Fatigue Life is a Statistical Quantity

# Fatigue Life is a Statistical Quantity - Fatigue Life is a...

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Fatigue Life is a Statistical Quantity Introduction to the Weibull distribution

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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 A: 726000 615000 508000 808000 755000 849000 384000 667000 515000 483000 529000 730000 651000 446000 343000 960000 730000 730000 973000 258000 Spring B
Objective You need a design where 90% of the springs last (at least) to 400 000 cycles.

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First Primitive Test: Average Lifetime Design A Design B 726000 615000 508000 808000 755000 849000 384000 667000 515000 483000 631000 529000 730000 651000 446000 343000 960000 730000 730000 973000 258000 635000 Average Design B looks better…. .
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%

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We now fit a smooth curve (2 nd order power) to the data and to find F (400000)
We repeat the same for Design B

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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.

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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 λ and κ are the Weibull parameters ( κ is the shape parameter also known to material scientists as the Weibull modulus and λ is the scale or length parameter. Note that λ 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.
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

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Fatigue Life is a Statistical Quantity - Fatigue Life is a...

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