The standard error SE of log_{10}(N) is used to adjust the fatigue life or the damage predicted to any given **probability of survival**. Fatigue life data always include some scatter, and at any given level of stress, the distribution of fatigue lives is assumed to be a log-normal distribution. Or stated differently, a Normal or Gauss distribution of the logarithm of the fatigue life.

## The standard error

In practice, if we want to make a life or damage prediction based on a particular percentage probability of survival, we use a lookup table (see Table 1 below) to determine the deviation from the mean (50%) life in terms of the number of standard errors.

The values in the table are calculated using the **Probability Density Function** (PDF) and the **Cumulative Density Function** (CDF) of the normal distribution.

The Probability Density Function (PDF) is the normal distribution curve shown in Figure 1 below:

With:

**μ**the mean of the data**σ**the standard deviation of the mean

The Cumulative Density Function is the integral of the PDF:

The percentage certainty of survival is then equal to **1 – CDF** (see also Figure 1).

Number of SD’s from the mean | % Certainty of Survival |
---|---|

-5 | 99.99997 |

-4 | 99.997 |

-3 | 99.87 |

-2 | 97.72 |

-1 | 84.13 |

0 | 50 |

1 | 15.87 |

2 | 2.28 |

3 | 0.13 |

4 | 0.003 |

5 | 0.00003 |

**Table 1.**Lookup table for Certainty Of Survival

## Calculation example

As an example, let’s say we have a component which is subject to a cyclic stress with constant amplitude, cycling between ±300 MPa. The component has a material for which we know the parameters of the S-N curve for a certainty of survival of 50%. Those parameters are:

- stress range intercept
**SRI**= 1300 MPa - the slope of the curve
**b**= -0.0612_{1} - standard error
**SE**= 0.12

The above parameters define an S-N curve in terms of the **stress range**, not the stress amplitude. The stress range S_{r} is a function of the number of cycles to failure N:

S_{r} = SRI ⋅ N^{b1}

In our case the stress range is 2 ⋅ 300 MPa = 600 MPa. The predicted number of cycles to failure for the component, based on a certainty of survival of 50% S-N curve, will be:

600 = 1300 ⋅ N_{50}^{-0.0612}

or **N _{50} = 306760 cycles**.

We now want to use the design S-N curve (typically a certainty of survival of 97.7% is used in most designs) and not the mean S-N curve. What is the predicted number of cycles at which the component will fail, based on a certainty of survival of 97.7% S-N curve?

First we need to find out the number of standard deviations *n* from the mean that corresponds to a certainty of survival of 97.7%. In Table 1 we find n = -2 for a certainty of survival of 97.7%.

The S-N curve is adjusted by shifting the curve to the left, or put differently, by reducing the number of cycles:

log_{10}(N) = log_{10}(N_{50}) – n ⋅ SE

or:

N = N_{50} ⋅ 10^{ – (n ⋅ SE)}

N_{97} = 306760 ⋅ 10^{ – (2 ⋅ 0.12)}

This gives us **N _{97} = 176522 cycles**, or a reduction of 42% of the fatigue life compared to N

_{50}.

## Our Services

**Fatigue Analysis** — Dutch | English**Finite Element Analysis** — Dutch | English

## Our Courses

We organize software-independent courses on **Finite Element Analysis** and **Fatigue Analysis**, in Dutch and English. If interested, find out more below:

- Practical Introduction to Fatigue Analysis with FEA — Dutch | English
- Practical Introduction to Finite Element Analysis — Dutch | English