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Using the Fault Tree Method to Analyze Dependent and Independent Failure ModesIn May's issue of the HotWire, a method was presented for modeling dependent and independent failure modes using a reliability block diagram. In this month's issue, we will revisit this topic. However, instead of using the reliability block diagram approach, the fault tree method will be incorporated utilizing BlockSim 6 FTI. Example The component fails if mode A, B or C occurs. If mode D, E or F occurs alone, the component does not fail; however, the component will fail if any two (or more) of these modes occur (i.e. D and E; D and F; E and F). Modes D, E and F have a constant rate of occurrence (exponential distribution) with mean times of occurrence of 200,000, 175,000 and 500,000 hours, respectively. Objective The objective of this example is to determine the following:
Mode A There are five independent events (submodes) associated with mode A: events S1, S2, T1, T2 and Y. It is assumed that events S1 and S2 each have a constant rate of occurrence with a probability of occurrence of 1 in 10,000 and 1 in 20,000, respectively, in a single year (8760 hours). Events T1 and T2 are more likely to occur in an older component than a newer one (i.e. they have an increasing rate of occurrence) and have a probability of occurrence of 1 in 10,000 and 1 in 20,000, respectively, in a single year and 1 in 1,000 and 1 in 3,000, respectively, after two years. Event Y also has a constant rate of occurrence with a probability of occurrence of 1 in 1,000 in a single year. There are three possible ways for mode A to manifest itself:
The fault tree for mode A is shown in Figure 1. Figure 1: Fault tree for mode A Each mode is identified as an event in the fault tree. Mode A Discussion The system reliability equation for this configuration (regardless of how it is drawn) is: R(t)=2R_{T2}R_{S1}R_{S2}R_{T1}R_{Y}+R_{T2}R_{S1}R_{S2}R_{T1}+R_{T2}R_{S1}R_{T1}R_{Y}+R_{T2}R_{S2}R_{T1}R_{Y} The distribution parameters for each mode are computed in the same manner as previously discussed in Part I of this article. Mode B
The fault tree for mode B is shown in Figure 2. Figure 2: Fault tree for mode B (using a load sharing gate unique to BlockSim FTI) Note that a "load sharing gate" is not a standard fault tree gate. BlockSim FTI introduces this gate to allow for representation of dependent events in a fault tree diagram. It behaves the same way as a load sharing container in an RBD. Mode B
Discussion Once the parameters have been obtained, the properties for each event for mode B are set. The load sharing container (if an RBD) or the gate (if a fault tree) properties for the events of mode B are shown in Figure 3. Figure 3: Arrheniusexponential lifestress relationship properties The reliability plot for this configuration is displayed in Figure 4. Figure 4: Reliability plot for mode B For details on the exact reliability equation formulation, please refer to ReliaSoft's System Analysis Reference: Reliability, Availability and Optimization (the load sharing section) that is available online at weibull.com and that accompanies BlockSim. Mode C There are two sequential events associated with mode C: CA and CB. Both events must occur for mode C to occur. Event CB will only occur if event CA has occurred. If event CA has not occurred, then event CB will not occur. Both events CA and CB occur based on a Weibull distribution. For event CA, beta = 2 and eta = 30,000 hours. For event CB, beta = 2 and eta = 10,000 hours. The fault tree for mode B is shown in Figure 5.
Mode C Discussion Figure 6: Failure distribution settings for event CA The failure distribution properties for event CB are set in the same manner. Modes D, E and F Modes D, E and F can all be represented using the exponential distribution. The failure distribution properties for modes D, E and F are presented next.
Component The last step is to set up the component based on the primary modes (A, B, C, D, E and F). Modes A, B and C can each be represented by single blocks that encapsulate the subdiagrams already created. The fault tree in Figure 7 represents the primary failure modes for the component. Figure 7: Fault tree of component The voting gate in the fault tree accomplishes a 2outof3 configuration. Subdiagrams are used for the submodes. Once the diagrams have been created, the reliability equation for the system can be obtained, as follows:
Where R_{A}, R_{B} and R_{C }are the reliability equations corresponding to the submodes.
Analysis
Figure 8: Reliability vs. time plot for component
Figure 9: Static reliability importance for each of the modes at t = 8,760 hours
Discussion
The component can be represented by an RBD, as shown in Part I of this article. Expanding on this concept, you can represent the component using an RBD where the subdiagrams are fault trees, as shown in Figure 11. Figure 11: RBD of component using fault trees as subdiagrams Now, using the same idea, the fault tree can contain transfers that link to reliability block diagrams, as shown in Figure 12. Figure 12: Fault tree of component using RBDs as subdiagrams The BlockSim 6 FTI project file (*.rb6) used for this example is available for download by clicking here. (In the File Download window that appears, click Save. You will then be prompted to specify the location to save the (*.rb6) file to. Once it has been saved, you can open the file in BlockSim 6 FTI. Please note that you must have BlockSim 6 FTI installed on your computer to open this file.) 

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