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A Bayesian inference process for system logistical planning is presented which provides a method for incorporating actual failures with prediction data for an ongoing and improving reliability estimates. The process uses the Weibull distribution, and provides a means for examining and updating logistical and maintenance support needs.
The incorporation of a number of additional capabilities into an existing Weibull analysis computer program and the results of Monte Carlo computer simulation study to evaluate the usefulness of the Weibull methods using samples with a very small number of failures and extensive censoring are discussed. Since the censoring mechanism inherent in the Space Shuttle Main Engine (SSME) data is hard to analyze, it was decided to use a random censoring model, generating censoring times from a uniform probability distribution. Some of the statistical techniques and computer programs that are used in the SSME Weibull analysis are described. The methods documented in were supplemented by adding computer calculations of approximate (using iteractive methods) confidence intervals for several parameters of interest. These calculations are based on a likelihood ratio statistic which is asymptotically a chisquared statistic with one degree of freedom. The assumptions built into the computer simulations are described. The simulation program and the techniques used in it are described there also. Simulation results are tabulated for various combinations of Weibull shape parameters and the numbers of failures in the samples.
Generally, it has been documented that the wearing of engine parts forms a failure distribution which can be approximated by a function developed by Weibull. The purpose here is to examine to what extent the Weibull distribution approximates failure data for designated engine parts of the Space Shuttle Main Engine (SSME). The current testing certification requirements will be examined in order to establish confidence levels. An examination of the failure history of SSME parts/assemblies (turbine blades, main combustion chamber, or high pressure fuel pump first stage impellers) which are limited in usage by time or starts will be done by using updated Weibull techniques. Efforts will be made by the investigator to predict failure trends by using Weibull techniques for SSME parts (turbine temperature sensors, chamber pressure transducers, actuators, and controllers) which are not severely limited by time or starts.
The calculation of shape and scale parameters of the two-parameter Weibull distribution is described using the least-squares analysis and maximum likelihood methods for volume- and surface-flaw-induced fracture in ceramics with complete and censored samples. Detailed procedures are given for evaluating 90 percent confidence intervals for maximum likelihood estimates of shape and scale parameters, the unbiased estimates of the shape parameters, and the Weibull mean values and corresponding standard deviations. Furthermore, the necessary steps are described for detecting outliers and for calculating the Kolmogorov-Smirnov and the Anderson-Darling goodness-of-fit statistics and 90 percent confidence bands about the Weibull distribution. It also shows how to calculate the Batdorf flaw-density constants by uing the Weibull distribution statistical parameters. The techniques described were verified with several example problems, from the open literature, and were coded. The techniques described were verified with several example problems from the open literature, and were coded in the Structural Ceramics Analysis and Reliability Evaluation (SCARE) design program.
The calculation of shape and scale parametes of the two-parameter Weibull distribution is described using the least-squares analysis and maximum likelihood methods for volume- and surface-flaw-induced fracture in ceramics with complete and censored samples. Detailed procedures are given for evaluating 90 percent confidence intervals for maximum likelihood estimates of shape and scale parameters, the unbiased estimates of the shape parameters, and the Weibull mean values and corresponding standard deviations. Furthermore, the necessary steps are described for detecting outliers and for calculating the Kolmogorov-Smirnov and the Anderson-Darling goodness-of-fit statistics and 90 percent confidence bands about the Weibull distribution. It also shows how to calculate the Batdorf flaw-density constants by using the Weibull distribution statistical parameters. The techniques described were verified with several example problems, from the open literature, and were coded in the Structural Ceramics Analysis and Reliability Evaluation (SCARE) design program.
The NASA Energy Efficient Engine (E(sup 3)-Engine) is used as the basis of a Weibull-based life and reliability analysis. Each component's life and thus the engine's life is defined by high-cycle fatigue (HCF) or low-cycle fatigue (LCF). Knowing the cumulative life distribution of each of the components making up the engine as represented by a Weibull slope is a prerequisite to predicting the life and reliability of the entire engine. As the engine Weibull slope increases, the predicted lives decrease. The predicted engine lives L(sub 5) (95 % probability of survival) of approximately 17,000 and 32,000 hr do correlate with current engine maintenance practices without and with refurbishment. respectively. The individual high pressure turbine (HPT) blade lives necessary to obtain a blade system life L(sub 0.1) (99.9 % probability of survival) of 9000 hr for Weibull slopes of 3, 6 and 9, are 47,391 and 20,652 and 15,658 hr, respectively. For a design life of the HPT disks having probable points of failure equal to or greater than 36,000 hr at a probability of survival of 99.9 %, the predicted disk system life L(sub 0.1) can vary from 9,408 to 24,911 hr.
