PSI - Issue 33

C. Mallor et al. / Procedia Structural Integrity 33 (2021) 391–401

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C. Mallor et. al. / Structural Integrity Procedia 00 (2020) 000 – 000

The probability distribution can be described in various forms, such as by the survival function (SF), by the cumulative distribution function (CDF) or by the probability density function (PDF). In the context of probabilistic fatigue crack growth life in railway axles, the SF is the function that gives the probability that an axle will survive beyond any specified time, number of cycles or kilometers travelled. Frequently, in engineering, the survival function is also known as the reliability function. Alternatively, the reliability function can also be evaluated for a given reliability percent obtaining the corresponding number of kilometers travelled. In other words, in this way it provides the minimum mileage travelled for a given surviving proportion of axles. Another name for the survival function is the complementary of the cumulative distribution function (CCDF). Moreover, as it is well known, the CDF and the PDF are closely related. Given these basic premises, the working approach selects a reliability level in such a way that a conservative lifespan balancing safety and economic issues is achieved. Notice that, the input uncertainties and scatter are implicitly in the output probability distribution provided by the Pr. Eq. and represented by its survival, cumulative distribution and probability density functions of fatigue life. The stated procedure is illustrated in Fig. 1 (a). As a result of the procedure, a conservative estimation of the lifespan is obtained, taking advantage of the knowledge available at the lower tail of the distribution of lives. Finally, instead of the deterministic lifespan calculation, the conservative lifespan estimation is considered as the FCG process (step 2) outcome, which is the basis for the subsequent steps oriented to the interval inspection definition. The idea for determining the periodicity of the non-destructive inspections (NDI) is depicted simply in Fig. 1 (b). First, based on the conservative lifespan estimation, the residual lifetime (step 4) is delimited. This portion of lifetime is denoted as in Fig. 1 (b) in reference to the lifetime for the definition of inspection intervals. The covers the propagation from to (steps 1 and 3), being the minimum and the maximum crack sizes considered for the lower and the higher lifetime bounds, respectively. Finally, the inspection interval is determined by dividing by a number of times that takes account of the number of times that the crack can be detected before a failure could occur. For example, the usual assumption considering equal to 2 or 3 [14], allows the crack to be observed at least twice or three times before it leads to catastrophic failure. This assumption is based on the fact that a crack could be missed at an inspection. It is, however, evident that even two or more inspections cannot ensure the crack detection. In particular, as it is not known exactly when crack growth is triggered by an accidental event, the component will always be subjected to inspection every km, and depending on the inspection method used, the cumulative probability of detecting (CPOD) a crack in the axle or its complementary cumulative probability of failure (CPOF) or simply referred to as probability of failure ( ) can be computed. It is important to recall the hypothesis made here, that is, the presence of a crack (step 1) and so the probability of failure equals the probability of not detecting the crack in due time throughout the axle lifetime. This must be distinguished from the probability of failure of an arbitrary axle in a fleet of trains since an existing defect an its nucleation to a crack of that size is very unlikely. To calculate the real probability of failure, the obtained in the damage tolerance analysis should be multiplied by the probability of having a defect on the axle and by the probability that a crack will nucleate from that defect and further grow during the service life. Therefore, the real probability of failure of an axle is, by orders of magnitude, smaller than the one obtained in a damage tolerance analysis. The calculation of the real probability of failure is beyond the scope of this work. Note that, in this context, damage tolerance does not mean that a crack detected during an inspection is considered acceptable even when its size is far from being critical. In some other applications, this is a possible option, but it should be handled with care especially for safety relevant applications, as it is the case of a railway axle. In consequence, preventive maintenance is a prevailing principle in the railway industry. In summary, the approach presented here extends the current damage tolerance principles in railway axles by means of improving the crack growth simulation (step 2), replacing the deterministic crack growth estimation by a probabilistic one. The damage tolerance assessment benefits from a better knowledge of the distribution of fatigue lifespan. As a result, it would give a more conservative recommendation for the definition of inspection intervals as it is based on a probabilistic fatigue propagation instead of on a deterministic one.