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According to Eq. (5.3), given the failure density function ft(t) it is a straight­forward task to derive the failure rate h(t). Furthermore, based on Eq. (5.3), the reliability can be computed directly from the failure rate as Ps(t) = exp   I h(x) dr 0   (5.10)   Substituting Eq. (5.10) into Eq. (5.3), the […]

The failure rate for many systems or components has a bathtub shape, as shown in Fig. 5.8, in that three distinct life periods can be identified (Harr, 1987). They are the early-life (or infant mortality) period, useful-life period, and wear-out — life period. Kapur (1989b) differentiates three types of failure that result in the bathtub […]

Similar to the cumulative distribution function (CDF), the cumulative hazard function can be obtained from integrating the instantaneous hazard function h(t) over time as H(t) = f h(t) dt J0 Referring to Eq. (5.3), the hazard function can be written as 1 d [pf (t)] 1 d [ps(t)] Ps (t) dt ps(t) dt Multiplying dt […]

The failure rate is defined as the number of failures occurring per unit time in a time interval (t, t + At ] per unit of the remaining population in operation at

Any system will fail eventually; it is just a matter of time. Owing to the presence of many uncertainties that affect the operation of a physical system, the time the system fails to perform its intended function satisfactorily is random. 5.1.1 Failure density function The probability distribution governing the time occurrence of failure is called […]

5.1 Basic Concept In preceding chapters, evaluations of reliability were based on analysis of the interaction between loads on the system and the resistance of the system. A system would perform its intended function satisfactorily within a specified time period if its capacity exceeds the load. Instead of considering detailed in­teractions of resistance and load […]

Consider a vector x 1 in an K-dimensional space to be used as one of the basis vectors. It is desirable to find the additional vectors, along with x 1, so that they would form K orthonormal basis vectors for the K-dimensional space. To do that, one can arbitrarily select K — 1 vectors in […]

The orthogonal transformation is an important tool for treating problems with correlated stochastic basic variables. The main objective of the transformation is to map correlated stochastic basic variables from their original space to a new domain in which they become uncorrelated. Hence the analysis is greatly simplified. Consider K multivariate stochastic basic variables X = […]

This appendix summarizes some commonly used numerical formulas for eval­uating the following integral: Detailed descriptions of these and other numerical integration procedures can be found in any numerical analysis textbook. 4A.1 Trapezoidal rule For a closed integral, Eq. (4A.1) can be approximated as 1 = hif 1 + 2 Щ fi + (4A.2a) where h […]

Considering only inherent hydrologic uncertainty. Traditionally, the risk associ­ated with the natural hydrologic randomness of flow or rainfall is explicitly considered in terms of a return period. By setting the resistance equal to the load with a return period of T years (that is, r* = lT), the annual reliability, without considering the uncertainty associated […]