A review of probabilistic fracture mechanics literature

Reliability Engineering3 (1982)423-448

A REVIEW OF PROBABILISTIC FRACTURE MECHANICS LITERATURE

G. O. JOHNSTON

Berkeley Nuclear Laboratories, CEGB, Berkeley, Gloucestershire GLI3 9PB, Great Britain (Received: 14 August, 1981)

ABSTRACT

Over recent years, the study of statistical methods applied to fracture and fatigue of materials has escalated. Many experimental programmes have been initiated to supply 'real' data, upon which estimates of failure probabilities may be based. There is a wealth of published literature which this paper attempts to cover. For easy reference, the papers have been classified where possible into broad categories. Initially, work of a more general nature is reviewed. The problems of input data are discussed with particular reference to the flaw distribution, the reliability of non-destructive testing, the fracture toughness properties of the material under consideration and the growth of defects by fatigue. Applications have been especially plentiful where high integrity is required, such as in the pressure vessel and off-shore industries. This is reflected by the large quantity of literature. Finally, a section is devoted to relevant mathematical techniques.

I.

INTRODUCTION

The interest in applying statistics to assess structural reliability has continued to increase in recent years. This is evident from the large amount of work in probabilistic fracture mechanics (PFM) and associated topics. In a previous review, 1 the literature was discussed in terms of the basic problem of designing against failure. This paper attempts to update the initial review following the same broad outline. For completeness, some background information is duplicated to clarify the problems. The subdivision of the subject of the papers remains the same (as discussed in the next few paragraphs) except that the design factor section has been replaced by one on off-shore structures. Few papers relate to deterministic design factors since they are gradually being replaced by probabilistic models. 423 Reliability Engineering 0143-8174/82/0003-0423/$02-75 © Applied Science Publishers Ltd, England, 1982 Printed in Great Britain

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Clearly there is some overlap between the sections and indeed the first contains references which are fairly general in their approach. Many factors contribute to reliability and these often change both systematically and at random. In the past, structures have been designed using a simple factor of safety approach to calculate a safe lifetime over which the applied loads will be sustained. Each variable in the equations governing the operation of the system (for example fracture mechanics or fatigue S - N relationships) was given a reasonable value drawn from past results or even just human intuition, and a predicted life obtained. This was then divided by a factor of safety which reduced it to give a 'safe' maximum operating life. It is conceivable that this is unrealistically conservative, but it is the price paid for simplicity ! There is a growing requirement to quantify the true safety of a structure. Guesses are no longer sufficient, so that there is a demand, particularly from industry, for a statistical justification behind the calculations, since many parameters may influence the behaviour of a structure and in practice each will be subject to both random and systematic variations. One approach is to use historical data from past failures and non-failures to evaluate the reliability of current productions. 2 This is a reasonably simple approach given sufficient data, but it suffers in that only specific structures are considered. The effect of a change in one of the key parameters often cannot be predicted as the required data probably do not exist. Thus, an alternative approach has been sought, namely the development and application of engineering models based on an understanding of the failure modes and statistical distributions of the controlling parameters. The latter distributions are not always known, so that errors may be introduced because of poor assumptions. Another disadvantage is the relative complexity of the approach which requires more input parameters where errors may again occur. Ideally, a combination of the two methods is necessary so that past experience and results plus specially designed laboratory tests can be used to obtain statistical distributions which best fit these data, but also, hopefully provide predictions about altered or new systems. In this review, some attempt has been made to sub-divide the broad scope of the subject. There is obviously some overlap, and indeed the first section contains references which are fairly general in their approach.

1.1. The basic problem The two main considerations are the distribution of crack-like defects (or flaws) introduced during the fabrication of a structure and the critical crack length distribution for the material under service conditions, the latter being controlled primarily by the fracture toughness. Several authors have considered this problem in specific cases and with varying assumptions. In general, there is a lack of data for the initial distribution of defects in a structure so the distribution is often assumed to be lognormal or exponential. F r o m a failure point of view, it is importanl~ to know

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how this distribution compares with that of the critical crack size, which is controlled by the material properties, and applied loading. The probability of failure is then given by the probability of interaction of the actual crack or flaw a i, and a critical crack length a c. 1 In the above, there has been an inherent assumption of independence between ai and ac. Marriott and Churchill 3 pose a warning that the presence of defects in regions of low toughness material may potentially increase the failure probability by several orders of magnitude. A preliminary examination of service failures suggests the possibility of a correlation effect may indeed be a real one, not to be ignored. 1.2. Further considerations Closely linked with the as-fabricated flaw size distribution is the role played by non-destructive testing in detecting defects which may, or may not, be repaired. Thus, if the initial flaw distribution and the probability of detecting a defect are known, then the distribution of flaws when the structure is put into service can be deduced. In situations where fatigue cracking occurs, the defect distribution will tend to alter with the number of applied cycles. This may have a profound effect on the interaction probability as the defects increase towards the critical crack size. This problem may be reduced by in-service inspection at fixed intervals of time, and by repairing the defects so found. Indeed, it is possible, though a little tedious, to compute optimum inspection intervals which are based on a maximum allowable failure probability. One main application of probabilistic fracture mechanics is in the estimation of the reliability of high integrity welded structures, such as nuclear power plants and off-shore structures. This appears to have encouraged a certain amount of work in these fields, some of which is referred to in Sections 7 and 8 Finally, a few references are given on useful mathematical techniques. Many of these are concerned with parameter estimation from available data. In particular, with reference to fatigue, the standard results require slight modifications in order that data from unbroken specimens may be included in the analysis. Additional references are added at the end of each section, which, though not reviewed, may be of interest in specific problems.

2.

GENERAL

The concept of design factors can be pursued in the probabilistic models where the factors themselves are defined in terms of the ratio of characteristic values of load and strength based on statistical models of the variables instead of the mean values.4 For example, the characteristic value for load may be the 90 percentile value of the load distribution and for strength the 5 percentile value of the strength distribution.

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(Here, '90 percentile value' indicates the value below which a load is expected to fall with 90 ~0 confidence.) The optimisation of the inherent cost may also be included. Alternatively, the uncertainties of the different loads may be combined statistically, 5 since only one safety factor should be sufficient since there is just one design condition, namely the specified reliability level. Other approaches include fault tree analysis which is essentially a deterministic approach whereby each event in a sequence is assigned a probability of success/ failure. Fukuda 6 considers this as a link between probabilistic and qualitative methods and explains its use with respect to improving the reliability of welded structures. However, no distribution functions are assumed and so there can be no real consideration of the uncertainties involved. The component probabilities are assumed known, though they may be ill-defined. On the other extreme, a branch of mathematics known as fuzzy set theory has developed to describe just such cases. It deals with the vagueness as distinct from the randomness. Subjective judgements are made of the uncertainties in the form of equations usedand data available, and these are used to calculate cumulative distribution functions for the system uncertainties. In essence, this is introducing design factors at earlier stages of an analysis. The basic ideas are given by Blockley and Ellison v including a prediction method for time to fatigue/creep failure. However, applications are still in their early stages of development. Other methods are more fully developed. In California, the influence function method 8 of quickly and accurately calculating stress intensity factors and subsequent residual life of cracked structures has gained favour, especially where there are high stress gradients. A powerful two- and three-dimensional elastic fracture mechanics computer program called B I G I F (boundary integral generated influence function) has been produced and is commercially available. A summary document has been prepared by Besuner e t al. 8 and this includes references to further explanatory papers. In a number of test cases with known crack analysis solutions, the method was found to produce more accurate answers more cheaply than the standard available finite element algorithms. 9 In the probabilistic treatment, the random variations are described using distributions of values for the parameters rather than one deterministic value. The distribution may be thought of as a curve combining the possible values and the likelihood (probability) they will occur. The introduction of distributions of values necessarily complicates an analysis, but the end result is likely to be more accurate and hence of more use. In some cases, neat analytical expressions for the reliability are impossible and numerical approximations are required. Here level 2 and 3 methods of reliability are appropriate, as described by Baker. 1° In the former, approximate interactive algebraic procedures are used to determine an estimate of the failure probability ( = 1 - reliability), whereas in the latter, the approximation is at a later stage, after all the exact probabilistic expressions have been obtained for the complete system; this is, of course, much harder, and the work involved may not

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be justified by the improvement gained. However, in a few special cases--some applicable to fracture mechanics--the analysis is sufficiently simple to permit a level 3 approach. The application of safety factors constitutes the level 1 method of reliability. Returning to the probabilistic assessment of fracture mechanics and fatigue, some comments on deterministic versus probabilistic models are highlighted by Jouris. ~1 He concludes the latter are valuable tools for assessing the likelihood of various critical events, but the figure obtained should be used as a guide for comparison purposes, rather than as an absolute measure. Lidiard ~2 reviewed the theoretical calculation of failure probabilities for engineering structures. The model is the same as that underlying the work in the Marshall Report, ~3 and is discussed with reference to the statistical distributions of parameters such as a i and a c applicable to nuclear pressure vessels and piping. This will be mentioned in more detail in Section 7. A similar procedure applies in terms of fatigue life where failure occurs when the accumulated number of cycles is too large. However, the general approach to the variability of the parameters remains the same. This is summarised by Bokalrud and Korsgren.~4 The sensitivity of the failure probability to changes in the parameters, namely stress range, crack propagation, C and m in the Paris law, and geometric misalignment of cruciform welded joints, was investigated. The scatter in the fatigue crack propagation parameters has also been investigated by Kanazawa et al.~5 in conjunction with tests to determine material toughness as described by crack opening displacement measurements. The problem of scarcity of data remains. At the University of California, a large programme is underway with respect to the combined effects of loss of coolant accident (LOCA) and earthquakes on nuclear plant safety. Traditionally, a worst case philosophy has been used, but recently there has been increasing concern that design for extreme loads may reduce the reliability of the normal plant operation. Lu and Streit 16 present the work so far, which details the fracture mechanics and fatigue models assumed, including preservice inspection, proof testing and inspection for leak detection--which are found to be important factors in reducing the probability of a LOCA. In many cases, the exact forms of the distributions modelling the statistical variability are unknown. The advantage of the sensitivity studies described earlier is that the result of changing assumed parameters may, or may not influence the final result. If the effect is minimal, then attention may be diverted to the more critical factors. Often the normal (Gaussian) distribution is chosen to be representative of the variation, simply because the techniques for analysing such distributions are reasonably well known. The disadvantage is that unrealistic values may be predicted by the model, for example, negative measurements of length. Where high reliability is required, the results may be very sensitive to changes in the tails of the distributions. Zemanick and Witt 17 report that often the assumption of normality seems to be reasonably conservative for engineering applications and they highlight specific examples with respect to stress and strength distributions.

