Fatigue Failure Theories Modified by Paul Meyer Fatigue ⚫ To this point – Static loads – Static stresses ⚫ Is this the case for mechanical design? ⚫ Can one apply a load that is less than the yield strength and still induce failure? 2 Modified by Paul Meyer Fatigue ⚫ History – Introduction of engines in 1800’s ⚫ – August Wohler (1870) studied axle failure ⚫ ⚫ 3 Railroads / steam-powered machinery described “endurance limit” for steels Developed the S-N diagram (still in use today) Modified by Paul Meyer Summary of Process ⚫ ⚫ ⚫ ⚫ ⚫ ⚫ 4 Define your problem Determine nominal stress free of stress concentrations Determine stress using stress concentrations – Slightly different for fatigue Determine your endurance limit – Marin factors (C factors) ** K in Shigley Construct your S-N diagram Plot your stress/number of cycles and figure factor of safety Modified by Paul Meyer Fatigue 5 Modified by Paul Meyer Fatigue ⚫ Reed et al. “The annual cost of fatigue of materials to the US economy corresponds to about 3% of the gross national product (GNP) ($100 billion).” Cost due to occurrence or prevention of fatigue failure for ground and rail vehicles, aircrafts, bridges, cranes, power plants, offshore structures, miscellaneous machinery ⚫ Windows in airplanes are round to avoid stress concentration (fatigue failure) ⚫ Fatalities from fatigue failure of aircraft fuselage due to pressurization/depressurization of cabin (1954, 1988) 6 Modified by Paul Meyer Mechanism of Fatigue Failure ⚫ Always begins at a crack ⚫ Cracks – – ⚫ Three stages of fatigue failure 1. 2. 3. 7 may form over time may be due to manufacturing process Crack initiation (may be skipped for brittle materials) Crack propagation Sudden fracture due to unstable crack growth Modified by Paul Meyer Mechanism of Fatigue Failure ⚫ Crack growth is due to tensile stress – – Because cracks grow under tension (positive stress half cycle) Cracks close (no effect) under compression (negative stress half cycle) P sin(wt) Crack growth P sin(wt) 8 0 Modified by Paul Meyer t Mechanism of Fatigue Failure Pmean+ P sin(wt) Pmean + P sin(wt) Pmean > 0 Pmean + P sin(wt) Pmean < 0 Crack growth Crack growth Pmean 0 0 Pmean t t Under compressive stresses, may not fatigue depending on value of mean stress ⚫ 9 Modified by Paul Meyer Mechanism of Fatigue Failure Pmean+ P sin(wt) Pmean + P sin(wt) Pmean > 0 Pmean + P sin(wt) Pmean < 0 Crack growth Crack growth Pmean 0 ⚫ 0 Pmean t Cracks grow about 10-8 to 10-4 inches per cycle – 10 Not much, but under many cycles, this does add up Modified by Paul Meyer t Fatigue-Failure Tests ⚫ ⚫ ⚫ ⚫ ⚫ 11 Test specimens are subjected to repeated stress while counting cycles to failure Most common test machine is R. R. Moore high-speed rotatingbeam machine Subjects specimen to pure bending with no transverse shear As specimen rotates at 1725 rpm, stress fluctuates between equal magnitudes of tension and compression, known as completely reversed stress cycling Specimen is carefully machined and polished Modified by Paul Meyer S-N Curves ⚫ Strength to Life Curves – – – – – S-N curves developed (bounding envelopes) “Knee” exists where cyclic loading allowed infinitely without failure Knee is at Se ~= 0.5 Sut Knee is at 106 cycles (1 million cycles) Materials without “knee” ⚫ 12 Aluminum, magnesium, copper, nickel alloys, some stainless steels, some carbon steels Modified by Paul Meyer S-N Curves For Ferrous Materials Construction ⚫ High-Cycle (start at 103) ⚫ “knee” at N=106 and S = Se ⚫ Adjust to set vertical