Probabilistic Fracture Mechanics Model Research Articles

Probabilistic fracture analysis for boiling water reactor vessels considering seismic loads during decommissioning transition period

During decommissioning transition period, the hazards from internal pressurization and thermal transients to a reactor vessel can be eliminated, but the effects of seismic loads on vessel integrity may be of concern. In this paper, we employed the Oak Ridge National Laboratory’s (ORNL) FAVOR code to perform probabilistic integrity assessment for the Taiwan domestic boiling water reactor (BWR) vessels of a decommissioning plant considering seismic loads. A manner to transform seismic loads as the equivalent internal pressure for FAVOR analysis was proposed. Two probabilistic fracture mechanics (PFM) models, the conservative model that U.S. NRC has used for BWR regulation, and the realistic model reflecting the entire vessel beltline configuration, were analyzed and compared. It is found that the circumferential welds dominate the fracture risk but still can be negligible. Present results could be the reference for weld inspection relief application of the RPV during the decommissioning transition period in Taiwan.

Effect of environmental modelling and inspection strategy on the optimal design of floating wind turbines

In order to reduce design conservatism and consequently the cost of energy, appropriate and cost-optimal safety factors should be derived, in light of environmental load uncertainties and lifetime costs. In the present work, a linearized dynamic model has been used together with Monte Carlo simulations and a numerical design optimization procedure to evaluate the impact of the description of wind and wave loads on the fatigue reliability and optimal design of a 10 MW spar floating wind turbine. Trade-offs between design costs and inspection costs with different design fatigue factors (DFFs) have also been assessed. The analyses have been performed for a realistic wind park site, where an environmental model has been developed based on hindcast data. Considering stochastic turbulence intensity, wind-wave misalignment, wind directional distribution, and a two-peak wave spectrum reduced the long-term fatigue damage by approximately two-thirds along the fatigue-critical part of the support structure compared to the base model. Re-designing the tower and platform with the full environmental model resulted in 11% reduction in CAPEX. However, due to the applied design optimization procedure, consistent reliability levels were achieved along the tower length, which resulted in important system side effects for the total structural reliability. Trade-offs between CAPEX and OPEX were derived based on a probabilistic fracture mechanics model and reliability updating through inspections. The necessary inspection intervals to achieve the same accumulated reliability after 20 years of operation were identified with different DFFs, and cost-optimal safety factors were computed with different OPEX costs and interest rates.

Probabilistic study on the macro-crack initiation of the rib-to-deck welded joint on orthotropic steel deck

The inherent randomness of the fatigue issues on the orthotropic steel deck (OSD) requires a probabilistic investigation. With respect to the unfavorable crack on the rib-to-deck welded joint of OSD, the macro-crack initiation, namely the process of the crack growth until it is detected, is studied using the probabilistic fracture mechanics (PFM). The PFM model is established based on the two-stage crack growth model and validated by test results. It is applied in the Monte-Carlo simulation (MCS) to reproduce the crack growth under the realistic traffic load obtained by the WIM data on an OSD for one year, suggesting that a lognormal function can be used in fitting the distribution of the life till the crack grows to half of the deck plate thickness. Based on the MCS results, further analysis is carried out on the macro-crack initiation by considering the probability of detection (PoD) of two different ways of non-destructive evaluation (NDE) method. Due to the lack of ability in detecting tiny cracks, it is stated that the endeavors in finding cracks in the early years could be inefficient, especially for the first 10 years. Fortunately, it is highly possible to detect the crack when it grows to a larger size by applying multiple times of inspections. With consideration of the repairs works for detected cracks, the probability that the crack grows to the critical size, e.g. half of the deck plate thickness, can be calculated. Accordingly, the inspections for the fatigue-prone sites can be optimized in a probabilistic way for the maintenance and management of the bridge.

