Volume 2, Issue 3

Reliability Edge Home

Analyzing Accelerated Test Data with Time-Dependent Test and Use Stress Profiles

Accelerated testing is becoming more widely used in order to obtain life data in a relatively short amount of time. The use of sophisticated tools such as ReliaSoft’s ALTA 6 PRO software allows the reliability engineer to design and analyze tests that would have been impossible to do a few years ago. This is particularly true if the test or use stress varies with time. ALTA 6 PRO is the only software package capable of using the cumulative damage model for such analyses. In this article, we will look at an analysis in which the test and the use stress levels have different time-dependent profiles.

Test Procedure with Time-Dependent Stress Profiles
This analysis involves accelerated life testing for tanks that are intended to be used with heat exchangers for automotive applications. A new design is being considered for the tank, and it is desired to estimate whether the new design will meet its intended target before releasing it to production. The target is a reliability of 90% at 2,000 hours of use, with a 90% confidence.

The stimulus that has the greatest effect on the life of the tanks is pressure, measured in pounds per square inch (psi). In order to perform an accelerated life test on these tanks, groups of tanks were put on a pressure cycling test. This test involves putting the tanks on a test fixture, pressurizing them to 5.5 psi and then rapidly increasing the pressure to either 15 psi or 21 psi. The tanks are held at the elevated pressure for an amount of time and then rapidly decreased to the original pressure. For each pressure cycle, the tank spends 80% of the time at the elevated pressure and 20% of the time at the “default” pressure of 5.5 psi. Since the cycle repeats every 10 hours, each cycle can be thought of as a square wave. Thus, for each cycle, the tank is held at 5.5 psi for 1 hour (one-tenth of the cycle), the pressure is rapidly increased to either 15 psi or 21 psi and held for 8 hours (eight-tenths of the cycle), then the pressure is rapidly reduced to 5.5 psi and held for an additional hour. The test fixture is capable of increasing and decreasing the pressure of the test units in a rapid fashion, so much so that the transitions between the pressure levels can be considered simultaneous for the purposes of this test. The stress profile for the 15 psi pressure cycling test is shown in Figure 1. The stress profile for the 21 psi test can be similarly constructed.

Stress profile for 15 psi pressure cycling test

Figure 1: Stress profile for 15 psi pressure cycling test

The test was conducted with two groups of test units: 22 units were tested with the 15 psi profile and 10 units were tested with the 21 psi profile. All of the units were tested to failure. That is, the stress profile was repeated until all of the units on test had failed. The failure data set is contained in the following table:

Table of time-to-failure data for the 15 psi and 21 psi profiles

As can be seen from the data, the range of failure times for the 15 psi pressure cycling test (2241 – 12774 hrs) is larger than the range for the 21 psi test (1175 – 7551 hrs), indicating an inverse relationship between life and stress. However, the data set will need to be formally analyzed to see if this conjecture is correct.

Analysis with ALTA 6 PRO
ReliaSoft’s ALTA 6 PRO will be used to analyze this data set, as it is capable of utilizing the cumulative damage model. This is necessary due to the fact that the test stress and the use stress both vary with time. Two choices must be made regarding the settings for the analysis using the cumulative damage model: choice of the underlying life distribution and choice of the cumulative damage relation. The choice of these settings will be largely based on the nature of the accelerating factor for the data, which in this case is mechanical (i.e., pressure). For this reason, the Weibull distribution is chosen as the underlying life distribution, because it is better suited to dealing with the mechanical or fatigue-related failures that would be encountered in this type of pressure cycling. Also, the power relationship is chosen for the cumulative damage relation, as this has been found to better model mechanical types of stress, not unlike the inverse power model. The other choice for the cumulative damage relation, exponential, is more suitable for analyses where temperature is the primary stress factor, similar to the Arrhenius model.