axis at 1,000 – 0.9Sut in bending Norton/Juvinall – 0.75Sut in axial loading Norton/Juvinall – 0.72Sut in pure torsion Juvinall Strength 0.9Sut 0.75Sut Sut failure Se safe cycles 100 13 103 106 Modified by Paul Meyer S-N Curves for Non-Ferrous Materials Construction ⚫ “point” at N=5*108 and S = Se – Line extrapolated after that Strength 0.9Sut 0.75Sut Sut failure Se safe 100 14 103 106 Modified by Paul Meyer 5*108 cycles Fatigue-Failure Corrections ⚫ We developed the SN curve for test specimens. ⚫ What needs to be done to correct this for our unique “real world” situations? ka kb kc kd ke kf in Shigley! Se = Se (Cload _ type Csize Csurface Creliability Ctemperature ) ⚫ Why do each of these things factor into our adjustment? ⚫ Let us consider each individually. 15 Modified by Paul Meyer Se` and Sf’ ⚫ ⚫ Unmodified endurance limit vs unmodified fatigue strength. Represents the endurance limit of laboratory specimen. – Typically a fraction of ultimate strength – Capped for high-strength steels, aluminums etc – N = 106 for steel, N = 5*108 for steel Se` or Sf` Steels Irons Sut 16 Modified by Paul Meyer Coppers Aluminums Surface Correction (Ka) ⚫ Csurface KL = Ka – Tests are done on polished specimens – “real” parts typically are not polished; thus have initial cracks – To determine • Use figures (Figure 6.26; Figure 6.27) • Or tables 6-3 b Csurf = A(Sut ) • But use highest ranking Csurf 1.0 Csurf = 1.0 17 Modified by Paul Meyer Size Correction (Kb) ⚫ Csize = Kb ⚫ For circular cross sections ⚫ 18 NOTE: the accuracy of these equations for nonferrous materials is questionable Modified by Paul Meyer Size Correction ⚫ Csize ⚫ For non-circular cross sections – adjust using A95 d eq= 0.0766 19 Modified by Paul Meyer Size Correction ⚫ ⚫ 20 Csize For non-circular cross sections – adjust using Modified by Paul Meyer A95 d eq= 0.0766 Loading Correction (Kc) ⚫ Cload (Kc) = 1.00 if in bending = 0.85 if in axial loading = 0.59 if in pure torsion* * =1.00 if combined stress (then use Von Mises for stress state) 21 Modified by Paul Meyer Temperature Correction (Kd) Shigley 10th Edition 22 Modified by Paul Meyer Temperature Correction (Kd) ⚫ Ctemperature – Creep becomes an issue at high temperatures – Temperature Models developed Celsius T 450 Ctemp = 1.0 450 T 550 Ctemp = 1 − 0.0058(T − 450) Fahrenheit T 840 Ctemp = 1.0 840 T 1020 Ctemp = 1 − 0.0032(T − 840) 23 Modified by Paul Meyer Reliability Correction (Ke) ⚫ Creliability – Our world is not deterministic (or at least too complex for us to make it deterministic) – Accuracy/Reliability are important (tolerances) Reliability (8% deviation) 24 Correction Factor 50% 1.000 90% 0.897 95% 0.868 99% 0.814 99.9% 0.753 99.99% 0.702 99.999% 0.659 99.9999% 0.620 Modified by Paul Meyer Miscellaneous Effects Factor ⚫ Kf – Corrosion Effect of Fresh Water Effect of Environment Effect of Saltwater 25 Modified by Paul Meyer Miscellaneous Effects Factor ⚫ Kf – – – – – – Effect of Nickel Plating Corrosion Electrolytic Plating Metal Spraying Cyclic Frequency Frettage Corrosion Kt Effect of Fresh Water Effect of Environment Effect of Saltwater 26 Modified by Paul Meyer Summary of Process ⚫ ⚫ ⚫ ⚫ ⚫ ⚫ 27 Define your problem Determine nominal stress free of stress concentrations Determine stress using stress concentrations – Slightly different for fatigue Determine your endurance limit – Marin factors (C factors) Construct your S-N diagram Plot your stress/number of cycles and figure factor of safety Modified by Paul Meyer Stress Concentration ⚫ Notch: – Any