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AppendicesLiterature: Selected papers 3. Short Description of Reliability Theory of STRUREL Programs3.1. Time-Invariant Component ReliabilityCOMREL-TI is designed to solve numerically the following problem. Let X=(X1,....,Xn)T be a vector of random variables with joint distribution function FX(x) and g(x) a state (performance) function or failure function such that g(x) > 0 denotes the safe state, g(x) = 0 the limit state and g(x) <= 0 the failure state. g(x) = 0 will also be denoted by failure surface. The failure probability then is  where the second formulation is valid if the probability density exists. The simplest problem of this kind is when failure occurs if a demand S on a system exceeds its capacity R. Then, the performance or state function is given by g(x) = r - s. If capacity and/or demand are random the probability of failure simply is a volume integral extended over the failure domain. This formulation does not only apply to structures but has many other applications, for example, in hydrology, mathematical statistics, control theory, econometrics and financial planning. Especially for large n and complex state function an exact evaluation by numerical integration can require considerable computational effort. Therefore, some special methods have been devised which can do the integration efficiently. These are FORM, SORM and various sampling techniques. 3.2. Time-Variant Component ReliabilityTime-variant reliability is much more difficult to compute than time-invariant component reliability. Note that one is hardly interested in the time dependent failure probability function Pf(t) where t is treated as a parameter but in quantities like the probability of first passage into the failure domain, the total duration of exceedances into the failure domain, the duration of individual exceedances and other related criteria. The quantity  will rather be denoted by non-availability so that A(t) = 1 - N(t) is the availability. Both quantities are easily determined. COMREL-TI can handle criteria of the first mentioned type. In principle, the basic formulation then is  where T is the random time of exit into the failure domain and [0,t] is the considered time interval. If the component does not fail at time tau = 0 failure occurs at a random time and the distribution function of T must be known. Unfortunately, this is rarely the case. Exceptions are the failure times of non-structural components (electronical or other) where often rich statistical material is available. Then it is also possible to use time-invariant reliability analysis by COMREL-TI because the limit state function then simply is g(x) = T - t < 0. In all other cases T must be inferred from the characteristics of the random processes affecting the performance of the component. T must be considered as a first passage time, i.e. is the time where the component enters the failure domain for the first time given that the component was in the safe state at time tau = 0. Exact first passage time distributions are known for only very few types of processes which generally are of little practical interest. In COMREL-TV the so-called outcrossing approach is implemented for the determination of the probability of first passage failure. As in time-invariant component reliability there exists a state function depending on random vectors and on random process variables. More specifically, COMREL-TV distinguishes between three types of variables: R is a vector of random variables as in time-invariant reliability. Its distribution parameters can be deterministic functions of time. This vector is used to model resistance variables. The most important characteristics of this type of variable is that they are non-ergodic. Q is a vector of stationary and ergodic sequences. Usually, it is used to model long term variations in time (traffic states, sea states, wind velocity regimes, etc.). Quite in general these variables determine the fluctuating parameters of the random process variables described next. S is a vector of (sufficiently mixing) not necessarily stationary random process variables whose parameters can depend on Q and/or R. The vector S is further subdivided into a vector J of rectangular wave renewal processes and a vector D of differentiable procceses (Gaussian and non-Gaussian). The safe state of the component is defined for g(r,q,s(t),t) > 0, the limit state for g(r,q,s(t),t) = 0 and the failure state for g(r,q,s(t),t) <= 0 , respectively. Note that the state function can contain time as a parameter, too. A rate of outcrossings into the failure domain conditional on q and r can be defined.  F denotes the failure domain {g(s(t),t/r,q) <= 0}. In order to exist it is necessary that the limiting operation can be performed. This excludes certain processes which fluctuate too rapidly in time. Further, in a small time interval there is at most one crossing. The probability of more than one crossing is negligible small. Then the process of crossings is called a regular point process. The mean number of crossings in the time interval [t1,t2] conditional on q and r can be determined from  Thus, the outcrossing rates are additive for a regular process. Then, it has been shown that an upper bound to the conditional failure probability is (Bolotin, 1981)  If further the process is strongly mixing it holds asymptotically for (Cramer/Leadbetter, 1967):  The conditions can be removed by integration. Whereas the expectation operation with respect to the condition on q can be performed inside the exponent by making use of the ergodicity property of Q it cannot be done with respect to the R-variables.  Schall et al. (1991) showed that for the upper bound solution  These are the basic formulae for time-variant component reliability. COMREL-TV offers for the general case upper bound solution together with a not always close lower bound solution. In most cases Pf(t1) is negligible small. It will always be calculated if the upper bound solution is chosen. For the upper bound solution R-variables are treated like Q-variables. For all computations a probability distribution transformation into standard space as in time-invariant reliability will be performed. The outcrossing approach cannot be improved easily (Engelund et al. 1995). 3.3. Time-Invariant System Reliability
Quite generally, a system in the reliability sense is a system
where the failure event is given as a union or intersection or combinations thereof
of componental failure events. One distinguishes basic types of systems depending
on the logical structure. In a parallel system the system failure event is the
intersection of the componential events. In a series system the system failure
event is the union of the componential events. Both types of elementary systems
can be combined to form either parallel systems in series or series systems in
parallel. In SYSREL systems must be given in terms of minimal unions of intersections,
i.e. by |