A Weibull analysis was performed of the strength distribution and size effects for 2000 specimens of H-451 nuclear-grade graphite. The data, generated elsewhere, measured the tensile and four-point-flexure room-temperature rupture strength of specimens excised from a single extruded graphite log. Strength variation was compared with specimen location, size, and orientation relative to the parent body. In our study, data were progressively and extensively pooled into larger data sets to discriminate overall trends from local variations and to investigate the strength distribution. The CARES/Life and WeibPar codes were used to investigate issues regarding the size effect, Weibull parameter consistency, and nonlinear stress-strain response. Overall, the Weibull distribution described the behavior of the pooled data very well. However, the issue regarding the smaller-than-expected size effect remained. This exercise illustrated that a conservative approach using a two-parameter Weibull distribution is best for designing graphite components with low probability of failure for the in-core structures in the proposed Generation IV (Gen IV) high-temperature gas-cooled nuclear reactors. This exercise also demonstrated the continuing need to better understand the mechanisms driving stochastic strength response. Extensive appendixes are provided with this report to show all aspects of the rupture data and analytical results.
Leonard Johnson published a methodology for establishing the confidence that two populations of data are different. Johnson's methodology is dependent on limited combinations of test parameters (Weibull slope, mean life ratio, and degrees of freedom) and a set of complex mathematical equations. In this report, a simplified algebraic equation for confidence numbers is derived based on the original work of Johnson. The confidence numbers calculated with this equation are compared to those obtained graphically by Johnson. Using the ratios of mean life, the resultant values of confidence numbers at the 99 percent level deviate less than 1 percent from those of Johnson. At a 90 percent confidence level, the calculated values differ between +2 and 4 percent. The simplified equation is used to rank the experimental lives of three aluminum alloys (AL 2024, AL 6061, and AL 7075), each tested at three stress levels in rotating beam fatigue, analyzed using the Johnson- Weibull method, and compared to the ASTM Standard (E739 91) method of comparison. The ASTM Standard did not statistically distinguish between AL 6061 and AL 7075. However, it is possible to rank the fatigue lives of different materials with a reasonable degree of statistical certainty based on combined confidence numbers using the Johnson- Weibull analysis. AL 2024 was found to have the longest fatigue life, followed by AL 7075, and then AL 6061. The ASTM Standard and the Johnson-Weibull analysis result in the same stress-life exponent p for each of the three aluminum alloys at the median, or L(sub 50), lives
The fracture strengths of two large batches of A357-T6 cast aluminum coupon specimens were compared by using two-parameter Weibull analysis. The minimum number of these specimens necessary to find the fracture strength of the material was determined. The applicability of three-parameter Weibull analysis was also investigated. A design methodology based on the combination of elementary stress analysis and Weibull statistical analysis is advanced and applied to the design of a spherical pressure vessel shell. The results from this design methodology are compared with results from the applicable ASME pressure vessel code.
The fracture strengths of two large batches of A357-T6 cast aluminum coupon specimens were compared by using two-parameter Weibull analysis. The minimum number of these specimens necessary to find the fracture strength of the material was determined. The applicability of three-parameter Weibull analysis was also investigated. A design methodolgy based on the combination of elementary stress analysis and Weibull statistical analysis is advanced and applied to the design of a spherical pressure vessel shell. The results from this design methodology are compared with results from the applicable ASME pressure vessel code.
This manual describes the operation and theory of the PC-CARES (Personal Computer-Ceramic Analysis and Reliability Evaluation of Structures) computer program for the IBM PC and compatibles running PC-DOS/MS-DOR OR IBM/MS-OS/2 (version 1.1 or higher) operating systems. The primary purpose of this code is to estimate Weibull material strength parameters, the Batdorf crack density coefficient, and other related statistical quantities. Included in the manual is the description of the calculation of shape and scale parameters of the two-parameter Weibull distribution using the least-squares analysis and maximum likelihood methods for volume- and surface-flaw-induced fracture in ceramics with complete and censored samples. The methods for detecting outliers and for calculating the Kolmogorov-Smirnov and the Anderson-Darling goodness-of-fit statistics and 90 percent confidence bands about the Weibull line, as well as the techniques for calculating the Batdorf flaw-density constants are also described.
