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 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGUbaabeaaaaa@3804@ 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:
Figure 1. Calculation of PSD Moments


Moments are calculated based on:

m n = k = 1 N f k n G k δ f MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGUbaabeaakiaaykW7cqGH9aqpcaaMc8+aaabCaeaacaWG MbWaa0baaSqaaiaadUgaaeaacaWGUbaaaOGaam4ramaaBaaaleaaca WGRbaabeaakiabes7aKjaadAgaaSqaaiaadUgacaaMi8Uaeyypa0Ja aGjcVlaaigdaaeaacaWGobaaniabggHiLdaaaa@4C99@

Where,
f k MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaDa aaleaacaWGRbaabaaaaaaa@37FB@
Frequency value.
G k MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGRbaabeaaaaa@37DB@
PSD response value at frequency f k MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaDa aaleaacaWGRbaabaaaaaaa@37FB@ .

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 P Δ S i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuamaabm aabaGaeyiLdqKaam4uamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaa wMcaaaaa@3BB2@ of a stress range occuring between Δ S i δS/2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHuoarcaWGtbWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVlabgkHiTiaa ykW7cqaH0oazcaWGtbGaai4laiaaikdaaiaawIcacaGLPaaaaaa@42CE@ and Δ S i +δS/2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHuoarcaWGtbWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVlabgUcaRiaa ykW7cqaH0oazcaWGtbGaai4laiaaikdaaiaawIcacaGLPaaaaaa@42C3@ is:

P Δ S i =pδ S i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaabm aabaGaeyiLdqKaam4uamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaa wMcaaiaaykW7cqGH9aqpcaaMc8UaamiCaiabes7aKjaadofadaWgaa WcbaGaamyAaaqabaaaaa@445C@

