4. Statistical estimation of damage

Mechanical damage is calculated using Miner’s linear accumulation rule.

The damage \(D\) caused by \(N\) cycles of half amplitude \(S\) is expressed by \(D\text{=}\frac{N}{\text{Nr}(S)}\)

where \(\text{Nr}(S)\) is the allowable number of cycles determined by the material’s endurance curve.

Mechanical damage is a random variable whose average is determined.

4.1. Average damage

The average damage is written in the form of the mathematical expectation:

\(E(D)\text{=}T\underset{{S}_{\text{min}}}{\overset{{S}_{\text{max}}}{\mathrm{\int }}}\frac{N(S)}{{N}_{r}(S)}\text{dS}\) where \(T\) is the duration of the signal

The two proposed counting methods calculate the number of cycles \(N(S)\) from positive amplitude constraints from where \({S}_{\text{min}}\text{=}0\) (except when the Wöhler curve is given in the form of « current zone », in which case \({S}_{\text{min}}\text{=}{S}_{l}\) with \({S}_{l}\) material endurance limit).

Moreover, the distribution laws used being continuous, \({S}_{\text{max}}\text{=}\mathrm{\infty }\). However, experience shows that the expression to be integrated attenuates quickly and we therefore take \({S}_{\text{max}}\mathrm{=}10{\sigma }_{S}\). where \({\sigma }_{S}\) is the standard deviation of the signal.

The calculation of \(E(D)\) is carried out by numerical integration, by the trapezium method using \(\frac{{S}_{\text{max}}\text{-}{S}_{\text{min}}}{300.}\) as an integration step.

Note:

In the case of the method for counting level breaches and for a Wöhler curve expressed in the mathematical form proposed by Basquin, the average damage per unit of time to an analytical expression (this expression is not used in the command POST_FATI_ALEA) .