Reliability Assessment of Machines and Structures with Weibull Type of Life Distribution

Abstract

The definition of the fleet reliability R' has been recently introduced by A. M. Freudenthal on the basis of the scatter factor defined as the ratio of the maximum likelihood estimator of the scale parameter of two-parameter Weibull distribution with a known shape parameter, assumedly describing the life distribution of structural elements or components, to the“time to first failure T1” among a fleet of nominally identical elements or components subjected to nominally identical operating conditions. Unfortunately, however, R' thus introduced does not represent rigorously the true fleet reliability R to be defined theoretically as R=P[T1≥t1*], where t1* is the service life (minimum life) specified for the fleet. Further, R' in itself is of statistical nature and therefore needs to be dealt with as an estimator of R.In this respect, a new estimator of R, termed as R in the case of known shape parameter, has been proposed in the present study. Statistical natures of R' and R have heen carefully examined and compared with the aid of Monte Carlo techniques, thus leading to the conclusion that R is much better than R'.The mathematical difficulty multiplies when the Weibull shape and scale parameters are both assumed to be unknown. However, this case is undoubtedly consistent with engineering reality. For these conditions, therefore, the reliability estimators R and R" have also been introduced as a natural extension of aforementioned R and R', respectively. Procedures involving Monte Carlo techniques have been established to evaluate the statistical properties of these estimators. Simulated results show that R is again much better than R". The effect of the size to be used in the fatigue test, of the fleet size and of the reliability level on the accuracy of such estimation has also been discussed.The fleet reliability can then be estimated based on either R in the case of known shape parameter or R unknown shape. This is believed to satisfy the essential part of the requirement in the reliability-based safety design of machines and structures.