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For brittle materials, the Weibull distribution may be used to characterise the failure probability, based on the weakest link theory. 18 This has been exploited for various practical examples by Stanley e t a / . 19'2° and Hunt and McCartney 2~ have shown that their developed theory for brittle materials is closely related to this classical Weibull theory. The final part of this section considers the analysis of risk, rather than just failure probability. Risk may be defined as the failure probability multiplied by the mean severity, which for example may be the cost of replacement of the failed part. Therefore, the risk of a large number of cheap items failing may be equivalent to that associated with one failure that has greater consequences. Tetelman and Besuner 22 discuss the problems associated with risk assessment in society with respect to car accidents, their causes and the subsequent injuries sustained. They extend the discussion to brittle fracture of steel bridges, in particular to a critical suspension bridge component, 23 where the Weibull distribution is used to model the fatigue life. The calculated best estimate was a failure rate of 1.9 × 10 6 failures per year which compared favourably with the figure of 0.95 × 10- 6 for all bridges calculated from the details of 10 in-service bridge failures. Risk analysis by PFM 24 and further examples involving the analysis of large steam turbine discs subject to inclusions near the bore and corrosion assisted cracking in the rim have been published by Rau et al. 25

Additional general literature is cited.26 -43

3.

FLAW DISTRIBUTION

Perhaps the greatest problem in assessing the reliability of a welded structure is that of data on defects which may be present. Ideally, there should be no defects, but in practice no structure is completely defect free, since no-one is perfect and the result can only be as accurate as the tools available permit, including the experience (or lack of experience) of human operators involved. The problem of developing flaw acceptance standards is discussed by Harrison 44 who includes the work of the WEE/37 Committee now published as a British Standards Institution document. 45 Data on service failures may be available, but rarely is sufficient information included to allow a statistical assessment to be made of the defects causing failure. Two separate parts of the problem may be identified, namely the position of defects and their respective sizes. Wiggins 46 presents a mathematical treatment, where the weld position, given by the left hand point of the defect, is assumed to be uniformly distributed along the weld length. He defines a crack damage parameter representing the fraction of total weld length containing cracks. However, in terms of fracture mechanics it is the crack depth that is of importance. Fearnehough and Jones 47 present such data for seamless and seam welded pipe, estimated from preservice pressure test failures. In recent years pipe

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supplies have been remarkably improved in quality, so alternative information has been obtained by sectioning and measuring all detected defects in a sample of nearly 3000 seamless pipes. The data are similar to those from the failure analysis, but still do not represent the true defect population of an existing pipeline since some defects may have been missed in the inspection and the pressure test will eliminate large unacceptable defects and some smaller insignificant ones. A way of helping to solve the problem of missing defects is described by Jagger and Manning. 48 The largest defect in each weld is recorded and an extreme value distribution used to model these data. From this, predictions may be made regarding the behaviour of the largest defects, which are expected to be most damaging. Weld defects found in high Jemperature steam pipe welds by nondestructive testing (NDT) techniques are analysed in terms of the defect length and depth through the wall thickness to obtain the appropriate distributions. A similar approach is taken by Becher and Hansen 49 who have performed a statistical analysis of the lengths of slag inclusions in welded steel based on 347 radiographs, and also of corrosion cracks in welded steel pressure vessels. Radiographs showing no slag inclusion were arbitrarily assigned lengths of 0.1 mm to denote the lower limit of resolution of the N D T technique. (Note this had no effect on the cumulative frequencies for observed inclusions.) The stress corrosion surface cracks were measured indirectly as the depth of grinding marks during their removal. In both cases, a lognormal distribution appeared to give reasonable graphical fits to the data. It is perhaps worth observing here that care is required in interpreting such data since the defects represent the largest in each radiograph. If these defect sizes are plotted as a histogram and then used in a probabilistic analysis, the results are likely to be over-conservative since all the defects present are then inherently assumed to be represented by these worst flaws which have been recorded. Additional flaw distribution literature is cited. 5°- 52

4.

NON-DESTRUCTIVETESTING(NDT)

After fabrication, a structure is usually inspected to check for large 'unacceptable' flaws which require repair before service. This inevitably changes the initial defect distribution by filtering some of the larger defects. A general appreciation of N D T is given by Hansen 53 including some of the difficulties encountered. No technique of non-destructive inspection is perfect, so that a probability of detection is associated with each, and this may well vary according to the defects of interest. In terms of fracture mechanics analysis the defect depth is essential, so that an estimate of the accuracy of sizing capabilities must be determined. Packman e t a l . 54 investigated penetrant systems of aluminium and titanium, which yield greater than 90 ~o ability to detect surface cracks of length greater than 0-939 mm. The probability of detection decreases for small cracks and also varies in

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accuracy depending on the inspectors' capabilities. Although two may find the same total number of defects (i.e. the same detection probability) one may have a higher percentage of false indications. A similar discussion by Herr and Marsh 55 includes data from magnetic particle, delta scan and ultrasonics as well as liquid penetrant inspections. In all cases, the effects of human factors were found to reduce the reliability of the N D T process, and it is suggested that an improvement will increase the confidence in the end product, since this may be causing the conflicting results of large round robin programmes mentioned later in this section. Conventional inspections on pressure vessels have used radiography to determine defective weld lengths requiring removal and repair. Forli e t al. 56 report on a pilot study in Denmark to compare radiography with ultrasonics, including the use of a number of operators. 75 m of test welds included lack of fusion, lack of penetration defects and slag inclusions. The data were analysed statistically to obtain operating characteristic (OC) curves describing the probability of accepting a single defect of a specific size. The OC curves were modelled by a modified tanh curve, but the results are only preliminary, to show the feasibility of a much larger research programme. For fracture mechanics applications, attention is often focussed on ultrasonics inspection which can be used to estimate the through thickness defect depth. C o f f e y , 57 however, proposes that the results will tend to depend on the limit of resolution of the equipment, and the metallurgical conditions, so that the probability of defect detection may be very low if the wrong beam angle and test sensitivity are chosen. Indeed, Tu and Seth 58 report underestimation of flaw depths by a factor of up to 2.5 compared with the actual depth. Certainly, the exact method used will affect the results since some of the new more sophisticated ultrasonics techniques will be given an improved detection and sizing capability as demonstrated by Adler e t al. 59 Their multi-transducer frequency analysis technique was developed and applied to characterise flaws in an 8 inch thick heavy section steel welded specimen. The initial results have been encouraging and further experiments are envisaged to include correlation with destructive examinations. The interest in inspection uncertainty has stimulated several large programmes o f work, including a recent one at Pacific Northwest laboratories. A summary by Becker 6° indicates that the aims are directed towards identification and quantification of inspection uncertainties with particular reference to primary piping systems. A round robin has also recently been initiated to evaluate the effectiveness of current Codes of practice. In the UK, a collaborative programme is in progress to examine the ultrasonics testing of ferritic steel weldments. A large quantity of data is available on a variety of welding defects, but the programme was not designed statistically to obtain detection probabilities. The initial results are for non-planar defects, and are more qualitative, 61 providing some comparison between conventional ultrasonics and improved techniques, for example the B scan display method which showed less tendency to undersize defect depth. Perhaps the most well known programme is the PISC study (Plate Inspection

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Steering Committee) which provided some insight into ultrasonics inspection capabilities. 62'63 The programme included 34 teams from 10 countries working on three test plates. 62 The defects comprised small volumetric defects, through thickness cracks and large sets of defects classed as one unacceptable defect. The defect detection probability, D D P , defined as: DDP =

Number of teams successfully detecting a particular defect Total number of teams

increased from 0.1 for 10 mm deep vertical cracks to 0.95 for 60 mm deep cracks, though these figures are based on a loglinear fit to five data points. The average error in positioning was approximately 1/25 of the plate thickness. In general, Code acceptable defects were oversized whereas rejectable defects were undersized. The group of true 'alternative' techniques showed a very significant improvement in performance. 63 In particular, the use of high sensitivity techniques, focussed beam probes and multiple orientations of the ultrasonic beams gave extremely good detection. One specific area requiring more consideration is that of defects approximately equal in size to the Code accept/reject level and it is hoped that proposals for future PISC extensions will include such an examination preferably for weld, HAZ and highly stressed, sensitive regions such as nozzle corners. An alternative approach to the explicit correlation between N D T and destructive examination has been proposed by Shaffer. 64 It is based on extreme value theory and requires maximum flaw size data to be collected as suggested earlier by some of the defect data collection exercises. Verification of the method remains to be demonstrated and this will require some destructive/non-destructive correlations. Once the appropriate detection probabilities are known, the effect on a flaw size distribution may be calculated. In service, several inspections may be carried out usually at equal time intervals to check the defects are not growing to unacceptable levels. It is then possible to calculate a figure for the reliability of a structure over its design life. However, assuming there is a reliability target, the analysis can be turned round to achieve an optimum inspection programme. One such approach has been pursued by Arnett, 6s though it is difficult to repeat since the analysis relies on a suite of computer programs. In a typical case of 3 in-service inspections, the optimum schedule shows inspections at 8, 27 and 52 ~o of the service life rather than the usual 25, 50 and 75 ~o. Additional N D T literature is cited. 66- 72

5.

