# Random Response Fatigue Analysis

The setup is similar to a Random Response Analysis setup, with an additional Fatigue subcase. The LCID field on the FATLOAD entry references the subcase ID of the Random Response Analysis subcase.

Power Spectral Density (PSD) results from the Random Response Analysis are used to calculate Moments (${m}_{n}$) that are used to generate the probability density function for the number of cycles versus the stress range.

The PSD Moments are calculated based on the Stress PSD generated from the Random Response Analysis, as shown below.

## Input

Calculates Random Response Fatigue.

### Power Spectral Density (PSD) Moments

PSD Moments (${m}_{n}$) are calculated based on the Stress PSD generated from the Random Response Analysis as:
The moments are calculated based on:(1) ${m}_{n}=\sum _{k=1}^{N}{f}_{k}^{n}{G}_{k}\delta f$
Where,
${f}_{k}$
Frequency value
${G}_{k}$
PSD response value at frequency ${f}_{k}$

The stability of ${m}_{0}$ can be checked by setting PARAM, CHKM0, YES. A warning is printed if the frequency interval must be further refined.

### Calculate Probability of Stress Range Occurrence

Calculation of the Probability of occurrence of a stress range between the initial and final stress range values within each bin section are user-defined.

The probability $P\left(\text{Δ}{S}_{i}\right)$ of a stress range occuring between $\left(\text{Δ}{S}_{i}-\delta S/2\right)$ and $\left(\text{Δ}{S}_{i}+\delta S/2\right)$ is $P\left(\text{Δ}{S}_{i}\right)={p}_{i}\delta S$.

### Probability Density Function (probability density of number of cycles versus stress range)

PSD Moments calculated as shown above are used in the generation of a Probability Density Function $f\left({m}_{n}\right)$ for the stress range. The function is based on the specified damage model on the RNDPDF continuation line on FATPARM. Currently, DIRLIK, LALANNE, NARROW, and THREE options are available to define the damage model. Multiple damage models are also supported (the worst damage is selected for output from the specified damage models).

DIRLIK (Default Damage Model):
DIRLIK postulated a closed form solution to the determination of the Probability Density Function as:(2) $p\left(S\right)=\frac{\frac{{D}_{1}}{Q}{e}^{\frac{-Z}{Q}}+\frac{{D}_{2}Z}{{R}^{2}}{e}^{\frac{-{Z}^{2}}{2{R}^{2}}}+{D}_{3}Z{e}^{\frac{-{Z}^{2}}{2}}}{2\sqrt{{m}_{0}}}$
Where,
${D}_{1}=\frac{2\left({x}_{m}-{\gamma }^{2}\right)}{1+{\gamma }^{2}}$
${D}_{2}=\frac{1-\gamma -{D}_{1}+{D}_{1}^{2}}{1-R}$
${D}_{3}=1-{D}_{1}-{D}_{2}$
$Z=\frac{S}{2\sqrt{{m}_{0}}}$
$Q=\frac{1.25\left(\gamma -{D}_{3}-{D}_{2}R\right)}{{D}_{1}}$
$R=\frac{\gamma -{x}_{m}-{D}_{1}^{2}}{1-\gamma -{D}_{1}+{D}_{1}^{2}}$
$\gamma =\frac{{m}_{2}}{\sqrt{{m}_{0}{m}_{4}}}$
Irregularity Factor
${x}_{m}=\frac{{m}_{1}}{{m}_{0}}\sqrt{\frac{{m}_{2}}{{m}_{4}}}$
$S$
Stress range
LALANNE:
The LALANNE Random Fatigue Damage model depicts the probability density function as: (3) $p\left(S\right)=\frac{1}{\sqrt{{m}_{0}}}\frac{\sqrt{1-{\gamma }^{2}}}{\sqrt{2\pi }}{e}^{\frac{-{S}^{2}}{8{m}_{0}\left(1-{\gamma }^{2}\right)}}+\frac{S\gamma }{4\sqrt{{m}_{0}}}\left(1+erf\left(\frac{S\gamma }{2\sqrt{2{m}_{0}\left(1-{\gamma }^{2}\right)}}\right)\right)$
Where,
$\gamma =\frac{{m}_{2}}{\sqrt{{m}_{0}{m}_{4}}}$
Irregularity factor
$S$
Stress range
NARROW:
The Narrow Band Random Fatigue Damage model uses the following probability functions:(4) $p\left(S\right)=\left(\frac{S}{4{m}_{0}}{e}^{-\left(\frac{{S}^{2}}{8{m}_{0}}\right)}\right)$

Where, $S$ is the stress range.

In the NARROW band model, if the irregularity factor is less than 0.95, then OptiStruct will issue a warning that the irregularity factor is small. Ideally, the irregularity factor should be 1.0 if the signal is NARROW band.

By default, OptiStruct uses number of zero crossings (${n}_{zcross}=\sqrt{{m}_{2}/{m}_{0}}$) instead of number of peaks (${n}_{peaks}=\sqrt{{m}_{4}/{m}_{2}}$) for NARROW band, because the numerical calculations involving ${m}_{4}$ sometimes may lead to unstable numerical behavior. If the signal is ideally NARROW band, the number of zero crossings and number of peaks should almost be equal. However, PARAM,NBZRCRS,NO can be used to switch OptiStruct to using number of peaks (${n}_{peaks}$ ) for NARROW band.

THREE:

The Steinberg 3-Band Random Fatigue Damage model uses the following probability function.

