Direct analysis method for probabilistic fracture mechanics

A new method for solving problems of probabilistic fracture mechanics (PFM) is proposed. A process of crack growth is reduced into an iterative integration equation with respect to the probabilistic distribution functions of crack geometry using approximate independence, which we have introduced. The integration equation which has a form of Stieltjes integral is solved by a numerical method. Some PFM problems are solved using the present method, and the results are compared with those by the MC method. Failure probabilities obtained from both calculations agree well. Execution time of the present method is shown to be remarkably short.     

Probabilistic fracture mechanics analyses of nuclear pressure vessels under PTS events

This paper describes some recent research activities on probabilistic fracture mechanics (PFM) for nuclear reactor pressure vessels (RPVs) performed by the RC111 research committee of the Japan Society of Mechanical Engineers (JSME) under a subcontract of the Japan Atomic Energy Research Institute (JAERI). To establish standard procedures for evaluating failure probabilities of nuclear RPVs, we have performed PFM analyses for aged RPV under pressurized thermal shock (PTS) events. The basic problems are chosen from some of US benchmark problems such as EPRI (Electric Power Research Institute) and US NRC (Nuclear Regulatory Commission) joint PTS benchmark problems and H.B. Robinson problems. Employed in this study are four PFM computer codes developed in Japan and in USA. Various sensitivity analyses are performed to quantitatively evaluate the influences of the input data, i.e. (a) initial crack shape, (b) the probabilistic distribution of initial crack depth, (c) cladding, (d) RT_NDT shift, (e) impurity content, (f) the through-wall distributions of material properties, (g) pre-service inspection (PSI) and (h) warm prestressing. It is clearly shown that in most cases, these data affect failure probabilities significantly. Therefore, we should use in the PFM analyses as reliable input data as possible. If any reliable data are not available, the data resulting in most conservative results could be chosen, referring the analysis results presented in this paper.     

Probabilistic fracture mechanics analysis of nuclear structural components: a review of recent Japanese activities

This paper describes a review of recent Japanese activities on probabilistic fracture mechanics (PFM) analyses. Japan Atomic Energy Research Institute (JAERI) has sponsored research committees on PFM organized by Japan Society of Mechanical Engineers (JSME) and Japan Welding Engineering Society (JWES) for more than 10 years. The purpose of the continuous activity is to establish standard procedures for evaluating failure probabilities of Japanese nuclear structural components such as PV&P and steam generator tube, combining the state-of-the-art knowledge on structural integrity of nuclear structural components and modern computer technology such as parallel processing. This paper shows two topics of the newest results of JWES committee, PFM analysis of aged reactor pressure vessel considering embedded cracks and PFM analysis of piping considering seismic loading, and one topic by JAERI itself, development of PTS analysis code for transient loading (PASCAL).     

Probabilistic fracture mechanics

This paper develops the methodology for probabilistic fracture mechanics analysis (PFM) of structural components with crack-like imperfections. Details are given for the development and application of both a simple nomographic method and a basic numerical tool for PFM applications. The tool is a computer program that uses Monte-Carlo simulation to predict the probability distribution of a structural performance parameter from known distributions of input parameters used to model the problem. The structural performance parameter might be the strength margin (strength minus stress), the life ratio (actual fatigue life divided by design fatigue life), or any other relevant model of the failure modes. Two illustrative applications based on linear elastic fracture mechanics are included to demonstrate the utility of PFM to problems of interest to the electric power generation industry.The first example selects the mean yield strength of an alloy in order to minimize the probability of failure for a hypothetical component with two failure modes, yielding and brittle fracture. The example shows that no single value of mean yield stress or of yield-related safety factor, such as specified as part of conventional engineering practice, suffices to minimize failure for all combinations of working stress and flaw size distribution. PFM analysis is required to compute the optimum value of mean yield stress for a given working stress and material quality (flaw size distribution). A second example is presented for which the residual life of a turbine rotor is assumed to be related to three parameters. The parameters are applied stress, material crack growth rate, and initial flaw size. Known variations of the input parameters are translated into variations in residual life. The residual life distribution is required to formulate improved fatigue design criteria. The effects on the turbine life distribution of mutual interdependence of the input random variables and of finite crack initiation life are examined. The second example points out the need for and current unavailability of required input data. It is recommended that data collection efforts be increased to quantify the variational characteristics of the required input parameters, as well as mean, typical, or worst-case values.     