MATERIAL PROPERTIES

Assuming the preservice flaw size distribution is known, with probability density function f ( x ) , then the failure probability of a structure is given by the interaction of an actual flaw length and a critical crack length a c.

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Through fracture mechanics, a c may be related to the fracture toughness of the material, usually assessed using Charpy, Ktc, crack opening displacement (COD) or J1c tests. The Charpy test is an impact test, which is quick and relatively cheap to perform, but it requires empirical correlations with K~c to give an indirect measure of fracture toughness. Hipkins and Dickinson 73 report on an experimental programme designed to investigate the scatter of results caused by inherent variations in the weld structure, its treatment and its composition. The study was for welds and materials representative of nuclear reactor pressure vessels, namely JTA101 and Coltuf 28 with manual metal arc and submerged arc weldments. Histograms of the data are presented and in the transition region bimodality is apparent. It is shown that the Charpy energy does vary with position through the weld, notch orientation and welding technique. In the United States, a large data base has been compiled for pressure vessel steels and appropriate weldments. 74 Most data are currently on A533B and A508 unirradiated material, though the data collection continues and some initial data on irradiated material are available, v5 This exercise has been part of a much larger programme including experimental and statistical work on fracture toughness assessments. 76 The fracture toughness, Ktc, of D6ac steel used in the aircraft industry has been statistically analysed by Coyle et al. v7 using compact tension and double cantilever bend specimen data. The Weibull distribution was found to fit the data, though reasonable results were also found for normal and lognormal distributions. Examples of data on COD and Jtc are given by Gulvin et al. 78 and Nilsson and Ostensson 79 respectively. In the latter paper, a statistical analysis was performed to compare different methods for fracture toughness evaluation, using two laboratories. No statistically significant difference was obtained from the two laboratories and there was only a small discrepancy between the three different methods used to calculate J. A large programme has been carried out in France 8° to analyse statistically the scatter in C O D results. Generally, no influence of plate thickness nor testing laboratory was observed. However, wide scatter was observed, which may mask any real effects. One advantage of the Weibull model of fracture toughness is its possible use in the prediction of large specimen behaviour from that of small tests. Landes and Shaffer sl explain the application of Weibull's weakest link theory, whereby a large specimen failing at its weakest point has a Weibull distribution, which is predicted from small scale specimen tests modelled by a similar distribution. An illustrative example is provided, based on some experimental test results which do not disprove the hypothesis. However, the method requires more data for verification. In the ASME Code, 82 a lower bound KIR curve is given for fracture toughness of nuclear pressure vessel steels, but this was based on the limited amount of data then available. Wullaert et al. s 3 discuss some of its shortcomings and propose improved reference toughness curves, which can be used to give statistically based lower

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bound fracture toughness curves. The exact formulation has changed as the work has progressed and application of the method has been simplified to the use of Charpy data from the material of interest. A tanh model

g=A+Btanh~)

/T-To\

(1)

is fitted to the Charpy data s4 and the parameters, A, B, To and C used to derive those for the tanh model of the toughness curve, ss In the continuing concern for safety, especially in nuclear and off-shore applications, there has been a move to impose tighter controls on chemical specifications in the hope that this will improve material properties. The Marshall Committee s6 has recommended upper limits on carbon, copper and sulphur contents of 0.2, 0.015 and 0.1 ~o respectively for pressure vessel materials and 0.2, 0.010 and 0.05 ~o for associated weldments. Initial results from recent German work by Kussmaul et al. 87 do indicate an improvement of properties with tighter chemical specifications. This is corroborated by preliminary analyses ss of some of the EPRI data TM though there are at present few data available on the 'new' materials. In a summary of his work Swindeman s9 claims that he has found that variability in material properties can be traced to differences in chemistry, fabrication and testing procedures. These correlate well with heat-to-heat, product-to-product and laboratory-to-laboratory variability, respectively. A final note of caution in the interpretation of the data is provided by Ostberg et al. 9° In structures where target failure probabilities are low, it is the extremes of the distributions that are of interest, including the possibility of inconceivable events producing 'outliers' to the distribution. Additional material properties literature is cited. 91 -loo

6.

FATIGUE

Failure by fatigue is a common occurrence in welded joints. Small cracks at weld toes (or the root) may propagate under cyclic loading. LittlC °1 reports on the work of the ASTM Subcommittee E09.06, which aims to bring statistical methods into the design and analysis of structures subjected to fatigue. The concepts must be brought within the grasp of practising engineers in order that they may be beneficial. Fong~ 02 emphasises this point and demonstrates that safety factors may be selected, not just arbitrarily, but by using plausible arguments based on the uncertainties and on risk/cost benefit analyses. It is usually assumed that the stress range versus endurance (S-N) curve is a straight line with slope between - 3 and - 4. For a particular stress level, S, it is the scatter in fatigue life, N, that is of interest. Therefore, a probability density function

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for N is required. A large quantity of data for structural titanium alloys and steels has been reviewed and analysed by Whittaker, 1°3 with respect to the reliability of aircraft structures. Some consideration of the model has been included, especially in comparing lognormal and two-parameter Weibull distributions. The latter was found to provide more conservative predictions on performance in the important region of early failure. Further discussion on the Weibull model may be found in the paper by Thomas 1°4 which uses it to predict jet engine turbine blade lives, as well as fatigue lives for other component parts. In terms of fracture mechanics analyses, it is the variation of the fatigue crack growth rate with the range of stress intensity factor that determines the growth of small defects to a critical size. The time to crack initiation is usually assumed small in comparison with that taken to propagate the crack. However, Talreja and Weibull 1°5 have considered both stages to be described by Weibull distributions. Experimental data give the parameters of the initial strength (Weibull) distribution from which predictions may be made about the fatigue lives. Salivar and Hoeppner ~°6 have collected and analysed crack propagation data from ASTM A533B material. The Weibull model was found to be reasonable, though inadequacies in the near threshold and high propagation rate data provided deviations from the usually accepted three-stage crack propagation curve. Only large experimental programmes such as that reported by Virkler et a l . l ° 7 for aluminium alloy will provide sufficient data for analysis, to give either the distribution of the number of cycles to fatigue to a certain crack length or that of crack propagation rate for a given stress range. In the absence of comprehensive data, one possibility is to group data from varying sources. This can be achieved by standardising all data to one value of m to obtain the variation of C in the Paris equation: da = C(AK) m dN

(2)

This assumes some relationship between C and m as for example given by Radhakrishnan ~°s for aluminium alloys or Gurney ~°9 for structural steels. Both expressions are of the form: log C = - q m + r

(3)

where q and r are material constants, which may incorporate a stress ratio or temperature effect. As mentioned earlier, the interpretation of test results requires some care. The difference between the regression analysis of experimental S - N results and the designer's requirements is explained by Hobbacher al° who also proposes a compromise (or 'in between' method of regression). Nevertheless, common practice still dictates that the first method is used, namely the equation is written as: log N = m log S + k

(4)

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and the sum of the squared deviations of log N is minimised. Further complications may be introduced by attempting to incorporate a 'cut-off' or endurance test limit, below which a specimen will not fail. In general, the mathematical description becomes harder, requiring computer programs for solution of equations. Spindel and Haibach ~1 have considered the method of maximum likelihood, but as in other cases the confidence which may be placed in the answers depends on the numbers of results available. Further thoughts on errors in curve fitting and in initial measurements themselves may be found in the paper by Wei et a1.112 A final complication considered in the literature is that of random loading, often either by wind or wave excitation. This may be simulated under test conditions by imposing stationary narrow band, random loads with normal probability density functions. Talreja ~~3 adopts a Poisson approximation to estimate the probability of failure in fatigue and comparison with experimental results shows gross conservativeness. A more theoretical approach is adopted by Wirsching and Haugen, 1~4 who extend their analysis to predict fatigue under wide band stresses. A limited amount of data only is required and the method is adaptable as new data become available. With particular reference to wind loading, Melbourne 115 reports on a number of studies on aero-elastic models tested in a wind tunnel. To establish an equivalent static design load, a peak factor may be defined as the number of standard deviations by which the peak hourly response exceeds the mean. For typical, normally distributed responses of a structure, the peak factor lies between 3.5 and 4.5. However, less regular responses may have peak factors which rise above 8.0. Additional fatigue literature is cited. 116 1 2 8 -

7.

