 Reliability HotWire

Issue 67, September 2006

Hot Topics

Analysis of Accelerated Life Tests with Competing Failure Modes

Accelerated tests that yield a mix of failure modes have long troubled reliability analysts. Oftentimes when dealing with accelerated test data, all failures, regardless of their cause or which part of the product they affected, are lumped under one category - failures. This approach does not provide a way to distinguish between failure modes and treat them separately. Also, lumping all failure types together might lead to the violation of a common accelerated life testing analysis assumption, the constant shape parameter.

This article presents a simple approach to analyze accelerated life tests with competing failure modes using ALTA, Weibull++ and BlockSim.

Example

The following table shows the failure times, at different accelerated stress levels, of a product that has two failure modes: A and B.

Table 1 - Failure Data (in hours) Obtained in an Accelerated Test of a Product with Two Failure Modes

Temp = 300K Temp=350K Temp=400K
 150 310 393 409 421 425 459 510 512 558 568 580 615 616 618 629 646 667 738 815 A A A A B B B B A B B B B A B B B B B B
 65 88 107 142 190 191 290 301 310 315 330 343 401 460 531 A A A A B A A A B A A B B B A
 21 48 55 56 81 82 89 155 160 164 200 212 215 225 290 A A A A A A A B B A B B B A B

The failure times are assumed to follow a Weibull distribution. The normal use condition is Temp = 290K.

We first perform a simple check to verify the assumption of constant shape parameter, which states that units/components will fail in the same manner across different stress levels. This requirement is upheld in accelerated life testing analysis in order to simplify extrapolations to use conditions and remove any requirements for additional modeling and assumptions about the behavior of the shape parameter across different stress levels (which would also result in larger sample sizes). For the Weibull distribution, the shape parameter is beta, β, and for the lognormal distribution, the shape parameter is the log-std, σT.

We use a graphical approach, contour plots, to verify the assumption of constant shape parameter. Contour plots are graphical representations of the possible solutions to the likelihood ratio equation; they show the variation of the parameter estimates with a certain confidence level. We enter the data set for each stress level in a different data sheet and fit a distribution to each one of them. We then create the contour plots of the estimated parameters of each data set at different stress levels using Weibull++. In this example, the Weibull distribution is fitted to the data set.   Figure 1 - Failure Data in Table 1 Entered in Weibull++

Note that in the above figures, the failure modes A and B are not separated (i.e. not treated differently). The only event that is modeled is Failure (regardless of whether it is an A failure or a B failure). The 85% confidence contour plots, which show the variation in the parameter estimate at a certain confidence level, are shown next. Figure 2 - Contour Plots of Failure Data (A and B not Separated) in Table 1

The above plot reveals that the constant shape parameter assumption has been violated, i.e. β is not the same at different stress levels. (One way to conclude whether or not β is the same at different stress levels is to try to draw one horizontal line that intersects with all the contour plots. If no such line exists, then the the β's are considered to be statistically different.) Since we cannot proceed with the common shape parameter assumption we need to consider alternative analysis methods. In this example, since we know that there are two failure modes in the data set, we can separate the data for each mode and check whether each mode exhibits a common shape at each stress level. (Note that even if the shape parameter is constant across the different stress levels, it is still recommended to separate the data by failure modes.)

In order to begin analyzing data sets with more than one competing failure mode, one must perform a separate analysis for each failure mode. During each of these analyses, the failure times for all other failure modes not being analyzed are considered to be suspensions. This is because the units under test would have failed at some time in the future due to the failure mode being analyzed, if the unrelated (not analyzed) mode had not occurred. Thus, in this case, the information available is that the mode under consideration did not occur and the unit under consideration accumulated test time without a failure due to the mode under consideration (or in other words, the unit is a suspension from the point of view of that mode). For example, the data used to analyze failure mode B at Temp = 300K is as follows (note that A is treated as a suspension). Figure 3 - Failure Data for Failure Mode B at Temp = 300K,  Used to Obtain Failure Model for Failure Mode B

The next figure looks at the contour plots at each stress level when the A and B failure modes are treated separately.  Figure 4 - Contour Plots of Failure Modes A and B Treated Separately

The above plot shows that the constant shape (β) assumption is valid for failure mode B if we separate the failure modes.

We now continue with our analysis using ALTA, which offers models that can be used to make statistical inferences about the reliability of the product at the normal use level (Temp=290K). As mentioned above, the proper way to treat failure data that contains different failure modes is by separating the analysis for each failure mode and obtaining a failure distribution for each failure mode.

The failure modes are separated using the approach described above. The following figure shows the failure data for A and B entered in ALTA. The data are analyzed using the Arrhenius model as the life-stress relationship, which is appropriate when the stress is of a thermal nature; while the Weibull distribution is used as the life distribution. Note that, due to window size limitation, the figures below show only a portion of the data.  Figure 5 - Accelerated Test Failure Data for Failure Modes A and B (Analyzed Separately)

The following plot shows the separated reliability models for each failure mode at the use stress level. Figure 6 - Reliability Plots for Failure Modes A and B

Once the analysis for each separate failure mode has been completed, the resulting reliability equation for the product with all its failure modes is the product of the reliability equation for each mode, or:

 R(t)=R1(t).R2(t).....Rn(t) (1)

where n is the total number of failure modes considered. This is the product rule for the reliability of a system with statistically independent in-series components, which states that the reliability for a series system is equal to the product of the reliability values of all the components comprising the system.

We now use the system reliability equation, Eqn. (1), to obtain the reliability of the product considering all the possible ways it could fail. In this example:

 R(t)=RA(t).RB(t) (2)

If we want to use the above equation to estimate the reliability at use level conditions, then we would first need to obtain the parameters of RA and RB at use level conditions.

For the A failure mode, the value of the η parameter at the use level Temp = 290K is 1423.4041 (as shown in Figure 5). Since the constant parameter assumption is validated (see Figure 4), the β parameter is assumed to be the same at the different stress levels used. Therefore, β at Temp = 290K should have the same value estimated in Figure 5. Thus:

ηA 290 = 1423.4041 hrs

βA 290  = 1.5362

In a similar fashion, we obtain the Weibull parameters for the B failure mode at the normal use level.

ηB 290  = 750.1948 hrs

βB 290  =  4.8053

We can now use Eqn. (2), or its corresponding reliability block diagram (RBD) equivalent, shown in Figure 7, to estimate R(t) at normal use conditions. Figure 7 - The Competing Failure Modes in a Graphical Representation

The above RBD can be created in BlockSim (or in Weibull++). The Weibull parameters of A and B, at normal use conditions, are entered in BlockSim as follows.  Figure 8 - Entering the Weibull Parameters of A and B at Normal Use Conditions

The following reliability plot, which shows the reliability of the product accounting for both types of failure modes, is obtained based on the RBD in Figure 7. Figure 9 - Reliability at Normal Use Conditions

Conclusion

This article presented a simple approach for treating accelerated life data containing different failure modes. It provides enhancements over the commonly used procedure in which all failure modes are treated as "equal" and the cause of failure is ignored. Using competing failure mode and system reliability concepts, we were able to derive failure distributions for each failure mode and combine the distributions to obtain a reliability model for the product. We also showed how separating failure modes data might help in analyzing an accelerated test data set that seems to violate the constant shape assumption, which is the basis for making inferences based on accelerated testing. 