# Scatter in Fatigue Material Data

Handle scatter in fatigue test results.

_{a}or range S

_{R}versus cycles to failure N, the relationship between S and N can be described by straight line segments. Normally, a one or two segment idealization is used.

S | Log (S) | Log (N) |
---|---|---|

2000.0 | 3.3 | 3.9 |

2000.0 | 3.3 | 3.7 |

2000.0 | 3.3 | 3.75 |

2000.0 | 3.3 | 3.79 |

2000.0 | 3.3 | 3.87 |

2000.0 | 3.3 | 3.9 |

The experimental scatter exists in both Stress Amplitude and Life data. The Standard Error of the scatter of log(N) is required as input (SE field for SN curve). The sample mean is provided by the SN curve as $\mathrm{log}({N}_{i}^{sm})$ whereas, the standard error is input via the SE field.

- Standard error of log(N) normal distribution SE
- Certainty of Survival required for this analysis

A normal distribution or gaussian distribution is a probability density function which implies that the total area under the curve is always equal to 1.0.

The SN curve data you defined is assumed as a normal distribution, which is typically characterized by the following Probability Density Function:

Where:

${x}_{s}$ is the data value ( $\mathrm{log}({N}_{i})$ ) in the sample you defined.

${\mu}_{s}$ is the sample mean ( $\mathrm{log}({N}_{i}^{sm})$ ).

${\sigma}_{s}$ is the standard deviation of the sample (which is unknown, as you input only Standard Error (SE).

The above distribution is the distribution of the sample you defined, and not the full population space. Since the true population mean is unknown, the range of the true population mean is estimated from the sample mean and the sample SE, and then the Certainty of Survival you defined is used to perturb the sample mean.

Standard Error is the standard deviation of the normal distribution created by all the sample means of samples drawn from the full population. From a single sample distribution data, the Standard Error is typically estimated as $SE=\left({\scriptscriptstyle \raisebox{1ex}{${\sigma}_{s}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{{n}_{s}}$}\right.}\right)$ , where ${\sigma}_{s}$ is the standard deviation of the sample, and ${n}_{s}$ is the number of data values in the sample. The mean of this distribution of all the sample means is actually the same as the true population mean. The certainty of survival you provided is applied on this distribution of all the sample means.

For the normal distribution of all the sample means, the mean of this distribution is the same as the true population mean $\mu $ , the range of which is what you want to estimate.

Statistically, you can estimate the range of true population mean as follows:

That is,

Since the value on the left side is more conservative, use the following equation to perturb the SN curve:

Where,

$\mathrm{log}({N}_{i}^{m})$ is the perturbed value

$\mathrm{log}({N}_{i}^{sm})$ is the sample mean you defined (SN curve)

$SE$ is the standard error (SE)

Z-values (calculated) | Certainty of Survival (Input) |
---|---|

0.0 | 50.0 |

-0.5 | 69.0 |

-1.0 | 84.0 |

-1.5 | 93.0 |

-2.0 | 97.7 |

-3.0 | 99.9 |

Notice how the SN curve is modified to the required certainty of survival and standard error input. This technique allows you to handle fatigue material data scatter using statistical methods and predict data for the required survival probability values.