travel

10004002500Cigarette Smoking

170503500Swimming

120801500Boating

1500 - 20005030000 - 40000Alpine Climbing

Typical Risk of Death (x10-6/year) (Rounded)

Typical Exposureb

(h/year)Approximate Death Ratea

(x 10-9 Deaths/h Exposure)Activity

a Adapted from Allen (1968) and CIRIA (1977)b For those involved in each activity (estimated values)

c Exposure for average person (estimated)

TABLE 2BROAD INDICATORS OF TOLERABLE RISK(5)

Not of great concern to average person; aware of hazard,but not of personal nature; act of God

10-6

People are warned of hazard (e.g. fire, drowning,firearms, poisons) also air travel avoidance

10-5

People spend money, especially public money to controlthe hazard (e.g. traffic signs, police, laws)

10-4

Uncommon accidents; immediate action is taken to reducethe hazard

10-3

Characteristic ResponseRisk of

Death/person/year

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TABLE 3TYPICAL COLLAPSE FAILURE RATES FOR BUILDINGS AND STRUCTURES(5)

10-2--AustraliaAll Bridges

10-3--USAAll Bridges

1.5 x 10-3--USACantilever and Suspended SpanBridge

3 x 10-34055World (1900 - 1960)Large Suspension Bridge

10-340-USA (<1900)Steel Railway Bridge

10-4--CanadaEngineered Structures

10-3505 x 106CanadaMixed Housing

10-5-145500Australia (New South Wales)Controlled Domestic Housing

5 x 10-4-2.5 x 106The Netherlands (1967 - 1968)Mixed Housing

3 x 10-7305 x 106DenmarkApartment Floors

EstimatedLifetime

(Pf)

AverageLife

(Years)

Number ofStructures(Estimated)

Data CoverStructure Type

TABLE 4HIERARCHY OF RELIABILITY METHODS(5)

Minimum cost,or maximum

benefitAny of the above, plus economic data4: Decision Methods

Failureprobability Pf

May be includedas randomvariables

Linear, orapproximatedas linear any

form

Related toequivalent

normaldistributionsfully used

Transformationnumerical

integration andsimulation

3: 'Exact Methods'

'Nominal'failure

probability PfN

May be includedas second

moment data

Linear, orapproximated

as linear

Normaldistributions

only

Secondmomentalgebra

2: 'Second MomentMethods'

Partial factorsArbitrary factorsLinear

functions(usually)

Not used

Calibration toexisting coderules usinglevel 2 or 3

1: Code LevelMethods

ResultUncertainty DataLimit StateFunctions

ProbabilityDistributions

CalculationMethodsLevel

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TABLE 5VARIABLES AND DISTRIBUTIONS FOR

YIELD/TENSILE RATIO ANALYSIS OF PIPELINES(7)

Lognormal60.01.001-2Longitudinal Surface Flaw Length

(mm)

Weibull60.01.001-2Longitudinal Surface Flaw Depth

(mm)

Lognormal60.01.00100Circumferential Surface Flaw

Length (mm)

Weibull60.01.001-2Circumferential Surface Flaw

Depth (mm)

Lognormal14.01.001.13Axial Flaw Model Error

(Multiplicative)

Shifted Lognormal71.01.000.34Circumferential Flaw Model Error

(Additive)

Deterministic-1.001.35Circumferential Flaw Model Error

(Multiplicative)

Normal1.01.0111.8Wall Thickness (mm)

Deterministic-1.00914Pipe Diameter (mm)

Normal4.01.17527-556Flow Stress (MPa)

Normal3.51.10483-542Yield Stress (MPa)

DistributionCOV(%)

Mean/NominalNominalVariable

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TABLE 6TYPICAL MODELLING UNCERTAINTIES(11)

-20 to 010 - 20Fatigue loads

-20 to 50(8)15 - 30Compliant offshore platforms

-10 to 10(7)20 - 40Fixed offshore platforms

15 - 25- extreme distribution

-30(6) to 05 - 10- initial distribution

Ship extreme bending

-10 to -30(5)10 - 20Live loads

5 - 10Dead loads

Factors affecting loads

50 to 150(4)30 - 70Fatigue strength

0 to 2015 - 20- general instability

-5 to 55 - 10- interframe collapse

Submarine pressure hull

0 to 30(1,3)20 - 40(2)- typical codes

-20 to 40(1)12 - 18(2)Stiffened cylinders - best

20 to 50(1)20 - 40Unstiffened cylinder - codes

0 to 2015 - 30- typical codes

-5 to 510 - 15Flat panel collapse - best

Factors affecting strength

Bias ζ Xm (%)Cov VXm (%)Item

Notes: (1) 'lower bound' curves are mainly used(2) for all load combinations(3) there are some much more extreme examples(4) large scatter on life predictions(5) no reliable data available but 'growth factor' essential(6) relates to overprediction of Linear Strip Theory(7) based on Morrison(8) based on diffraction modelling studiesCOV Co-efficient variation = standard deviation/mean (%)

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TABLE 7TYPICAL CHARACTERISTICS OF STRENGTH RANDOM

VARIABLES FOR SHIP STRUCTURE ANALYSIS(41)

