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# The Parametric Models for Common Cause Failure Analysis

Common Cause Failures (CCF) occur when multiple (usually identical) components fail due to shared causes. Typical examples of shared causes include impact, vibration, temperature, contaminants, miscalibration and improper maintenance. In the September 2010 issues of HotWire, the Reliability Basics article explained how to implement the Beta factor model using the duty cycle in BlockSim. In this article, we will show how to apply these models via mirror blocks in BlockSim.

The five parametric models for CCF are listed in Table 1. The Basic Parameter Model (BPM) is used to estimate the basic event probabilities directly. The other four models are each a reparameterization of the BPM. The main characteristics of the models are summarized in the table. For more details, please refer to .

Table 1 - Main characteristics of five popular parametric models for CFF

 Model Parameters General Form for Multiple Component Failure Frequency Basic Parameter Model (BPM) where is the probability of a basic event involving k specific components in a common cause component group of size m. Beta Factor Qt , β where: Qt is the total probability of each component failing due to all independent and common cause events. β is a constant fraction of the component failure probability that can be associated with common cause events shared by other components in a common cause component group. Multiple Greek Letters (MGL) Qt , β, γ, ... where: Qt is the total probability of each component failing due to all independent and common cause events. β is the conditional probability that the cause of a component failure will be shared by one or more additional components, given that a specific component has failed. γ is the conditional probability that the cause of a component failure that is shared by one or more components will be shared by two or more components, given that two specific components have failed. Alpha Factor Qt , α1, α2, α3, ..., αm where: Qt is the total probability of each component failing due to all independent and common cause events. αk is the probability that when a common cause basic event occurs in a common cause group of size m, it involves the failure of k components. Non-staggered testing (all components tested simultaneously): Staggered testing (components tested sequentially): where:  Binomial Failure Rate (BFR) Qt , μ, ρ, ω where: Qt is the independent failure frequency for each component. μ is the frequency of occurrence of nonlethal shocks. ρ is the conditional probability of failure of each component, given a nonlethal shock. ω is the frequency of occurrence of lethal shocks. From Table 1, we can see that the only difference between these five models is the way in which they calculate the common factor failure probability, . Once we have , it is easy to model the system reliability via mirror blocks in BlockSim.

## Mirror Blocks

Mirror blocks allow you to place duplicates of an original block in multiple locations within a reliability block diagram (RBD) or fault tree. This can be useful for many purposes, such as modeling bi-directional paths or common cause failures. The duplicate block behaves in the same way as the original block. In BlockSim, a gray box will appear in the upper left corner of the block to indicate that the block is a mirror block, as shown next. ## Example

Let’s consider a system with three identical components A, B and C which form a common cause component group of size 3 (m = 3). The success criterion we will consider is that one of the three components must function. Thus, for the system to fail, all three components must fail.

We will initially use the BPM and assume we can obtain the different failure probabilities directly. We will then illustrate how the Alpha Factor parameterization of the BPM would be used. The other methods can be used in a similar way.

Using the BPM, we define the total failure of component A with the following equation: where:

• At is the total failure of component A from any cause.
• AI is the failure of component A from independent causes.
• SAB is the failure of A and B from common causes.
• SAC is the failure of A and C from common causes.
• SABC is the failure of A, B and C from common causes.

Similarly, we define total failure of components B and C by the following equations:  The terms in these equations are defined analogously to the ones for component A.

Table 2 lists the failure probabilities and the CCF parameters for the Alpha Factor model.

Table 2 - Parameters By using the staggered testing equations for the Alpha Factor model in Table 1, we can calculate , and as follows:   Table 3 lists the basic event failure probabilities and reliabilities.

Table 3 - Basic event probabilities The fault tree in BlockSim is shown in Figure 1. Note that event S_AB under "B_Fails" is a duplicate of S_AB under "A Fails." Also, the events named S_ABC under "B Fails" and "C Fails" are duplicates of S_ABC under "A Fails." Events S_AC and S_BC follow a similar logic. Figure 1 - Fault tree for a system with three components

Now we can obtain the reliability of the system using the Quick Calculation Pad, as shown in Figure 2. The system reliability is 0.9995. Figure 2 - QCP results

## Conclusion

This article briefly reviewed five parametric models for common cause failure analysis and illustrated how one of these models can be applied in BlockSim. After parameterization of the BPM using any one of the other four models, it is very easy to model the common cause failure via mirror blocks in BlockSim. Note that while these models make simplifying assumptions (for example, that all components are identical), using mirror blocks does not always require these assumptions.

## References

 A. Mosleh, K. Fleming, G. Parry, H. Paula, D. Worledge, and D. Rasmuson, "Procedures for treating common cause failure," Safety and Reliability Studies, NUREG/CR4780 EPRI NP-5613, vol. 1, Jan. 1988.

 ReliaSoft Corporation. (2005, Aug.). "Treating common cause failure in fault trees," Reliability HotWire. 2005. Available: https://www.weibull.com/hotwire/issue54/relbasics54.htm 