Unlike the other damage models, for THREE band, the following values are probability (and not probability density). This is also evident in the use of upper case $P\left(S\right)$ compared to the lower case $p\left(S\right)$ for the other damage models. For the THREE damage model, the following probabilities are directly used to calculate the number of cycles by multiplying $P\left(S\right)$ with the total number of zero-crossings in the entire time history (for other damage models (except THREE), the probability density values are first multiplied by DS (bin size) to get the probability).(5)

Where, $S$ is the stress range.

The probability density function can be adjusted based on the following parameters defined on the RANDOM continuation line of FATPARM:
FACSREND
Calculates the upper limit of the stress range (SREND). This is calculated as SREND = 2*RMS Stress*FACSREND. The RMS Stress is output from Random Response Subcase. The stress ranges of interest are limited by SREND. Any stresses beyond SREND are not considered in Random Fatigue Damage calculations.
SREND
Directly specifies the upper limit of the stress range (if SREND is blank, then the SREND calculated based on FACSREND is used).
NBIN
Calculates the width of the stress range (DS = $\delta S$) for which the probability is calculated (Figure 2). The default is 100 and the first bin starts from 0.0 to $\delta S$. The width of the stress range is calculated as DS=SREND/NBIN.
DS
Directly defines the width of the stress ranges ($\delta S$). (if DS is blank, then the DS calculated based on NBIN is used).

### Calculate Probability of Stress Range Occurrence

Calculation of the Probability of occurrence of a stress range between the initial and final stress range values within each bin section are based on the damage models.

DIRLIK, LALANNE, and NARROW Damage Models

The probability $P\left({S}_{i}\right)$ of a stress range occurring between $\left({S}_{i}-\delta S/2\right)$ and $\left({S}_{i}+\delta S/2\right)$ is $P\left({S}_{i}\right)={p}_{i}\left({S}_{i}\right)\delta S$.

THREE Damage Model
The probability is directly defined using the probability function defined above. It is being repeated here for clarity.(6)

Where, $S$ is stress range.

For the THREE damage model, there are only three bins. The number of cycles at each stress range (2*RMS, 4*RMS, and 6*RMS) are calculated by directly multiplying the corresponding probabilities with the total number of zero-crossings (refer to section below regarding calculation of number of zero-crossings).

### Select Damage Models

DIRLIK, LALANNE, NARROW, and THREE are available for selection on the PDFi fields of the RNDPDF continuation line on the FATPARM Bulk Data Entry. The following information may provide additional understanding to help choose the damage model for an OptiStruct run.
1. You can see from the previous sections, that the PSD moments of stress are used to calculated corresponding moments, which are used to determine the probability density function for the stress-range.
2. DIRLIK and LALANNE models generate probabilities across a wider distribution of the stress-range spectrum. Therefore, these models should be used when the input random signal consists of a variety of stress-ranges across multiple frequencies. Therefore, the information in the probability density function better captures the wider range in stress-range distribution if DIRLIK and LALANNE are used.
3. The NARROW model is intended for random signals for which the stress range is expected to be closely associated with a high probability of particular certain stress range distribution. Therefore, if you know that the input random data does not have a wide range of stress-range distribution, and that the distribution is mainly concentrated about a particular stress range, then you should select NARROW, since it expects the highest probability of stress-ranges to lie at or around this particular stress range.
4. The THREE model is like the NARROW model, except that it expects the distribution of the random signal to contain, in addition to the association with 1*RMS, associations (albeit smaller) with 2*RMS, and 3*RMS. Therefore, if your input random data is mainly clustered around stress ranges in 1*RMS, and to a lesser extent, 2*RMS, and 3*RMS, then you should select THREE.

### Number of Peaks and Zero Crossings

NARROW and THREE Damage Models
The number of zero crossings per second in the original time-domain random loading (from which the Frequency based Random PSD load is generated) is determined as:(7) ${n}_{zcross}=\sqrt{\frac{{m}_{2}}{{m}_{0}}}$
DIRLIK and LALANNE Damage Models
The number of peaks per second in the original time-domain random loading (from which the Frequency based Random PSD load is generated) is determined as:(8) ${n}_{peaks}=\sqrt{\frac{{m}_{4}}{{m}_{2}}}$

Where, ${m}_{n}$ is the corresponding moments calculated, as shown in Power Spectral Density (PSD) Moments.

The total number of cycles for THREE band and NARROW band is calculated as:(9) ${N}_{T}={n}_{zcross}T$
The total number of cycles for DIRLIK, LALANNE, and NARROW (with PARAM,NBZRCRS,NO is calculated as):(10) ${N}_{T}={n}_{peaks}T$

Where, $T$ is Total exposure time given by the T# fields on the FATSEQ Bulk Data Entry.

Total Number of Cycles of a Particular Stress Range
The total number of cycles with Stress range $\text{Δ}{S}_{i}$ is calculated as:(11) ${N}_{i}=P\left({S}_{i}\right){N}_{T}$

### Fatigue Life and Damage

Fatigue Life (maximum number of cycles of a particular stress range ${S}_{i}$ for the material prior to failure) is calculated based on the Material SN curve as:(12) ${N}_{f}\left({S}_{i}\right)={\left(\frac{{S}_{i}}{{S}_{f}}\right)}^{\frac{1}{b}}$
Total Fatigue Damage as a result of the applied Random Loading is calculated based on:(13) $D=\sum _{i=1}^{N}\frac{{N}_{i}}{{N}_{f}\left({S}_{i}\right)}$

To account for the mean stress correction with any loading that leads to a mean stress different from zero, you can define a static subcase that consists of such loading (typically gravity loads). This static subcase can be referenced on the STSUBID field of the RANDOM continuation line.