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 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamyBamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaaaaa @3A82@ 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:
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@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaabm aabaGaam4uaaGaayjkaiaawMcaaiaaykW7cqGH9aqpcaaMc8+aaSaa aeaadaWcaaqaaiaadseadaWgaaWcbaGaaGymaaqabaaakeaacaWGrb aaaiaadwgadaahaaWcbeqaamaalaaabaGaeyOeI0IaamOwaaqaaiaa dgfaaaaaaOGaaGPaVlabgUcaRiaaykW7daWcaaqaaiaadseadaWgaa WcbaGaaGOmaaqabaGccaWGAbaabaGaamOuamaaBaaaleaacaaIYaaa beaaaaGccaWGLbWaaWbaaSqabeaadaWcaaqaaiabgkHiTiaadQfada ahaaadbeqaaiaaikdaaaaaleaacaaIYaGaamOuamaaCaaameqabaGa aGOmaaaaaaaaaOGaaGPaVlabgUcaRiaaykW7caWGebWaaSbaaSqaai aaiodaaeqaaOGaamOwaiaadwgadaWcaaqaaiabgkHiTiaadQfadaah aaWcbeqaaiaaikdaaaaakeaacaaIYaaaaaqaaiaaikdadaGcaaqaai aad2gadaWgaaWcbaGaaGimaaqabaaabeaaaaaaaa@606E@
Where,
D 1 = 2 x m γ 2 1 + γ 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaakiaaykW7cqGH9aqpcaaMc8+aaSaaaeaacaaI YaWaaeWaaeaacaWG4bWaaSbaaSqaaiaad2gaaeqaaOGaaGPaVlabgk HiTiaaykW7cqaHZoWzdaahaaWcbeqaaiaaikdaaaaakiaawIcacaGL PaaaaeaacaaIXaGaaGPaVlabgUcaRiaaykW7cqaHZoWzdaahaaWcbe qaaiaaikdaaaaaaaaa@4E23@
D 2 = 1 γ D 1 + D 1 2 1 R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIYaaabeaakiaaykW7cqGH9aqpcaaMc8+aaSaaaeaacaaI XaGaaGPaVlabgkHiTiaaykW7cqaHZoWzcaaMc8UaeyOeI0IaaGPaVl aadseadaWgaaWcbaGaaGymaaqabaGccaaMc8Uaey4kaSIaaGPaVlaa dseadaqhaaWcbaGaaGymaaqaaiaaikdaaaaakeaacaaIXaGaaGPaVl abgkHiTiaaykW7caWGsbaaaaaa@5400@
D 3 = 1 D 1 D 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIZaaabeaakiaaykW7cqGH9aqpcaaMc8UaaGymaiaaykW7 cqGHsislcaaMc8UaamiramaaBaaaleaacaaIXaaabeaakiaaykW7cq GHsislcaaMc8UaamiramaaBaaaleaacaaIYaaabeaaaaa@47F7@
Z = S 2 m 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwaiaayk W7cqGH9aqpcaaMc8+aaSaaaeaacaWGtbaabaGaaGOmamaakaaabaGa amyBamaaBaaaleaacaaIWaaabeaaaeqaaaaaaaa@3E7A@
Q = 1.25 γ D 3 D 2 R D 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuaiaayk W7cqGH9aqpcaaMc8+aaSaaaeaacaaIXaGaaiOlaiaaikdacaaI1aWa aeWaaeaacqaHZoWzcaaMc8UaeyOeI0IaaGPaVlaadseadaWgaaWcba GaaG4maaqabaGccaaMc8UaeyOeI0IaaGPaVlaadseadaWgaaWcbaGa aGOmaaqabaGccaWGsbaacaGLOaGaayzkaaaabaGaamiramaaBaaale aacaaIXaaabeaaaaaaaa@4F11@
R = γ x m D 1 2 1 γ D 1 + D 1 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaayk W7cqGH9aqpcaaMc8+aaSaaaeaacqaHZoWzcaaMc8UaeyOeI0IaaGPa VlaadIhadaWgaaWcbaGaamyBaaqabaGccaaMc8UaeyOeI0IaaGPaVl aadseadaqhaaWcbaGaaGymaaqaaiaaikdaaaaakeaacaaIXaGaaGPa VlabgkHiTiaaykW7cqaHZoWzcaaMc8UaeyOeI0IaaGPaVlaadseada WgaaWcbaGaaGymaaqabaGccaaMc8Uaey4kaSIaaGPaVlaadseadaqh aaWcbaGaaGymaaqaaiaaikdaaaaaaaaa@5BC6@
x m = m 1 m 0 m 2 m 4 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGTbaabeaakiaaykW7cqGH9aqpcaaMc8+aaSaaaeaacaWG TbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamyBamaaBaaaleaacaaIWa aabeaaaaGcdaGcaaqaamaalaaabaGaamyBamaaBaaaleaacaaIYaaa beaaaOqaaiaad2gadaWgaaWcbaGaaGinaaqabaaaaaqabaaaaa@43E9@
γ = m 2 m 0 m 4 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaaG PaVlabg2da9iaaykW7daWcaaqaaiaad2gadaWgaaWcbaGaaGOmaaqa baaakeaadaGcaaqaaiaad2gadaWgaaWcbaGaaGimaaqabaGccaWGTb WaaSbaaSqaaiaaisdaaeqaaaqabaaaaaaa@4178@
Irregularity factor.
S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CB@
Stress range.
LALANNE
The LALANNE random fatigue damage model depicts the probability density function as:
p S = 1 m 0 1 γ 2 2 π e S 2 8 m 0 1 γ 2 + S γ 4 m 0 1 + e r f S γ 2 2 m 0 1 γ 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaabm aabaGaam4uaaGaayjkaiaawMcaaiaaykW7cqGH9aqpcaaMc8+aaSaa aeaacaaIXaaabaWaaOaaaeaacaWGTbWaaSbaaSqaaiaaicdaaeqaaa qabaaaaOGaaGPaVpaalaaabaWaaOaaaeaacaaIXaGaaGPaVlabgkHi TiaaykW7cqaHZoWzdaahaaWcbeqaaiaaikdaaaaabeaaaOqaamaaka aabaGaaGOmaiabec8aWbWcbeaaaaGccaWGLbWaaWbaaSqabeaadaWc aaqaaiabgkHiTiaadofadaahaaadbeqaaiaaikdaaaaaleaacaaI4a GaamyBamaaBaaameaacaaIWaaabeaalmaabmaabaGaaGymaiaaykW7 cqGHsislcaaMc8Uaeq4SdC2aaWbaaWqabeaacaaIYaaaaaWccaGLOa GaayzkaaaaaaaakiaaykW7cqGHRaWkcaaMc8+aaSaaaeaacaWGtbWa aSbaaSqaaiabeo7aNbqabaaakeaacaaI0aWaaOaaaeaacaWGTbWaaS baaSqaaiaaicdaaeqaaaqabaaaaOWaaeWaaeaacaaIXaGaaGPaVlab gUcaRiaaykW7caWGLbGaamOCaiaadAgadaqadaqaamaalaaabaGaam 4uaiabeo7aNbqaaiaaikdadaGcaaqaaiaaikdacaWGTbWaaSbaaSqa aiaaicdaaeqaaOWaaeWaaeaacaaIXaGaaGPaVlabgkHiTiaaykW7cq aHZoWzdaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaaaSqabaaa aaGccaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@7E24@
Where,
γ = m 2 m 0 m 4 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaaG PaVlabg2da9iaaykW7daWcaaqaaiaad2gadaWgaaWcbaGaaGOmaaqa baaakeaadaGcaaqaaiaad2gadaWgaaWcbaGaaGimaaqabaGccaWGTb WaaSbaaSqaaiaaisdaaeqaaaqabaaaaaaa@4178@
Irregularity factor.
S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CB@
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 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaabm aabaGaam4uaaGaayjkaiaawMcaaiaaykW7cqGH9aqpcaaMc8+aaeWa aeaadaWcaaqaaiaadofaaeaacaaI0aGaamyBamaaBaaaleaacaaIWa aabeaaaaGccaWGLbWaaWbaaSqabeaacqGHsisldaqadaqaamaalaaa baGaam4uamaaCaaameqabaGaaGOmaaaaaSqaaiaaiIdacaWGTbWaaS baaWqaaiaaicdaaeqaaaaaaSGaayjkaiaawMcaaaaaaOGaayjkaiaa wMcaaaaa@4A90@
Where,
S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CB@
Stress range.
By default, the number of zero crossings n zcross = m 2 / m 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWG6bGaam4yaiaadkhacaWGVbGaam4CaiaadohaaeqaaOGa aGPaVlabg2da9iaaykW7daGcaaqaamaalyaabaGaamyBamaaBaaale aacaaIYaaabeaaaOqaaiaad2gadaWgaaWcbaGaaGimaaqabaaaaaqa baaaaa@44DC@ is used instead of number of peaks n p e a k s = m 4 / m 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGWbGaamyzaiaadggacaWGRbGaam4CaaqabaGccaaMc8Ua eyypa0JaaGPaVpaakaaabaWaaSGbaeaacaWGTbWaaSbaaSqaaiaais daaeqaaaGcbaGaamyBamaaBaaaleaacaaIYaaabeaaaaaabeaaaaa@43CB@ for NARROW band, because the numerical calculations involving m 4 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaI0aaabeaaaaa@37CF@ 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:
P S = 0.683 a t 2 m 0 0.271 a t 4 m 0 0.043 a t 6 m 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaabm aabaGaam4uaaGaayjkaiaawMcaaiaaykW7cqGH9aqpcaaMc8+aaiqa aeaafaqabeWabaaabaGaaGimaiaac6cacaaI2aGaaGioaiaaiodaca aMe8UaamyyaiaadshacaaMe8UaaGOmamaakaaabaGaamyBamaaBaaa leaacaaIWaaabeaaaeqaaaGcbaGaaGimaiaac6cacaaIYaGaaG4nai aaigdacaaMe8UaamyyaiaadshacaaMe8UaaGinamaakaaabaGaamyB amaaBaaaleaacaaIWaaabeaaaeqaaaGcbaGaaGimaiaac6cacaaIWa GaaGinaiaaiodacaaMe8UaamyyaiaadshacaaMe8UaaGOnamaakaaa baGaamyBamaaBaaaleaacaaIWaaabeaaaeqaaaaaaOGaay5Eaaaaaa@6059@
Where,
S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36CB@
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 S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaabm aabaGaam4uaaGaayjkaiaawMcaaaaa@3929@ compared to the lower case p S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaabm aabaGaam4uaaGaayjkaiaawMcaaaaa@3949@ for the other damage models.
For the THREE damage model, these probabilities are directly used to calculate the number of cycles by multiplying P S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaabm aabaGaam4uaaGaayjkaiaawMcaaaaa@3929@ 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 D S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiaado faaaa@3794@ (bin size) to get the probability.
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 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 D S = δ S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiaado facaaMc8Uaeyypa0JaaGPaVlabes7aKjaabofaaaa@3E2B@ for which the probability is calculated (see Figure 2). The default is 100 and the first bin starts from 0.0 to δS MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaae 4uaaaa@386E@ . 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 ( δ S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaae 4uaaaa@386E@ ).