The parameter estimates resulting from this analysis are:

beta = 2.3773
a = 3.8165E+13
n = 0.3094

where beta is the Weibull slope or shape parameter, and a and n are parameters of the life-stress relationship model. The fact that n is positive confirms the hypothesis that life is inversely proportional to stress. This is illustrated by Figure 2, which displays the life vs. stress plot for the analysis. The negative slope of the lines in the graph indicates that life decreases as pressure increases.

Life vs. stress plot for pressure cycling data

Figure 2: Life vs. stress plot for pressure cycling data

Time-Dependent Use Stress Profile
As we continue the analysis and make our reliability estimate to determine whether the product meets the specified reliability target, we must take into account that the use stress will not be constant. We will have to develop a time-dependent stress profile in order to make a realistic prediction of the reliability of the tanks at 2,000 hours of use.

In order to develop the use stress profile, it is necessary to incorporate information regarding the actual usage of the heat exchanger tanks in the field. For this particular automotive application, the pressure in the tank is directly proportional to the engine RPM. Fortunately, the manufacturer has plenty of information regarding the amount of time the engines are expected to spend at various engine speed levels, which was obtained from numerous customer usage profiling programs. Based on this information, the engineers are able to develop a simplified pressure profile that represents the stress that a 98th percentile customer would inflict on a tank. That is, 98% of the drivers would operate their vehicles in such a way that the stress on the heat exchanger tank would be equal to or less than that represented by the stress profile. Simply put, the stress profile indicates that the tank will experience a pressure of 6 psi for 66.7% of its life, 11 psi for 25% of its life and 14 psi for 8.3% of its life. For our 2,000 hour period of interest, this means that the tank will see 1,333 hours at 6 psi, 500 hours at 11 psi, and 167 hours at 14 psi.

There is no distinction, however, regarding when each tank will see these particular stresses in relation to its life-span, and it is assumed that the different stress levels are evenly distributed over the life of the product. The question then arises of how to construct the use stress profile to reflect the usage the tanks will see in the field. One option is to begin with the lowest stress, stepping up the stress at the appropriate times like a typical step-stress test profile. Alternatively, one could construct the use stress profile with the higher, more damaging stress first and step down to the lower stress levels. As another option, one could split the difference between these options and construct the profile so that the highest stress is in the middle of the profile, similar to the pressure cycles that were used for the test stress profiles. The plots in Figure 3 illustrate these three possible use stress profile concepts.

Step Up Use Stress ProfileStep Down Use Stress Profile

Center Cycle Use Stress Profile

Figure 3: Three time-dependent stress profiles that could represent the use stress profile for the analysis

Reliability Calculations
As it turns out, for the purposes of our analysis, it does not matter which use stress profile is used. Since this analysis only requires the reliability at one point in time, the way in which the stress is accumulated does not affect the result. This is illustrated in Figure 4. As can be seen, the reliability at 2,000 hours is the same for all three stress profiles, even though the path that the reliability plot follows differs from profile to profile. The reliability result is the same because at 2,000 hours, the tanks will have encountered 1,333 hours of use at 6 psi, 500 hours of use at 11 psi, and 167 hours of use at 14 psi under all three stress profiles.

Reliability vs. Time plot for various profiles

Figure 4: Reliability vs. Time plot for various profiles

Note that from zero to 667 hours, the reliability plots for the “step down” and “center cycle” profiles are identical, as they are operating at the same stress level and accumulating the same amount of damage. Similarly, the reliability plots for the “center cycle” and “step down” profiles are identical from 1,333 to 2,000 hours. This is because these profiles have accumulated the same amount of damage at 1,333 hours and continue to operate at the same stress level for the rest of the analysis life. If we were to extend the analysis past 2,000 hours by repeating or cycling the stress profiles, we would see that the reliability plots will always intersect at multiples of 2,000 hours.

We can now address the issue of the reliability requirement. Using ALTA’s QCP to calculate the reliability, as shown in Figure 5, the 90% lower, one-sided reliability is 91.02%. This exceeds the requirement of 90% and the design can be released to production.

ALTA's QCP with reliability results at 2,000 hours

Figure 5: ALTA's QCP calculates reliability at 2,000 hours


--End of Reliability Edge Article--


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