Treatment of the thermal–hydraulic uncertainties in the pressurized thermal shock analysis

The pressurized thermal shock (PTS) analysis is a quantitative analysis to calculate the vessel failure probability of the embrittled reactor pressure vessel. The PTS analysis consists of three major parts, such as the probabilistic safety analysis (PSA), the thermal–hydraulic analysis (T/H), and the probabilistic fracture mechanics (PFM) analysis. Because each analysis involves many parameters and assumptions associated with the uncertainties, it is important to identify and incorporate them into the analysis. Though the PSA and PFM analysis can be easily treated statistically, the thermal–hydraulic analysis results are very difficult to be treated statistically. Instead, sensitivity analyses of the thermal–hydraulic inputs were performed to understand the significance of the variation in the thermal–hydraulic inputs to the PFM analysis. In this study, the existing PFM code was modified to incorporate the uncertainties in the thermal–hydraulic inputs for the PFM analysis. The effects of the uncertainties in the thermal–hydraulic inputs for the vessel failure probabilities were evaluated using the modified code. The results showed the effects of uncertainties in the thermal–hydraulic inputs on the vessel failure probabilities are not significant for the ranges of the transient types. Even for the larger uncertainties, the effects on the vessel failure probabilities are small. Also, the effects of the thermal–hydraulic uncertainties vary depending on the transient characteristics such that the effects are greatest for the pressure dominant transient. Within the transient, the relative increases in the failure probabilities are greatest for the circumferentially oriented semi-elliptical flaws. It was found that the results of the sensitivity analysis using one standard deviation are conservative enough to bound the analysis results considering the uncertainties in the thermal–hydraulic inputs.     

Critical crack depth diagram of reactor vessel for pressurized thermal shock

During the operation of a pressurized water reactor, a certain type of transients could induce rapid cooldown of the reactor pressure vessel (RPV) with relatively high or increasing system pressure. This induces a high tensile stress at the inner surface of the RPV, which is called the pressurized thermal shock (PTS). The structural integrity of the RPV during PTS should be evaluated assuming the existence of a flaw at the vessel. For the quantitative evaluation of the vessel failure risk associated with PTS, the probabilistic fracture mechanics (PFM) analysis technique has been widely used. But along with PFM analysis, deterministic analysis is also required to determine the critical time interval in the transient during which mitigating action can be effective. In this study, therefore, the procedure for the deterministic fracture mechanics analysis of RPV during PTS is investigated using the critical crack depth diagram and the computer program to generate it is developed. Four transients of typical PTS, steam generator tube rupture, small break loss of coolant accident and steam line break are analyzed, and their response characteristics such as critical crack depth and critical time interval from the initiation of the transient are investigated.     

Critical parameter study of PWR primary coolant pipe leak failure using probabilistic fracture mechanics

This paper discusses the critical parameters which influence the failure probabilities of a PWR primary coolant loop. Probabilistic fracture mechanics (PFM) is applied for the parametric study, using the Monte Carlo program PRAISE to predict the failure probabilities of a PWR primary coolant loop from various distributions of input parameters. Parameters such as nondetection probability of preservice and inservice inspection, vibratory stress, residual stresses, and their correlations are extensively studied. Critical crack depth which causes immediate failure are calculated in the presence of various vibratory stresses with and without residual stresses. Crack growth schemes are determined with various initial defect depth and depth-length ratio as a function of plant operation time. The results show quantitatively how PWR primary coolant loop reliability can be greatly improved by increasing the sensitivity and decreasing the uncertainty of preservice nondestructive inspection.     

Failure probability assessment of wall-thinned nuclear pipes using probabilistic fracture mechanics

The integrity of nuclear piping system has to be maintained during operation. In order to maintain the integrity, reliable assessment procedures including fracture mechanics analysis, etc., are required. Up to now, this has been performed using conventional deterministic approaches even though there are many uncertainties to hinder a rational evaluation. In this respect, probabilistic approaches are considered as an appropriate method for piping system evaluation. The objectives of this paper are to estimate the failure probabilities of wall-thinned pipes in nuclear secondary systems and to propose limited operating conditions under different types of loadings. To do this, a probabilistic assessment program using reliability index and simulation techniques was developed and applied to evaluate failure probabilities of wall-thinned pipes subjected to internal pressure, bending moment and combined loading of them. The sensitivity analysis results as well as prototypal integrity assessment results showed a promising applicability of the probabilistic assessment program, necessity of practical evaluation reflecting combined loading condition and operation considering limited condition.     