Effect of Corrosive Environment Conditions on Austenitic Structure Reliability

In this paper, the work is based on the application of probabilistic fracture mechanics models (PFM) to predict the reliability of nuclear reactors pipes under pressure. Cracking simulation of a stainless steel piping under the conditions of intergranular stress corrosion cracking (IG-SCC) is based on the general methodology recommended in the modified software M-PRAISE. IG-SCC is characterized by a unique damage parameter depending on residual stresses, environmental conditions, and sensitization degree. This parameter can be used to evaluate the structural reliability and identify the majority of efficient approaches to improve the piping reliability: effect of a corrosive medium on the reliability, which is analyzed in this present work.

Environmental effects on the reliability of an AISI 304 structure

This work is based on the application of probabilistic fracture mechanics (MRP) models to predict the reliability of pressure pipelines in nuclear power reactors. The simulation of cracking of stainless steel piping under the conditions of intergranular stress corrosion cracking (IGSCC) is based on the improved modified piping reliability analysis including seismic events (PRAISE) code. The IGSCC is characterized by a unique damage parameter that depends on residual stresses and environmental conditions. Case studies of IGSCC failures in nuclear power plants are presented and the contribution of environmental effects on crack initiation and leakage is discussed. For small damages we observed that the change in temperature or oxygen concentration does not affect the initiation process but their decrease contribute favourably to the decrease in the leakage probabilities.

The Combination of Probabilistic Method and Fracture Mechanics Model in Multiple Cracks Analysis

Probabilistic method and fracture mechanics model were combined to analysis the failure probability of structure with multiple site cracks. Comparing the result with probabilistic model showed that, accounting the actual failure mode and the interaction of cracks, combination of probabilistic method and fracture mechanics model get better approaching to determine failure probability during structural integrity analysis.

Probabilistic fracture mechanics of irradiation assisted stress corrosion cracking in stainless steels

A probabilistic fracture mechanics model is applied to irradiation assisted stress corrosion cracking effect, assuming that the oxidized part of stainless steel sample plays an essential role in the crack propagation and sample failure. The Weibull statistical distribution of time-to-failures, estimated through the correlation with the statistical distribution of oxide strengths, fully accounts for the experimental results/scatter obtained in the constant load time-to-failure tests. Large failure uncertainties in these type of tests originate from intrinsic stochastic behavior of the oxide cracking due to subcritical crack propagation process.

Probabilistic fracture mechanics analysis of spalling during edge indentation in ice

The design of marine structures for ice environments requires ice load estimates, which are greatly influenced by fracture and associated scale effects. During edge indentation of natural ice, fracture processes are highly influenced by variability associated with flaw structure and contact conditions. A probabilistic framework is the most appropriate for modeling such interactions. In the present analysis, a probabilistic fracture mechanics model has been developed to model localized fracture events for ice specimens with ice edge taper angles ranging from 0° to 45°. Crack instability criteria have been formulated for compressive and tensile states of stress. This approach varies from traditional probabilistic failure models which assume that zones of compressive stress have zero failure probability and include only tensile zones in the calculation of failure probability. For compressive edge loading, experimental results indicate that local edge fractures (spalls) are most likely to emanate from regions of the ice where cracks would be subjected to axial compression and lateral tension. Compressive contact loads required to propagate fractures from zones of bi-axial tension or bi-axial compression were found to be considerably higher and are much less probable. Estimates of the mean pressure required to trigger a fracture event as a function of ice thickness were generated for thicknesses in the range of 0.2–2.0m. These results demonstrate that observed scale effects in ice pressure data can be explained by probabilistic aspects of fracture. Simulations of fracture probabilities as a function of loading eccentricity confirmed that extreme loads of interest in engineering design are most likely to correspond to high pressure zones located near the center of thickness of the ice sheet. Results indicate that fractures occur more readily for flat ice edges and higher pressures are required to propagate fractures for edges with more highly tapered edges. This result provides a theoretical basis to explore the interplay between fracture and crushing, both of which occur during compressive ice failure: fractures create tapered edges that localize contact in manner that supports crushing; crushing, over several cycles, flattens the ice edge in a manner that in turn promotes fracture. While the application presented here is focused on ice mechanics, the analysis methodology could readily be applied to other brittle materials.

