FORMULA SHEET: Fatigue, Fracture Mechanics Structural Mechanics σ i = M x y i I xx − M y x i I yy + F A ε x = σ x E − ν E ( σ y +σ z ) + αΔT Fatigue: Machined Components σ ar m N=Cσ ar , 1 m N 1 =σ ar ,2 m N 2 m= log ( N 2 N 1 ) log ( σ ar , 1 σ ar , 2 ) N R = { ( σ ar ,1 σ ar ) m N 1 0.9 σ u ≥σ ar ≥σ er ∞ σ ar <σ er OR σ ar N b =Cσ ar , 1 N 1 b =σ ar ,2 N 2 b b= log ( σ ar , 1 σ ar , 2 ) log ( N 2 N 1 ) N R = { ( σ ar ,1 σ ar ) 1 b N 1 0.9 f ut ≥σ ar ≥S e ∞ σ ar < S e Endurance limit estimates (see Dowling Table 10.1) σ erb = { 0.25BHN ksi for BHN ≤ 400 100 ksi forBHN >400 Steel σ erb = { 0.5 σ u for σ u ≤ 200 ksi ( 1400 MPa 100 ksi ( 700 MPa) for σ u >200 ksi ( 1400 MPa Cast Iron + Cast Steels: σ erb = { 0.45 σ u for σ u ≤ 600 MPa 275 MPa forσ u >600 MPa Stress concentrations K f : σ sc =K t S K t = σ sc S K f = σ erb (un−notched ) σ erb ( nothced ) K f =1+ K t −1 ( 1+ α r ) Approximations for α: α= [ 300 σ u [ ksi ] ] 1.8 × 10 −3 ∈. ¿ [ 300 × 6.89 σ u [ MPa ] ] 1.8 × 10 −3 × 25.4 mm Peterson for steels (Dowling, 2013, p. 498): log α =2.654 × 10 −7 σ u 2 −1.309 × 10 −3 σ u + 0.01103 α [ ¿ mm ]=10 log α ( 345 ≤σ u ≤ 2070 MPa ) ¿ 10 2.654 ×10 −7 σ u 2 −1.309 × 10 −3 σ u +0.01103 Typical values are: α= { 0.51 mm aluminiumalloys 0.25 mm annealed∨normalizedlow −carbo 0.064 mm quenched∧temperedsteel K f ' :