PRESSURE VESSEL APPLICATIONS

In view of the continuing interest in safety, especially in relation to nuclear power plant, much effort has been concentrated in this field. Many studies relate specifically to the properties of A533B and A508 steel and it will become apparent that there is a large degree of overlap between this section and those preceding. The initial problem is to define what is meant by an 'acceptable' level of risk. This is discussed by O'Neil, ~z9 who compares failure statistics for a variety of common everyday accidents. In the nuclear field, it is releases of radioactivity (in particular iodine-131) that are of concern when considering the safety of the reactor pressure vessel and primary piping, valves and pumps. This is exemplified by the computational effort described by Fryer, 13° which permits the calculation of the consequences of releasing radioactive material to the atmosphere. Both human and environmental effects are included. Other computer programs are also available for calculating the reliability of pressure vessels subject to given pressure transients-see for example Vesely et al. 131 and Dufresne et al. 132

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As indicated in the earlier section on materials properties, there has been a trend towards the improvement of pressure vessel materials in order to try to increase their fracture toughness. This can only be demonstrated by an experimental programme, which investigates the effects of the various chemical elements present to seek some correlation. The development of pressurised water reactor (PWR) pressure vessel steels is discussed by Druce and Edwards ~33 who have conducted such a programme with respect to Charpy energy. The effect of irradiation on material properties is even less well documented. Many data exist on unirradiated material ~4,~34 so current research efforts are being concentrated on irradiation effects for PWR water environments at 288 °C. Preliminary fatigue crack growth rate results for A533B and A508 are reported by Cullen et al. ~a 5 The irradiation effects appear to be small in comparison with those obtained by reducing the cyclic testing frequency from 1 to 0-017 Hz. Since most fatigue failures occur from cracks in welded material, there is a requirement to produce high quality welds with limitations on defect sizes remaining after non-destructive testing. Yapp 136 considers the optimisation of welding procedures in an attempt to achieve this goal. Schmitt and Wellein ~37 extend this quality assurance approach to the inspection of primary components of a PWR. So far, the effect of human (operator) error has been ignored. The error may be caused by carelessness due to a lack of interest in the work or simply through inexperience. Bush 138 reports on a variety of piping failures and it appears that catastrophic failures are more likely to occur from operator error and design and construction errors, rather than from conventional flaw initiation and growth by fatigue. A pilot study has recently been conducted by Beyers and Marriott 139 where the manufacture of a pressure vessel was closely monitored from emergence of the specification from the planning department, to the commissioning stage. Initially, the design stage was omitted for simplicity. The results were qualitative, though some recommendations could be made to improve the 'demonstrable risk' level for example, all material properties must conform to the specification, for the whole plate. Where data are not available for a specific application, extrapolation from other sources may be attempted. In particular, data may be limited to Charpy energies for unirradiated plate and irradiated 'surveillance capsules' placed in the reactor prior to start-up. Jouris and Shaffer 14° have devised a method of weighting such data so as to reflect the variance composition in order to obtain more realistic estimates for the probability of fracture. I n the absence of sufficient data, an alternative approach is to use sensitivity analyses to determine which parameters most influence the final failure probability. In this way, data generation may be directed to the important features. One such study by Harris ~4~ shows that for stress corrosion cracking in austenitic stainless steel piping in boiling water reactors (BWR), it is the residual stress due to welding that is the most influential factor. In-service inspections were found to produce minimal reductions in the failure probabilities. In this case, the results were found to be insensitive to the input parameters, which were not

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accurately known. This constituted a considerable saving, since otherwise they might have been investigated in more detail. The interrelation between nuclear pressure vessel weld integrity and the in-service inspection parameters, such as sample size, frequency and detection efficiency has been investigated by Solomon e t a l . 142 The report is long, but provides a comprehensive study, including a variety of sensitivity analyses and a consideration of optimal weld inspection programmes consistent with a minimum cost criterion. The calculations are performed for defects under both low and high stress, yielding distinct procedures. The approach taken by the Marshall Committee to pressure vessel reliability has been extended further by Harrop and Lidiard'43 to include sensitivity analyses. The work shows the effect of variations in mean toughness and predicts reduced failure rates (by a factor of 100) when complete in-service inspection is carried out. Few data are yet available on the exact nature of the average number of cracks per vessel. Failure rates were found to be sensitive to assumptions about crack shape, but in general they were small under normal operating conditions. Additional literature on pressure vessel applications is cited.144-157

8.

OFF-SHORE APPLICATIONS

The safety of off-shore structures is influenced by many factors, which are discussed in some detail by Flint. 158 These include environment, structural properties, errors in design, construction or operation, and commercial and political pressures. Target levels are proposed suggesting that an annual failure probability of l0 6 might be appropriate for a major North Sea platform, assuming the risk of environmental forces predominates. For operational conditions values up to 10-3 might be acceptable. It is envisaged that amendments may be made to the design codes to take account of factors influencing structural safety. One of the main areas of concern is the fatigue life of the welded joints which are subjected to wind, wave, and perhaps even earthquake loading. This is an application of some of the work mentioned earlier in the fatigue section. WirschingX 59 presents a probability model based on lognormal assumptions for the (cumulative) damage at failure and various other parameters. The model should be viewed as a basis for incorporating more realistic assumptions on ocean wave characteristics and fatigue stresses. Extreme value distributions have also attracted some interest in describing wave and wind behaviour and their application is discussed by Ochi ~6° and Baker and Wyatt, 161 respectively. The latter paper compares various methods for calculating reliability and their associated errors, with particular reference to complex jacket platforms. The level 2 methods referred to earlier a° are shown to be extremely powerful. The final document in this section is in fact a large, preliminary report prepared -

5--10

-

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by Jensen e t al., 162 on risk levels in Norwegian off-shore petroleum activities. The study concentrates primarily on human safety and secondarily on pollution and economic risk. The report gives a qualitative assessment of risk and points to the areas which require further work (to be included in the 'main' project), in order to obtain increased safety. Additional literature on off-shore applications is cited.163- 165 9.

MATHEMATICAL TECHNIQUES

The use of statistics in data collection and analysis is discussed qualitatively by Hahn, 166 including some of the limitations with respect to applications in fatigue and fracture. If the data are insufficient, statistics cannot provide high precision results. Some of the more basic ideas in reliability are defined and explained by Byers and Weibe 16~ in their 'pocket handbook'. These include single component reliability, the use of test data and the Weibull model for failure in life testing. The problem of obtaining sufficient data, upon which statistical predictions for a component may be based, is a real one. Where previous information is available, estimates of sample size may be made using the (previous) sample standard deviation (see Cochran and Cox 168.) However, more often, there are no data readily available. Either a guess must be made for the expected variance of the results or sequential testing may be utilised. Smith 169 explains how the method is based on the desired and minimum acceptable reliabilities and the risk involved. After each test, the number of failures and successes is reviewed and a decision made to accept or reject the component or to continue sampling. In general, even when the sample size can be estimated, fewer tests are required using sequential testing. An approximation to the amount of confidence which can be placed in the minimum of a small sample of results is given by Wilks.l vo The equation: /3" =

1 -

~

(5)

predicts the percentage, fl, of data (from the population) which would be expected to lie above the minimum of n sample results with a confidence of ~. Therefore, for three test results and a 90 ~o confidence, 46 ~o of the population are forecast to lie above the minimum value of the three. The general theory of extreme values, of which the Weibull model is a special case, has been known for some time, as is evident from a series of lectures given by Gumbel 171 in 1951 and compiled from his work over the preceding 20 years. The theory is developed and applied to a number of practical examples so that readers may understand how to apply the methods to their own problems. For example, in 1947 Epstein 172 used the same methods in the description of the statistical aspects of fracture. In fatigue too, the Weibull distribution may be used to describe the life of a component and even data on unbroken specimens may be incorporated into the analysis to obtain tolerance limits.173

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Once data have been obtained and an appropriate model chosen, the parameters of the model must be estimated from the data. Graphical methods are very popular because of their simplicity, but care may be required when using extended scales on probability plots. Deviations from a straight line fit may appear deceptively large, causing poor estimates of the parameters. More consideration of this problem for the Weibull distribution is given by Hinds et al.* 74 in a fairly mathematical paper. Other techniques available include the method of moments, maximum likelihood estimation and Monte Carlo methods. Mazumdar et al.* 75 compare these methods in terms of the approximations required and the resulting errors. In cases where the expected failure probability is very small, say I in 106 or 107, Monte Carlo computer programs become very time consuming and hence costly since a large number of simulations are required. If 108 simulations are obtained, then only 10 failures are predicted if the failure probability is 1 0 - 7 . In this case, importance sampling can be very useful to reduce the variance of estimators. A systematic method has been proposed by Spanier 176 in order to apply this technique to accomplish the optimisation of the choice of parameters. Once the chosen models have been fitted to the data, it remains to test how well they represent that particular set of data. Of course, for large quantities of data the problem is somewhat easier than for the more realistic samples. Standard tests are available for cases where the model parameters are not estimated from the data 177 but often the data stand alone. Tests for normality have been proposed by Stephens ~78 and Lilliefors 179 and both are modifications of the standard Kolmogorov-Smirnov test. Tables have been produced using Monte Carlo simulation so that the test is easy to apply, requiring only the calculation of a parameter from the experimental data and its comparison with a tabulated value. The test will also work for lognormal distributions since the logarithms of the values follow a normal distribution. For other distributions, the problem transgresses into the realms of statistical research as shown by Atkinson's paper.18° Additional mathematical literature is cited. ~81 - 195

10.

CONCLUSION

Probabilistic methods are increasingly being applied to assess the reliability of structures. Though an analysis may not exactly model the real situation, it will often yield useful approximations or orders of magnitude for the failure probability. Obviously, the hardest, but most interesting requirement, is to obtain information regarding the as-fabricated flaw distribution and the associated reliabilities of non-destructive examination techniques. Data on fracture toughness distributions are easier to obtain as shown by the various experimental programmes in progress. Where fatigue damage occurs, the initial flaw distribution may change substantially, so increasing the failure pr~)bability.