Probabilistic fracture mechanics of nuclear structural components: consideration of transition from embedded crack to surface crack

This paper describes a probabilistic fracture mechanics (PFM) analysis of aged nuclear reactor pressure vessel (RPV) material. New interpolation formulas of three-dimensional stress intensity factors are presented for both embedded elliptical surface cracks and semi-elliptical surface cracks. To investigate effects of transition from embedded crack to surface crack in PFM analyses, one of the PFM round-robin problems set by JSME-RC111 committee (i.e. aged RPV under normal and upset operating conditions) is solved, employing the interpolation formulas.     

Random thermal fatigue in fast breeder reactor: a narrow-band spectrum

Thermal fatigue crack growth in a fast breeder reactor is theoretically investigated with the aid of probabilistic fracture mechanics (PFM) under the conditions that (i) the temperature variation is a narrow-band stationary process and (ii) the crack grows owing only to the peak stress variation. First, a statistical property of residual life of the component with single crack is derived in an analytical form with the aid of an extended Markov approximation method, which is an efficient mathematical technique in PFM. Next, discussion is carried out on the generalization of the primitive model to the case with plural cracks, where a stress relaxation factor is introduced to express a stress intensity factor of each crack. Finally, a numerical example is shown to examine the quantitative behavior of the component's residual life, and sensitivity analysis is performed with respect to some model parameters.     

Case studies of risk-informed inservice inspection of nuclear piping systems

In the present paper, a probabilistic failure analysis is used to find failure probabilities of piping segments, and a probabilistic risk assessment model is employed to obtain risks to a nuclear power plant should these failures occur. The multiplication of the piping failure probability and the consequence for that particular failure results in the risk contribution of the pipe. The degrees of risk for different piping segments can then be ranked, and their results can be used as the basis for planning a risk-informed inservice inspection program. Numerical studies are offered with special emphases on: (1) the status and experience with RI-ISI applications in Taiwan; (2) the comparison of risk-rankings performed with three different methods developed in the US; (3) aspects of the probabilistic fracture mechanics calculation including the flaw size distributions and stress corrosion cracking model. The results indicate the proposed method can indeed be adopted for planning a cost effective inservice inspection program.     

A comparison of the praise code and the paris code for the evaluation of the failure probability of crack-containing components

Two computer codes developed for the calculation of failure probabilities of crack-containing structures are compared with each other. The basic fracture mechanical, statistical, and numerical models used in the codes are described with special emphasis on probabilistic leak-before-break analysis. Sample problems taken from nuclear applications show that very small failure probabilities can be calculated with sufficient numerical accuracy.     

Probabilistic fracture assessment of piping systems based on FITNET FFS procedure

A probabilistic method based on the fracture module of the FITNET FFS procedure is developed to perform the structural integrity analysis for piping systems. Monte Carlo simulation is used to calculate the failure probability of the whole piping system, as well as that of separate defects by considering the random variables in the method. Both the sensitivity of uncertainties of variables and the model sensitivity are analyzed to identify the most important parameters that affect the failure probability of piping systems, thereby providing an insight into the countermeasures against the failure risk. The results show that the outer diameter of the pipe has the strongest influence on the failure probability of a piping system having a circumferential crack of 0.757 rad, followed by the bending moment, the piping wall thickness, the fracture toughness, the crack angle, the axial force, the ultimate tensile strength and the yield stress.     

