Failure Weibull distribution

As ean be seen from the above equation, for brittle materials like glass and eeramies, we ean seale the strength for a proposed design from a test speeimen analysis. In a more useful form for the 2-parameter Weibull distribution, the probability of failure is a funetion of the applied stress, L. 

Repair and maintenance records were analyzed to determine failure rates and distribution of failure modes. Preliminary findings are reported which include the Weibull distribution characteristics. Failure mode distributions are approximate. Overall mean-time-between-failure is given for the kiln, leach tank, screwfeeder, tank pump, tank gearbox, and kiln gearbox. The study was confined to an analysis of unscheduled repairs and failures. 

Statistical Methods for Nonelectronic Reliability, Reliability Specifications, Special Application Methods for Reliability Prediction Part Failure Characteristics, and Reliability Demonstration Tests. Data is located in section 5.0 on Part Failure Characteristics. This section describes the results of the statistical analyses of failure data from more than 250 distinct nonelectronic parts collected from recent commercial and military projects. This data was collected in-house (from operations and maintenance reports) and from industry wide sources. Tables, alphabetized by part class/ part type, are presented for easy reference to part failure rates assuminng that the part lives are exponentially distributed (as in previous editions of this notebook, the majority of data available included total operating time, and total number of failures only). For parts for which the actual life times for each part under test were included in the database, further tables are presented which describe the results of testing the fit of the exponential and Weibull distributions. 

Weibull distribution This distribution has been useful in a variety of reliability applications. The Weibull distribution is described by three parameters, and it can assume many shapes depending upon the values of the parameters. It can be used to model decreasing, increasing, and constant failure rates. 

A plant accident is more likely to luippen during tlie startup of a new plant or a retro fit process because new equipment usually experiences a high failure rate during the early or break-in period. The overall process is best described by the Weibull distribution. The Weibull Distribution or tlie batlitub curve is a tliree-... 

The Weibull distribution provides a iiiatliematical model of all tliree stages of the batlitub curve. Tliis is now discussed. An assumption about failure rate tliat reflects all tliree stages of tlie batlitub curve is... 

Equation (20.4.3) defines tlie pdf of the Weibull distribution. Tlie exponential distribution, whose pdf is given in Eq. (20.4.1), is a special case of the Weibull distribution witli p = 1. Tlie variety of assumptions about failure rate and tlie probability distribution of time to failure tliat can be accommodated by the Weibull distribution make it especially attractive in describing failure time distributions in industrial and process plant applications. 

To illustrate probability calculations involving tlie exponential and Weibull distributions introduced in conjunction willi llie batlitub curve of failure rate, consider first llie case of a mansistor having a constant rate of failure of 0.01 per tliousand hours. To find the probability tliat llie transistor will operate for at least 25,000 hours, substitute tlie failure rate... 

As anollier example of probability calculations - lliis time involving the Weibull distribution - consider a component whose time to failure T in hours lias a Weibull pdf with parameters a = 0.01 and p = 0.50. To find llie probability that llie component will operate for at least 8100 hours, substitute a = 0,01 and p = 0.50 in Eq. (20.4.3), Tliis gives... 

Tlie life of an automobile seal lias a Weibull distribution with failure rate Z(t) =, where t is measured in years. What is tlie probability that the... 

Theodore et al. employed Monte Carlo metliods in conjunction with the binomial and Weibull distributions to estimate out-of-compliance probabilities for electrostatic precipitators on tlie basis of observed bus section failures. The following definitions apply (see Fig. 21.6.1). 

For each of the 36 bus sections tliat had not already failed, the Weibull distribution was used to detennine tlie probability of failure before tlie next outage. Under assumption (a), tliis probability is P(T < 3301T > 209) i.e., tlie conditional probability of failure before 330 days, given tliat tlie bus section lias survived 209 days. Under assumption (b), tlie corresponding probability is P(T < 330 T > 230). For part (b), tlie estimates of the Weibull distribution parameters used in part (a) were modified to take into consideration tlie absence of failures for 3 additional weeks. 

Assume tlie time to failure T of a bus section lias a Weibull distribution with a = 1.3 X 10 and p = 0.77. 

The investigation of failures of manufactured components and systems, especially in the electronics and aerospace industries, has generated a variety of statistical models on which data analysis may be based. Each model uses a specific distibution of failure probabilities, and it is important to select a model that matches the actual distribution inherent in the product concerned. In the case of dielectric breakdown, where a large number of quite different modes of failure are known to occur, sometimes even together, the application of a particular statistical failure model must be approached with great caution. Nevertheless, one treatment, based on a Weibull distribution of failure probability, has taken root, and is most generally used in practice. For a dielectric, the Weibull failure probability function has the form... 

Figure 7.13 Failure probability of 1 m and 10m gage length optical glass fibers. The long lengths of optical glass fibers have multiple flaw populations, i.e. there is more than one source of flaws, thus they do not follow the simple Weibull distribution (after Maurer, 1985).

We can regard a fiber as consisting of a chain of links. We assume that fiber failure occurs when the weakest link fails. This is called the weakest-link assumption. It turns out that such a weak-link material is well described by the statistical distribution known as the Weibull distribution (Weibull, 1939,1951). We first describe the general Weibull treatment for brittle materials and then describe its application for fibers. 

Let us say that a series of identical samples are tested to failure. From such tests we can obtain the fraction of identical samples, each of volume that survives when loaded to a given stress, a. Let us call this According to the Weibull distribution, this survival probability is given by... 

A statistical analysis of the liber tensile strength values determined on a series of fiber samples can be easily made by using the two-parameter Weibull distribution described above. Using the form of the Weibull expression given in Eq. (10.2), we can write the probability of failure F(o) of the liber at a stress a, as... 

Using the procedure described in Section 20.4, obtain graplrical estimates of the parameters a and P for tlie Weibull distribution of time to failure on die basis of the data in Table 21.6.1. 

In case of disorder correlations, or for example at the criticality (at P = Pc)i the probability g l) of a defect cluster of size I decreases following a power law g l) Using then the relation (1.24), connecting the failure stress a with the size I of the crack, one gets p(a), which in turn, when put in (1.22) for the cumulative failure probability F a) gives the Weibull distribution (Weibull 1951, Ray and Chakrabarti 1985a)... 

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