Fatigue Life is a Statistical Quantity

1. Fatigue Life is a Statistical Quantity Introduction to the Weibull distribution

2. Consider 2 competing Spring Designs, called A and B. 10 Samples of each spring are tested in fatigue The number of cycles to failure are recorded. Spring B 529000 730000 651000 446000 343000 960000 730000 730000 973000 258000

3. Objective You need a design where 90% of the springs last (at least) to 400 000 cycles.

4. First Primitive Test: Average Lifetime Design A Design B 529000730000651000446000343000960000730000730000973000258000 635000 726000615000508000808000755000849000384000667000515000 483000 631000 Average Design B looks better…..

5. Next test, a bit more sophisticated: We plot fraction of springs failed vs number of cycles. We use this to get estimates of reliability of Design A and Design B as a function of cyles Example: If first failure in design A is at 200 000 cycles, reliability above 200 000 cycles is reduced from 100% to 90%

6. We now fit a smooth curve (2nd order power) to the data and to find F (400000)

7. We repeat the same for Design B

9. The question is : Is our method to decide the best there is ? (best meaning “standing up in court”) Sadly no But it is not a bad method to make a crude estimate of what the Weibull distribution predicts.

10. The Weibull distribution was found by Weibull strictly by trial and error. He tried to model the distribution of failure strength of steels and derive probabilities for a high reliability (such as 99.9 %) from a limited set of test data. After settled on this distribution: X is the variable (here the number of cycles to failure) and  and  are the Weibull parameters ( is the shape parameter also known to material scientists as the Weibull modulus and  is the scale or length parameter. Note that  has a physical meaning. If a tensile specimen is twice as long, the probability for a flaw terminating its fatigue life is twice as high. Size matters.

11. There is nothing god given about the Weibull distribution. There are other reliability distributions - all invented before Weibull. But it does fit, experimentally, a very wide variety of phenomena. Weibull wrote a famous paper demonstrating it fit the size distribution of beans, the height distribution of the population on an island (I forgot which one) and so on, i.e. Biological phenomena as well as steels, with a total of 10 examples The paper is a classic - I will put it on the website. The reason is that depending on how the modulus is picked it fits both infant mortality and wear out phenomena. And fatigue failure is a kind of wear out phenomena. MOST IMPORTANTLY IT HAS BECOME A DE FACTO STANDARD FOR ENGINEERS. TO PREVAIL IN COURT, YOU BETTER SHOW YOU USED IT – PROPERLY !

12. From Wiki • The Weibull distribution is used • In survival analysis[6] • In reliability engineering and failure analysis • In industrial engineering to represent manufacturing and delivery times • In extreme value theory • In weather forecasting (To describe wind speed distributions, as the natural distribution often matches the Weibull shape[7] Fitted cumulative Weibull distribution to maximum one-day rainfalls) • In communications systems engineering (In radar systems to model the dispersion of the received signals level produced by some types of clutters. To model fading channels in wireless communications, as the Weibull fading model seems to exhibit good fit to experimental fading channel measurements) • In General insurance to model the size of Reinsurance claims, and the cumulative development of Asbestosis losses • In forecasting technological change (also known as the Sharif-Islam model)[citation needed]

13. In hydrology the Weibull distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. • In describing the size of particles generated by grinding, milling and crushing operations, the 2-Parameter Weibull distribution is used, and in these applications it is sometimes known as the Rosin-Rammler distribution. (In this context it predicts fewer fine particles than the Log-normal distribution and it is generally most accurate for narrow particle size distributions).[ Stolen From Wiki

14. So we need to learn to do the proper Weibull analysis This involves three steps “Massaging” the data point . (we simply took the first spring to fail as the 10% probability for reliability –not a good idea ) Plotting the massaged data point in a “double ln” plot to get the Weibull parameters Learning on how to put confidence limits on the results.

15. Massaging the data point The data are plotted in increasing sequence (ranking low to high) The equivalent of our failure percentage becomes approximately This will do for most engineering problems. One can do this better by looking up the F distribution. There is a nifty website; “teach yourself statistics” which might come in handy when you are in industry (Modern industry runs on statistics)

17. Once you have done the ranking, it is just plotting : And, from the double ln plot to extract the values for K and  So here is the plot :

19. From which you get the following results: This does not look all that earth shattering different from our primitive estimate which yielded 0.8807 0. 74866 Which shows you that it is a good idea to make a primitive estimate first before setting out to do a “state of the art Weibull with an F distribution)

20. Now, we should put confidence limits on our answer… But as this is not a statistics course, I will leave it at this. But before you use a statistics package to analyze your data it is a good idea to make the primitive plot we started out and decide what other curves might “reasonably” be drawn to the data. This will give you a rough idea as to the confidence limits you can put on the data.