Spot Weld Fatigue Analysis

Allows for the study of fatigue performance of spot welds in structures.

Currently, only stress-life (SN) based spot weld fatigue analysis is supported. The spot weld location is defined by three attributes, sheet 1, sheet 2, and the nugget.
Figure 1. Spot Weld Fatigue


Implementation

Fatigue analysis for spot welds involves examining the weld at three distinct locations, the sheets and nugget, and is based on a paper by Rupp et al. The cross-sectional forces and moments at the nugget location are determined and used to calculate corresponding stresses at the sheets and the nugget. These stresses are then used to calculate Fatigue Damage using Rainflow counting and the SN approach.

The following sections illustrate how stresses and subsequently damage are calculated at each of the three locations.

Sheet Location (1 or 2)

Figure 2. Forces and Moments of Interest at Sheet Locations


Radial stresses are calculated at the sheet by considering forces and moments at the nugget. The radial stresses σ ( θ ) are calculated as a function of θ for each point in the load-time history as:

σ(θ)= σ max ( f y )cosθ σ max ( f z )sinθ+σ( f x )+ σ max ( m y )sinθ σ max ( m z )cosθ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaai ikaiabeI7aXjaacMcacqGH9aqpcqGHsislcqaHdpWCdaWgaaWcbaGa ciyBaiaacggacaGG4baabeaakiaacIcacaWGMbWaaSbaaSqaaiaadM haaeqaaOGaaiykaiGacogacaGGVbGaai4CaiabeI7aXjabgkHiTiab eo8aZnaaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaaiikaiaadA gadaWgaaWcbaGaamOEaaqabaGccaGGPaGaci4CaiaacMgacaGGUbGa eqiUdeNaey4kaSIaeq4WdmNaaiikaiaadAgadaWgaaWcbaGaamiEaa qabaGccaGGPaGaey4kaSIaeq4Wdm3aaSbaaSqaaiGac2gacaGGHbGa aiiEaaqabaGccaGGOaGaamyBamaaBaaaleaacaWG5baabeaakiaacM caciGGZbGaaiyAaiaac6gacqaH4oqCcqGHsislcqaHdpWCdaWgaaWc baGaciyBaiaacggacaGG4baabeaakiaacIcacaWGTbWaaSbaaSqaai aadQhaaeqaaOGaaiykaiGacogacaGGVbGaai4CaiabeI7aXbaa@78E3@