Research and development related to PFM for aged nuclear components

At the Japan Atomic Energy Research Institute (JAERI), research activities related to probabilistic fracture mechanics (PFM) have been conducted as a part of the research program on aging and structural integrity of LWR components. This paper describes the outline of two activities related to PFM, i.e. the development of a PFM code and a contract research on ‘Application of PFM Methodology to Reliability Assessment of Nuclear Components’ implemented by the Japan Welding Engineering Society (JWES). In the former research, a new PFM code PASCAL (PFM Analysis of Structural Components in Aging LWR) was developed. This code has some new functions in models of semi-elliptical crack extension, elastic–plastic fracture analysis based on R6 method and options for the evaluation of overlay cladding and warm pre-stress (WPS) effect. Besides, the code has the function to evaluate the effect of irradiation embrittlement recovery by thermal annealing of a reactor pressure vessel and re-irradiation embrittlement. Based on the analyses on benchmark problem conducted by USNRC/EPRI, performance and functions introduced in the code were examined. Some case studies were also carried out to investigate the influence of various parameters. On the other hand, JAERI has been sponsoring the PFM related activities in relation to the structural integrity of LWR components. These activities have been conducted at JSME and JWES. The objective of this activity has been to provide for the future need of PFM methodology.     

Small punch testing for assessing the fracture properties of the reactor vessel steel with different thicknesses

Small punch (SP) tests have been performed at room temperature on round specimens with six kinds of thicknesses. Experimental results show that the SP energy, the fracture strain and the fracture toughness increase with the thickness of the SP samples. Fracture surface displays typical ductile fracture, and the outer surface of the hemispherical bulk part punched off is full of little bowings around which there are many microcracks caused by the stretch stress under biaxial strain/stress state. From the view point of the energy dissipation, the fundamental theoretical model of fracture toughness is proposed, according to which the ductile fracture toughness is obtained from the SP energy, the plastic deformation and the experimental fractography. The fracture strain and the fracture energy density criteria are introduced into the local failure model. With the continuum damage model, the FEA simulations provide the results of the crack propagation process which agree with the experimental results and can verify the fracture model.     

Time dependent ductile crack extension in A533B class I steel

Four wide plate specimens manufactured in A533B Class I, 90 mm thick by 500 mm wide containing through-thickness or semi-elliptical surface fatigue cracks were tested at +70°C. These specimens were subjected to a series of increasing applied loads, each of 100 h duration, until failure. Testing was performed using a computer interactive 40 MN load controlled tensile testing rig. Values of the fracture toughness parameters J and crack tip opening displacement (CTOD) were derived from the recorded values of applied load, plate extension and crack mouth opening displacement.The influence of loading rate, degree of yield containment and crack orientation on the time dependent behaviour is assessed and compared with data obtained from wide plate and bend tests under monotonic loading and from bend tests conducted with a variable loading rate, with hold periods, under crack mouth opening control. Interpretation of the results provides a clearer understanding of low temperature time dependent ductile crack extension and enables the identification of the conditions under which this phenomenon is apparent, to allow the necessary adjustments to failure assessments.     

Probability of crack-induced failure in BWR recirculation piping

The Lawrence Livermore National Laboratory (LLNL) has estimated the probability of double-ended guillotine break (DEGB) in the reactor coolant piping of Mark I boiling water reactor (BWR) plants. Two causes of pipe break are considered: crack growth at welded joints and the seismically-induced failure of component supports. For the former a probabilistic fracture mechanics model is used, for the latter a probabilistic support reliability model. This paper describes a probabilistic model developed to account for effects of intergranular stress corrosion cracking (IGSCC). The IGSCC model, based on experimental and field data compiled from several sources, correlates times to crack initiation and crack growth rates for Types 304 and 316NG stainless steel against material-specific ‘damage parameters’ which consilidate the separate effects of coolant environment (temperature, dissolved oxygen content, level of impurities), stress (including residual stress), and degree of sensitization. Application of this model to actual BWR recirculation piping shows that IGSCC clearly dominates the probability of failure in 304SS piping, mainly due to cracks that initiate within a few years after plant operation has begun. Replacing Type 304 piping with 316NG reduces failure probabilities by several orders of magnitude.     

