Failure Probability Distribution Research Articles

For How Long Are Newly Chartered Banks Financially Fragile?

We examine the financial performance of 1,664 commercial banks chartered between 1980 and 1985, a period of intense chartering activity just preceding the banking recession of the late-1980s. We compare new banks to a benchmark sample of 2,047 small established banks. Using a split population duration model, we estimate the probability distribution of long-run failure for both sets of banks over a 14 year period, and assess how regulatory, environmental, and bank specific conditions affect that probability distribution. We find that the fragility of a new bank varies over time in a fairly regular 'life cycle' pattern, but that how this basic life cycle pattern is positioned vis a vis the business cycle also matters. On average, new banks are initially less likely to fail than established banks; after about four years they become more likely to fail than stablished banks; and as time passes and new banks mature they fail at rates similar to established banks. But banks chartered just prior to the banking recession failed at the highest rates, and their estimated hazard functions followed an extreme life cycle shape. State laws restricting the acquisition of de novo banks are associated with higher rates of new bank failure, but easy-entry chartering policies are not. We find that de novo failure is more sensitive to capital levels than established bank failure, evidence that Justifies recent increases in minimum capital requirements for de novo banks. Finally, our results suggest that early warning signals may be easier to identify for de novo banks than for established banks, perhaps because banks in the early stages of their life cycles are less heterogeneous and hence simpler to model than mature banks.

Dimensionality and size effects on the strength of fiber-reinforced composites

The tensile failure of fiber-reinforced composites in 2D (1-ply) and thin multi-ply samples has been studied numerically in order to assess the role of dimensionality on the absolute magnitude, reliability, and size scaling of composite strength. Single-ply tapes, often used in model calculations because of the simplifications attending the reduced dimensionality, and 2-ply tapes are shown to have (i) lower strengths than 4-ply specimens, with the latter approaching ‘bulk’ response, and (ii) broader failure probability distributions than 4-ply samples. These two effects together indicate that using a 1-ply or 2-ply geometry to investigate composite strength size effects and reliability, or to compare against real composites, is not quantitatively accurate. Calculations are also performed on the same systems but with fiber ‘pull-out’ loads, i.e. the residual load carrying capability of broken fibers in the composite, essentially eliminated. Such a model approximates the ‘chain of bundles’ models widely developed for prediction of composite failure. In the absence of fiber pull-out, the composite strength is much lower and again more broadly distributed. Overall, the present results suggest that simplified analytic and computational models based on a 2D tape geometry and/or neglecting fiber ‘pull-out’ must be viewed with caution in applications to real composite systems.

On the probabilistic failure assessment diagram

Reliability analysis of structures containing defects carried out directly by practising engineers is quite complex and costly. Therefore, a simple and convenient reliability analysis method should be investigated and established. In this paper, the principle and building procedure of a reliability analysis method are described and the feasibility of the method are also explicitly clarified. Based on the assumptions that: (1) the load is a deterministic variable; (2) the yield strength and fracture toughness satisfy normal distribution and their variational coefficients are constant; (3) the initial crack length remains unchangeable, a set of probabilistic failure assessment diagrams (PFAD) called elementary reliability assessment method are established under the simplified option 1 curve of the 3rd version of CEGB R6. The investigation shows that the distribution of random safety factor and failure probability of a structure only depend on the coordinates of the expectation of P( L r, K r) in the failure assessment diagram of the simplified option 1 curve and is not influenced by the other factors. The investigation further shows that the failure probability will decrease with the increase of the initial crack length if the expectation of P( L r, K r) keeps unchanged. A conservative result will be given when using the PFAD as an elementary reliability analysis method to assess the integrity of a structure and only a simple procedure is required like the deterministic assessment method given in CEGB R6. When the secondary stress exists, the conservativeness of the result assessed will increase with the increase of parameter ρ although the PFAD is only established based on considering primary stress.

