## Fault Tree Analysis

Fault tree analysis (FTA) [86, 42, 107] is an analytical tool that uses deductive reasoning to determine the occurrence of an undesired event (called "Top" Event). The FTA, along with component failure data and human reliability data, can enable determination of the frequency of occurrence of an accidental event. It yields both qualitative as well as quantitative information.

A logical relationship between Top events and the corresponding contributory events are conveniently represented as a 'fault tree'. The Top event is taken as the root of a tree of logic. Then, each situation that could cause that effect is added to the tree as a series of logic expressions. Basic events at the bottom of the fault tree are linked via logic symbols (known as gates) to the Top events. The logical connections in the fault tree are generally represented by two types of gates, the "OR" and the "AND". If components from several barriers have to fail for the undesired event to occur, these are combined with the initiating event by an "AND" gate. If several of these combination exist, they are input into an "OR" gate, just as contributions from different initiating events to the undesired events.

Quantitative evaluation of a fault tree requires quantitative reliability data for equipments as well as human error. When fault trees are labelled with actual numbers about probabilities or frequencies, a computer program can be used to calculate failure probabilities from fault trees. Fault trees for complex system normally must be evaluated with the aid of computer program. There are three methods available for this purpose: (1) direct simulation of the fault tree, (2) minimal cut set calculation using a simulation procedure, and (3) minimal cut set calculation by analytical methods. The last procedure was used to calculate undesired events of the study objects. A brief description of the method is to be discussed in the following section, and the detailed description of the program is presented in the Appendix F.

### 4.4.1.1 Analytical Approach

As described before, the evaluation of a fault tree for a complex system requires the aid of computer programs. Some of the programs are readily available as commercials software, such as CARA Fault Tree Application from SINTEF, AvSim+ from Isograph, and so on. The study used the fault tree analysis (FTA) program developed by Hauptmanns (1978). The FTA program basically consists of two following sub-programs, i.e. (1) determination of minimal cut set with an analytical approach, and (2) fault tree evaluation.

The first sub-program is used to determine minimal cut sets with an analytical method. The method is made up of Boolean algebra operations in order to transform the tree in such a way that it is expressed in term of its minimal cut sets. In contrast to other methods, this method does not require reliability data for obtaining the minimal cut sets of the tree [86]. These are only needed for calculating the probability of the undesired events. Basically two approaches may be used in the method, i.e. the "Top-down" approach, in which the algorithm starts with the undesired event represented by the Top gate working its way down to the basic events, and the "Bottom-up", where the calculation is initiated at the level of basic events, and ends with the undesired event. The latest is, however, not implemented in the program.

After finding the minimal cut sets the procedure may be continued with the evaluation of the fault tree, e.g. calculation of the expected frequency of the undesired event (Top event). This is calculated by forming the expectation of the structure function given in Eq. F-1 (Appendix F). With eliminating of the powers of binary variables in the Eq.F-1, which are equal to the binary variables themselves (law of idempotencies), the general relationship of the structure function can be described as [86]:

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