geometric characteristic that disrupts the “force flow” ⚫ Stress concentrations in static loading – Only brittle material considered – Ductile material would yield at the local stress concentration and lower the stresses to acceptable levels ⚫ Theoretical Stress Concentrations – Kt (see Appendices or other books for values) – only geometric ⚫ Fatigue Stress Concentration Factor Kf – Considers both the Kt and the notch sensitivity of material 28 Modified by Paul Meyer Stress Concentrations ⚫ For dynamic loading – Stress concentration factors should be adjusted – Neuber’s equation Theoretical Stress Concentration See Figures Fatigue Stress Concentration K f = 1 + q(K t − 1) Material Based q= 29 1 a 1+ r Notch radius Modified by Paul Meyer Stress Concentrations ⚫ 30 Neuber’s Constant for aluminums Modified by Paul Meyer Stress Concentrations ⚫ 31 Notch sensitivity – q for bending or axial loading Modified by Paul Meyer Stress Concentrations ⚫ 32 Notch sensitivity – q for reverse torsional loading Modified by Paul Meyer Application of Kf for Fluctuating Stresses ⚫ For fluctuating loads at points with stress concentration, the best approach is to design to avoid all localized plastic strain. ⚫ In this case, Kf should be applied to both alternating and midrange stress components. ⚫ When localized strain does occur, some methods (e.g. nominal mean stress method and residual stress method) recommend only applying Kf to the alternating stress. 33 Modified by Paul Meyer Fatigue Diagrams 34 Modified by Paul Meyer Modified-Goodman Diagram ⚫ ⚫ 35 Probably most common and simple to use is the plot of sa vs sm Modified Goodman line from Se to Sut is one simple representation of the limiting boundary for infinite life Modified by Paul Meyer 36 Modified by Paul Meyer Fluctuating Stresses General Fluctuating Repeated Completely Reversed 37 Modified by Paul Meyer Characterizing Fluctuating Stresses ⚫ ⚫ ⚫ Fluctuating stresses shown as minimum and maximum stresses, smin and smax sm - midrange steady stress component (sometimes called mean stress) sa - amplitude of alternating stress component 38 Modified by Paul Meyer Modified-Goodman Diagram ⚫ ⚫ Loads may vary the Mean Loads may vary the Amplitude Amplitude Stress sa Sy Yielding limit Se -Syc 39 Sy Modified by Paul Meyer Mean Stress sm Sut Constant Amplitude Case 1 ⚫ Amplitude Constant ⚫ Mean Varies ⚫ N= Sm/sm Amplitude Stress sa Sy Yielding limit Se -Syc 40 sm Modified by Paul Meyer Sm Sy Mean Stress sm Sut Constant Midrange Case 2 ⚫ Amplitude Varies ⚫ Mean Constant ⚫ N= Sa/sa Amplitude Stress sa Sy Yielding limit Se Sa sa -Syc 41 Sy Modified by Paul Meyer Mean Stress sm Sut Proportionally increasing Load Case 3 ⚫ Amplitude and Mean vary together Amplitude Stress sa Sy Yielding limit Se -Syc 42 Sy Modified by Paul Meyer Mean Stress sm Sut Independently increasing Loads Case 4 ⚫ Amplitude and Mean vary independent Amplitude Stress sa Sy Yielding limit Se -Syc 43 s 'm @ S = s 'a@ S = − S 2f + Sut2 (s ' ) + S Sf m@ S Sut ZS = (s ' OZ = (s 'm ) + (s 'a ) Nf = f m − s 'm @ S ) + (s 'a − s 'a@ S ) 2 2 2 OZ + ZS OZ Shortest distance (perpendicular) Sy Modified by Paul Meyer Sut ( S 2f − S f s 'a + Suts 'm ) Mean Stress sm Sut 2 Shigley Equation Tables ⚫ ⚫ ⚫ Shigley also provides 3 tables of specific equations for Modified Goodman, Gerber, and Elliptic Approaches(Tables 6-6 through 6-8) Equations Below for proportionally increasing loads Shigley provides additional equations for Sa, Sm (6-40 through 6-44) Shigley 10th Edition 44 Modified by Paul Meyer