Test sequencing problems arising in test planning and design for testability

We consider four test sequencing problems that frequently arise in test planning and design for testability (DFT) processes. Specifically, we consider the following problems: (1) how to determine a test sequence that does not depend on the failure probability distribution; (2) how to determine a test sequence that minimizes expected testing cost while not exceeding a given testing time; (3) how to determine a test sequence that does not utilize more than a given number of tests, while minimizing the average ambiguity group size; and (4) how to determine a test sequence that minimizes the storage cost of tests in the diagnostic strategy. We present various solution approaches to solve the above problems and illustrate the usefulness of the proposed algorithms.

Apparent fracture toughness master curve of a zirconia—alumina composite

Recently, the existence of specimen-size-independent Weibull master curves characterizing strength and failure of macroscopically homogeneous, brittle materials, if the Weibull modulus m > 1, has been demonstrated by the present author. Furthermore, an apparent fracture toughness master curve M(a) has been derived from experimental Weibull master curves, which can even be applied to materials undergoing an amount of stable crack growth prior to failure. The objective of the present paper is to show that another type of specimen-size-independent Weibull master curves K(y,m), which is obtained by scaling the cumulative failure probability distribution function P(sigma) with the physically, highly significant mean stress = integral(0)(1) sigma dP, can be constructed for every Weibull modulus m > 0. Moreover, an apparent fracture toughness master curve T(b), being physically highly significant and evident, can be derived from the Weibull master curves K(y, m) for every m > 0. T(b) provides characteristic magnitudes, which facilitate a thorough comprehension of the fracture toughness of the investigated solids. Therefore the apparent fracture toughness master curve T(b) of an 8 wt% yttria partially stabilized zirconia-20vol.% beta-alumina composite has been evaluated, using data from three-point bend tests performed at room temperature.

ＣＲＴ用ガラスの強度特性の統計的性質

It is well known that the strength characteristics of the glass used for the cathode ray tubes usually has a distinct scatter under the influence of surface flaws. Therefore, it is important to consider the variations in the static and delayed fracture strength of glass for the reliable design of vacuum cathode ray tubes under the atmospheric pressure. In this study, static and delayed fracture strength tests of glass were carried out using polished, abraded and scratched surface specimens in atmosphere in order to investigate the influence of the surface conditions on the strength characteristics statistically. From the goodness of fit test by the chi-square (x2) methed, the validity of the assumed distribution of the cumulative failure probability on the static and delayed fracture strength corresponding to each surface condition are discussed. Moreover, the statistical properties of the relationships between the static and delayed strengths are examined.

Mixture model of the power law

This paper deals with a new degradation model, a power-law mixture-model which extends the power law in accelerated life testing. Some types of insulation deteriorate at a rate determined by the stress. If the stresses are classified into distinct groups, a total degradation model could be a mixture or several degradation models. If the breakdown voltage (V/sub B/) has a certain failure probability distribution and the degradation follows a power law, then a mixture model of the power law can be constructed. The cluster analysis is done by maximum likelihood estimation. For instance, by using the intermittently-inspected V/sub B/ test-data of electrical insulation removed from active power-lines, the remaining life of the insulation can be assessed based on this mixture model. Application of the model to the V/sub B/ data of cross-linked polyethylene insulated cables is demonstrated: the breakdown data are classified into groups using the Akaike Information Criterion and the generalized likelihood ratio test method. The remaining life of the insulation can be estimated with approximate standard deviation, for Japanese 22 kV and 33 kV insulation-class inspected cables, samples are clustered into 4 groups and a positive threshold stress is not found.

Alternative scaling of cumulative Weibull failure probability distribution functions

Reference CRPP-ARTICLE-1997-032doi:10.1023/A:1018539611942View record in Web of Science Record created on 2008-04-16, modified on 2017-05-12

Master curve for brittle cleavage fracture toughness testing

The scatter distribution of the fracture toughness of macroscopically homogeneous, brittle materials is usually described by a cumulative failure probability distribution function, which is dependent on the material, the specimen-size and the testing-temperature. In this work, a master curve is derived which is material, specimen-size and testing-temperature independent, as long as the fracture is of brittle nature. With the help of this master curve statistical, macroscopic homogeneity, respectively quality of any investigated solid undergoing brittle cleavage fracture, can be readily checked by fracture toughness testing.

