Failures probability

Fig. 10. Reliability and failure probability computations for components ia (a) series linkages where the failure of either component adds to the total system failure, and (b) parallel linkages where failure of the system requires the failure of both components. There is no convenient way to combine the failure rate...

Conditional Failure Probability. The concept of conditional piobabihty of faiuie is useful to predict the chances of survival for a device that has been in operation for a period of time and is not in a failed state. Such information is helpful for maintenance planning. 

A logic model that graphically portrays the combinations of failures that can lead to a particular main failure (TOP event) or accident of interest. Given appropriate data, fault tree models can be quantitatively solved for an array of system performance characteristics (mean time between failures, probability of failure on demand, etc.)... 

The chalk with which I write on the blackboard when I teach is a brittle solid. Some sticks of chalk are weaker than others. On average, I find (to my slight irritation), that about 3 out of 10 sticks break as soon as I start to write with them the other 7 survive. The failure probability, Pf, for this chalk, loaded in bending under my (standard) writing load is 3/10, that is... 

When you write on a blackboard with chalk, you are not unduly inconvenienced if 3 pieces in 10 break while you are using it but if 1 in 2 broke, you might seek an alternative supplier. So the failure probability, Pf, of 0.3 is acceptable (just barely). If the component were a ceramic cutting tool, a failure probability of 1 in 100 (Pf= 10 ) might be acceptable, because a tool is easily replaced. But if it were the window of a vacuum system, the failure of which can cause injury, one might aim for a Pf of lO and for a ceramic protective tile on the re-entry vehicle of a space shuttle, when one failure in any one of 10,000 tiles could be fatal, you might calculate that a Pf of 10 was needed. 

This, then, is our final design equation. It shows how the survival probability depends on both the stress (rand the volume V of the component. In using it, the first step is to fix on an acceptable failure probability, Pp 0.3 for chalk, 10 for the cutting tool, 10 for the vacuum-chamber window. The survival probability is then given by P = 1 -. ... 

The material properties of window glass are summarised in Table 18.1. To use these data to calculate a safe design load, we must assign an acceptable failure probability to the window, and decide on its design life. Failure could cause injury, so the window is a critical component we choose a failure probability of 10The vacuum system is designed for intermittent use and is seldom under vacuum for more than 1 hour, so the design life under load is 1000 hours. 

In a probabilistie approaeh, design deeisions must reduee the probability of unwanted performanee to aeeeptable levels. In a deterministie approaeh, the designer ean only assure that the performanee remains within an aeeeptable domain (Ben-Haim, 1994). As the deterministie approaeh provides no firm basis for dealing with variability, it is not pertinent to a reliability approaeh (Morrison, 1997 Shigley and Misehke, 1989). As an example of this, Haugen (1968) shows that the failure probability for a partieular... 

A meehanieal eomponent is eonsidered safe and reliable when the strength of the eomponent, S, exeeeds the value of loading stress, L, on it (Rao, 1992). When the loading stress exeeeds the strength, failure oeeurs, the reliability of the part, R, being related to this failure probability, P, by equation 4.26 ... 

Figure 4.31 Failure probability (per application of load) versus safety margin for various loading roughness values (adapted from Carter, 1997)...

However there are NV such defects in the specimen, where N is the number of pores per unit volume and V is the specimen volume. Each has a survival probability of one minus its failure probability, i.e.,... 

The Burchell model s prediction of the tensile failure probability distribution for grade H-451 graphite, from the "SIFTING" code, is shown in Fig. 23. The predicted distribution (elosed cireles in Fig. 23) is a good representation of the experimental distribution (open cireles in Fig. 23)[19], especially at the mean strength (50% failure probability). Moreover, the predicted standard deviation of 1.1 MPa con ares favorably with the experimental distribution standard deviation of 1.6 MPa, indicating the predicted normal distribution has approximately the correct shape. 

As described above, the code "SIFTING" requires several microstructural inputs in order to ealculate a failure probability distribution. We are thus able to assess the physieal soundness of the Burchell model by determining the change in the predicted distribution when microstructural input parameters, such as particle or pore size, are varied in the "SIFTING" code. Each microstructural input parameter... 

Fig. 23. Predicted and experimental tensile failure probability distributions for grade H-451 graphite.

Fig. 28. A comparison of experimental and predicted tensile failure probabilities for four graphites with widely different textures AGX, H-451, IG-110 and AXF-5Q.

A lrLL[uently encountered problem requires estimating a failure probability based on the number of failures, M, in N tests. These updates are assumed to be binomially distributed (equation 2.4-10) as p r N). Conjugate to the binomial distribution is the beta prior (equation 2.6-20), where / IS the probability of failure. 

Suppose a designer does not know the failure probability of something, but i.s confident it is less than some probability/ . 

When a risk or reliability analysis has been performed, it is appropriate to inquire into the sensitivity of the results to uncertainties in data. One type of sensitivity analysis is the effect on system reliability that results from a small change in a component s failure probability. A problem in doing this is determining the amount of data uncertainty that is reasonable. The amount of change... 

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