(na)normal0.20.74nac

(1.0)lognormal0.18FyZFyZpMp

(1.0)lognormal0.15FyZFyZMy

(1.04)lognormal0.051.04 ZrZrSection Modulus Z

(1)00.30.3Poisson Ration v

(1.03)normal0.021.024EEE

(1.05)normal0.051.05 FuFuFu

(1.22)lognormal0.091.22 FyFyHigh Strength Fy

(1.11)lognormal0.071.11 FyFyOrdinary Strength Fy

normal0.010BShip Breadth B

normal0.010DShip Depth D

normal0.080LShip Length L

normal0.090bPlate Size b

normal0.110aPlate Size a

normal0.020tThickness t

DistributionType

StandardDeviation

MeanDistribution

TypeCOVMean

(Bias) or Error InformationStatistical InformationNominalValue

Random Variable

na = not available, Mp = plastic moment, Zp = plastic Z, Zr = rules Z

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TABLE 8RECOMMENDED PARTIAL SAFETY FACTORS FOR DIFFERENT

TARGET RELIABILITIES AND VARIABILITY OF INPUT DATA BASED ON FAD(76,80)

Notes: γσ is a multiplier to the mean stress of a normal distributionγa is a multiplier to the mean flaw height of a normal distributionγΚ or γδ are dividers to the mean minus one standard deviation value offracture toughness of a Weibull distributionγΜ is a divider to the mean minus two standard deviation value of yieldstrength of a log-normal distributionCOV = Co-efficient of variation (= standard deviation/mean)

1.201.101.051.000.1(on min. spec.)

γΜγΜγΜγΜ(COV)MYield Strength

NPNP8.001.000.6

10.006.753.201.000.4

2.892.251.691.000.2

(min. of 3)

γδγδγδγδ(COV)dToughness δ

NPNP2.851.000.3

3.202.601.801.000.2

1.701.501.301.000.1

(min. of 3)

γΚγΚγΚγΚ(COV)KToughness Κ

2.101.851.701.150.5

1.901.651.501.080.3

1.801.551.451.050.2

1.701.501.401.000.1

γaγaγaγa(COV)a

Flaw Size

1.601.501.401.120.3Live

1.401.351.251.100.2Dead + Res

1.301.251.201.050.1Extreme

γσγσγσγσ(COV)sStress

β = 4.27β = 3.8β = 3.09β = 0.739

p(F) 10-5p(F) 7 x 10-5p(F) 10-3p(F) 2.3 x 10-1

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TABLE 9RECOMMENDED PARTIAL SAFETY FACTORS FOR DIFFERENT

TARGET RELIABILITIES AND VARIABILITY OF INPUT DATA BASED ONFAD WITH MODELLING UNCERTAINTY INCLUDED(80)

Notes: γσ is a multiplier to the mean stress of a normal distributionγa is a multiplier to the mean flaw height of a normal distributionγΚ or γδ are dividers to the mean minus one standard deviation value offracture toughness of a Weibull distributionγΜ is a divider to the mean minus two standard deviation value of yieldstrength of a log-normal distributionCOV = Co-efficient of variation (= standard deviation/mean)

1.21.11.0510.1(on min. spec.)

γΜγΜγΜγΜ(COV)MYield Strength

NPNP4.8410.6

6.764.842.2510.4

2.251.961.4410.2

(min. of 3)

γδγδγδγδ(COV)dToughness δ

NPNP2.210.3

2.62.21.510.2

1.51.41.210.1

(min. of 3)

γΚγΚγΚγΚ(COV)KToughness Κ

2.11.851.71.150.5

1.91.651.51.080.3

1.81.551.451.050.2

1.71.51.41.000.1

γaγaγaγa(COV)a

Flaw Size

1.501.411.301.120.3Live

1.351.281.201.10.2Dead + Res

1.251.201.141.050.1Extreme

γσγσγσγσ(COV)sStress

β = 4.27β = 3.8β = 3.09β = 0.739

p(F) 10-5p(F) 7 x 10-5p(F) 10-3p(F) 2.3 x 10-1

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TABLE 10EFFECT OF MODELLING UNCERTAINTY ON FAILURE PROBABILITY

ESTIMATES IN DIFFERENT REGIONS OF FAD(80)

3.73E-3†5.16E-3†2.35E-23: (Collapse Dominated)

2.72E-34.40E-31.06E-22: (Elastic-Plastic)

1.53E-21.61E-35.94E-21: (Fracture Dominated)

GeneralUncertainty

Region SpecificUncertainty

No ModellingUncertainty (Mu = 0)

Failure Probability (Pf)FAD Region

† Results from MONTE, analysis using FORM did not converge

TABLE 11TARGET RELIABILITY INDICES (β) OF ISO 2394

4.3 (10-5)3.8 (7 x 10-5)3.1 (10-3)2.3 (10-2)Low

C 3.8 (7 x 10-5)3.1 (10-3)2.3 (10-2)1.3 (10-1)Moderate

B 3.1 (10-3)2.3 (10-2)A 1.50 (100)High

GreatModerateSomeSmall

Consequence of FailureRelative Costs ofSafety Measures

( ) = Equivalent failure probability for each β value.

Suggestions for Cases A, B, C in ISO 2394 are.