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In the a b s e n c e o f sufficient d a t a u p o n w h i c h to base c o m p l e t e analyses, sensitivity studies m a y be c a r r i e d out. T h e s e will h a v e to d r a w o n c u r r e n t l y a v a i l a b l e d a t a and the e n g i n e e r s ' k n o w l e d g e o f t h e c o m p a r a t i v e b e h a v i o u r o f v a r i o u s s t r u c t u r e s , to p r e d i c t the effect o f c h a n g i n g the initial a s s u m p t i o n s o f an analysis. T h e n a t t e n t i o n m a y be f o c u s s e d o n the p a r a m e t e r s h a v i n g the m a x i m u m beneficial effect. P r o b a b i l i s t i c f r a c t u r e m e c h a n i c s is b e i n g d e v e l o p e d in a n s w e r to a g r o w i n g d e m a n d for statistically b a s e d a n a l y s e s o f failure p r o b a b i l i t i e s a n d it is to be h o p e d t h a t it will, in time, aid t h e f a b r i c a t o r s in i m p r o v i n g their designs.

ACKNOWLEDGEMENT T h i s p a p e r is p u b l i s h e d by p e r m i s s i o n o f the C e n t r a l Electricity G e n e r a t i n g B o a r d .

REFERENCES General 1. JOHNSTON,G. O. What are the chances of failure ?, Weld. Inst. Res. Bull., 19 (2, 3) (1978), pp. 36~40, 78-82. 2. WEARNE,S. H. A review of reports of failures, Proc. Inst. Mech. Engrs., 193 (20) (1979), pp. 125-36. 3. MARRIOTT,D. L. and CHURCHILL,A. R. Effects of a correlation between defects and material properties on component failure probability, Proc. 5th Int. Conf. SMiR T, Berlin 1979, paper M8/7. 4. LEICESTER,R. H. Load factors for design and testing, Proc. Conf. Prevention of Fracture, Melbourne, Australia, 1977. 5. ANG, A. H-S. A comprehensive basis for reliability analysis and design, Proc. US-Japan Seminar, Reliability Approach in Structural Engineering, Tokyo, 1974. 6. FUKUDA,S. A consideration on usefulness of fault tree analysis technique in improving reliability of welded structures, Trans. Japan Weld. Soc., 10(l) (1979), pp. 18 23. 7. BLOCKLEY,O. I. and ELLISON,E. G. A new technique for estimating system uncertainty in design, Proc. Inst. Mech. Eng., 193(5) (1979), pp. 159-68. 8. BESUNER,P. M., PETERS,D. C. and CIPOLLA,R. C., B1GIF, Fracture mechanics code for structures, EPRI Report NP-838, RP 700-1, Palo Alto, California, 1978. 9. BESUNER,P. M. An engineering fracture mechanics analysis of the Pilgrim 1 nozzle-to-pressure vessel weld discontinuities, Nucl. Engng. Des., 43 (1977), pp. 15~60. 10. BArrieR,M. J. Reliability theory and statistical treatment of data, Proc. C1RIA Seminar, Structural Codes, London, 1976, pp. 39-48. l 1. JOURIS,G. M. Some comments on the probabilistic approach to safety analysis, Nucl. Engng. Des., 511 (1978), pp. 169-72. 12. LIDIARD, A. B. Probabilistic fracture mechanics, Proc, Syrup. Fracture Mechanics--Current Aspects, Future Prospects, Cambridge, 1979. 13. MARSHALL,W. An assessment ofthe integrity of PWR pressure vessels--a UK study group report. Proc. 4th Int. Conf. SMiRT, California, 1977, paper G6/1. 14. BOKALRUD,T. and KORSG~N, P. Some aspects of the application of probabilistic fracture mechanics for design purposes, Proc. Colloq. Practical Application of Fracture Mechanics, Bratislava, 1979. 15. KANAZAWA,T., ITAGAK1,H., MACHIDA,S. and MIYATA,T. Outline of JWES standard for critical assessment of defects with regard to brittle fracture, and some case studies. Proc. Colloq. Practical Application of Fracture Mechanics, Bratislava, 1979, paper Dl0, pp. 274-84. 16. Lu, S. C. and STilT, R. D. Probabilistic assessment of decoupling the effects of large LOCA and earthquake in nuclear power plant design, Proc. 8th Water Reactor Safety Research Information Meeting, Gaithersburg, Maryland, 1980.

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17. ZEMANICK,P. P. and WITT, F. J. An engineering assessment of probabilistic structural reliability analyses, Nucl. Engng. Des., 50 (1978), pp. 173-83. 18. BATDORF,S. B. New light on Weibull theory, Univ. of California Report, UCLA-ENG-7762, 1977. 19. STANLEY,P., SIV1LL, A. O. and FESSLER, H. The application and confirmation of a predictive technique for the fracture of brittle components, Proc. 5th Int. Conf. ExperimentalStress Analysis, Udine, Italy, 1974, Paper 22, pp. 2.42-2.51. 20. STANLEY,P., SIVILL,A. O. and FESSLER,H. Applications of the four function Weibull equation in the design of brittle components. In: Fracture mechanics of ceramics, R. C. Bradt, D. P. H. Hasselman and F. F. Lange (eds), Plenum Press, New York-London, Vol. 3 Flaws and Testing, 1978. 21. HUNT, R. A. and MCCARTNEY, L. N. A new approach to Weibull's statistical theory of brittle fracture, Int. J. Fract., 15(4) (1979), pp. 365-75. 22. TETELMAN, A. S. and BESUNER, P. M. How safe is safe enough? In: Social consequences of engineering, L. S. Hager (ed.), Dun-Donnelly Publishing Corp., ca 1978. 23. TETELMAN,A. S. and BESUNER, P. M. The application of risk analysis to the brittle fracture and fatigue of steel structures. Proc. 4th Int. Conf. Fracture, Waterloo, Canada, 1977, pp. 137-56. 24. RAU,C. A. and BESUNER,P. M. Risk analysis by probabilistic fracture mechanics, J. Engng. Mater. Tech., 102 (1980), pp. 56-63. 25. RAU, C. A. JR., BESUNER, P. M. and SORENSON, K. G. The role of micromechanics models in risk analysis, Proc. Conf. Micromechanisms of Crack Extension, Cambridge, 1980, Paper 21. A d d i t i o n a l reading 26. ALMOND, E. A. Strength of hardmetals. Metal Sci., (Dec.) (1978), pp. 587-92. 27. AUGUSTLG. Probabilistic methods in plastic structural analysis, Nucl. Engng. Design, 57(2) (1980), pp. 403-15. 28. BESUNER, P. M., TETELMAN, A. S. and SORENSON, K. G. An engineering analysis of the risk associated with windshield wiper pivot assembly failures on 1971-73 model year Capris, Failure Analysis Associates Report FAA-77-8-2, 1977. 29. BONDI,A. A. Reliability as a materials property, J. Engng. Mater. Tech., 100(1) (t979), pp. 27-33. 30. DAVtDGE,R. W. Combination of fracture mechanics, probability and micromechanical models of crack growth in ceramic systems, Proc. Conf. Micromechanisms of Crack Extension, Cambridge, 1980, paper 20. 31. ELLINGWOOD,B. R. An improved basis for formulating acceptance criteria for structural welds, J. Press. Vess. Tech., 100(1) (1978), pp. 85-90. 32. FORD, D. G., Reliability and structural fatigue in one-crack models, Proc. Conf. Prevention of Fracture, Melbourne, Australia, 1977, pp. 54-68. 33. HOOKE, F. H. Survival and chance--the principles and application of reliability and risk analysis, Proc. Conf. Prevention of Fracture, Melbourne, Australia, 1977, pp. 1-16. 34. KELLER, A. Z. and KAMATH, A. R. R. Semi-conductor failure analysis, Proc. Syrup. of Soc. of Environmental Engineers, London, 1979. 35. MCCARTNEY,L. N. Statistics for the static strength of fibre bundles, Proc. Conf. Micromechanisms of Crack Extension, Cambridge, 1980. 36. MARGETSON,J. and STANLEY, P. Stress and failure probability analysis for a transversely isotropic, brittle, elastic cylinder subjected to internal pressure and axisymmetric thermal loading, Int. J. Mech. Sci., 18 (1976), pp. 561-70. 37. STANLEY,P., FESSLER,H. and SIVILL,A. D. An engineer's approach to the prediction of failure probability of brittle components, Proc. Brit. Ceramic Soc., 22 (1973), pp. 453-87. 38. STANLEY,P., SIVILL,A. D. and FESSLER,H. A design exercise for a ceramic turbine disc subjected to centrifugal stresses, J. Strain Analysis, 13(2) (1978), pp. 103-13. 39. STRATING,J. The interpretation of test results for a level-1 Code. Int. Inst. Weld. Document IIW XV-462-80, 1980. 40. TOAKLEY, A. R. Risk levels in structures, Proc. Conf. Applications of Probability Theory to Structural Design, Melbourne, Australia (1974), pp. 170-7. 41. WEN, Y-K. Probability of extreme load combination, Proc. 4th Int. Conf. SMiRT, San Francisco, 1977, paper M9/3. 42. WHITTAKER, I. C. and SAUNDERS,S. C. Application of reliability analysis to aircraft structures subject to fatigue crack growth and periodic structural inspection, Air Force Materials Laboratory Report AFML TR-73-92, Ohio, 1973.

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43. YAO,J. T. P. and ANDERSON,C. A. Reliability analysis and assessment of structural systems, Proc. 4th Int. Conf. SMiRT, San Francisco, 1977, paper M9/9.

