Random Response Fatigue Analysis

The study of fatigue life of structures under Random Loading.

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGUbaabeaaaaa@3807@ ) 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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGUbaabeaaaaa@3807@ ) are calculated based on the Stress PSD generated from the Random Response Analysis as:
Figure 1. Calculation of PSD Moments


The moments are calculated based on:

m n = k = 1 N f k n G k δ f

Where,
f k
Frequency value
G k
PSD response value at frequency f k

The stability of m 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIWaaabeaaaaa@37CC@ 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 ( Δ S i ) of a stress range occuring between ( Δ S i δ S / 2 ) and ( Δ S i + δ S / 2 ) is P ( Δ S i ) = p i δ 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 ( m n ) 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:

p( S )= D 1 Q e Z Q + D 2 Z R 2 e Z 2 2 R 2 + D 3 Z e Z 2 2 2 m 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaacaWGtbaacaGLOa GaayzkaaGaeyypa0ZaaSaaaeaadaWcaaqaaiaadseadaWgaaWcbaGa aGymaaqabaaakeaacaWGrbaaaiaadwgadaahaaWcbeqaamaalaaaba GaeyOeI0IaamOwaaqaaiaadgfaaaaaaOGaey4kaSYaaSaaaeaacaWG ebWaaSbaaSqaaiaaikdaaeqaaOGaamOwaaqaaiaadkfadaahaaWcbe qaaiaaikdaaaaaaOGaamyzamaaCaaaleqabaWaaSaaaeaacqGHsisl caWGAbWaaWbaaWqabeaacaaIYaaaaaWcbaGaaGOmaiaadkfadaahaa adbeqaaiaaikdaaaaaaaaakiabgUcaRiaadseadaWgaaWcbaGaaG4m aaqabaGccaWGAbGaamyzamaaCaaaleqabaWaaSaaaeaacqGHsislca WGAbWaaWbaaWqabeaacaaIYaaaaaWcbaGaaGOmaaaaaaaakeaacaaI YaWaaOaaaeaacaWGTbWaaSbaaSqaaiaaicdaaeqaaaqabaaaaaaa@532E@

Where,
D 1 = 2 ( x m γ 2 ) 1 + γ 2
D 2 = 1 γ D 1 + D 1 2 1 R
D 3 = 1 D 1 D 2
Z = S 2 m 0
Q = 1.25 ( γ D 3 D 2 R ) D 1
R = γ x m D 1 2 1 γ D 1 + D 1 2
γ = m 2 m 0 m 4
Irregularity Factor
x m = m 1 m 0 m 2 m 4
S
Stress range
LALANNE:

The LALANNE Random Fatigue Damage model depicts the probability density function as:

p S = 1 2 m 0 1 γ 2 2 π e S 2 8 m 0 1 γ 2 + S γ 4 m 0 e S 2 8 m 0 1 + E r f S γ 2 2 m 0 1 γ 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaacaWGtbaacaGLOa GaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmamaakaaabaGa amyBamaaBaaaleaacaaIWaaabeaaaeqaaaaakmaabmaabaWaaSaaae aadaGcaaqaaiaaigdacqGHsislcqaHZoWzdaahaaWcbeqaaiaaikda aaaabeaaaOqaamaakaaabaGaaGOmaiabec8aWbWcbeaaaaGccaWGLb WaaWbaaSqabeaadaWcaaqaaiabgkHiTiaadofadaahaaadbeqaaiaa ikdaaaaaleaacaaI4aGaamyBamaaBaaameaacaaIWaaabeaalmaabm aabaGaaGymaiabgkHiTiabeo7aNnaaCaaameqabaGaaGOmaaaaaSGa ayjkaiaawMcaaaaaaaGccqGHRaWkdaWcaaqaaiaadofacqaHZoWzae aacaaI0aWaaOaaaeaacaWGTbWaaSbaaSqaaiaaicdaaeqaaaqabaaa aOGaamyzamaaCaaaleqabaWaaSaaaeaacqGHsislcaWGtbWaaWbaaW qabeaacaaIYaaaaaWcbaGaaGioaiaad2gadaWgaaadbaGaaGimaaqa baaaaaaakmaabmaabaGaaGymaiabgUcaRiaadweacaWGYbGaamOzam aabmaabaWaaSaaaeaacaWGtbGaeq4SdCgabaGaaGOmamaakaaabaGa aGOmaiaad2gadaWgaaWcbaGaaGimaaqabaGcdaqadaqaaiaaigdacq GHsislcqaHZoWzdaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaa aSqabaaaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaaacaGLOaGaay zkaaaaaa@6E4F@