A: For serviceability limit states, use β = 0 for reversible and β = 1.5 for irreversible limitstates.

B: For fatigue limit states, use β = 2.3 to β = 3.1, depending on the possibility ofinspection.

C: For ultimate limit states design, use β = 3.1, 3.8 and 4.3.

TABLE 12MAXIMUM TARGET FAILURE PROBABILITIES FOR DIFFERENT

FAILURE CONSEQUENCES IN ACCORDANCE WITH BS 7910

10-57 x 10-5Very Severe

7 x 10-510-3Severe

10-32.3 x 10-1Moderate

Non-Redundant ComponentRedundant ComponentFailure Consequences

Target Probability of Failure (Events/Year)

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TABLE 13TARGET RELIABILITY AND Pf VALUES IN VARIOUS INDUSTRIES

1 x 10-7 (/yr)5.2Incredibility of Failure event(22)Nuclear RPVs

1 x 10-5 - 1 x 10-6

(/km/yr)4.2 - 4.7Ultimate limit state

1 x 10-2 - 1 x 10-3

(/km/yr)2.3 - 3.1Serviceability limit state(96)Offshore Pipelines

1 x 10-75.2Leaks in areas with schools/hospitals1 x 10-64.7Dangerous gas leaks

(71)HSE Pipeline Requirements

1 x 10-6 (death/yr)4.7Onshore, gas(95)Pipelines in Netherlands1 x 10-4 (/km/yr)3.7Gas pipeline, remote areas1 x 10-5 (/km/yr)4.3Gas pipeline, moderate population density1 x 10-4 (/km/yr)3.7Average North American (oil and gas)

(70)Pipelines in USA

2 x 10-5 (2 x 10-7)4.0 (5)Hull girder collapse

1 x 10-3 (2 x 10-5)3.0 (4)Cat 3: fatigue crack compromises safety,

severe economic and environmentalconsequences

6 x 10-3 (1 x 10-3)2.5 (3)Cat 2: fatigue crack not of immediatedanger but repair costly

2 x 10-2 (6 x 10-3)2.0 (2.5)Cat 1: fatigue crack has little effect onsafety/crew/environmental

(91)Tanker ships and MilitaryShips [Shown as ( )] for

Fatigue Limit States

2 x 10-5 (per year)4.0Earthquake failure (annual)

5 x 10-2 (per event)1.8Earthquake design (given earthquakeoccurs)(94)

6 x 10-32.5Temporary buildings1 x 10-33.0Ordinary buildings

(89)

USA Building Codes

1 x 10-64.8Overall structural average target(93)UK Steel Bridges1 x 10-43.7Overall structural average target(92)UK Offshore2 x 10-32.8Min.1 x 10-43.6Mean

(89)DNV Rules

2 x 10-21.9Min.3 x 10-43.3Mean

(89)API RP2A

2 x 10-43.5Hull girder strength(44)FPSO1 x 10-43.7Primary structures1 x 10-33.0Secondary structures

(89)North Sea TLP

2 x 10-43.5Very serious consequences6 x 10-32.5Tertiary structure (plates)

1 x 10-33.0Secondary structure (stiffened panelsbetween bulkheads)

2 x 10-54.0Primary structure (hull)

(91)Tanker Ships, Ultimate LimitStates

1 x 10-54.3Very severe consequence, non-redundantstructure

7 x 10-53.8Severe consequence, non-redundantstructure

(77)BS7910: Fracture

7 x 10-53.8Ultimate limit state with majorconsequences(87)EC3: Bridges and Buildings

1 x 10-33.1Great consequence/high cost of mitigation1 x 10-54.3Great consequence/low cost of mitigation

7 x 10-53.8Moderate consequence/low cost ofmitigation

(83)General Structures: ISO2394

Aim maximumFailure Probability

Target ReliabilityIndex (β)Conditions and ConsequencesReferenceApplication

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TABLE 14SUMMARY OF VARIOUS GENERAL PURPOSE AND FRACTURE SPECIFIC SOFTWARE

DOS- based system for PD6493 level 2 FAD.Largely supersededFORM/MCSFracture/collapseTWI (UK)FORM/MONTE

Details not availableFORMFracture/collapseUMIST (UK)UMFRAP

Wide user base, links with R6 code. Extensiveapplication in nuclear industryAllFracture/collapse

(Nuclear)British Energy (UK)STAR 6

Emphasis on material property and flaw size asrandom variables. Stresses treated

deterministically. Limited SIF Solutions

MCS andFORMFracture/collapse

DNV (SAQ)/SINTAPConsortium

(Sweden/EU)proSINTAP

Specifically aimed at design of structures tomeet certain target reliabilitiesMCS-ISGeneral +FEUniversity Innsbruck

(Austria)COSSAN

System reliability. Suitable for high non-linearitywhere FORM would not converge to a solutionMCS-ISGeneralUniversity Innsbruck

(Austria)ISPUD

Includes FE interface. Many random variables.Extensively used for creep analysisAllGeneral +FE

