Weibull distribution using the fatigue test as an example (survival/failure/reliability analysis)
The Weibull distribution is frequently used in failure analysis to describe the breakdown of mechanical or electronic components. In the Wohler test, the durability of materials is examined by subjecting samples to alternating tensile and compressive stress. The number of load cycles until failure varies due to material deviations. The results are statistically evaluated to predict lifespan. Cumulative frequency describes how many samples have failed up to a certain number of load cycles, while a histogram shows the frequency of failures in specific load cycle intervals.
The Weibull distribution mathematically models the lifespan of components. It is defined by the shape parameter k and the scale parameter T. The shape parameter determines the curve's form, while the scale parameter influences its position. The probability of a sample failing can be calculated using the Weibull distribution function. The scale parameter corresponds to the number of load cycles at which 63.2% of the samples have failed. To determine the distribution parameters, experimental data is transferred into a Weibull plot, allowing for a linear representation. The slope of the resulting line corresponds to the shape parameter, while the intercept is used to calculate the scale parameter.
The Weibull distribution enables the calculation of the mean time to failure, determined by the expected value of the distribution. Additionally, the standard deviation is calculated as a measure of data dispersion. The survival probability describes the likelihood that a component will remain functional for a given time without failure. The failure rate indicates how many samples fail per unit of time, distinguishing between absolute and relative failure rates. The relative failure rate refers to the still functional samples and increases with longer operating time.
The Weibull distribution can model different failure behaviors. When k smaller 1, the failure rate decreases over time, whereas for k greater 1, it increases. A special case is the bathtub curve, which describes three phases of a component’s lifespan: an initial phase with high early failures, a phase with a constant failure rate, and a phase with an increasing failure rate due to wear. Adjustments to the Weibull function allow for the consideration of a failure-free period, such as in components that initially wear down before reaching a critical failure threshold.
00:00 Stress-cycle curve (Wöhler curve)
02:18 Cumulative frequency
03:48 Frequency (histogram)
04:48 Relationship between frequency and cumulative frequency
05:24 Relative frequency
06:23 Probability
07:00 Corrected probability (population and sample)
10:00 Weibull distribution
12:00 Determination of the probability
13:47 Determination of the Weibull modulus and the scale parameter
15:35 Evaluation of the data (Weibull plot)
18:04 Characteristic lifetime
18:48 Weibull density function
20:16 Mean time to failure (empirical expected value)
21:12 Sample variance (empirical standard deviation)
22:21 Expected value and standard deviation
23:20 Probability of survival (reliability)
25:04 Absolute failure rate
26:29 Relative failure rate (hazard function)
28:32 Derivation of the hazard function
30:55 Selected Weibull distribution functions in comparison
32:00 Bathtub curve
33:39 Weibull distribution with failure free time