Where,
γ = m 2 m 0 m 4
Irregularity factor
S
Stress range
NARROW:

The Narrow Band Random Fatigue Damage model uses the following probability functions:

p ( S ) = ( S 4 m 0 e ( S 2 8 m 0 ) )

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 z c r o s s = m 2 / m 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadQhacaWGJb GaamOCaiaad+gacaWGZbGaam4CaaqabaGccqGH9aqpdaGcaaqaamaa lyaabaGaamyBamaaBaaaleaacaaIYaaabeaaaOqaaiaad2gadaWgaa WcbaGaaGimaaqabaaaaaqabaaaaa@3D8E@ ) instead of number of peaks ( n p e a k s = m 4 / m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadchacaWGLb GaamyyaiaadUgacaWGZbaabeaakiabg2da9maakaaabaWaaSGbaeaa caWGTbWaaSbaaSqaaiaaisdaaeqaaaGcbaGaamyBamaaBaaaleaaca aIYaaabeaaaaaabeaaaaa@3C7D@ ) for NARROW band, because the numerical calculations involving m 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGTbWaaSbaaSqaaiaaisdaaeqaaa aa@3397@ 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 p e a k s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadchacaWGLb GaamyyaiaadUgacaWGZbaabeaaaaa@3787@ ) 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 ( S ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGqbWaaeWaaeaacaWGtbaacaGLOa Gaayzkaaaaaa@34F1@ compared to the lower case p ( S ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGqbWaaeWaaeaacaWGtbaacaGLOa Gaayzkaaaaaa@34F1@ 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 ( S ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGqbWaaeWaaeaacaWGtbaacaGLOa Gaayzkaaaaaa@34F1@ 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).

P ( S ) = { 0.683  at  2 m 0 0.271  at 4 m 0 0.043  at 6 m 0

Where, S is the stress range.

Figure 2. Probability Density Function (probability density of number of cycles versus 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 = δ S ) for which the probability is calculated (Figure 2). The default is 100 and the first bin starts from 0.0 to δ S . The width of the stress range is calculated as DS=SREND/NBIN.
DS
Directly defines the width of the stress ranges ( δ 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 ( S i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGqbWaaeWaaeaacaWGtbWaaSbaaS qaaiaadMgaaeqaaaGccaGLOaGaayzkaaaaaa@3615@ of a stress range occurring between ( S i δ S / 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadofadaWgaaWcbaGaam yAaaqabaGccqGHsisldaWcgaqaaiabes7aKjaadofaaeaacaaIYaaa aaGaayjkaiaawMcaaaaa@397C@ and ( S i + δ S / 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadofadaWgaaWcbaGaam yAaaqabaGccqGHRaWkdaWcgaqaaiabes7aKjaadofaaeaacaaIYaaa aaGaayjkaiaawMcaaaaa@3971@ is P ( S i ) = p i ( S i ) δ S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGqbWaaeWaaeaacaWGtbWaaSbaaS qaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaamiCamaaBaaa leaacaWGPbaabeaakiaacIcacaWGtbWaaSbaaSqaaiaadMgaaeqaaO Gaaiykaiabes7aKjaadofaaaa@3F06@ .

THREE Damage Model

The probability is directly defined using the probability function defined above. It is being repeated here for clarity.