Probabilistic failure analysis is essential when analysis of stress-life (S-N) curves is inconclusive in determining the relative ranking of two or more materials. In 1964, L. Johnson published a methodology for establishing the confidence that two populations of data are different. Simplified algebraic equations for confidence numbers were derived based on the original work of L. Johnson. Using the ratios of mean life, the resultant values of confidence numbers deviated less than one percent from those of Johnson. It is possible to rank the fatigue lives of different materials with a reasonable degree of statistical certainty based on combined confidence numbers. These equations were applied to rotating beam fatigue tests that were conducted on three aluminum alloys at three stress levels each. These alloys were AL 2024, AL 6061, and AL 7075. The results were analyzed and compared using ASTM Standard E739-91 and the Johnson-Weibull analysis. The ASTM method did not statistically distinguish between AL 6010 and AL 7075. Based on the Johnson-Weibull analysis confidence numbers greater than 99 percent, AL 2024 was found to have the longest fatigue life, followed by AL 7075, and then AL 6061. The ASTM Standard and the Johnson-Weibull analysis result in the same stress-life exponent p for each of the three aluminum alloys at the median or L(sub 50) lives.
A Fortran computer program has been written for performing statistical analyses of fatigue-test data that are assumed to be adequately represented by a two-parameter Weibull distribution. This program calculates the following: (1) Maximum-likelihood estimates of the Weibull distribution; (2) Data for contour plots of relative likelihood for two parameters; (3) Data for contour plots of joint confidence regions; (4) Data for the profile likelihood of the Weibull-distribution parameters; (5) Data for the profile likelihood of any percentile of the distribution; and (6) Likelihood-based confidence intervals for parameters and/or percentiles of the distribution. The program can account for tests that are suspended without failure (the statistical term for such suspension of tests is "censoring"). The analytical approach followed in this program for the software is valid for type-I censoring, which is the removal of unfailed units at pre-specified times. Confidence regions and intervals are calculated by use of the likelihood-ratio method.
The statistical analysis of strength and fracture of materials in general, with application to fiber composites are discussed. The weakest link model is considered in a fairly general form, and the resulting equations are demonstrated by using a Weibull distribution for flaws. This distribution appears naturally in a variety of problems, and therefore additional attention is devoted to analysis and statistical estimation connected with this distribution. Special working charts are included to facilitate interpretation of observed data and estimation of parameters. Implications of the size effect are considered for various kinds of flaw distributions. Failure and damage in a fiber-reinforced system are described. Some useful graphs are included for predicting the strength of such a system. Recent data on organic-fiber (PRD 49) composite material is analyzed by the Weibull distribution with the methods presented.
A statistical characterization for multicomponent brittle fibers in presented. The method, which is an extension of usual Weibull distribution procedures, statistically considers the components making up a fiber (e.g., substrate, sheath, and surface) as separate entities and taken together as in a fiber. Tensile data for silicon carbide fiber and for an experimental carbon-boron alloy fiber are evaluated in terms of the proposed multicomponent Weibull characterization.
Necessary to the development and understanding of brittle fiber reinforced composites is a means to statistically describe fiber strength and strain-to-failure behavior. A statistical characterization for multicomponent brittle fibers is presented. The method, which is an extension of usual Weibull distribution procedures, statistically considers the components making up a fiber (e.g., substrate, sheath, and surface) as separate entities and taken together as in a fiber. Tensile data for silicon carbide fiber and for an experimental carbon-boron alloy fiber are evaluated in terms of the proposed multicomponent Weibull characterization.
The suitability of the two-parameter Weibull distribution for describing highly censored cat motion sickness latency data was evaluated by estimating the parameters with the maximum likelihood method and testing for goodness of fit with the Kolmogorov-Smirnov statistic. A procedure for determining confidence levels and testing for significance of the difference between Weibull parameters is described. Computer programs for these procedures may be obtained from an archival source.
A structural ceramic analysis and reliability evaluation code has recently been developed encompassing volume and surface flaw induced fracture, modeled by the two-parameter Weibull probability density function. A segment of the software involves computing the Weibull polydimensional stress state crack density coefficient from uniaxial stress experimental fracture data. The relationship of the polydimensional stress coefficient to the uniaxial stress coefficient is derived for a shear-insensitive material with a random surface flaw population.