F l a w distribution 44. HARRISON,J. D. Developments in flaw analysis for pressurised components. In: Developments in pressure vessel technology--I, R. W. Nichols (ed.), Applied Science Publishers Ltd, London, 1979. 45. BSI. Guidance on some methods for the derivation of acceptance levels for defects infusion welded joints. PD 6493. 46. WIGGINS,m. D. Statistical estimation of a crack damage parameter in manufacturing processes, Q. J. Appl. Math., (Oct.) (1974), pp. 347-50. 47. FEARNEHOUGH,G. D. and JONES, D. G. An approach to defect tolerance in pipelines, Proc. Conf. Defect Tolerance of Pressure Vessels, I. Mech. E., 1978. 48. JAGGER,M. and MANNING, P. V. C. The statistical analysis of defect distribution and growth data obtained from high temperature steam pipe welds in the CEGB and their comparison with materials data, Proc. BSSM 14th Conf. The Application of Materials Testing to Experimental Stress Analysis, Bradford, 1978. 49. BECHER,P. E. and HANSEN,B. Statistical evaluation of defects in welds and design implications. Danish Welding Institute Report, 1974. A d d i t i o n a l reading 50. MOORE,J. C. Statistical comment on data on distribution of cracks found on inspection of steel vessel. In: Canvey--an investigation of potential hazards from operations in the Canvey lsland/Thurrock area, HSE, Appendix 22, pp. 188-9. 51. PASSOJA,O. E. and HILL, D.C. Comparison of inclusion distributions on fracture surfaces and in the bulk of carbon-manganese weldments. In: Fractography~microscopic cracking processes, ASTM, STP 600, 1976, pp. 30-46. 52. YURIOKA, N., OKUMURA,M. et al. An approach to weld defect evaluation by decision analysis, Proc. Colloq. Practical Application of Fracture Mechanics, Bratislava, 1979, paper D9, pp. 259-65. Non-destructive testing 53. HANSEN, B. Sense and nonsense in the application of non-destructive inspection. International Institute of Welding, Van Ouwerkerk Lecture, 1977. 54. PACKMAN,P. F., MALPAN1,J. K., WELLS, F. and YEE, B. W. E. Reliability of defect detection in welded structures. Proc. 2nd Nat. Congress Pressure Vessels and Piping, ASME, San Francisco, 1975, pp. 15-28. 55. HERR, J. C. and MARSH, G. L. NDT reliability and human factors, Mater. Eval., (Dec.) (1978), pp. 41~. 56. FORLI,O., nANSEN,B., SANDBERG,S. and ASTROM,T. A comparison of radiographic and ultrasonic NDE, Danish Welding Institute Report, NORDTEST Project 72-6, 1978. 57. COFFEY,J. M. Quantitative assessment of the reliability of ultrasonics for detecting and measuring defects in thick-section welds, Proc. 3rd Int. Conf. Pressure Vessel Technology, I. Mech. E., 1978, paper C85/78, pp. 57-63, 292-4. 58. Tu, L. K. L. and SETH, B. B. Non-destructive/destructive test correlations and fracture mechanics analysis, J. Test. Eval., 5(5) (1977), pp. 361-8. 59. ADLER, L., COOK, K. V., WHALEY, H. L. and MCCLUNG, R. W. Flaw size measurement in a weld sample by ultrasonic frequency analysis, Mater. Eval., (March) (1977), pp. 44-50. 60. BECKER,F. L. Reliability of in-service ultrasonic flaw detection for primary piping systems. Proc. 8th Water Reactor Safety Research Information Meeting, Gaithersburg, Maryland, 1980. 61. JESSOP,T. J. and MUDGE, P. J. Size measurement andcharacterisation of weM defects by ultrasonic testing, Part 1, non-planar defects in ferritic steels, Welding Institute Report 3527/4/77, 1979. 62. PISC. Report on the ultrasonic examination of three test plates using the "PISC" procedure based on the ASME XI Code, Nuclear Energy Agency, OECD, 1979. 63. PISC. Analysis of the PISC trials--results for alternative procedures, Report No. 6, OECD, 1980. 64. SHAFFER,D. H. A method for estimating flaw detection probability from inspection data, Proc. IAEA Int. Syrup., Vienna, 1977, paper IAEA-SM-218/40, pp, 331-9. 65. ARNETT,L. M. Optimisation ofinservice inspection of pressure vessels, USAEC Report DP-1428, 1976.

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Additional reading 66. BusH, S. H. The impact of inservice inspection on the reliability of pressure vessels and piping, 2nd Nat. Congr. Pressure Vessels and Piping, ASME, San Francisco, 1975. 67. HAINES,N. F. The reliability of ultrasonic inspection, Proc. IAEA Syrup., Vienna, 1977, paper IAEA-SM-218/41, pp. 341-57. 68. JOHNSON,D. P. Determination of non-destructive inspection reliability using field or production data, Mater. Eval. (Jan.) (1978), pp. 78-84. 69. Jougls, G. M. and SHAFFER,D. H. A procedure for estimating the probability of flaw nondeteetion. Proc. 4th Int. Conf. SMiRT, San Francisco, 1977, paper G6/4. 70. ROGOVSKY,A. J. and ROSE, J. L. Ultrasonic testing sensitivity selection based on probabilistic techniques, Mater. Eval., 37 (March) (1979), pp. 47-55. 71. TAKASE,S., KIKUCHI, S. et al. A convenient high performance manual ultrasonic flaw detector, Proc. Int. Conf. Quality Control and NDT in Welding, WI/IQA/NDT Soc. GB, London, 1974, paper 19, pp.98-106, 187. 72. TROITSKII,V. A., DAVIDENKO,V. F. and DONIN,A. R. The automated non-destructive inspection of the quality of welded joints, Art. Svarka., (8) (1978), pp. 34-8, 1 (translated in Auto Weld). Material properties 73. HIPKINS,M. G. and DICKINSON,F. S. Influence of notch orientation and position of notch ductility of automatic and manual welds in mild steel, Weld. Metal Fabric (May-June) (1965), pp. 216-27, 228-50. 74. SERVER,W. L. and OLDEIELO,W. Nuclearpressure vesselsteeldata base, EPRI Report NP-933, RP 886-1, Palo Alto, California, 1978. 75. STAHLKOPF,K. E., ODETTE,G. R. and MARSTON,T. U. Radiation damage saturation in reactor pressure vessel steel: data and preliminary model, Proc. 4th Int. Conf. Pressure Vessel Technology, I. Mech. E. (1980), paper C40/80, pp. 265-72. 76. SERVER,W. L., OLDFIELD,W. and WULLAERT,g. A. Experimental and statistical requirements for developing a well-defined KtR curve, EPRI Report NP 372. RP 696-1, Palo Alto, California, 1977. 77. COYLE,J., GRANDAGE,J. M. and FORD, D. G. The distribution offracture toughness data for D6ac steel, Aeronautical Research Laboratories, Structures note 429, Melbourne, Australia. 78. GULVIN,T. F., TERRY,P. and WEBSTER,S. E. The prediction of failure of welded steel plates for pressure vessels, Proc. 4th Int. Conf. Pressure Vessel Technology, I. Mech. E. (1980), paper C 17/80, pp. 71-84. 79. NILSSON,F. and OSTENSSON,B. J ~c-testingof A533 B--statistical evaluation of some different testing techniques, Engng. Fract. Mech., 10 (1978), pp. 223-32. 80. Statistical analysis of the scatter of crack opening displacement measurements, Proc. Colloq. Practical Application of Fracture Mechanics, Bratislava, 1979, paper D5, pp. 222-32. 81. LANDES,J. D. and SHAFFER,D. H. Statistical characterisation offracture in the transition region, Westinghouse R & D Scientific paper 79-1D3 JINTF P4, 1979. 82. ASME Boiler and Pressure Vessel Code, Section III, Division l, Appendix G, article G-2110. 83. WULLAERT,R. A., SERVER, W. L., OLDFIELD, W. and STAHLKOPF, K. E. Development of a statistically-based lower bound fracture toughness curve, Proc. 4th Int. Conf. SMiRT, San Francisco, 1977, paper G6/5. 84. OLDF1ELD,W. Fitting curves to toughness data, J. Test. Eval., 7(6) (1979), pp. 326-33. 85. OLDEIELD,W. Fracturetoughnesscorrelationsandreferencecurves, MRCSReport, EPRI Contract RP1021-4, 1980. 86. MARSHALL,W. UKAEA Report on the integrity of PWR pressure vessels, 1976. 87. KUSSMAUL,K., EWALD,J., MAIER,G. and SCHELLHAMMER,W. Enhancement of the quality of the R PV used in light water power plants by advanced material, fabrication and testing technologies, Int. J. Pres. Ves. & Piping, 8 (1980), pp. 323-46. 88. JOHNSTON,G. O. Improvedfracturetoughnessrequirementsfornuclearpressurevesselsteelsandthe contribution of reference toughness curves, CEGB Report RD/B/5017N81, 1981. 89. SWlNDEMAN,R. W. Heat-to-heat material property variability and sampling methods, Proc. Conf. Energy Technology, Houston, Texas, 1977. 90. OSTBERG,G., HOFFSTEDT,H. et al. Inconceivable events in handling material in a heavy mechanical engineering industry, National Defence Research Institute Report FTL Al 6:71 E, Stockholm 1977.

