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Fatigue analysis: How damage is calculated |
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Fatigue analysis: How damage is calculated |
For each load case and fatigue point, OrcaFlex calculates damage values as follows:
The S-N curve defines the number of cycles to failure, $N(S)$, for stress range $S$, and also defines an endurance limit, $F_\mathrm{L}$, below which no damage occurs. OrcaFlex uses these to calculate a damage value given by \begin{equation} D(S) = \begin{cases} 1/N(S) & \text{if $S \gt F_\mathrm{L}$} \\ 0 & \text{if $S \leq F_\mathrm{L}$} \end{cases} \label{Damage} \end{equation} The damage value $D$ can be thought of as the proportion of the fatigue life that is used up by one cycle of stress range $S$.
S-N data are usually generated assuming that the mean stress, $\Sm$ is zero. The precise definition of mean stress, $\Sm$, depends on the analysis method being used:
A non-zero value of mean stress will, generally speaking, require a modification of the stress range $S$ used in the damage equation (\ref{Damage}). This modified value, the effective stress $\Se$, can be calculated by a number of different methods as follows.
The Goodman and Soderberg models are given by \begin{equation} \Se = \begin{cases} \cfrac{\Sr}{1 - \Sm/\Sref} & \text{if $0 \lt \Sm \lt \Sref$} \\ \Sr & \text{if $-\Sref \lt \Sm \leq 0$} \end{cases} \end{equation} where $\Sr$ is the true stress range, $\Sm$ is the mean stress and $\Sref$ is the reference stress as specified in the S-N data. For the Goodman model $\Sref$ is the tensile strength SMTS and for the Soderberg model $\Sref$ is the yield strength SMYS.
The Gerber model is \begin{equation} \Se = \frac{\Sr}{1 - (\Sm/\Sref)^2} \textrm{ if $-\Sref \lt \Sm \lt \Sref$} \end{equation} where $\Sref$ is the tensile strength SMTS.
The Smith-Watson-Topper model is defined in terms of the R ratio \begin{equation} \Se = \Sr\sqrt{\frac{2}{1-R}} \end{equation} where \begin{equation} \begin{aligned} R &= \begin{cases} \cfrac{\Smin}{\Smax} & \text{if $\Smax \neq 0$} \\ 0 & \text{otherwise} \\ \end{cases} \\ \Smin &= \begin{cases} S_1 & \text{if $|S_1| \lt |S_2|$} \\ S_2 & \text{otherwise} \\ \end{cases} \\ \Smax &= \begin{cases} S_2 & \text{if $|S_1| \lt |S_2|$} \\ S_1 & \text{otherwise} \\ \end{cases} \\ S_1 &= \Sm + \Sres + \Sr/2 \\ S_2 &= \Sm + \Sres - \Sr/2 \\ \Sres &= \textrm{residual stress as defined by the S-N data} \\ \end{aligned} \end{equation}
For a regular analysis, the stress range $S$ is defined by $S = \Smax-\Smin$, where $\Smax$ and $\Smin$ are the maximum and minimum values of stress over the last simulated wave cycle. The associated single-occurrence load case damage value (\ref{Damage}) is then given by $D(\kappa S)$, where $\kappa$ is the product of the stress concentration factor and the thickness correction factor. If one of the mean stress effects methods is used, then the equivalent stress range $\Se$ is used for $S$.
For rainflow analysis, the stress time history is analysed using the rainflow cycle counting method. This gives a number of stress ranges for half-cycles, say $S_i$ where $i$ runs from 1 to the number of stress ranges. The associated single-occurrence load case damage value (\ref{Damage}) is then given by \begin{equation} \frac{1}{2} \sum D(\kappa S_i) \end{equation} where the summation is over all the half-cycles and $\kappa$ is as for regular analysis. Note that the factor of $\small{1/2}$ arises because the rainflow algorithm counts half-cycles rather than full-cycles. If one of the mean stress effects methods is used, then the equivalent stress range is used for each $S_i$.
For either of the spectral analysis methods, damage is calculated in the frequency domain by statistical methods. In either case, the calculation requires a power spectral density function (PSD) for response, obtained as follows.
For a frequency domain analysis, the PSD is obtained from a frequency domain simulation. This PSD is then used to calculate damage using either Dirlik's formula or the Rayleigh distribution. The stress concentration factor, thickness correction factor and mean stress effects are all accounted for in the spectral damage calculation.
For a response RAOs analysis, the response PSD is obtained from either a time domain response calculation or a frequency domain simulation. OrcaFlex calculates response RAOs which are then combined with the load case wave spectrum to give the response PSD. Once the response PSD is calculated, damage is calculated in exactly the same manner as for frequency domain spectral analysis.
For detailed references on how spectral fatigue analysis calculates damage from stress PSDs, we recommend
T-N curves are handled in an analogous way to S-N curves. A T-N curve defines the number of cycles to failure, $N(T)$, for effective tension range $T$.
As for S-N curves, OrcaFlex defines damage as \begin{equation} D(T) = \frac{1}{N(T)} \end{equation} The summation of damage is then performed in an identical manner to that for S-N curves.
SHEAR7 fatigue is rather different from the other fatigue methods, in that the damage itself is calculated not by OrcaFlex, but by SHEAR7. OrcaFlex merely provides a means to collate, sum and plot the damage from a number of different SHEAR7 load cases in a convenient manner. This is done based on the damage rate output in the SHEAR7 .plt file; this value is multiplied by the load case exposure time to produce the damage associated with that load case.