
Calculating Equivalent Acceleration Factors for Random Stresses
In accelerated life tests, when a component has been tested for a number of hours under the stressed condition, we want to know the equivalent operation time at the use stress condition. If the use stress is a constant value, this is easy to calculate. However, a stress at the use condition is usually random, so how can we calculate the equivalent acceleration factor for a random stress? In this article, we will illustrate one simple method.
Let's assume the stress is temperature. The following Arrhenius model is used:
(1) 
where:
 L(T) is the life at a temperature of T.
 T is the temperature in kelvins.
 E_{a} is the activation energy.
 K is the Boltzman’s constant (8.617 x 10^{5}).
If the use condition temperature is T_{U} and the accelerated temperature is T_{A}, then the acceleration factor (AF) can be calculated by:
(2) 
Assume E_{a} = 0.7, T_{A} = 90 degrees Celsius and T_{U} = 30 degrees Celsius. By converting the temperatures to kelvins, the AF is:
(3) 
Therefore, 1 hour of testing at the accelerated temperature is equivalent to 83.7 hours at the use condition. From Eqn. (2), it can be seen that we only need to know the value of E_{a} in order to calculate AF, if T_{A} and T_{U} are provided. If E_{a} is unknown, it can be estimated from a quantitative accelerated life test. In quantitative accelerated testing, at least two different temperature values should be used. For more details on quantitative accelerated life data analysis, please see ALTA.
Now, for a situation where the use stress is not constant, let's use a simple example to examine how the equivalent acceleration factor and test time can be calculated. Assume the use stress is random and can be quantified using the following table.
Temperature (C)  Temperature (K)  Probability 
< 20  < 293.15  0.05 
[20, 40]  [293.15, 313.15]  0.7 
[40, 50]  [313.15, 323.15  0.2 
> 50  > 323.15  0.05 
Consider a demonstration test in which 10 components are tested for 500 hours at a temperature of 90 degrees Celsius (363.15 kelvins) without failure. What are the equivalent hours under the use condition, and what is the demonstrated reliability at the use condition? A simple way to determine this is to calculate the AF for each temperature range. Let's assume that we have prior engineering knowledge that E_{a} = 0.920299. Using Eqn. (2), the calculated AFs are:
Temperature (C)  Temperature (K)  Probability  AF 
< 20  < 293.15  0.05  1121.3 
[20, 40]  [293.15, 313.15]  0.7  337.1 
[40, 50]  [313.15, 323.15  0.2  64 
> 50  > 323.15  0.05  38.1 
Note: For a temperature range, the middle point is used for the AF calculation.
Since the operation time at the usage level consists of the time at different temperature, we have:
It means if a product operated for Time_{U} hours at the usage condition, the equivalent test time at the accelerated condition is Time_{A}.P_{i} is the probability in the above table. Define the equivalent AF as:
Then
(4) 
For the case in the above Table AF_{e} = 152.47.Therefore, 500 hours of testing at a temperature of 90 degrees Celsius is equivalent to 500 x 152.47 = 76,237 hours of operation.
The next question is the demonstrated reliability at the use condition based on the results of the accelerated life test. For a random use stress condition, this question can be answered using the binomial equation for the zerofailure demonstration test:
(5) 
where CL is the confidence level, R is the demonstrated reliability and n is the sample size (i.e., the number of units). For this example, if we set CL = 0.8, then R = 0.85134.
Therefore, if 10 components are tested without failure at a temperature of 90 degrees Celsius for 500 hours, the demonstrated reliability is 85.13%. This is the same as saying that the reliability at 76,237 hours under the use condition is 0.85134 at a confidence level of 0.8.
Conclusion
In this article, we discussed how to calculate the equivalent acceleration factor (AF) for a random stress. When the stress at the use condition is random, it can be quantified by its possible ranges and the probability that the use stress falls within each range. The AF for each range can be calculated; the overall equivalent AF will then be the weighted average of the AFs from all the ranges. Once the equivalent AF is obtained, the equivalent operation time under the use condition can be calculated. From the sample size, number of failures and confidence level, the demonstrated reliability also can be calculated.
Although temperature and the Arrhenius model are used for illustration in this article, the methods used here can also be applied to other stresses and models.