Where,
σ max ( f y ) = f y π D T × C f y z × D d e f y z × T t e f y z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaGccaGGOaGaamOzamaaBaaa leaacaWG5baabeaakiaacMcacqGH9aqpdaWcaaqaaiaadAgadaWgaa WcbaGaamyEaaqabaaakeaacqaHapaCcaWGebGaamivaaaacaaMc8Ua ey41aqRaaGPaVlaadoeadaWgaaWcbaGaamOzaiaadMhacaWG6baabe aakiaaykW7cqGHxdaTcaaMc8UaamiramaaCaaaleqabaGaamizaiaa dwgacaWGMbGaamyEaiaadQhaaaGccaaMc8Uaey41aqRaaGPaVlaads fadaahaaWcbeqaaiaadshacaWGLbGaamOzaiaadMhacaWG6baaaaaa @63C6@
σ max ( f z ) = f z π D T × C f y z × D d e f y z × T t e f y z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaGccaGGOaGaamOzamaaBaaa leaacaWG6baabeaakiaacMcacqGH9aqpdaWcaaqaaiaadAgadaWgaa WcbaGaamOEaaqabaaakeaacqaHapaCcaWGebGaamivaaaacaaMc8Ua ey41aqRaaGPaVlaadoeadaWgaaWcbaGaamOzaiaadMhacaWG6baabe aakiaaykW7cqGHxdaTcaaMc8UaamiramaaCaaaleqabaGaamizaiaa dwgacaWGMbGaamyEaiaadQhaaaGccaaMc8Uaey41aqRaaGPaVlaads fadaahaaWcbeqaaiaadshacaWGLbGaamOzaiaadMhacaWG6baaaaaa @63C8@
σ ( f x ) = 1.744 f x T 2 × C f x × D d e f x × T t e f x for f x > 0.0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaai ikaiaadAgadaWgaaWcbaGaamiEaaqabaGccaGGPaGaeyypa0ZaaeWa aeaadaWcaaqaaiaaigdacaGGUaGaaG4naiaaisdacaaI0aGaamOzam aaBaaaleaacaWG4baabeaaaOqaaiaadsfadaahaaWcbeqaaiaaikda aaaaaaGccaGLOaGaayzkaaGaaGPaVlabgEna0kaaykW7caWGdbWaaS baaSqaaiaadAgacaWG4baabeaakiaaykW7cqGHxdaTcaaMc8Uaamir amaaCaaaleqabaGaamizaiaadwgacaWGMbGaamiEaaaakiaaykW7cq GHxdaTcaaMc8UaamivamaaCaaaleqabaGaamiDaiaadwgacaWGMbGa amiEaaaakiaaywW7caqGMbGaae4BaiaabkhacaaMf8UaamOzamaaBa aaleaacaWG4baabeaakiaaysW7cqGH+aGpcaaMe8UaaGimaiaac6ca caaIWaaaaa@6FB5@
f x   =   0.0 for f x 0.0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaqa aaaaaaaaWdbiaadAgadaWgaaWcbaGaamiEaaqabaaak8aacaGLOaGa ayzkaaWdbiaabccacqGH9aqpcaqGGaGaaGimaiaac6cacaaIWaGaaG zbVlaabAgacaqGVbGaaeOCaiaaywW7caWGMbWaaSbaaSqaaiaadIha aeqaaOGaaGjbVlabgwMiZkaaysW7caaIWaGaaiOlaiaaicdaaaa@4D5B@
σ max ( m y ) = 1.872 m y D T 2 × C m y z × D d e m y z × T t e m y z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaGccaGGOaGaamyBamaaBaaa leaacaWG5baabeaakiaacMcacqGH9aqpdaqadaqaamaalaaabaGaaG ymaiaac6cacaaI4aGaaG4naiaaikdacaWGTbWaaSbaaSqaaiaadMha aeqaaaGcbaGaamiraiaadsfadaahaaWcbeqaaiaaikdaaaaaaaGcca GLOaGaayzkaaGaaGPaVlabgEna0kaaykW7caWGdbWaaSbaaSqaaiaa d2gacaWG5bGaamOEaaqabaGccaaMc8Uaey41aqRaaGPaVlaadseada ahaaWcbeqaaiaadsgacaWGLbGaamyBaiaadMhacaWG6baaaOGaaGPa VlabgEna0kaaykW7caWGubWaaWbaaSqabeaacaWG0bGaamyzaiaad2 gacaWG5bGaamOEaaaaaaa@6854@
σ max ( m z ) = 1.872 m z D T 2 × C m y z × D d e m y z × T t e m y z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaGccaGGOaGaamyBamaaBaaa leaacaWG6baabeaakiaacMcacqGH9aqpdaqadaqaamaalaaabaGaaG ymaiaac6cacaaI4aGaaG4naiaaikdacaWGTbWaaSbaaSqaaiaadQha aeqaaaGcbaGaamiraiaadsfadaahaaWcbeqaaiaaikdaaaaaaaGcca GLOaGaayzkaaGaaGPaVlabgEna0kaaykW7caWGdbWaaSbaaSqaaiaa d2gacaWG5bGaamOEaaqabaGccaaMc8Uaey41aqRaaGPaVlaadseada ahaaWcbeqaaiaadsgacaWGLbGaamyBaiaadMhacaWG6baaaOGaaGPa VlabgEna0kaaykW7caWGubWaaWbaaSqabeaacaWG0bGaamyzaiaad2 gacaWG5bGaamOEaaaaaaa@6856@
D
Diameter of the weld element
T
Thickness of the sheet under consideration for damage calculation
C f y z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGMbGaamyEaiaadQhaaeqaaaaa@39D0@ , C m y z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGTbGaamyEaiaadQhaaeqaaaaa@39D7@ , C f x MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGMbGaamiEaaqabaaaaa@38D0@
Scale factors
d e f y z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamizaiaadw gacaWGMbGaamyEaiaadQhaaaa@3AB0@ , d e m y z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamizaiaadw gacaWGTbGaamyEaiaadQhaaaa@3AB7@ , d e f x MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamizaiaadw gacaWGMbGaamiEaaaa@39B0@
Diameter exponents
t e f y z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamiDaiaadw gacaWGMbGaamyEaiaadQhaaaa@3AC0@ , t e m y z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamiDaiaadw gacaWGTbGaamyEaiaadQhaaaa@3AC7@ , t e f x MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamiDaiaadw gacaWGMbGaamiEaaaa@39C0@
Thickness exponents

To be equivalent to the Rupp method:

C f y z = 1 , d e f y z = 0 , t e f y z = 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGMbGaamyEaiaadQhaaeqaaOGaaGPaVlabg2da9iaaykW7 caaIXaGaaiilaiaaykW7caaMf8UaamizaiaadwgacaWGMbGaamyEai aadQhacaaMc8Uaeyypa0JaaGPaVlaaicdacaGGSaGaaGzbVlaadsha caWGLbGaamOzaiaadMhacaWG6bGaaGPaVlabg2da9iaaykW7caaIWa aaaa@57E9@
C m y z = 0.6 , d e m y z = 0 , t e m y z = 0.5 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGTbGaamyEaiaadQhaaeqaaOGaaGPaVlabg2da9iaaykW7 caaIWaGaaiOlaiaaiAdacaGGSaGaaGPaVlaaywW7caWGKbGaamyzai aad2gacaWG5bGaamOEaiaaykW7cqGH9aqpcaaMc8UaaGimaiaacYca caaMf8UaamiDaiaadwgacaWGTbGaamyEaiaadQhacaaMc8Uaeyypa0 JaaGPaVlaaicdacaGGUaGaaGynaaaa@5AE0@
C fx =0.6,defx=0,tefx=0.5 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGMbGaamiEaaqabaGccaaMc8Uaeyypa0JaaGPaVlaaicda caGGUaGaaGOnaiaacYcacaaMc8UaaGzbVlaadsgacaWGLbGaamOzai aadIhacaaMc8Uaeyypa0JaaGPaVlaaicdacaGGSaGaaGzbVlaadsha caWGLbGaamOzaiaadIhacaaMc8Uaeyypa0JaaGPaVlaaicdacaGGUa GaaGynaaaa@57CB@

The equivalent radial stresses are calculated at intervals of θ (Default =18 degrees). The value of θ can be modified by varying the Number of angles field in the spot weld solution settings. Subsequently, Rainflow cycle counting is used to calculate fatigue life and damage at each angle ( θ ). The worst damage value is then picked for output. A similar approach is conducted for the other sheet.

Nugget Location

Figure 3. Forces and Moments of Interest at Nugget Cross-Section


The absolute maximum principal stresses are calculated using the shear stress and bending stress of the beam element as a function of θ for each point in the load-time history as:

τ ( θ ) = τ max ( f y ) sin θ + τ max ( f z ) cos θ
σ(θ)=σ( f x )+ σ max ( m y )sinθ σ max ( m z )cosθ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaai ikaiabeI7aXjaacMcacqGH9aqpcqaHdpWCcaGGOaGaamOzamaaBaaa leaacaWG4baabeaakiaacMcacqGHRaWkcqaHdpWCdaWgaaWcbaGaci yBaiaacggacaGG4baabeaakiaacIcacaWGTbWaaSbaaSqaaiaadMha aeqaaOGaaiykaiGacohacaGGPbGaaiOBaiabeI7aXjabgkHiTiabeo 8aZnaaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaaiikaiaad2ga daWgaaWcbaGaamOEaaqabaGccaGGPaGaci4yaiaac+gacaGGZbGaeq iUdehaaa@5C85@

Where,
τ max ( f y ) = 16 f y 3 π D 2
τ max ( f z ) = 16 f z 3 π D 2
σ f x = 4 f x π D 2 for f x > 0.0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aae WaaeaacaWGMbWaaSbaaSqaaiaadIhaaeqaaaGccaGLOaGaayzkaaGa aGjbVlabg2da9iaaysW7daWcaaqaaiaaisdacaWGMbWaaSbaaSqaai aadIhaaeqaaaGcbaGaeqiWdaNaamiramaaCaaaleqabaGaaGOmaaaa aaGccaaMf8UaaeOzaiaab+gacaqGYbGaaGzbVlaadAgadaWgaaWcba GaamiEaaqabaGccaaMc8UaeyOpa4JaaGPaVlaaicdacaGGUaGaaGim aaaa@5430@
σ f x = 0.0 for f x 0.0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aae WaaeaacaWGMbWaaSbaaSqaaiaadIhaaeqaaaGccaGLOaGaayzkaaGa aGjbVlabg2da9iaaysW7caaIWaGaaiOlaiaaicdacaaMf8UaaeOzai aab+gacaqGYbGaaGzbVlaadAgadaWgaaWcbaGaamiEaaqabaGccaaM c8UaeyizImQaaGPaVlaaicdacaGGUaGaaGimaaaa@509E@
σ max ( m y ) = 32 m y π D 3
σ max ( m z ) = 32 m z π D 3
D
Diameter of the weld element
T
Thickness of the sheet under consideration for damage calculation

The equivalent maximum absolute principal stresses are calculated for each θ from τ ( θ ) and σ ( θ ) . These stresses are used for subsequent fatigue analysis. Rainflow cycle counting is used to calculate fatigue life and damage at each angle ( θ ). The worst damage value is then picked for output.