Earthquake size‐frequency relations from an analytical stochastic rupture model

An analytical stochastic model of earthquake rupture is presented. Although the general formulation makes no a priori self‐similarity assumption, scale independent earthquake size‐frequency (ESF) statistics emerge from the interaction between fault strength heterogeneity and rupture mechanics. The model also gives theoretical derivations for scale dependent ESF statistics, including gradual frequency saturation for small earthquakes, and a continuum of possible transitions in the statistics of large earthquakes. A model fault is characterized as a surface composed of a large number of asperities, defined as small, discrete contact surfaces of finite strength. An earthquake rupture grows via successive asperity‐breaking subevents. The probability of rupture to a certain size earthquake or greater is the total product of incremental rupture growth probabilities. These are governed by a new asperity failure probability distribution, derived as a mixture of Weibull distributions. By including fracture mechanics, the resulting mixture distribution, characterized by the scale and shape parameters Λ and Γ, is written in terms of the rupture size. The important difference between the mixture distribution and the Weibull distribution is that the mixture results in power law ESF statistics. Self‐similarity occurs when material heterogeneity and rupture mechanics are mutually “tuned” to give Λ = Γ = 1. Frequency saturation for small earthquakes is shown to be a simple consequence of a general aseismic nucleation process. The nucleation magnitude mν defines the roll‐off point in ESF statistics that occurs when the area of seismic rupture equals the area of precursory, aseismic rupture growth. By fitting the model to the Parkfield earthquake data set, mν = 0.9 is obtained. The change in ESF scaling for the transition from moderate to large earthquakes depends on the changes in stress at the growing rupture front. The Northern California Earthquake Data Center seismicity data set is fitted for a transition at magnitude m ≃ 5 with Λ′ = 0.4 and Γ′ = 0.75. This result indicates that the change in the ESF relation associated with the transition from moderate to large earthquakes in California and Nevada is caused by surface rupture accompanied by partial width and slip saturation.

A statistical model for cleavage fracture of low alloy steel

A new statistical model for cleavage fracture of the low alloy steel is proposed. This model is based on a recently suggested physical model and takes account of the effect of the preceding loading processes. This statistical model satisfactorily describes the failure probability distribution of 42 precracked specimens fractured at various loads at a test temperature of − 100°C. The micromechanisms of cleavage fracture of low alloy steel are also further discussed.

Projection Method for Moment Bounds on Order Statistics from Restricted Families: I. Dependent Case

We present a method of projections onto convex cones for establishing the sharp bounds in terms of the first two moments for the expectations ofL-estimates based on samples from restricted families. In this part, we consider the case of possibly dependent identically distributed parent random variables. For the classes of decreasing failure probability, DFR, and symmetric unimodal marginal distributions, we first determine parametric subclasses which contain the distributions attaining the extreme expectations for allL-estimates. Then we derive the bounds for single order statistics. The results provide some new characterizations of uniform and exponential distributions

Fracture statistics of a unidirectional composite

We propose a model dealing with the prediction of the failure stress of a unidirectional composite 0°; it is based on a probabilistic micro-macro approach. Experimental tests have been carried out on specimens (unidirectional composite 0° T300/914) with different gauge lengths in order to estimate the scale effect in the failure probability distribution.

Failure of emergency diesel generators: a population analysis using empirical Bayes methods

Abstract This paper presents analysis of failure data from a large population of emergency diesel generators (EDGs) at nuclear power plants. Empirical Bayes methods are used to estimate the population distribution and the individual failure probabilities from EDG's failure to start and load-run data over 4 years for 195 EDGs at 63 plant sites. Each diesel failure probability is treated as being a sample value from an underlying population distribution. The statistical characteristics of the failure probability distributions (mean, variance, standard defiation and others) are estimated. The estimates of the failure probability distributions are useful for evaluation of EDG reliability and can be used as prior distributions and in uncertainty propagations in probabilistic risk assessments (PRAs).