Application of first and second order reliability methods in the safety assessment of cracked steam generator tubing

The First- and Second Order Reliability Methods (FORM and SORM) have been applied in the safety assessment of steam generator tubes with through-wall axial stress corrosion cracks. The underlying probabilistic fracture mechanics model takes into account the scatter in tube geometry, material properties and stable crack propagation. Also, the effect of the maintenance strategy has been considered. A realistic numerical example has been given to compare the failure probabilities calculated by FORM and SORM to those obtained by different versions of Monte Carlo simulations. The relative errors of the numerical methods employed have been analysed, which has shown that FORM performs in an acceptable and SORM in an excellent manner. Some changes in failure surface properties, caused by different maintenance strategies, are indicated and a sensitivity analysis of influencing parameters is made. The results obtained demonstrate the applicability of FORM and SORM in the safety assessment of stress corrosion cracked steam generator tubing.     

Investigation of the ductile fracture properties of type 304 stainless steel plate, welds, and 4-inch pipe

J-integral fracture toughness tests were performed on welded 304 stainless steel 2-inch plate and 4-inch diameter pipe. The 2-inch plate was welded using a hot-wire automatic gas tungsten arc process. This weldment was machined into 1T and 2T compact specimens for single specimen unloading compliance J-integral tests. The specimens were cut to measure the fracure toughness of the base metal, weld metal and the heat affected zone (HAZ). The tests were performed at 550°F, 300°F and room temperature. The results of the J-integral tests indicate that the J_Ic of the base plate ranged from 4400 to 6100 in lbs/in² at 550°F. The J_Ic values for the tests performed at 300°F and room temperature were beyond the measurement capacity of the specimens and appear to indicate that J_Ic was greater than 8000 in lb/in². The J-integral tests performed on the weld metal specimens indicate that the J_Ic values ranged from 930 to 2150 in lbs/in² at 550°F. The J_Ic values of the weld metal specimens tested at 300°F and room temperature were 2300 and 3000 in lbs/in² respectively. One HAZ specimen was tested at 550°F and found to have a J_Ic value of 2980 in lbs/in² which indicates that the HAZ is an average of the base metal and weld metal thoughness. These test results indicate that there is a significant reduction in the initiation fracture toughness as a result of welding.The second phase of this task dealt with the fracture toughness testing of 4-inch diameter 304 stainless steel pipes containing a gas tungsten arc weld. The pipes were tested at 550°F in four point bending. Three tests were performed, two with a through wall flaw growing circumferentially and the third pipe had a part through radial flaw in combination with the circumferential flaw. These tests were performed using unloading compliance and d.c. potential drop crack length estimate methods. The results of these test indicate that the presence of a complex crack (radial and circumferential) reduces in the initiation toughness and the tearing modulus of the pipe material compared to a pipe with only a circumferentially growing crack.     

Some advances in fracture studies under the heavy-section steel technology program

Recent results are summarized from HSST studies in three major areas that relate to assessing nuclear reactor pressure vessel integrity under pressurized-thermal-shock (PTS) conditions. These areas are irradiation effects on the fracture properties of stainless steel cladding, crack run-arrest behavior under non-isothermal conditions, and fracture behavior of a thick-wall vessel under combined thermal and pressure loadings.Since a layer of tough stainless steel weld overlay cladding on the interior of a pressure vessel could assist in limiting surface crack extension under PTS conditions, its resistance to radiation embrittlement was examined. A stainless steel overlay cladding, applied by a submerged arc, single-wire, oscillating-electrode method, was irradiated to 2 × 10²³ neutrons/m² (> 1 MeV) at 288°C. Yield strength increases up to 27% and a slight increase in ductility were observed. Charpy V-Notch data showed a ductile-to-brittle transition behavior caused by temperature-dependent failure of the 8-ferrite phase. The type 308 cladding, microstructurally typical of that in reactor pressure vessels, showed very little degradation in either upper-shelf energy or transition temperature due to irradiation.Crack-arrest behavior of A533 grade B class 1 steel was examined for temperatures extending above the onset of Charpy upper-shelf. Crack-arrest experiments that use wide-plate specimens have shown crack arrest occurring prior to transition to tearing or tensile instability. High values of crack-arrest toughness have been recorded (static values above 400 MPa that are well above the maximum value that safety assessment criteria assume such materials can exhibit.A validation experiment was performed by exposing an intentionally flawed HSST intermediate test vessel to combined pressure and thermal transients. The experiment addressed warm-prestressing phenomena, crack propagation from brittle to ductile regions, and crack stabilization in ductile regions. Test and analysis results are summarized.