Fault Tree AND Gates

Solution The fault tree can be applied quantitatively. Since both power sources must fail for the system to fail, Equation 5-5 can be used. 

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Figure 15.23 shows a fault tree and Gate based on the first standard example. The gate has two inputs failure of P-IOIA, which is steam driven, and failure of P-IOIB, which is electrically driven. (Pump A is normally operating, with B being on standby.) It is assumed that the two pumps have failure modes that are totally independent of one another, i.e., the failure of one is completely independent of the failure of the other. Pump 101-A has a predicted failure rate of once in 2 years, or 0.5 yr Pump 101-B has a predicted probability of failure on demand (PFD) of 1 in 10 or 0.1. 

The error rates can be modified with a recovery factor which allows for corrective action to be taken before the consequences of the error affects the overall system performance (corresponding to the fault tree and gate). For example, if there is a 40% chance that the operator will take immediate corrective action on... 

FTA, fault free analyzer module, uses SETS and FTAP to reduce fault trees and generate minimal cutsets for storage as minimal cutset libraries. Cutset control uses truncation hy probability or order. The user chooses the codes according to the personal computer s capabilities. The FTA module uses OR, AND, N/M, switch gates and supercomponents. 

Fault trees and event trees can be combined. The gates in the event tree are treated as top events of multiple fault trees. So, for example, one of the event tree gates could be Loss of Electrical Power. That term then becomes the top event of a Loss of Electrical Power fault tree. When using fault trees as subsets of event trees it is important to identify the common cause or interdependent events and to enter them separately into the tree s structure. 

Fault Trees and Reliability Block Diagrams are both methods of showing probability combinations. There have been a number of solution techniques developed to solve probability combinations. These include Cut Sets, Tie sets. Event Space, Decomposition Method, Gate Solution Method, and many others. In this appendix three examples will be shown - the Event Space method and the Cut Set method, and the Gate Solution Method. Details and full development of the methods can be found in (Ref. 1) Chapter 5. 

Consider the case of a pressure switch and a process connection for that pressure switch (Figure C-6. Pressure Sensor Subsystem). If the pressure switch has a PFD of 0.005 and the process connection has a PFD of 0.02, the PFD of the system could be modeled with a fault tree OR gate. 

The above example may seem obvious, but this effect can easily be hidden when working with large fault trees across multiple pages. Most fault tree programs can handle repeated events across a fault tree and calculate the correct probability. However, the software can only apply this if the basic event entered on both sides of the tree is entered as the same event. If they have different event identifiers, the software will treat them as different events and allow the incorrect calculation of probability for the gate and the top-level event. 

Fault Trees and Event Trees are formalized diagrammatical models that trace paths from a top Hazard (unwanted event) to the basic input events via logic gates and the development of a Hazard through to the consequences resulting from success or failure of systems intended to mitigate effects of the initiating event. 

Figure 2.6-5 is a very simple example. Fault trees for actual nuclear power plant systems commonly involve hundreds of logic gates and hundreds of base events. Nevertheless, Figure 2.6-5 can be used to illustrate the process undertaken to solve fault trees and event trees. The first step is to fi-id the minimal combinations of events that lead to system failure. These are called minimal cut sets for... 

These two events are related to each other through an OR gate, ie, the top event can occur if either road debris or tire failure occurs. Another type of gate is the AND gate, where the output occurs if and only if both inputs occur. OR gates are much more common in fault trees than AND gates, ie, most failures are related in OR gate fashion. 

The resulting fault tree is shown in Figure 6, in which the top event is defined in terms of two intermediate events failure of the tank system or failure of the pumping system. Failure in either system would contribute to the overall system failure. The intermediate events are then further defined in terms of basic events. All of the basic events are related by AND gates because the overall system failure requires the failure of all of the individual components. Failures of the tanks and pumps are basic events because, without additional information, these events cannot be resolved any further. 

The immediate causes of the TOP event are shown in the fault tree with thek relationship to the TOP event. If any one of the immediate causes results dkecdy in the TOP event, the causes are connected to the TOP event with an OR logic gate. If all the immediate causes are requked for TOP event occurrence, then the causes are connected to the TOP event with an AND logic gate. Each of the immediate causes is then treated in the same manner as the TOP event, and its immediate, necessary, and sufficient causes are determined and shown on the fault tree with the appropriate logic gate. This development continues until all intermediate fault events have been developed into thek basic causes. 

Fault Tree Analysis Faiilt tree analysis permits the hazardous incident (called the top event) frequency to be estimated from a logic model of the failure mechanisms of a system. The top event is traced downward to more basic failures using logic gates to determine its causes and hkelihood. The model is based on the combinations of fail-... 

Consider the example fault tree (Figure 3.4,4-4) the Boolean equations taken, gate at a time,. ire T = EI E2, El = A+E3, E3 = B+C, E2 = C+E4, E4=A B. Start with the 7-cquation, substitute, and expand until the minimal cutset expression for the top event is obtaineil Subst ituting for El and E2 gives ... 

Initially, a system s hierarchy is identified for subsystems, sub-subsystems and so on to the components for which data must be found. The top event specifies system failure subsystems required for operation of the system in the mode specified are input to the top event s OR gate. Redundancy is represented by the redundant systems inputting an AND gate. This process of grouping subsystems under OR gates, if they can individually fail a function, or under AND gates if concurrent failures are necessary, is continued to the component or support system level until the tree is completed. This process grades the hierarchy from top to bottom, down the fault tree. 

Gates look alike the are only distinguished by the gate label. Each event box is labeled to fully describe the postulated fault according to the what and when about the event, The precise statement enhances the analysis by focusing on the event, and aiding the fault tree revic. . ... 

Comptete all gate inputs. Beginning a new gate before completing a previous gate leads to confusion and incompleteness. These rules are made concrete by an example of fault tree analysis. 

Using de Morgan s theorem (Table 2.1-1), the fault tree in Figure 3.4.4-9 is converted into the success tree shown in Figure 3.4.4-10. Notice that the effect of the theorem is to reverse tlie logic "AND" gates become "OR" gates and vice versa failure becomes success and vice versa. 

NsiiT /]n uL/ " gates. iMethod of generating i )ns cutsets Fault tree truncation/ jOther features Other outDuts Computer language and availalality... 

MOCUS implemented the Fussell algorithm (Fussell, 1974) for top-down solutions of the fault tree. This algorithm was used in ALLCUT and was modified to be bottom-up in WAMCE T. While cutsets are valuable for qualitative and quantitative purposes, they are not compact. They are thousands of trains (AND sequences) connected to one OR gate. 

The fault tree (Figure 7.4-1) has "Pre.ssure Tank Rupture" as the top event (gate G1). This may result from random failure of the tank under load (BEl), OR the gate G2, "Tank ruptures due to overpressure" which is made of BE6 "Relief valve fails to open" AND G3, "Pump motor operates too long." This is made of BE2, "Timer contacts fail to open," AND G4, "Negative feedback loop inactive" which is composed of BE3, "Pressure gauge stuck," OR BE4, "Operator fails to open switch," OR "BE5, "Switch fails to open,"... 

The overall frequency of the top event is calculated by combining together the constituent probabilities and frequencies of the various events in the fault tree using the appropriate logical relationships described by the AND and OR gates (the detailed calculation is given in CCPS, 1989b). 

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