A study on the effect of flaw detection probability assumptions on risk reduction achieved by non-destructive inspection

Leakages and ruptures of piping components lead to reduction or loss of the pressure retaining capability of the system, and thus contribute to the overall risk associated with nuclear power plants. In-service inspection (ISI) aims at verifying that defects are not present in components of the pressure boundary or, if defects are present, ensuring that these are detected before they affect the safe operation of the plant. Reliability estimates of piping are needed e.g., in probabilistic safety assessment (PSA) studies, risk-informed ISI (RI-ISI) applications, and other structural reliability assessments. Probabilistic fracture mechanics models can account for ISI reliability, but a quantitative estimate for the latter is needed. This is normally expressed in terms of probability of detection (POD) curves, which correlate the probability of detecting a flaw with flaw size. A detailed POD curve is often difficult (or practically impossible) to obtain. If sufficient risk reduction can be shown by using simplified (but reasonably conservative) POD estimates, more complex PODs are not needed. This paper summarises the results of a study on the effect of piping inspection reliability assumptions on failure probability using structural reliability models. The main interest was to investigate whether it is justifiable to use a simplified POD curve. Further, the study compared various structural reliability calculation approaches for a set of analysis cases. The results indicate that the use of a simplified POD could be justifiable in RI-ISI applications.

Probabilistic fracture analysis for boiling water reactor pressure vessels subjected to low temperature over-pressure event

This paper performs the probabilistic fracture analyses for boiling water reactor pressure vessels subjected to the low temperature over-pressure event. Using the FAVOR code, which was developed by the Oak Ridge National Laboratory in America, a conservative model with the plant specific data of domestic BWRs in Taiwan to the USNRC requirements is built. A hypothetical transient of low temperature over-pressure event which could severely challenge the integrity of reactor pressure vessel of BWR is assumed as the loading condition for the probabilistic fracture mechanics analyses. The effects of performing inservice inspection on the probability of failure are also simulated. The computed low probability of failure indicates that the analyzed reactor pressure vessel can maintain sufficient reliability even without performing any inservice inspections for the circumferential welds. It also indicates that extensive inservice inspections on shell welds cannot promote the compensating level of safety significantly. Present results can be regarded as the risk incremental factors compared with the safety regulation requirements on reactor pressure vessel degradation. The developed probabilistic fracture mechanics model is helpful for the subsequent BWR regulation in Taiwan.

A reliability assessment method for structural metallic component with inherent flaws based on finite element analysis and probabilistic fracture mechanics model

Reliability assessment is an essential step to promote advanced materials and components into applications. In this paper, a general reliability assessment framework was proposed to predict the lifetime distribution of a structural steel component with inherent flaws. By combining materials information, finite element analysis of the stress field and probabilistic fracture mechanics model, the distribution of failure probability subjected to fatigue loads was predicted. The local failure probability distributions identify the critical regions of the component visually. Both the global failure probability and the local failure probability distribution can be considered as essential and fundamental data in structure design and system maintenance. Focus was placed on the probabilistic fracture mechanics model and fatigue crack growth model.

A probabilistic fracture mechanics model including 3D ductile tearing of bi-axially loaded pipes with surface cracks

This paper presents a probabilistic fracture mechanics model established from three-dimensional FEM analyses of surface cracked pipes subjected to tension load in combination with internal pressure. The models are particularly interesting for offshore pipelines under operational conditions or during laying, where inelastic deformations may occur. In the numerical models, the plastic deformations, including ductile tearing effects, are accounted for by use of the Gurson–Tvergaard–Needleman model. This model is calibrated to represent a typical X65 pipeline steel behaviour under ductile crack growth and collapse. Several parameters are taken into account, such as crack depth, crack length and material hardening. Another important topic is the examination of the influence of bi-axial loading due to internal pressure on capacity. From the results of the deterministic analyses a probabilistic fracture mechanics model is established using the response surface methodology. Two failure criteria are examined to represent the structural capacity. Based on the established model, we illustrate the methodology by examples employing the two different failure criteria solved with first and second order reliability methods.