(Aerospace)Southwest Research

Institute (USA)NESSUS

Library of failure modes. Links to FEAllGeneral +FEMartec Ltd. (Canada)COMPASS

Component and System reliabilities. Reliabilityupdating, multiple simultaneous failure modes,

conditional probability, links to SESAMAllGeneral

(Offshore/ships)DNV (Norway)PROBAN

Library of probability density functions.Facilities for sensitivity analysis, Interface to FEAllGeneral Links to

FEUniversity Berkeley

(USA)CALREL

Component and system, time-variant andinvariant, wide user experienceAllGeneral + FERCP/University

Munich (Germany)STRUREL

Main attributes/advantagesMethodsApplicationOwner/DeveloperTitle

SL/WEM

/R/M8663/5/01/C

47

Value

Load(Stress)

Fre

quen

cy

Resistance(Strength)

Area (Probabilityof failure)

α

(a) Generalised Probability Case (2d)

(b) Generalised Probability Case (3d)(Acknowledgements to Professor M. Chryssanthopoulos, University Surrey)

FIG. 1(a and b) SCHEMATIC REPRESENTATION OF FAILURE (D0271H08)PROBABILITY BASED ON LOAD AND RESISTANCE DISTRIBUTIONS

SL/WEM/R/M8663/5/01/C

F1

1E-06

1E-05

1E-04

1E-03

1E-02

1E-01

1 10 100 1000 10000

Deaths (N )

Fre

qu

en

cy

Ev

en

t (N

/YR

) LocalTolerability

Local Scrutiny

Negligible

Intolerable

ALARP Region

Negligible

(a) Definitions of Risk Tolerability

1E-09

1E-08

1E-07

1E-06

1E-05

1E-04

1E-03

1E-02

1E-01

1E+00

1 10 100 1000 10000

Persons at Risk

Fai

lure

Pro

bab

ilit

y

PublicAssemblies

Houses/Offices

Bridges

Masts/Offshore

(b) Probability of Failure v Societal Consequencefor Common Structures and Events

FIG. 2(a and b) RISK CLASSIFICATION AS A (D0271H08)FUNCTION OF CONSEQUENCE

(Acknowledgement to Professor F.M. Burdekin, UMIST)

SL/WEM/R/M8663/5/01/C

F2

Fracture

Collapse BucklingBurst

Fatigue SCC

Corrosion Creep

Crack Tip Processes

Net-SectionProcesses

Time InvariantProcesses

Time VariantProcesses

Fracture

Collapse BucklingBurst

Fatigue SCC

Corrosion Creep

Crack Tip Processes

Net-SectionProcesses

Time InvariantProcesses

Time VariantProcesses

FIG. 3 DIVISION OF FAILURE PROCESS (D0271H08)ACCORDING TO SCALE AND TIME

SL/WEM/R/M8663/5/01/C

F3

Economic climate Political climate

Time

Education

DesignDesignguidance

Research

Clients requirements

Communication

Identification

MaterialsStructuralsystem

Analyticalmethod

Calculationaccuracy

StrengthDuctility Thicknesses

Metallurgy

Componentshape

Erection

Basis of contract

Tolerances

Fabrication

Load data

Indicates control pointsHeavily framed boxesindicate matters relatedto safety factors

Load control

Design oftemporary

works

Structural reliability

Reliability of components

Number of components

Maintenance

Execution

Workmanship

Imperfections

Scheme

Industrial climate

Economic constraints Social constraints

FIG. 4 FACTORS AFFECTING STRUCTURAL RELIABILITY(6)

SL/WEM/R/M8663/5/01/C

F4

Failure probability

Maximumoperatingpressure

Data on pipeproperties and

dimensions

InspectionData

pipeproperties

corrosioncharacteristics

modeluncertainties

corrosiongrowth rates

Corrosion modeland test results

Measurementuncertainties

Data fromrepetitive

inspections

Yield stress (MPa)

Prob. densityProb. density

Frequency

Flaw depth (mm) Growth Rate (mm/yr)Model results

Test results

FIG. 5 SCHEMATIC OF RELIABILITY ANALYSIS FOR (D0271H08)FAILURE OF A CORRODED PIPE(7)

LoadM L

SL

MR

Resistance

SR

r,l

fR(r), fL(l)

R = Random Variable Describing ResistancefR(r) = Probability Density Function for R(Likewise L, fL(l) are for load)ML = Mean of Load, MR = Mean of ResistanceSL = Standard Deviation of Load, SR = Standard Deviation of ResistanceIf L and R are normally distributed and are independent random variables

∴ Reliability index = b = MR − ML

ª(SR2 − SL

2 )and Pf = 1 - F (b)

where Φ ( ) is the cumulative distribution function of the Standard Normal Distribution

FIG. 6 THE GENERAL RELIABILITY CASE AND (D0271H08)ASSOCIATED DEFINITIONS

SL/WEM/R/M8663/5/01/C

F5

1 2 3 4 5 6 7 81E-15

1E-13

1E-11

1E-9

1E-7

1E-5

1E-3

1E-1

Reliability Index (BETA)

Probability of Failure (Pf)

Pf = Φ (-β)or

β = -Φ -1 (Pf)

6.05.24.33.73.12.31.3β

10-910-710-510-410-310-210-1Pf

FIG. 7 FAILURE PROBABILITY v GENERALISED (D0271H08)RELIABILITY INDEX (β)

SL/WEM/R/M8663/5/01/C

F6

X1

X2

MCS

MCS - IS

a b

d

c

NO FAIL

Increasing N

Contours ofincreasingnumber ofsimulations

FAIL

Limit state

Pf = N Points in Area abcao

N Points in Area abcdao

FIG. 8 SCHEMATIC OF MCS AND MCS-IS METHODS(11) (D0271H08)

STANDARDNORMALSPACE

Contours of pdf are circleswith radius and centre atpoint O.