P ( S ) = { 0.683  at  2 m 0 0.271  at 4 m 0 0.043  at 6 m 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGqbWaaeWaaeaacaWGtbaacaGLOa GaayzkaaGaeyypa0ZaaiqaaeaafaqabeWabaaabaGaaGimaiaac6ca caaI2aGaaGioaiaaiodacaqGGaGaaeyyaiaabshacaqGGaGaaGOmam aakaaabaGaamyBamaaBaaaleaacaaIWaaabeaaaeqaaaGcbaGaaGim aiaac6cacaaIYaGaaG4naiaaigdacaqGGaGaaeyyaiaabshacaqGGa GaaeinamaakaaabaGaamyBamaaBaaaleaacaaIWaaabeaaaeqaaaGc baGaaGimaiaac6cacaaIWaGaaGinaiaaiodacaqGGaGaaeyyaiaabs hacaqGGaGaaeOnamaakaaabaGaamyBamaaBaaaleaacaaIWaaabeaa aeqaaaaaaOGaay5Eaaaaaa@5375@

Where, S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGtbaaaa@3293@ 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:

n z c r o s s = m 2 m 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadQhacaWGJb GaamOCaiaad+gacaWGZbGaam4CaaqabaGccqGH9aqpdaGcaaqaamaa laaabaGaamyBamaaBaaaleaacaaIYaaabeaaaOqaaiaad2gadaWgaa WcbaGaaGimaaqabaaaaaqabaaaaa@3D88@

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:

n p e a k s = m 4 m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaadchacaWGLb GaamyyaiaadUgacaWGZbaabeaakiabg2da9maakaaabaWaaSaaaeaa caWGTbWaaSbaaSqaaiaaisdaaeqaaaGcbaGaamyBamaaBaaaleaaca aIYaaabeaaaaaabeaaaaa@3C77@

Where, m n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGUbaabeaaaaa@3807@ 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:

N T = n z c r o s s T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGobWaaSbaaSqaaiaadsfaaeqaaO Gaeyypa0JaamOBamaaBaaaleaacaWG6bGaam4yaiaadkhacaWGVbGa am4CaiaadohaaeqaaOGaamivaaaa@3C67@

The total number of cycles for DIRLIK, LALANNE, and NARROW (with PARAM,NBZRCRS,NO is calculated as):

N T = n p e a k s 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 Δ S i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqqHuoarcaWGtbWaaSbaaSqaaiaadM gaaeqaaaaa@3513@ is calculated as:

N i = P ( S i ) N T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGobWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaamiuamaabmaabaGaam4uamaaBaaaleaacaWGPbaabeaa aOGaayjkaiaawMcaaiaad6eadaWgaaWcbaGaamivaaqabaaaaa@3AE9@

Fatigue Life and Damage

Fatigue Life (maximum number of cycles of a particular stress range S i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgaaeqaaa aa@33AD@ for the material prior to failure) is calculated based on the Material SN curve as:

N f ( S i ) = ( S i S f ) 1 b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGobWaaSbaaSqaaiaadAgaaeqaaO WaaeWaaeaacaWGtbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzk aaGaeyypa0ZaaeWaaeaadaWcaaqaaiaadofadaWgaaWcbaGaamyAaa qabaaakeaacaWGtbWaaSbaaSqaaiaadAgaaeqaaaaaaOGaayjkaiaa wMcaamaaCaaaleqabaWaaSaaaeaacaaIXaaabaGaamOyaaaaaaaaaa@3FA7@

Total Fatigue Damage as a result of the applied Random Loading is calculated based on:

D= i=1 N N i N f ( S i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGebGaeyypa0ZaaabCaeaadaWcaa qaaiaad6eadaWgaaWcbaGaamyAaaqabaaakeaacaWGobWaaSbaaSqa aiaadAgaaeqaaOWaaeWaaeaacaWGtbWaaSbaaSqaaiaadMgaaeqaaa GccaGLOaGaayzkaaaaaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWG obaaniabggHiLdaaaa@40CE@

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.