Role of probabilistic analysis in integrity assessments of reactor pressure vessels exposed to pressurized thermal-shock conditions

In recent years, probabilistic analyses have been playing an increasing role in safety assessments for nuclear power plants (NPPs) in the USA. An important example is the treatment of integrity analyses for the primary reactor pressure vessel (RPV) when it is exposed to pressurized thermal-shock (PTS) events. The U.S. Nuclear Regulatory Commission’s (NRC) approach to ensuring PTS integrity over the RPV lifetime has been to establish limits that ensure that the likelihood of through-wall-crack (TWC) development is kept below a prescribed very low level. The key elements for the probabilistic fracture mechanics (PFM) analysis package include thermal-hydraulic analysis input, neutron exposure models, embrittlement models for RPV steels, representations of fracture properties data, flaw characterization models, PFM models, and computational computer codes. The state of knowledge that existed when the NRC initially formulated the PTS rule in the 1980s necessitated that the rule and its technical bases use conservative treatment of parameters and models. Advancements made in the ensuing 20 years have motivated the NRC to undertake a comprehensive update of these technical bases to support potential revisions to the rule. The current status of the key technical elements has now been established, and an updated PFM methodology is being drafted and used for preliminary analyses of three U.S. PWRs. The frequencies of vessel failure predicted by these analyses have been compared with those calculated under the early PTS methodology and are encouraging that a revision to the PTS rule may be forthcoming. Because the PFM calculations utilize a series of deterministic LEFM fracture analyses, it is important to know the degree to which the LEFM methods are applicable to the analysis of thick-wall vessels. U.S. efforts to establish this applicability have been extensive and included a significant number of fracture experiments using large-scale vessels to validate analysis methods and demonstrate margins within design codes. This paper includes a summary of results from several of those key experiments.

Probabilistic fracture mechanics analysis of nuclear piping considering variation of seismic loading

In conventional probabilistic fracture mechanics (PFM) analyses, seismic loading is considered as a large deterministic value, although there exists the variation of the seismic load as well as response of building and components. On the other hand, such stochastic behaviours have already been taken into account in the field of seismic probabilistic safety assessment. This paper proposes a new PFM model for nuclear piping that takes into account the variation of seismic loading. The distribution in ground acceleration is modelled with the seismic hazard curve. The distribution in piping response during a seismic event is modelled with a log–normal distribution. Since the seismic load has large variation, when not adopting an upper limit to the distribution in seismic stress, the break probability calculated from the present PFM analysis becomes equal to the probability that the seismic stress exceeds the collapse stress of a sound pipe. This implies that the existence of a crack has no effect in these PFM analyses, and this result does not satisfy the purpose of PFM analysis to evaluate the failure probability per crack. Therefore, the seismic stress was limited to the collapse stress of a sound pipe in the present PFM analysis to evaluate the conditional break probability per crack.

Random Outcome and Stochastic Analysis of Some Fatigue Crack Growth Data

ABSTRACTFatigue crack propagation data of a batch of AISI 4340 steel specimens are released in the present paper. The statistical nature of the data is specially emphasized, and a probabilistic fracture mechanics model is introduced to analyze the data. The stochastic differential equation associated with the model is then solved. The solution gives us the crack exceedance probability as well as the probability distribution of the random time to reach a specified crack size. These quantities are useful in the reliability assessment of structures made of the tested material. Comparing the analytical result with the experimental result, it is found that the proposed probabilistic fracture mechanics model can reasonably explain the experimental data. For those data that cannot be fitted well by the proposed model, methods of improvement are proposed in the present paper as well.