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 P S i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuamaabm aabaGaam4uamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaaa @3A4B@ of a stress range occuring between Δ S i δ S / 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHuoarcaWGtbWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVlabgkHiTiaa ykW7cqaH0oazcaWGtbGaai4laiaaikdaaiaawIcacaGLPaaaaaa@42CE@ and Δ S i + δ S / 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHuoarcaWGtbWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVlabgUcaRiaa ykW7cqaH0oazcaWGtbGaai4laiaaikdaaiaawIcacaGLPaaaaaa@42C3@ is:
P S i = p i S i δS MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaabm aabaGaam4uamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaiaa ykW7cqGH9aqpcaaMc8UaamiCamaaBaaaleaacaWGPbaabeaakmaabm aabaGaam4uamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaiab es7aKjaadofaaaa@4684@
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:
n zcross = m 2 m 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWG6bGaam4yaiaadkhacaWGVbGaam4CaiaadohaaeqaaOGa aGPaVlabg2da9iaaykW7daGcaaqaamaalaaabaGaamyBamaaBaaale aacaaIYaaabeaaaOqaaiaad2gadaWgaaWcbaGaaGimaaqabaaaaaqa baaaaa@44D6@
Where,
m n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGUbaabeaaaaa@3804@
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:
n p e a k s = m 4 m 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGWbGaamyzaiaadggacaWGRbGaam4CaaqabaGccaaMc8Ua eyypa0JaaGPaVpaakaaabaWaaSaaaeaacaWGTbWaaSbaaSqaaiaais daaeqaaaGcbaGaamyBamaaBaaaleaacaaIYaaabeaaaaaabeaaaaa@43C5@
Where,
m n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGUbaabeaaaaa@3804@
Corresponding moments calculated.

