Cumulative probability of failure

Unlike that for the classical linear responses of such solids, the extreme nature of the breakdown statistics, nucleating from the weakest point of the sample, gives rise to a non-self-averaging property. We will discuss these distribution functions F a), or F(/), or F E) giving the cumulative probability of failure of a disordered sample of linear size L. We show that the generic form of the function F a) can be either the Weibull (1951) form... 

In what follows, we will show that, similar to that for the electrical breakdown (Section 2.2.2(c)), the cumulative probability of failure or fracture... 

Lq = standard gage length m — Weibull modulus cr = tensile strength (To = a scaling parameter (T = an arbitrary parameter normally set to 0 F= cumulative probability of failure... 

Measured strengths of identically prepared glass fibers always show a distribution. Although without any fundamental basis, it is customary to plot the measured strength distribution on a Weibull plot where the ordinate is ln(ln [l/(l — F)] and the abscissa is In S. Here P S) is the cumulative probability of failure for strengths less than or equal to... 

Because the probability of finding a flaw of a given severity increases with increase in the volume or surface of a fiber sample, the average strength tends to decrease with increase in length or diameter. Using the weakest link model, the cumulative probability of failure of a fiber can be shown to be (Hunt and McCartney, 1979) ... 

For the quantitative comparison of the critical stress for maincrack formation under thermal shock, oth, with mechanical loading, oc, the volume effect of strength should be considered. According to Weibull statistics, the cumulative probability of failure of brittle materials is written in the following simplified form. 

For most lifetests it is impractical to monitor the performance of the specimens while the test is running, so it is not usually possible to detect failures at the instant they occur. Instead, the lifetests are interrupted at frequent intervals to permit assessment of the performance of the specimens, and the number of failures at stages during the test is thereby determined. There is then a choice of several equations which can be used to calculate a statistically unbiased estimate of the cumulative probability of failures. One of the most popular of these equations is... 

Knowing the initial instantaneous probability of failure and the up-crossing rate v(t), it is possible to compute an upper bound of the cumulative probability of failure of mechanical products within the time interval [0, 7 by (Mejri et al. 2010) ... 

It is found from Figure 5 that (i) the upper bound of cumulative probability of failure gradually increases with the increasement of n (ii) the lower bound of the reliability of the pin gradually decreases with the increasement of n (hi) the upper bound of cumulative probability of failure and the lower bound of the reliability are respective 0.4626 and 0.5374 within the interval [0, n]. 

Manual proof testing reduces the probability of failure on demand because it effectively resets the clock on the cumulative probability of a failure. After a proof test has been successfully carried out the cumulative probability figure is very low and begins to rise with time. Hence the average probability between proof tests is lower than the untested and cumulative probability of failure (which would eventually approach 100%). 

The cumulative probability of failure of a single channel device is p(t) = 1 - e This is shown on the next diagram. 

In evaluating the performance of products, several statistical concepts are utilized. Reliability and the occurrence of failures are expressed in terms of probabilities. The reliability function R t) is the time-dependant reliability, or a survival probability up to a time t. In terms of failures, F t) is the cumulative probability of failure at time t these two terms are interrelated. F(t) is a monotonically increasing function of time and, as a probability, takes on values between 0 and 1 over time. Geometrically, F(t) is the area under the probability density function (/). Thus, F(t) is the probability of failures occurring before or at time t. 

The three-parameter Weibull cumulative distribution fimction, F t), that predicts the cumulative probability of failure up to a specific time, t, is mathematically expressed by Equation 6.11. The probability density fimction,/(t), which is a derivative of the cumulative distribution fimction, is expressed by Equation 6.12 ... 

The cumulative binomial distribution is given by equation 2.5-33, where M is the number of f ailures out of items each having a probability of failure p. This can be worked backH tirLh lo find tlic implied value of p for a specified P(M, p,... 

The probability density function, fit), is defined as the probability of failure in any time interval df. The cumulative distribution function, F(t), is the integral of/(f). 

It is clear that with the assumptions made the probability of failure is very low. For a normal distribution a value of /Iuk of 7.2 corresponds to a probability of failure of 3.8 x 10", and for = 4.2, the figure is about 10". These are such low figures that slight differences in P cause relatively large differences in the probability of failure. A cumulative distribution function for the distribution of Z in the ultimate limit state was plotted from a histogram generated by the Monte Carlo process and showed an approximately normal distribution with a slight tendency to deviate from this at the tail (Fig. 5.10). 

The normalising factor is L S(y) 0.02 + 0.15 + 0,315 +. .. + 0i)15 = 4.35 and the cumulative distribution function is obtained by dividing each of these figures by 4.35 and adding cumulatively. The resulting distribution is shown in Fig. 6.6. A measure of the safety of the column is the probability < 1 ] = 0.715. This means the probability that working stress is less than, or equal to the permissible stress is less than or equal to 0.715. The probability of failure in this defined limit state is 1 — 0.715 = 0.282. Of course, in a real example this figure would be much smaller. 

Modeling. In order to carry out the analysis of the nature of the operational phenomena in facilities and equipment, it is very useful to use statistics as a support for the quantification of the parameters. The phenomena s historical behavior is characterized based on operation and failure periods that have occurred since the commissioning time. The conditions that characterize the equipment operational time data are so numerous that it is not possible to say when exactly the next failure will occur. However, it is possible to express which will be the probability that the equipment is in operation or out of service at any given time. These times are associated with a cumulative distribution function of the random variable, which is defined as the addition of the probabilities of possible values of the variable that are lower or equal to a preset value. The mentioned random variable is constituted by the operating times and downtime of equipment or system in a given period. For its parameterization Weibull distribution is very appropriate as it is very effective and relatively simple to use in the reliability evaluation of a system by quantifying the probability of failure in the performance of the system s duties from the failure probabilities of its components based on the operation times. There are three different parameters ... 

The reliability of a component or system can be defined as the probability that a functioning product at time zero will function in the desired service environment for a specified amount of time. Without these three parameters, the question Is x reliable cannot be answered yes or no. Since reliability describes the probability that the product is still functioning, it is related to the cumulative number of failures. Mathematically, the reliability of an object at time r can be stated as... 

Since the standard normal space is rotational symmetric, probability of failure can be directly obtained using the reliability index, Pj =<6(-y, where <6 is the standard normal cumulative probability function. 

The probability of failure of structures, systems or components as the result of external events is computed as the product of the full range hazard curve of the external events convoluted with the derivative of the fragility of the structure, system or component under consideration, as shown in Section 4. The fragility of structures, systems and components is defined as the cumulative conditional Pp (unacceptable performance) versus the selected hazard parameter. The hazard parameter is typically represented by factors such as the peak ground acceleration (PGA) for earthquakes, the water depth for floods and the maximum wind speed for winds. 

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