Newsletter Artikel 28/2019

Endurance Safety and associated Probability of Failure

In an endurance safety factor analysis, the safety factor SF is defined as the quotient of the local fatigue strength σD and the locally occurring stress amplitude σA: SF = σ/ σA 

What does a safety factor of 1 mean? First of all, that the sustainable amplitude is completely exhausted. For a further (statistical) interpretation, the survival probability pü of the S/N curve is taken into account: in 100 * pü% of cases, the load can be sustained, in 100*(1-pü)% not. For example, with the 97.5% survival probability typical of the FKM guideline, this means that out of 1000 parts, 975 hold, 25 not.

In the real application, however, one will demand significantly higher survival probabilities. To take account of the desired / required survival probability, the statistical influence factor can be activated in FEMFAT.

Now, how is a node with safety other than 1 different in terms of survival probability?

Some Theory:

To answer this question, we need some tools from probability theory, which are summarized below.

 

According to Haibach, the probability of failure pA is calculated for logarithmically normally distributed stress amplitudes using the cumulative distribution function Φ(u) of the standard normal distribution

 

A typical measure of variability - especially in mechanical engineering - is the range of dispersion TS. The range of dispersion is defined as the ratio of component fatigue strength at 10% probability of survival and component fatigue strength at 90% probability of survival.

The range of dispersion can be specified in FEMFAT in the node characteristics menu and corresponds by default to the value recommended by Haibach of 1.26.

For the conversion to other survival probabilities the statistical influencing factor fstat is calculated in FEMFAT. The value (1/fstat) is used for the scaling of the local endurance limit.

With the initial survival probability (the material S/N curve) pü1, the corresponding fatigue strength value of the specimen σD, specimen and the fatigue strength value with desired survival probability pü2 we get with the respective quantiles ucomponent and uspecimen:

 

How can a probability of failure be calculated for any safety factors other than 0?

Based on the given survival probability pü1 we get for the safety factor

We are looking now for the survival probability pü2, s.t.

 

Dividing equation (2) by equation (3) gives

Solving this equation for u2 results in

From this, the probability of failure pA is now obtained directly with equation (1)

 

With a suitable approximation formula (for example, Waissi and Rossin*) for the cumulative distribution function of the standard normal distribution, this integral and thus the local probability of failure can be calculated in a comfortable way with the Results Manager.