Number of Cycles

NARROW band, THREE band
The total number of cycles is calculated as:
N T = n zcross T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGubaabeaakiaaykW7cqGH9aqpcaaMc8UaamOBamaaBaaa leaacaWG6bGaam4yaiaadkhacaWGVbGaam4CaiaadohaaeqaaOGaam ivaaaa@43B5@
Where,
T
Total exposure time.
DIRLIK, LALANNE
The total number of cycles is calculated as:
N T = n peaks T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGubaabeaakiaaykW7cqGH9aqpcaaMc8UaamOBamaaBaaa leaacaWGWbGaamyzaiaadggacaWGRbGaam4CaaqabaGccaWGubaaaa@42A0@
Where,
T
Total exposure time.
Total Number of Cycles of Particular Stress Range
The total number of cycles with with stress range Δ S i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam 4uamaaBaaaleaacaWGPbaabeaaaaa@394C@ is calculated as:
N i =P Δ S i N T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGPbaabeaakiaaykW7cqGH9aqpcaaMc8Uaamiuamaabmaa baGaeyiLdqKaam4uamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawM caaiaad6eadaWgaaWcbaGaamivaaqabaaaaa@439F@

Fatigue Life and Damage

Fatigue life (maximum number of cycles of a particular stress range S i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbaabeaaaaa@37E5@ 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@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGMbaabeaakmaabmaabaGaam4uamaaBaaaleaacaWGPbaa beaaaOGaayjkaiaawMcaaiaaykW7cqGH9aqpcaaMc8+aaeWaaeaada WcaaqaaiaadofadaWgaaWcbaGaamyAaaqabaaakeaacaWGtbWaaSba aSqaaiaadAgaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaWaaS aaaeaacaaIXaaabaGaamOyaaaaaaaaaa@46F5@

Total fatigue damage as a result of the applied random loading is calculated as:

D= i=1 N N i N f S i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiaayk W7cqGH9aqpcaaMc8+aaabCaeaadaWcaaqaaiaad6eadaWgaaWcbaGa amyAaaqabaaakeaacaWGobWaaSbaaSqaaiaadAgaaeqaaOWaaeWaae aacaWGtbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaaaaWc baGaamyAaiaaykW7cqGH9aqpcaaMc8UaaGymaaqaaiaad6eaa0Gaey yeIuoaaaa@4B32@

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.