THE STATISTICAL TREATMENT OF SMALL SPECIMEN FRACTURE TOUGHNESS DATA IN THE TRANSITION REGION

Abstract— The paper deals with the fracture toughness data gained from small Charpy‐COD specimens of CrNiMoV steel in both unirradiated and irradiated states. The new single transition temperature is defined as the quantile of the Pui distribution. The Weibull (2‐ and 3‐parameter) and Neville models are used for the statistical treatment of the data. The lower part of the failure probability distribution can well be described by the models, the upper part, less steep, needs a new approach. The change between these two conditions is considered to be caused by failure of the small scale yielding condition. The effect is more significant for irradiated materials. A new definition of the limit for Jmax values reflecting the effect of both flow stress and work hardening exponent is suggested.

Macroscopic nature of brittle fracture

The macroscopic nature of brittle fracture has been examined based on simple micromechanism based statistical models. Based on the examination the following conclusions can be drawn regarding the macroscopic nature of brittle fracture: -The fracture toughness has, in the case of brittle fracture a specific scatter distribution which can be described by the cumulative failure probability distribution Pf=1-exp-{const.*B*(K-Kmin) 4 }. -The specific scatter distribution indicate a distinct specimen size (crack front length) effect

Failure rate coupling effects in uncertainty analysis of redundant systems

Abstract A simple method is described for deriving system failure probability distributions from component failure rate uncertainties which obey either gamma or log normal distributions. The formalism is applicable to series-parallel systems and includes the effects of coupling between the failure rates of individual components. Coupling means that the failure rates of identical components under the same conditions scatter significantly less than the failure rates of just any two components of the same type. Examples are used to illustrate that there are various interpretations for the coupling phenomenon and alternative ways to use observed failure data in reliability and uncertainty analysis. Both the mean value and the variance of the distribution of the failure probability of a redundant system tend to increase with increasing coupling.

ワイブルパラメータの推定値の信頼度

Not only in the field of Electrical Engineering, but in the fields of Mechanical Engineering, Metal Engineering, Pharmacology and Medical Science, the Weibull distribution is widely used in discussing the probability distribution of breakdown, failure and lifetime. Therefore, it is very important to estimate the Weibull parameters and percentile points from the actual data.It has been often used the Weibull probability chart to estimate the parameters of the Weibull distribution. That is, the graphical method is taken. This method has a good advantage that we can estimate the parameters easily. On the contrary, there are some disadvantages such as the inaccuracy in estimation and having no confidence. Thus, we have to estimate the parameters with accuracy by the analytical methods such as maximum likelihood estimation method, and then to calculate the reliability of the estimates.Here, I mainly describe the calculated results of reliability of the estimates of the parameters and percentile points when all the three parameters are unknown, along with the explanation of the theoretical background. In addition, I describe the confidence limit and confidence region with the method of asymptotic normal theory of maximum likelihood estimation comparing with the method of likelihood ratio test.

Reliability analysis of a two-unit parallel system with dissimilar units and general distributions

We study the reliability characteristics of a parallel system attended by a single repair facility and general failure and repair time probability distribution functions (pdf). The basic mathematical problem is the solution of a pair of simultaneous Wiener-Hopf integral equations.A formal solution procedure of the equations has been discussed in the case of unspecified characteristic functions (cf).Fairly explicit solutions are performed if at least one of the failure time pdf's has a rational cf.We compare some of the results presented in previous literature.

Thermal shock resistance of ceramic materials in melt immersion tests

Abstract A melt immersion test is applied to determine the relative resistance of ceramic materials to thermal shock failure under high heat flux conditions. The testing method is demonstrated mainly for Al2O3 pellets, while AlN is included to represent elevated thermal shock resistance. In order to quantify the resistance to crack formation, the critical temperature difference ΔTc between sample and metal melt is determined from the failure probability distribution of a set of pellets. In quenching tests correspondence of ΔTc with the thermal shock parameter R = σ(1 − μ) E α was found, if the initial surface temperature of the sample was correctly estimated. This assessment was the main concern of the evaluation work. ΔTc resulting from heating tests was correlated with the maximum tensile stress in the sample by modeling calculations. The stress limits determined show that the ultimate bending strength could serve as a rough approximation for the materials tested.