β

O

Z2

Straight line in Z - Z plane1 2

p = (- )Φ β

β

SAFE

Z = N(O,I)2

Z = N(O,I)1Z1 FAIL

FIG. 9 FORM ANALYSIS WITH LINEAR LIMIT STATE(11) (D0271H08)

SL/WEM/R/M8663/5/01/C

F7

Major contributionto Pƒ

StandardNormalSpace

tangent

β1

P = (- )ƒ Φ β

g( ) = 0x

FIG. 10 FORM ANALYSIS WITH NON-LINEAR LIMIT STATE(11) (D0271H08)

0 0.5 1 Lrmax

Lr

Safe K = f(L )r r

K = K /KL = P /P (a, )L = /

r 1 mat

r 1 L y

r y

σσmax σ

Fail

0.5

1

Kr

FIG. 11 BASIC PRINCIPLE OF THE FAILURE (D0271H08)ASSESSMENT DIAGRAM (FAD)

SL/WEM/R/M8663/5/01/C

F8

1

0.1

0.01

0.001

0.0001

10-5

10-6

10-7

10-8

10-9

10-10

0 100 200 300 400 500Stress

PF

Advan. Proc.

'True' PF

Bad Proc.

1

0.1

0.01

0.001

0.0001

10-5

10-6

10-7

10-8

10-9

10-10

0 100 200 300 400 500

F

Advan. Proc.

'True' PF

Bad Proc. Note:'True' Pf = StandardDeviation on flaw size = 0

FIG. 12 EFFECT OF NDE QUALITY ON FAILURE PROBABILITY(13) (D0271H08)

Coefficient of Variation (%)

Fai

lure

Pro

babi

lity

5

10-10

10-8

10-6

10-4

10-2

10 15

standard deviationmeanCOV =

FIG. 13 EFFECT OF SCATTER IN FRACTURE TOUGHNESS (D0271H08)ON RESULTANT FAILURE PROBABILITY

FOR A FRACTURE CONTROLLED CASE(24)

SL/WEM/R/M8663/5/01/C

F9

Mean load trend (e.g. increasing stress)Mean resistance trend (e.g. embrittlement)

Time

R, L

t0 t1 t2

Resultant failureprobability

R(Toughness)

L(Stress)

FIG. 14 EFFECT OF TIME DEPENDENCY OF LOAD AND RESISTANCE (D0271H08)FACTORS ON FAILURE PROBABILITY THROUGHOUT SERVICE LIFE

FORMSORMRound-Robin MCS

01E -08

1E -07

1E -06

5 10 15 20 25 30 35 40Operation Years

Cum

ulat

ive

Failu

re P

roba

bilit

ies

(1/c

rack

)

FIG. 15 RESULTS OF BENCHMARK STUDIES ON TIME-DEPENDANT (D0271H08)FAILURE PROBABILITY FOR FRACTURE FROM

CRACKS GROWING IN A REACTOR PRESSURE VESSEL(25)

SL/WEM/R/M8663/5/01/C

F10

0102030405060708090

100

0 5 10 15 20 25 30 35 40

POD

95% Confidence

Crack Depth (mm)

POD (%)

(a) MPI Defect Depth Dependant POD

0102030405060708090

100

0 5 10 15 20 25 30 35 40

POD

95% Confidence

Crack Depth (mm)

POD (%)

(b) ACFM Defect Depth Dependant POD

FIG. 16(a and b) EFFECTS OF INSPECTION METHODS ON (D0271H08)PROBABILITY OF FLAW DETECTION (POD)(28)

SL/WEM/R/M8663/5/01/C

F11

Without Inspection

Servicelife (Years)

Pƒ

TargetP = 0.04ƒ

Pr = 1

0 10 20 30 4010-4

10-3

10-2

10-1

100

FIG. 17 INSPECTION INTERVALS REQUIRED TO RETAIN FAILURE (D0271H08)PROBABILITY UNDER A PREDETERMINED TARGET LEVEL (Pf = 0.04)

(ASSUMING SUCCESSFUL INSPECTION) IN AN OFFSHORE JACKET STRUCTURE(31)

Original reliability

Reliability of the sample joints

Reliability after first update

Reliability after second update

Reliability after third update

Reliability after fourth update

Inspection time

Time (Years)

Saf

ety

Inde

x

10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3 5 7 9 11 13 15 17 19 21 23 25 27 29

FIG. 18 RELIABILITY v TIME FOR A NON-INSPECTED JOINT-IN (D0271H08)OFFSHORE JACKET FOLLOWING PERIODIC UPDATING FROM

INSPECTION OF A SAMPLE OF FIVE JOINTS(33)

