# Random Response Fatigue Analysis

The study of fatigue life of structures under random loading.

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 PSD stresses generated from the random response analysis.

## Power Spectral Density (PSD) Moments

PSD Moments are calculated based on the Stress PSD generated from the random response analysis as:
Moments are calculated based on:(1)
${m}_{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sum _{k\text{ }=\text{ }1}^{N}{f}_{k}^{n}{G}_{k}\delta f$
Where,
${f}_{k}^{}$
Frequency value.
${G}_{k}$
PSD response value at frequency ${f}_{k}^{}$ .

## Calculate Probability of Stress Range Occurence

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 $\text{P}\left(\Delta {S}_{i}\right)$ of a stress range occuring between $\left(\Delta {S}_{i}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\delta S/2\right)$ and $\left(\Delta {S}_{i}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\delta S/2\right)$ is: (2)
$P\left(\Delta {S}_{i}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}p\delta {S}_{i}$

## 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. DIRLIK, LALANNE, NARROW, and Steinberg 3 band (THREE) options are available to define the damage model.

DIRLIK
DIRLIK postulated a closed form solution to the determination of the Probability Density Function as:(3)
$p\left(S\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\frac{{D}_{1}}{Q}{e}^{\frac{-Z}{Q}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{{D}_{2}Z}{{R}_{2}}{e}^{\frac{-{Z}^{2}}{2{R}^{2}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{D}_{3}Ze\frac{-{Z}^{2}}{2}}{2\sqrt{{m}_{0}}}$
Where,
${D}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{2\left({x}_{m}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\gamma }^{2}\right)}{1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\gamma }^{2}}$
${D}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{D}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{D}_{1}^{2}}{1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}R}$
${D}_{3}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{D}_{1}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{D}_{2}$
$Z\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{S}{2\sqrt{{m}_{0}}}$
$Q\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1.25\left(\gamma \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{D}_{3}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{D}_{2}R\right)}{{D}_{1}}$
$R\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\gamma \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{x}_{m}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{D}_{1}^{2}}{1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{D}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{D}_{1}^{2}}$
${x}_{m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{m}_{1}}{{m}_{0}}\sqrt{\frac{{m}_{2}}{{m}_{4}}}$
$\gamma \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{m}_{2}}{\sqrt{{m}_{0}{m}_{4}}}$
Irregularity factor.
$S$
Stress range.
LALANNE
The LALANNE random fatigue damage model depicts the probability density function as:(4)
$p\left(S\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{\sqrt{{m}_{0}}}\text{\hspace{0.17em}}\frac{\sqrt{1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\gamma }^{2}}}{\sqrt{2\pi }}{e}^{\frac{-{S}^{2}}{8{m}_{0}\left(1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\gamma }^{2}\right)}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{{S}_{\gamma }}{4\sqrt{{m}_{0}}}\left(1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}erf\left(\frac{S\gamma }{2\sqrt{2{m}_{0}\left(1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\gamma }^{2}\right)}}\right)\right)$
Where,
$\gamma \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{m}_{2}}{\sqrt{{m}_{0}{m}_{4}}}$
Irregularity factor.
$S$
Stress range.
NARROW
The Narrow Band random fatigue damage model uses the following probability functions:(5)
$p\left(S\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(\frac{S}{4{m}_{0}}{e}^{-\left(\frac{{S}^{2}}{8{m}_{0}}\right)}\right)$
Where,
$S$
Stress range.
By default, the number of zero crossings ${n}_{zcross}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{{m}_{2}/{m}_{0}}$ is used instead of number of peaks ${n}_{peaks}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{{m}_{4}/{m}_{2}}$ for NARROW band, because the numerical calculations involving ${m}_{4}$ can sometimes lead to unstable numerical behavior. If the signal is ideally NARROW band, the number of zero crossings and number of peaks should be almost equal.
THREE
The Steinberg 3-band random fatigue damage model uses the following probability function:(6)
$P\left(S\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left\{\begin{array}{c}0.683\text{ }at\text{ }2\sqrt{{m}_{0}}\\ 0.271\text{ }at\text{ }4\sqrt{{m}_{0}}\\ 0.043\text{ }at\text{ }6\sqrt{{m}_{0}}\end{array}\right\$
Where,
$S$
Stress range.
Unlike the other damage models, for THREE band, these 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, these 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.
The probability density function can be adjusted based on the following parameters defined in the random response fatigue solution settings.

## Upper Stress Range Factor

Calculates the upper limit of the stress range as:

upper limit of the stress range = 2*RMS Stress*Upper Stress Range factor

The RMS Stress is output from random response subcase. The stress ranges of interest are limited by the upper limit of the stress range. Any stresses beyond the upper limit are not considered in random fatigue damage calculations.

## Upper Stress Range

Directly specify the upper stress range.

## Number of Bins

Calculates the width of the stress range $DS\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\delta \text{S}$ for which the probability is calculated (see Figure 2). The default is 100 and the first bin starts from 0.0 to $\delta \text{S}$ . The width of the stress range is calculated as DS=Upper stress range/Number of bins.

## Stress Range Width

Directly defines the width of the stress ranges ( $\delta \text{S}$ ).

## Calculate Probability of Stress Range Occurence

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, NARROW
The probability $\text{P}\left({S}_{i}\right)$ of a stress range occuring between $\left(\Delta {S}_{i}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\delta S/2\right)$ and $\left(\Delta {S}_{i}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\delta S/2\right)$ is: (7)
$P\left({S}_{i}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{p}_{i}\left({S}_{i}\right)\delta S$
THREE
See Equation 6.
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

The following information may help assist in choosing the damage model.
1. 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. 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 stress range distribution. Therefore, if you know the input random data does not have a wide range of stress-range distribution, and the distribution is mainly concentrated about a particular stress range, you should select NARROW. This model 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 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, THREE
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:(8)
${n}_{zcross}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{\frac{{m}_{2}}{{m}_{0}}}$
Where,
${m}_{n}$
Corresponding moments calculated.
DIRLIK, LALANNE
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:(9)
${n}_{peaks}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{\frac{{m}_{4}}{{m}_{2}}}$
Where,
${m}_{n}$
Corresponding moments calculated.

## Number of Cycles

NARROW band, THREE band
The total number of cycles is calculated as:(10)
${N}_{T}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{n}_{zcross}T$
Where,
T
Total exposure time.
DIRLIK, LALANNE
The total number of cycles is calculated as:(11)
${N}_{T}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{n}_{peaks}T$
Where,
T
Total exposure time.
Total Number of Cycles of Particular Stress Range
The total number of cycles with with stress range $\Delta {S}_{i}$ is calculated as:(12)
${N}_{i}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}P\left(\Delta {S}_{i}\right){N}_{T}$

## 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:(13)
${N}_{f}\left({S}_{i}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\left(\frac{{S}_{i}}{{S}_{f}}\right)}^{\frac{1}{b}}$
Total fatigue damage as a result of the applied random loading is calculated as: (14)
$D\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{N}\frac{{N}_{i}}{{N}_{f}\left({S}_{i}\right)}$

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 in random fatigue analysis event setup.