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Additional reading 91. ARONE, R. A statistical strength criterion for low temperature brittle fracture of metals, Engng. Fract. Mech., 9(2) (1977), pp. 241-9. 92. AWAJI, H. and SATO, S. A statistical theory for the fracture of brittle solids under multiaxial stresses, Int. J. Fract., 14(1)(1978), pp. R13-RI6. 93. BATDOgF, S. B. Weibuil statistics for polyaxial stress states, J. Am. Ceram. Soc., 57(1) (1974), pp. 44-5. 94. BATOOgF, S. B. Fracture statistics of brittle materials with intergranular cracks, Nucl. Engng. Design, 35(3) (1975), pp. 349-60. 95. CnIgIGOS, J. N. and MEYER, T. A. Influence of material property variations on the assessment of structural integrity of nuclear components, J. Test. Eval., 6(5) (1978), pp. 289-95. 96. GRUNDY, P. Probabilistic aspects of stability, Proc. Conf. Applications of Probability Theory to Structural Design, Melbourne, Australia, 1974, pp. 138-47. 97. KOONS, G. F. Statistical analysis of notch toughness data, J. Test. Eval., 7(2) (1979), pp. 96-102. 98. OLDFXELD,W. and MA~rorq, T. U. Revised fracture toughness reference curves, Proc. 5th Int. Conf. SMiRT, Berlin, 1979, paper G2/7. 99. OLDFIELD,W., WULLAERT,R. A. et al. Statistically defined reference toughness curves, Proc. 3rd Int. Conf. Pressure Vessel Technology, Tokyo, 1977, pp. 703-15. 100. WULL~RT, R. A. and SERVER,W. L. Small specimen predictions of fracture toughness for nuclear pressure vessel steels, Nucl. Engng. Design, 57(1) (1980), pp. 153-73. Fatigue 101. LITTLE, R. E. Statistical aspects of fatigue, ASTM Stand. News, (Feb.) (1980), pp. 23~5. 102. FONG, J. T. Uncertainties in fatigue life prediction and a rational definition of safety factors, Nucl. Engng. Des., 51(1) (1978), pp. 45-54. 103. WHITTAKER,I. C. Development of titanium and steel fatigue variability model for application of reliability analysis approach to aircraft structures. Air Force Materials Laboratory Report AFML-TR-72-236, Ohio, 1972. 104. THOMAS, F. H. Why Weibull? Unclassified Rolls Royce Report, 1979. 105. TALREJA,R. and WEIBULL, W. Probability of fatigue failure based on residual strength, Proc. 4th Int. Conf. Fracture, Waterloo, Canada, 1977, pp. 1125-31. 106. SALIVAS,G. C. and HOEPPt~ER,D. W. A Weibull analysis of fatigue crack propagation data from a nuclear pressure vessel steel, Engng. Fract. Mech., 12(2) (1979), pp. 181~1. 107. VIRKLER, D. A., HILLBERRY, B. M. and GOEL, P. K. The statistical nature of fatigue crack propagation, J. Engng. Mater. Tech., 101(2) (1979), pp. 148-53. 108. RADHAKRISHNAN,V. M. Quantifying the parameters in fatigue crack propagation, Engng. Fract. Mech., 13(1) (1980), pp. 129~1. 109. GURNEY, T. R. Fatigue of welded structures, Cambridge University Press, London, 1979. 110. HOBBACHER,A. On evaluation of fatigue test data, International Institute of Welding Document IIW-JWG-XIII-XV- 15-75, 1975. 111. SPINDEL,J. E. and HAIBACH,E. Some considerations in the statistical determination of the shape of S/ N curves, Proc. Symp. Statistical Analysis of Fatigue Data, ASTM, Pittsburgh, 1979. 112. WEt, R. P., WE1, W. and MILLER, G. A. Effect of measurement precision and data processing procedures on variability in fatigue crack growth rate data, J. Test. Eval., 7(2) (1979), pp. 90-5. 113. TALgEJA, R. On fatigue reliability under random loads, Engng. Fract. Mech., 11(4) (1979), pp. 717-32. 114. WIgSCmNG, P. H. and HAUGEN, E. B. A general statistical model for random fatigue, J. Engng. Mater. Tech., 96(1) (1974), pp. 34-40. 115. Mt%BOURNE,W.H. Probability distributions associated with the wind loading ofstructures, Trans. Aust. Cir. Engrs., 19(1) (1977), pp. 58-67. Additional reading 116. FONG, J. T. Statistical aspects of fatigue at microscopic, specimen and component levels, Proc. ASTM-NSF-NBS Symp. Fatigue Mechanisms, 1978. 117. FlU/UDEtqTHAL,A. M. Reliability of reactor components and systems subject to fatigue and creep, Nucl. Engng. Design, 28(2) (1974), pp. 196-217. 118. HOOKE, F. H. Probabilistic design and structural fatigue, Proc. Conf. Applications of Probability Theory to Structural Design, Melbourne, Australia, 1974, pp. 149-64.

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119. IHARA,G. and TSUR1,A. Fatigue of metals as stochastic phenomena, J. Engng. Mater. Tech., 99(1) (1977), pp. 26-8. 120. KRAUSZ,A. S. The random walk theory of crack propagation, Engng. Fract. Mech., 12 (1979), pp. 499-504. 121. LARDNER,R. W. Crack propagation under random loading, J. Mech. Phys. Solids, 14(3) (1966), pp. 141-50. 122. NISHIJIMA,S. and TAKEUCm, E. Statistical fatigue properties of SM50A steel plates for welded structures, Trans. Nat. Res. Inst. Metals, 21(2) (1979), pp. 74-84. 123. OH, K. P. A weakest link model for the prediction of fatigue crack growth rate, J. Engng. Mat. Tech., 100(2) (1978), pp. 170-4. 124. RITTER, JR., J. E., BANDYOPADHYAY,N. and JAKUS, K., Statistical reproducibility of the crack propagation parameter Nin dynamic fatigue tests, J. Am. Ceram. Soc., 62(9-10) (! 979), pp. 542-3. 125. SllqGH, B. R. and THOMAS, F. H. A method for predicting turbine blade lives using unfailed specimens and analytical techniques, Proc. 2nd Nat. Centre Syst. Rel. Conf., Birmingham, 1979, paper 103. 126. SPINDEL,J. E. and HAIBAeH, E. The method of maximum likelihood applied to the statistical analysis of fatigue data including run-outs, Int. J. Fatigue, 1(2) (1979), pp. 81-8. 127. STAGG,A. M. Parameter estimation for the lognormal parent population of fatigue failures from a sample containing both failed and non-failed members, Royal Aircraft Establishment Report 70145, Structures YSE/B/0275, 1970. 128. TALR~A, R. Fatigue reliability under multiple-amplitude loads, Engng. Fract. Mech., 11 (1979), pp. 83949.

Pressure vessel applications 129. O'NEIL, R. Safety and reliability criteria, Proc. IAEA Int. Symp., Vienna, 1977, paper IAEA-SM-218/i, pp. 3-20. 130. FREER, L. S. A guide to TIRION 4--a computer code for calculating the consequences of releasing radioactive material to the atmosphere, UKAEA Report SRD R 120, HMSO, 1978. 131. VESELY,W. E., LYNy, E. K. and GOLDBERG, F. F. OCTAVIA--a computer Code to calculate the probability of pressure vessel failure from pressure-transient occurrences, Proc. IAEA Int. Symp., Vienna, Vol. 1, 1978. 132. DUFI~SNE, M. J., LANORE,J. M. et al. PWR reactor pressure vessel failure probabilities, Proc. 4th Int. Conf. Pressure Vessel Technology, I. Mech. E., 1980, paper C13/80, pp. 37-44. 133. DRUCE,S. G. and EowAgos, B. C. Development of PWR pressure vessel steels, Nucl. Energy, 19(5) (1980), pp. 347-60. 134. CULLEN, W. H. and TORRONEN, K. A review of fatigue crack growth of pressure vessel and piping steels in high-temperature, pressurised reactor-grade water, Naval Research Laboratory Report 4298. NUREG/CR-1576, Washington, 1980. 135. CULLEIq,W. H., LOSS, F. J. et al. Influence of critical variables on environmentally-assisted crack growth rates of PVP materials in PWR coolant environments, Proc. 8th Water Reactor Safety Research Information Meeting, Gaithersburg, Maryland, 1980. 136. YAPP, D. Optimisation and reliability in welding of nuclear plants, Proc. Brit. Nucl. Energy Soc. Conf. Welding and Fabrication in the Nuclear Industry, London, 1979, paper 36, pp. 275-81. 137. SCHMITT,W. and WELLEIN,R. Methods to determine the influence of quality assurance on the reliabihty of primary components of a PWR, Proc. 5th Int. Conf. SMiRT, Berlin, 1979, paper M8/6. 138. BUSH, S. H. Reliability of piping in light-water reactors, Proc. 1AEA Int. Symp., Vienna, 1977, paper IAEA-SM-218/! 1. 139. BEYERS,C. J. E. and MARRIOTT,D. L. Quantitative analysis of human error in the production of a pressure vessel, Proc. 4th Int. Conf. Pressure Vessel Technology, I. Mech. E., 1980. 140. JOURIS, G. M. and SHAFFEg, D. H. The use of vessel specific data in estimating brittle fracture probabilities, Proc. 4th Int. Conf. SMiRT, San Francisco, 1977, paper G6/3. 141. HARRIS, D. O. The influence of crack growth kinetics and inspection on the integrity of sensitised BWR piping welds, EPRI Report NP-1163, RP 1325-2, Palo Alto, California, 1979. 142. SOLOMON,K. A., OKRENT, D. and EAS~NBERG, W. E. Pressure vessel integrity and weld inspection procedure, Nucl. Engng. Design, 35 (1975), pp. 87-153. 143. HARROP, L. P. and LIDIARD, A. B. The probability of failure of PWR pressure vessels, evaluated using elastic-plastic failure criteria, Proc. Specialist Meeting, Elastoplastic Fracture Mechanics, OECD, Daresbury, 1978.