SL/WEM/R/M8663/5/01/C

F12

20 - 25

InspectionSample Size

Ti/Ts: Inspection Time/Intended Service Time

Orig

inal

Saf

ety

Inde

x of

the

not I

nspe

cted

Joi

nt

15 - 20

10 - 15

5 - 10

0 - 5

0.5 0.6 0.7 0.8 0.9 10.2 0.3 0.40.1

1.2

1

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

FIG. 19 REQUIRED SIZE OF INSPECTION SAMPLE REQUIRED TO (D0271H08) UPDATE ORIGINAL RELIABILITY OF AN UNINSPECTED JOINT TO 3.0(33)

Beam modelsof Jackets

Distributionof Joint Life

Nominal Stress Rangein Jacket Braces

Hot SpotStress Range

Stress Range asWeibull Distribution

Fracture MechanicsAnalysis

Probabilistic FractureMechanics Analysis

Wave ExceedanceData

Uncertainty Factors forProbabilistic Analysis

Complex Joint FiniteElement Analysis

FIG. 20 FLOWCHART FOR DETERMINATION OF FATIGUE LIFE (D0271H08)PDFS BASED ON LEFM FATIGUE ANALYSIS WITH

PROBABILISTIC WAVE AND STRESS DATA(36)

SL/WEM/R/M8663/5/01/C

F13

Best fit

1E+3 1E+4 1E+5Life (miles)

1E+6

Pro

babi

lity

of F

ailu

re (

%)

0.1

0.01

1

0.0001

100

0.001

10

FIG. 21 OUTPUT OF MONTE-CARLO SIMULATIONS OF (D0271H07da/dN FATIGUE LIFE OF RAIL BOGEY(39)

Target cost

Reduced detectability

Increased detectability

Reduced inspection cost

Reference

1E+05

1E+07

1E+08

Inspection Interval (Miles)

1E+06

3000 4000 5000 6000 7000 8000 9000

1E+09

Cos

t per

Life

Sav

ed (

£)

FIG. 22 EFFECT OF CHANGING DETECTABILITY AND (D0271H08)INTERVAL ON INSPECTION COST OF RAIL BOGEY(39)

SL/WEM/R/M8663/5/01/C

F14

0 10 20 30

1988

Ris

k of

Fai

lure

(P)

F

40 50 60 70 80

Years After Bridge Opening

10-5

10-4

10-3

10-2

10-1

P = 2.4%F

1960 1970

16 24 33 41 48 56 63 70 76

1980 1990 2000 2010 2020 2030 2040

FIG. 23 RISK ANALYSIS OF YELLOW MILL POND BRIDGE, (D0271H08)CONNECTICUT, SHOWING EFFECT OF

INSPECTION ON CALCULATED RISK OF FAILURE(46)

S1

0 1.0 2.0

Period (s)

3.0 4.01.2

1.4

1.6

1.8β

2.0

2.2

2.4

S2

S3

FIG. 24 EFFECT OF EARTHQUAKE VIBRATION PERIOD (D0271H08)ON RELIABILITY OF A STEEL BRIDGE INTHREE DIFFERENT LOCATIONS (S1 - 3)(50)

SL/WEM/R/M8663/5/01/C

F15

Towers

0 5 10 15 20 25 30 35

Others/Unknown

Heaters/Boilers

Heat Exchangers

Pumps/Compressors

Reactors

Drums

Percent of Losses

Tanks

Piping Systems

FIG. 25 EQUIPMENT INVOLVED IN MAJOR PROPERTY LOSSES (D0271H08)IN THE OIL REFINING AND PETROCHEMICAL PROCESS

INDUSTRIES DURING THE PERIOD 1960 TO 1990(57)

Maintenance Planning

RBI

Preventativemaintenance

Predeterminedmaintenance

Conditionmonitoring

Continuousmonitoring

PeriodicInspection

Calendarbased

Op. timebased

Correctivemaintenance

Unplannedcorrective

Plannedcorrective

Predictivemaintenance

FIG. 26 AREAS ADDRESSED IN RIMAP PROJECT(61) (D0271H08)

SL/WEM/R/M8663/5/01/C

F16

Data

00

0.005

0.02

0.015

0.01

100 200 300 400 500Outside Force (kN)

Pro

babi

lity

Den

sity

MOM Gamma

FIG. 27 STATISTICS OF OUTSIDE FORCE ASSOCIATED WITH (D0271H08)THIRD PARTY DAMAGE TO PIPELINES(68)

0 0.21.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

Pro

babi

lity

of F

ailu

re (

/km

-yr)

0.4 0.6

Design Factor ( / )v a

0.8 1.0 1.2 1.4

FIG. 28 PROBABILITY OF FAILURE PER km-YEAR FOR THIRD PARTYDAMAGE TO PIPELINES OPERATING AT DIFFERENT DESIGN FACTORS(68)

SL/WEM/R/M8663/5/01/C

F17

(a) Yielding

(b) Corrosion

(c) Fatigue Crack Growth

Operating Pressure (Barg)

90

Pro

babi

lity

of F

ailu

re

110100 120 130 140 150 160 170


100

Pro

babi

lity

of F

ailu

re

110 120 130 140 150


100

Pro

babi

lity

of F

ailu

re

110 120 130 140 150

FIG. 29(a - c) SENSITIVITY OF FAILURE PROBABILITY TO FAILURE (D0271H08)MODE AND PIPELINE OPERATING PRESSURE(69)