Effect of through-wall stress gradients on piping failure probabilities

An approach has been developed that predicts leak and rupture probabilities of reactor piping in a structured parametric format. This approach applies the probabilistic fracture mechanics code pc-PRAISE (Piping Reliability Analysis Including Seismic Events) to address the mechanical and thermal fatigue life of piping. The probabilistic fracture mechanics model is applied to predict the relative effects of uniform stresses and through-thickness stress gradients on the reliability of stainless steel piping welds. Results generated using the numerical technique revealed that the calculated leak probabilities can be sensitive to the different types of stress gradients and to local stress concentrations.

確率論的破壊力学に基づく軽水炉配管のＬＢＢ成立性評価

This paper describes Probabilistic Fracture Mechanics (PFM) analyses for quantitative evaluation of the likelihood of Leak-Before-Break (LBB) of Light Water Reactor's (LWR's) piping. The PFM analyses in general assume probabilistic distributions of initial crack size, applied stress cycles, crack growth laws, fracture criteria, leakage detection capability, defect inspection capability and so on. Referring to the deterministic procedure for LBB evaluation, most appropriate PFM models and data for LBB evaluation are discussed. Here the LBB index is newly proposed in order to quantitatively evaluate the likelihood of LBB. Through intensive sensitivity analyses, it is clarified that the LBB is more likely to occur for larger diameter pipe; the performance of leakage detection significantly affects the LBB likelihood; the LBB likelihood increases with plant's aging even conservatively assuming leak detection capability; the R6 method (Category 1, Option 1) for fracture criterion gives very conservative results; and In-Service Inspection (ISI) reduces the increase rate of failure probability than the failure probability itself.

On the risk-based steam generator lifetime optimisation

A probabilistic fracture mechanics model is employed to estimate the failure probability of axially cracked steam generator tubes. The model estimates the failure probability from the random changes of the influencing parameters such as tube and crack geometry, material properties and non-destructive examination results, reliability and sizing accuracy and stable crack propagation. The performance of the model is illustrated by a numerical example. A steam generator tubing severely affected by the stress corrosion cracking is studied during most unfavourable accidental conditions. Two different plugging approaches are analyzed and the quality is compared, showing the superior performance of crack length oriented approach over tube wall thickness reduction both in terms of SG failure probability and extent of plugging. Thus, apart from setting the acceptable SG failure probability, all elements for the risk-based SG lifetime optimisation are provided on the example of stress corrosion cracking in the tube expansion transition zone.

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.

On the application of probabilistic fracture mechanics to the reliability and inspection of pressure vessels

A probabilistic fracture mechanics model was developed to analyse the failure probability of a typical high power density reactor pressure vessel. The major causes for the nuclear reactor pressure vessel failure include fatigue, corrosion fatigue and brittle failure. All these causes are greatly affected by the stress loading conditions, material properties (aged by neutron damage), and defects embedded in the structure. Both an analytical first-order second-moment approximation and a hybrid methodology were employed in this study. In addition to the static scatter of the pre-existing cracks and material properties, a random walk model based on the operating history was introduced to represent the random occurrence of the abnormal transient stresses. The failure mode is defined as the brittle failure caused by a critical crack, meaning the stress intensity factor around a critical crack exceeding the fracture toughness of the pressure vessel material. Through a sample study on a typical high power density nuclear power plant, it was found that the vessel failure probability is about 4 × 10 −4 at the 40th year of operation and the failure rate is in the order of 5 × 10 −6 per vessel per year, which had reasonable agreement with the value of 10 −4−10 −6 as reported based on real-world statistics. In addition to the failure probability caused by fatigue crack growth, the reliability of a Low Temperature Overpressure incident was also evaluated.