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BlockSim 7 - Case Studies

This is the same example as Case Study RC4. For this case study, the Fault Tree Analysis (FTA) approach is integrated into the analysis.  

Case Study RC4 used a reliability block diagram (RBD) approach to analyze a component and its associated failure modes. The example considers both independent modes (i.e. if one mode occurs, the rest are not more likely to occur) and dependent modes (i.e. if one mode occurs, the rest are more likely to occur). In this study we will repeat the same example using a combined fault tree and RBD methodology. 

Example
Assume that a component can fail due to six independent primary failure modes: A, B, C, D, E and F. Some of these primary modes can be broken down further into the events that can cause them, or sub-modes. Furthermore, assume that once a mode occurs, the "event" also occurs and the mode does not go away. Specifically:

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. The rates of occurrence for modes A, B and C depend on their sub-modes.

Objective

The objective of this example is to determine the following:

1. The reliability of the component after 1 year (8760 hrs).

2. The B10 life of the component.

3. The mean time to failure (MTTF) of the component.

4. Rank the modes in order of importance at 1 year.

5. Re-calculate results 1, 2 and 3 assuming mode B is eliminated.

Mode A

• Events S1 and S2 both occur.
• Event T1 or T2 occurs.
• Event Y and either event S1 or event S2 occur (i.e. events Y and S1 or events Y and S2).

RBD Solution
The RBD that satisfies the conditions for mode A is shown in Figure 1.

Figure 1: Reliability block diagram for mode A

Each mode is identified in the RBD. Furthermore, two additional items are included: a starting block (NF) and an end node (2/2). The starting block and the end node are set so they cannot fail and, therefore, will not affect the results. The end node is used to define a 2-out-of-2 k-out-of n configuration (i.e. both paths leading into the node must work).

Fault Tree Solution
The Fault Tree for mode A is shown in Figure 2.

Figure 2: 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

Based on the given probabilities, distribution parameters are computed for each block (either RBD block or the fault tree event block). These were computed in Case Study RC4.

Mode B

There are three dependent events associated with mode B: events BA, BB and BC. Two out of the three events must occur for mode B to occur. Events BA, BB and BC have an exponential distribution with a mean of 50,000 hrs. The events are dependent (i.e. if BA, BB or BC occurs, the remaining events are more likely to occur). Specifically, when one event occurs, the MTTF of the remaining events is cut in half. This is basically a load sharing configuration. The reliability function for each block will change depending on the other events. Therefore, the reliability of each block is not only dependent on time, but also on the stress (load) that the block sees.

RBD Solution
The reliability block diagram for mode B is shown in Figure 3.

Figure 3: Reliability block diagram for mode B

Fault Tree Solution
The fault tree for mode B is shown in Figure 4.

Figure 4: Fault Tree diagram for mode B (using a load sharing gate unique to BlockSim).

Note that a "load sharing gate" is not a standard fault tree gate. BlockSim introduces this gate to allow for representation of dependent events in a fault tree diagram. It behaves in exactly the same way as a load sharing container in an RBD.

Mode B Discussion
To describe the dependency, we need a model that describes how a life characteristic (in this case, the mean) changes as the events occur. Life-stress relationships used in accelerated testing provide a very good way to describe the effects of stress (load) on life. Since the failure rate is constant, the exponential distribution applies. Any standard life-stress relationship (i.e. an exponential curve or power curve) would apply equally because the function is only being evaluated at the two loads of interest and not necessarily extrapolating or interpolating between these two points. For simplicity, the Arrhenius life-stress relationship will be used.

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 5.

Figure 5: Arrhenius-exponential life-stress relationship properties

The reliability plot for this configuration is displayed in Figure 6.

Figure 6: 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 on-line 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.

RBD Solution
To model this, you can think of a scenario similar to standby redundancy. Basically, if CA occurs then CB gets initiated. A standby container can be used to model this, as shown in Figure 7.

Figure 7: Standby container for mode C 

In this case, event CA is set as the active component and CB as the standby. If event CA occurs, CB will be initiated. For this analysis, a perfect switch is assumed. The properties are set in BlockSim as follows:

• Contained Items

  • CA: Active failure distribution, Weibull distribution (beta = 2, eta = 30,000).

  • CA: Quiescent failure distribution: None, cannot fail or age in this mode.

  • CB: Active failure distribution, Weibull distribution (beta = 2, eta = 10,000).

  • CB: Quiescent failure distribution: None, cannot fail or age in this mode.

• Switch

  • Active Switching: Always works (100% reliability) and instant switch (no delays).

  • Quiescent Switch failure distribution: None, cannot fail or age in this mode.

Fault Tree Solution
The fault tree for mode C is:

Figure 8: Sequence enforcing (standby) gate for mode C 

Mode C Discussion
The failure distribution settings for event CA are shown in Figure 9.

Figure 9: 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.

• D: MTTF = 200,000 hours

• E: MTTF = 175,000 hours

• F: MTTF = 500,000 hours

Component
The last step is to set up the model for 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 RBD in Figure 10 represents the primary failure modes for the component while the fault tree in Figure 11 illustrates the same.

RBD Solution

Figure 10: RBD of Component

Fault Tree Solution

Figure 11: Fault Tree of Component

The node represented by 2/3 in the RBD indicates a 2-out-of-3 configuration. The voting gate in the fault tree accomplishes the same. Subdiagrams are used in both configurations for the sub-modes. 

Once the diagrams have been created, the reliability equation for the system can be obtained, as follows:

R(t)_System  = R_A·R_B·R_F·R_D·R_C+R_A·R_B·R_F·R_C·R_E+R_A·R_B·R_D·R_C·R_E-2(R_A·R_B·R_F·R_D·R_C·R_E)

Where R_A, R_B and R_C are the reliability equations corresponding to the sub-modes.

Analysis
The answers to the questions posed earlier can be answered using BlockSim.  Regardless of the approach used (i.e. RBD or FTA), the answers are the same.  

1) The reliability of the component at 1 year (8760 hours) can be calculated using the Analytical Quick Calculation Pad (QCP) or by viewing the reliability vs. time plot, as displayed in Figure 12.

Figure 12: Reliability vs. time plot for component

Therefore, R(t = 8760) = 86.4975%.

2) Using the Analytical QCP, the B10 life of the component is equal to 7,373.94 hours.

3) Using the Analytical QCP, the mean life of the component is equal to 21,659.68 hours.

4) The ranking of the modes after 1 year can be shown via the static reliability importance plot, as shown in Figure 13.

Figure 13: Static reliability importance for each of the modes at t = 8760 hours

5) Re-computing the results for 1, 2 and 3 assuming mode B is removed:

• R = 98.72%

• B10 = 16,928.38 hours

• MTTF = 34,552.89 hours

Discussion
There are many options for modeling systems with fault trees and RBDs in BlockSim. The following figures illustrate some of these options.

Figure 14: Fault tree for the component without using subdiagrams (transfers)

Figure 15: RBD for the component using fault trees as subdiagrams.

Figure 16: Fault tree for the component using RBDs as subdiagrams.

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