SL/WEM/R/M8663/5/01/C

F18

Ann

ual P

roba

bilit

yof

Fai

lure

per

Def

ect

Time (years)

1.E-080 5 10 15 20 25 30

1.E-071.E-06

1.E-05

1.E-04

1.E-031.E-02

1.E-011.E+00

Design FactorCorrosion Depth Growth Rate

Severe Moderate Low

0.85

0.72

(a) Annual Failure Probability of Corroded Pipe

Dent depth = 20 mm

Con

ditio

nal P

roba

bilit

yof

Fai

lure

per

Def

ect

(Giv

en D

amag

e)

Wall Thickness (inches)0.400 0.500 0.600 0.700 0.800 0.900

1.E-081.E-07

1.E-06

1.E-051.E-041.E-031.E-021.E-011.E+00

Design PressureGouge Depth

1.5 mm 0.5 mm

1050 psi

1550 psi

(b) Conditional Failure Probability of Dent/Gouged Pipe

FIG. 30(a and b) EFFECT OF CORROSION AND DENT/GOUGE (D0271H08)DAMAGE ON FAILURE PROBABILITY(70)

SL/WEM/R/M8663/5/01/C

F19

0 0.2 0.4 0.6

sr

0.8 1.0 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1.0

1.2

ª dr

No Failure

Failure

FIG. 31 MONTE-CARLO SIMULATION FOR FAILURE PROBABILITY (D0271H08)FROM A 1.5 mm DEEP FLAW DURING EXPANSION

OF X60 LINEPIPE (1000 SIMULATIONS)(73)

Manual input

Inputs:CompositionHydrogen scaleWelding conditions

Calculate:Hardness MeanValue - Transformto a normaldistribution with this as the meanvalue and astandard deviationof 28HV

Compare thiswith distributionof criticalhardness for thehydrogen scalespecified

Display probability of cracking

Calculate:Area of overlap of hardnessdistributions to determinethe probability of cracking

Calculate:t/5Carbon Equivalentsg

Process

Connector

FIG. 32 FLOWCHART FOR PROBABILISTIC ANALYSIS (D0271H08)OF HAZ HYDROGEN CRACKING(75)

SL/WEM/R/M8663/5/01/C

F20

`

0

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0NC

NC

NC

0

43

6

6

C C

C

C

W

Numbers indicate number of cracked faces out of six.

C = CrackedNC = Not Cracked

Heat Input (kJ/mm)

Pro

bab

ility

of

Cra

ckin

g

tg/5 1/(R) Nomogram½

W 1.57 1.21 1.70

X 3.49 2.60 2.56

Y 3.82 3.86 4.08

Z 2.47 1.65 1.68

Minimum Heat Input(kJ/mm) for avoidance

of cracking :

X

Y Z

C66

6

000

0

0

0 0.5 1 1.5 2 2.5 3

0

0

26

FIG. 33 PROBABILITY OF HYDROGEN CRACKING DETERMINED BY (D0271H08)STRUREL FOR FOUR CTS SERIES, SELECTED TO HAVE DIFFERENT

TRANSITIONS FROM CRACK TO NO CRACK THRESHOLDS(75)

Data

00

0.5

1

1.5

2

2.5dr^0

.5

3

3.5

4

4.5

5

5.5

6

0.5 1 1.5

Sr

2 2.5 3 3.5 4

Level 2 FAD

FIG. 34 COMPARISON OF FULL SCALE TEST RESULTS WITH FAD (D0271H08)PREDICTIONS (CTOD - BASED TOUGHNESS)(81)

SL/WEM/R/M8663/5/01/C

F21

Level 2 FAD

Modelling uncertainty, Mu

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

1

00 1.2

1.2

1.4

1.4

1.6

1.8

2

Sr

A

R

M : R-rM : R/r

u

u1

r

K o

r r

r√δ

FIG. 35 DEFINITION OF SAFETY FACTOR (R/r) AND ANGLE θ IN (D0271H08)RELATION TO FAD AND ANALYSIS POINT A(81)

h,°

00

0.5

1

1.5

2

2.5

3

3.5

4

4.5

R/r

10 20 30 40 50 60 70 80 90

FIG. 36 FAD SAFETY FACTOR AS A FUNCTION OF POSITION IN (D0271H08)FAD (θ = 90° = BRITTLE FRACTURE, θ = 0° = PLASTIC COLLAPSE)(81)

SL/WEM/R/M8663/5/01/C

F22

Increasing SD on a from2 to 5 mm (a = Flaw size)

0 0.2 0.4 0.6S r

0.8 1.0 1.2 1.40

0.2

0.4

0.6

()

δ r0.

5 0.8

1.0

1.2

1.4

P = 0.2f

P = 0.1f

P = 5 x 10f-2

P = 10f-2

FIG. 37 CALCULATED FAILURE PROBABILITY FOR VARYING (D0271H08)STANDARD DEVIATION ON FLAW DEPTH COMPARED WITH FAD(82)

Define scope

Select calibrationpoints [length, loads,

properties etc.]