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A d d i t i o n a l reading 144. ADAMS,N. J. and MACLEOO,J. S. Criteria relevant to the determination of reliability, Proc. IAEA Int. Syrup., Vienna, 1977, paper IAEA-SM-218/I. 145. APOSTOLAKlS,G. Methodology and application of probabilistic evaluation to thermal reactor safety, EPRI Report NP-1443, RP 297-1, 1980. 146. BASU,S. and ZEMDEGS,R. Method of reliability analysis of control systems for nuclear power plants, Microelectronics and Reliability, 17 (1978), pp. 105-16. 147. FUKUDA,S. An application of graph theory to the safety and reliability analysis of a pressure vessel, Proe. 4th Int. Conf. Pressure Vessel Technology, London, 1980. 148. HASHM~, M. F. Reliability of large equipment and systems of nuclear power plants. Microelectronics and Reliability, 17 (1978), pp. 99-104. 149. HAUETER,R. L. Current programs on plant availability, Proc. ASME Conf. Energy Technology, Houston, 1977. 150. HILL,R. B. and Fox, K. Parametric study and sensitivity analysis of variables affecting probabilistic reliability of a pressure vessel. Babcock and Wilcox (Operations) Ltd. Report NEDR/06/0070, ca 1977. 151. HOLT,A. B. The probability of catastrophic failure of reactor primary system components, Nucl. Engng. Design, 28(2) (1974), pp. 239-51. 152. LUCIA,A. C., ELBAZ,J. and BRUYr~HUnER,R. COVASTOL : A computer Code for the estimation of pressure vessel failure probability, Proc. 5th Int. Conf. SMiRT, Berlin, 1979, paper M8/5. 153. MARRIOTT, D. L. and HUDSOy, J. M. Prediction of failure risk in a pressure vessel due to a manufacturing defect, Proc. 1AEA Int. Syrup., Vienna, 1977, paper 1AEA-SM-218/33. 154. RAVINDRA,M. K. Probability-based design criteria for nuclear structures, Nucl. Engng. Design, 59 (1980), pp. 197-204. 155. REINHARr,E. R. EPRI program to improve nuclear code inspection methods. Mater. Eval., 36(6) (1978), pp. 36-42. 156. SCI~M~TT,W., ROHRICH, E. and WELLEIN,R. Application of probabilistic fracture mechanics to the reliability analysis of pressure-bearing reactor components, Proc. 4th Int. Conf. SMiRT, San Francisco, 1977, paper G6/2. 157. SHAFFER,D. H., BAMFORD,W. H. and JouRIs, G. M. Probabilistic assessment of flaw evaluation procedures for pressure vessel integrity, Proc. 4th Int. Conf. Pressure Vessel Technology, London, 1980, paper C14/80.

O f f s h o r e applications 158. FL~NT,A. R. The treatment of safety for marine structures design codes, Proc. Int. Symp. Advances in Marine Technology, ca 1979, pp. 671-90. 159. WIRSCmNG,P. H. Fatigue reliability in welded joints of off-shore structures, Proc. Conf. Off-Shore Technology, Houston, 1979, paper 3380, pp. 197-206. 160. OcHI, M. K. Probabilistic extreme values and their implication for off-shore structure design, Proc. Conf. Off-Shore Technology, Houston, 1978, paper 3161, pp. 987-92. 161. BA~ER,M. J. and WYATT,T. A. Methods of reliability analysis for jacket platforms, 2nd Int. Conf. Behaviour of Off-Shore Structures, London, 1979, paper 84, 499-520. 162. JENSEN,S. B., VEDELER,B. and WULFF,E. Risk assessment--a study of risk levels within Norwegian off-shore petroleum activities, Royal Norwegian Council for Scientific and Industrial Research Report 26-27/2, 1979.

A d d i t i o n a l reading 163. BOUMA,A. L., MONNIER,TH. and VROUWENVELDER,A. Probabilistic reliability analysis, Proc. 2nd. Int. Conf. Behaviour of Off-shore Structures, London, 1979, paper 85, pp. 521-42. 164. Vos, CIr. J., VAN DER POT, B. J. G. and VRIJL1NG,J. K. Future safety considerations--a matter of scarcity and probabilistic approach, Proe. 2nd Int. Conf. Behaviour of Off-Shore Structures, London, 1979, paper 87, pp. 557 70. 165. WADE,B. G., LANGLEY,E. and Wu, S. C. The effects of current on probabilistic fatigue analysis of fixed off-shore platforms, Proc. lOth Conf. Off-Shore Technology, Houston, 1978, paper 3164, pp. 1011-18.

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M a t h e m a t i c a l techniques 166. HAHN, G. J. Statistical methods for creep, fatigue, and fracture data analysis, J. Engng. Mater. Tech., 101(4) (1979), pp. 344-8. 167. BVERS, J. K. and WEIBE, H. A. Pocket handbook on reliability, USA AVS COM-TR-77-16, 1975. 168. COCnRAN, W. G. and Cox, G. M. Experimental designs, John Wiley and Sons Inc., New York, 1957. 169. SMITh, C. O. Introduction to reliability in design, McGraw-Hill, New York, 1976. 170. WILKS,S. S. Statistical prediction with special reference to the problem of tolerance limits, Ann. Math. Stat., 13 (1942), pp. 400-9. 171. GUMBEL,E. J. Statistical theory of extreme values and some practical applications, Appl. Math. Series, 33 (1954). 172. EPSTEIN, B. Statistical aspects of fracture problems, J. Appl. Phys., 19(1) (1948), pp. 140-7. 173. LIttLE, R. E. Statistical tolerance limits for censored Weibull data, J. Test. Eval., 5(4) (1977), pp. 303-8. 174. HINDS, P. R., NEWrOS, D. W. and J^RDINE, A. K. S. Problems of Weibull parameter estimation from small samples, Proc. 1st Nat. Centre Syst. Rel. Conf., Birmingham, 1977, paper 5/3. 175. MAZUMDAR,M., M A ~ L L , J. A. and CHAr, S. C. Propagation of uncertainties in problems of structural reliability, Nucl. Engng. Design, 50 (1978), pp. 163-7. 176. SPANIER,J. A new multi-stage procedure for systematic variance reduction in Monte Carlo, North American Rockwell Science Center, California, 1971. 177. KENDALL,H. G. and STUART,A. The advanced theory of statistics, Vol. 2, Griffin and Co. Ltd., 1961, Section 30.49. 178. STEPHErqS,M. A. Tests for normality, Stanford University Department of Statistics Report 152, 1969. 179. LILLIEFORS, H. W. On the Kolmogorov-Smirnov test for normality with mean and variance unknown. J. Am. Statist. Ass., 62 (1967), pp. 399-402. 180. ATKINSON,A. C. A method for discriminating between models, J. Roy. Statist. Soc. (B), 32 (1970), pp. 323 53.

A d d i t i o n a l reading 181. BATDORr,S. B. and CROSE,J. G. A statistical theory for the fracture of brittle structures subjected to nonuniform polyaxial stresses, J. Appl. Mech., (June) (1974), pp. 459-64. 182. CLARK,R. M. Non-parametric estimation of a smooth regression function, J. Roy. Statist. Assoc. (B), 39(1) (1977), pp. 107-13. 183. DURBIN,J. Kolmogoro~Smirnov tests when parameters are estimated with applications to tests of exponentiality and tests on spacings, Biometrika, 1 (1975), pp. 5-22. 184. EpsTEIrq,B. Application of the theory of extreme values in fracture problems. J. Am. Statist. Ass., 43 (1948), pp. 403-12. 185. HALD,A. Statistical theory with engineering applications, Wiley and Sons Inc., New York, 1952. 186. HANNUS, M. Numerical analysis of structural reliability, Technical Research Centre of Finland Report, Helsinki, 1973. 187. HASOFER,A. M. Second moment reliability of engineering structures. Proc. Conf. Applications of Probability Theory to Structural Design, Melbourne, Australia, 1974, pp. 9-16. 188. KASHVAP,R. L. A Bayesian comparison of different classes of dynamic models using empirical data, Trans. Auto. Control. AC-22(5)(1977), pp. 715-27. 189. KELLER,A. Z. and KAMATH,A. R. R. The bimodal Weibull distribution and its application to failure analysis, Proc. 14th Conf. Statistics, Computer Science and Operations Research, Egypt, 1979. 190. LITTLE,R. E. Statistical tolerance limits for censored lognormal data, J. Test. Eval., 8(2) (1980), pp. 80-4. 191. M~STRAN,D. V. and SIrqGPUaWALLA,N. D. A Bayesian estimation of the reliability of coherent structures, Ops. Res., 26(4)(1978), pp. 663-72. 192. MEIER,P. Estimation of a distribution function from incomplete observations. In: Perspectives in

probability and statistics--papers in honour of M. S. Bartlett on the occasion of his 65th birthday, J. Gani (ed.), Applied Prob. Trust Acad. Press.

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193. PALOHEIMO,E. and HANNUS,M. Structural design based on weighted factiles, J. Struct. Div. Am. Soc. Cir. Engnrs., 100(ST 7) (1974), pp. 1367 78. 194. PAYNE,A. O. and GRAHAM,A. D. Reliability analysis for optimum design, Engng. Fract. Mech., 12(3) (1979), pp. 329-46. 195. REILLY, P. M. The numerical computation of posterior distributions in Bayesian statistical inference, Appl. Stat., 25(3) (1976), pp. 201-9.