Use existing designcode for each

Calibration Point togive a member size

Calculate safety indexfor each member

Select target safety index

Select (or modify) partialfactors for new code format

Use new design code foreach calibration point to give

a member size

Test for closeness to target safetyindex, using weighting factors

Calculate safety indexfor each member

Calculate partial factorformat implicit in

existing mode

New code partial factors

Define limit states(strength models)

Determine usageweighting factors

Define basicvariables

Statistical propertiesfor basic variables

Select new codepartial factor format

Insu

ffici

ent c

lose

ness

to T

arge

t

FIG. 38 METHOD FOR RELIABILITY-BASEDCALIBRATION OF DESIGN CODES(5)

SL/WEM/R/M8663/5/01/C

F23

USA Buildingsin earthquakes

UK warships

USA conventional buildings

Offshore Structures(DofE, API, DNV)

FPSOUltimate Collapse

ISO2394 targetModerate consequences

ISO2394 targetGreat consequences

UK Bridges

Nuclear

Offshore pipelines(SLS)

Onshore gas pipelineshigh population density

Pf=Φ(-β)

Tanker-hull collapse

1 2 3 4 5 61E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

Target Reliability Index (BETA)

Target Max. Probability of Lifetime Failure

FIG. 39 TARGET RELIABILITY LEVELS FOR DIFFERENT (D0271H08)STRUCTURES AND DESIGN CODES

SL/WEM/R/M8663/5/01/C

F24

APPENDIX 1

STRUCTURE OF VARIOUS RELIABILITY SOFTWARE PACKAGES

SL/WEM/R/M8663/5/01/C

A1/1

FIG. A1.1 LAYOUT OF STRUREL SOFTWARE (D0271H10)

FIG. A1.2 ProSINTAP INPUT TAB FOR MATERIAL (D0271H10)

SL/WEM/R/M8663/5/01/C

A1/F1

FIG. A1.3 ProSINTAP INPUT TAB FOR CALCULATION TYPE (D0271H10)

problemdefinition

analysisroutines

userroutines

graphicoutputs

elementlibrary

problemsolution

programcontrol

programcontrol

problemdefinition

TRUSS

MEMB

3D

2D

BEAM

UGFUNFORM

Calrel Interface Feap

MONT

UDGXSORM

SENS

UDDDIRS

BOUN

PNET

FIG. A1.4 CALREL SOFTWARE LAYOUT (D0271H10)

SL/WEM/R/M8663/5/01/C

A1/F2

Manager

BRIDGES TO CAD/CAE

Structural Analysis

Environmental Analysis

PrefemPlatework

Preframe

Stofat

Xtract

Installjac Wadam

WaveshipWajac

Sestra Splice

Advance Usfos

Mimosa

Pretube

Frame-work

PostrespPresel

Profast

Viewer

Submod

Associated

Proban

Cutres

Concode

Pre-processing

generalstructures

probabilisticrisk and

sensitivity concretedesign

shell/platefatigue

presentationof statistical

response

linearstatics anddynamics

launchingof jackets

mooringanalysis

progressivecollapse

wave loadson ships

framestructures

frame design

graphicpresentation

of results

non-linearstatics anddynamics

wave loadson framestructures

super-element

assembly

tubularjoints probabilistic

fatigue andinspection

structurepile-soil

interaction

wave loadson generalstructures

presentationof sectional

resultssub-

modelling

model andresults

visualisation

platedesign

Post-processing

FIG. A1.5 RELATIONSHIP BETWEEN DNV SESAM (D0271H10)SOFTWARE AND PROBAN PROGRAM

M AP

S

S

C

O

FAST PROBABILITY INTEGRATORS& ADVANCED SIMULATION SCHEMES

COMPONENT/SYSTEMRELIABILITY ANALYSIS

PROBABILISTIC SENSITIVITY STUDIESAND PARAMETRIC STUDIES

PROBABILISTICRISK ASSESSMENT

FAILURE MODESLIBRARY

PROBABILISTICDATA CHARACTERISATION

EFFICIENTFEA INTERFACE

GRAPHICALUSER INTERFACE

FIG. A1.6 LAYOUT OF COMPASS SOFTWARE (D0271H10)

SL/WEM/R/M8663/5/01/C

A1/F3

Primitiverandom variables

Most probablepoints

FiniteElementAnalysis

Reliabilitymodel

response

CD

F

Response

Fre

quen

cy

Variable

Structuralresponse

Reliability model(damage models)

Fast ProbabilityIntegration

module

User

Information

Function

Key Probabilitydatabase

NESSUSrisk

Risk

Randomvariable

definitions

NESSUS 5.0

FIG. A1.7 LAYOUT OF NESSUS SOFTWARE (D0271H10)

DynamicAnalysis

DamageAnalysis

TOOLS

DeterministicFinite Elements

ReliabilityAssessment

Monte CarloSimulation

ResponseSurface Method

NonlinearProgramming

Models of Fractureand Fatigue

StochasticFinite Elements

SystemIdentification

Reliability BasedOptimization

SystemsModelling

FIG. A1.8 STRUCTURE OF COSSAN SOFTWARE (D0271H10)

SL/WEM/R/M8663/5/01/C

A1/F4

methods, appls and software for structural reliability

Documents